Patent Application
20190229972
The present patent proposes for designing receive methods, receive apparatus, transmitting method, transmitting apparatus, and transmitter-receiver systems.
There are many kinds of techniques about communication systems; These techniques have also led to technical changes in communication; The inventor has been proposed a transmitter-receiver system using time-division and frequency-division schemes (e.g., Patent list [1]-[6] and Non-Patent list [1]-[32]):
The inventor remarked that the time-frequency shift operator (TFSO) representing the non-commutative property (NCP) of time-frequency shift (TFS)s in the time-frequency plane (TFP) plays an important role in signal detection and estimation and concluded that the NCP of the TFSO should be noticed.
To solve the above problem, in accordance with an embodiment of the present invention, a proposed method mainly depends on this remark and its aim is to give efficient receive method, receiver, transmission method, transmitter, and transmitter-receiver system.
To solve the above problem, a method, in accordance with an embodiment of the present invention, for receiving a signal is a method for receiving a signal, comprising an estimation step for estimating a time shift and a frequency shift that are embedded in a received signal, wherein the estimation step refers to a non-commutative time-frequency shift parameter space of co-dimension 2.
To solve the above problem, a receiver apparatus in accordance with an embodiment of the present invention is a receiver apparatus for receiving a signal, comprising an estimation part for estimating a time shift and a frequency shift that are embedded in a received signal, with reference to a non-commutative shift parameter space of co-dimension 2.
To solve the above problem, a method, in accordance with an embodiment of the present invention, for transmitting a signal is a method for transmitting a signal, comprising a shift step for time-frequency shifting the signal to be transmitted, with reference to a noncommutative time-frequency shift parameter space of co-dimension 2.
To solve the above problem, a transmitter apparatus in accordance with an embodiment of the present invention is a transmitter apparatus for transmitting a signal, comprising a time-frequency shift part for time-frequency shifting the signal to be transmitted, with reference to a non-commutative time-frequency shift parameter space of co-dimension 2.
To solve the above problem, a method, in accordance with an embodiment of the present invention, for receiving an image signal is a method for receiving an image signal, comprising an estimation step for estimating a space shift and a spatial frequency shift that are embedded in the received image, with reference to a parameter space, wherein each of the space shift and the spatial frequency shift has dimension 2.
To solve the above problem, a method, in accordance with an embodiment of the present invention, for transmitting an image signal is a method for transmitting an image signal, comprising a shift step for space-spatial frequency shifting the image signal to be transmitted, with reference to a parameter space, wherein each of the space shift and the spatial frequency shift has dimension 2.
The present disclosure enables one to realize an efficient receive method, a receiver, a transmission method, a transmitter, and a transmitter-receiver system.
Fig. shows three kinds of divisions of the time-frequency plane (TFP): (a) indicates time division, (b) frequency division, and (c) Gabor division [1]. The solid lines in (a) show divisions of data of time duration T; the thin lines subdivisions by Time Domain (TD)-Phase code (PC); The dotted lines in (b) show divisions of data of bandwidth F; the dashed lines subdivisions by Frequency Domain (FD)-PC.
Tτ,0·T0,υ=e−i2πτυT0,υTτ,0: Its LHS corresponds to the triangle in the figure; the RHS to the square; the Phase Distortion (PD) e−i2πτυ appears; the circle in the figure shows the Symmetrical Time-Frequency Shift Operator (TFSO)[26]
τ,υ
e
iπτυ
r,0·0,υ.
{circumflex over (t)}
d
,{circumflex over (f)}
D,
and convergent values td, fD.
u
m′
TD[k], 0≤m′≤N′−1,
and generates the TD signature v[k] in (25), (67).
U
m
FD[], 0≤m≤N−1.
and generates the FD signature in (25), (67)
V[].
{dp,p′}p,p′=1P,P′
generating the associated output, TD-Complex Envelope (CE) in (27), (71)
ψ[k].
{dp,p′}p,p′=1P,P′
generating the output, FD-CE in (27), (71)
Ψ[].
{dp,p′}p=1P, 1≤p′≤P′.
{dp,p′}p′=1P′, 1≤p≤P.
-ary PSK communication and its transmitter (encoder) capable of jointly estimating delay and Doppler efficiently and precisely.
-ary PSK communication and its receiver-synchronizer (decoder) capable of jointly estimating delay and Doppler efficiently and precisely.
(kd(0),D(0)),(kd(1),
D(1)),(kd(2),
D(2)),(kd(3),
D(3))
and their associated 2-D PC codes according to the division
χ(i).
In one embodiment of the invention, referring to Figures, the inventor explains the transmitter-receiver system. The inventor starts by expressing the theoretical issues behind the proposed method and an example of embodiment. Next, the inventor gives a correspondence between the proposed method and the content of the scope of claims described below.
This disclosure cites several references by referring to patent references and non-patent references and hence the citation is within this disclosure.
These references are listed to cite technical terms and to refer the problems to be solved and the background, relating to this disclosure. Thus the citation does not affect the patentability of the present disclosure.
It is not easy to perform synchronization of TD- and FD-Division communication systems, that are designed to convey data symbol of duration Ts and bandwidth Fs efficiently. The radar problem of estimating delay td and Doppler fD from a received echo signal remains unsolved. These difficulties come from phase distortion (PD)s
e
j2piT
F
and
e
j2pit
f
that are generated by TFSOs and are similar as well-known algebraic relations for the position and momentum operators in quantum mechanics[4].
The basic purpose of a radar is to detect the presence of an object of interest, called target detection, and provide information concerning the object's location, motion, and other parameters, referred to as parameter estimation. So, radars are based on the statistical testing of hypotheses for signal detection and estimation. The determination of delay and Doppler is an estimation problem from the PDs containing two unknowns. Neither delay nor Doppler is successfully detected and estimated with high precision from a noisy received signal without the help of the Weyl-Heisenberg group (WHG) theory. The description of the invention is summarized as the following 5 items.
(summary1) When one designs a transmit signal, one should treat a signal in the TD and its Fourier transform (FT), i.e., a signal in the FD symmetrically. In addition, the symmetrical time and frequency shift operator (TFSO) satisfying time-frequency symmetrical property (TFSP) is proven to be an operator such that the address of a multiplexed signal is manifested by the non-commutative property (NCP) of TFSs (cf. (39), (44). (51), (56)).
(summary2) Time-frequency shifted Gaussian pulses, i.e., Gabor functions by 2-D binary phase-shift-keying (BPSK) modulation by TD- and FD-PCs of periods N,N′ are shown to be useful for maximum likelihood estimate (MLE)s of parameters td, fD among N′,N hypotheses in FD- and TD-likelihood functional (LF)s, respectively.
TD- and FD-PCs are usually called “2-D spread spectrum (SS) codes (the adjective “SS” is something of a misnomer) to the contrary, the inventor calls it a 2-D PC for BPSK modulation; Note that BPSK modulation has two functions: merits and demerits. Many researchers have not been really aware of the important roles of BPSK modulation. The present disclosure gives the pros and cons of the BPSK modulation that have not been noticed as follows.
The TD- and FD-CEs (27), wideband signals, called “signatures”, contain PDs due to the BPSK modulation. The modulation makes a situation that N TD-template CE of type-3 (29) (or of type-1 (49)) with its support [0,Ts]×[0,LΔf] (or [0,LΔt]×[0, Fs]) and N′ FD-template CE of type-4 (33) (or of type-2 (54)) with support [0,LΔt]×[0, Fs] (or [0, Ts]×[0,LΔf]) are automatically embedded into these TD- and FD-CEs as indications of matching. Hence these PDs play an important role in hypotheses-testing by the use of CCRs (see (30), Proposition 4 and (35), Proposition 5).
(summary3) Phase information of a signal has not been effectively used in ordinary MLEs to the contrary, the inventor defines 4 kinds of TD- and FD-cross-correlation function (CCF)s between a received signal and templates: TD-template CE of type 3 (or of type 1) and FD-template CE of type 4 (or of type 2) as a kind of optimum receivers. These CCFs are proven to have rigorous expressions in the product form of the ambiguity function (AF) and several twiddle factors, defined by a discretized signal of time- and frequency-samplings Δt, Δf=(LΔt)−1, when co-operatively using BPSK modulation with TD- and FD-PCs, and denoted as
The PDs due to the NCP of TFSs can be evaluated in the powers of the twiddle factors, i.e., the summation of the PDs over the chip-address can be rigorously represented in the form of DFT and IDFT; Thus this expression has a product form of three functions (see (41) in Lemma 2, (45) in Lemma 4, and (52), (57)). In accordance with an embodiment of the present invention, an efficient computation by DSP can be guaranteed.
(summary4) Using the Youla's signal reconstruction method[22], the inventor gives a proof of the APT-based Phase Updating Loop (PUL) algorithm, defined and introduced in the patent reference, patent[1].
First, define Ts (or LΔt)-time limited (TL) TD space, E3 (or E1) and Fs (or LΔf)-band limited (BL) FD space E4 (or E2), as subspaces of the Hilbert space.
Secondly, according to the arrays of N′ TD-CCFs and N FD-CCFs, define 4 projection operator (PO)s orthogonal projecting onto E3 (or E1) and onto E4 (or E2), denoted by P3 (or P1) and P4 (or P2), respectively.
Thirdly, in accordance with an embodiment of the present invention, the inventor defines the alternative projection theorem operator (APTO) based on the alternative projection theorem (APT), defined as
3
−1,d
4
d (or 4
d
3
−1,d),
where F−1,d,Fd denote the IDFT, DFT. Fourthly, the inventor gives an expression (59) for updating MLEs of a gain factor Aeiκ of the channel as a function of estimates
{circumflex over (t)}
d
,{circumflex over (f)}
D
and two expressions (60), (61) for updating MLE of
{circumflex over (t)}
d
,{circumflex over (f)}
D.
Using these three expressions for updating MLEs, the inventor concludes that (td, fD) are estimated within the convergence region of the APT operator, i.e., the rectangle of chip-pulse duration LΔt and chip bandwidth LΔf with chip and data addresses ((ρ,ρ′),p→) and proves that MLEs
{circumflex over (t)}
d
,{circumflex over (f)}
D
are estimated within LΔt×LΔf and its computional complexity order is
(N+N′),
in place of
(N·N′).
That is, this APT operator singles out some rectangle in the time-frequency plane (TFP) and filters out other regions. Such an operator is referred to as a phase-space (or time-frequency) localization operator[7] and thus plays an important role of filters, in place of conventional sharp filters usually used in DSP.
A Gaussian function is not employed in most of communication systems primarily because it does not satisfy the Nyquist condition. However, several favourable properties of Gaussians in the TFP are shown to be of benefit to our (td, fD)-estimation problem and hence Gaussians are shown to be of crucial importance.
The PUL algorithm is an iteration for searching (td, fD) with no restriction of the range of (td, fD) if the resource of time duration PTs and bandwidth P′Fs, for the data-level address
{right arrow over (p)}=(p,p′)
can be available. Accordingly, it is shown that the use of a combination of a transmitter of a 2-D PC modulated signal of TD-, FD-Gaussian functions and a receiver in which the PUL is implemented in TD-, FD-CCF arrays makes it possible to provide a communication system which is capable of high-precision and high-speed parameter estimation. In other words, the use of the above-described configuration presents a paradigm shift in communication systems utilizing NCP.
(summary5) the inventor gives an encoding-decoding system for a high
-PSK communication in cooperation with establishing synchronization.
-PSK communication is available to an automotive radar capable of both estimating delay-Doppler and communicating to another object with data. It is not easy to transmit an
-PSK signal
primarily because the identification of the phase
is difficult in the midst of phase errors and phase noise; but is an important modulation for the sake of efficient use of radio resources (the number
log2 -bit
is transmitted at once). It is, however, known that the realization of this modulation/demodulation is difficult.
To solve this problem, as shown in the lower part (the intermediate block between the Switches 1-1 and 1-2) in
1) decomposes an integer (“information”) k given as
2) divides the delay-Doppler parameter space, called the target space equally into
sub-parameter spaces and assigns 2-D PC (TD- and FD-PCs) to each sub-space, where
(≥
0).
Furthermore, the transmitter 1) 2-D BPSK modulates a chip pulse by the
-th 2-D PC; 2) combines it to form a
-code-multiplexed signal;
3) time-frequency-shifts the resultant signal (called a signature) about by around the center of the j-th sub-parameter space, i.e., by the time delay
k
d
(j)
and by the frequency delay
D
(j)
where
shift(kd(j),D(j)))
is referred to as the shift of the j-th Artificial Channel (AC);
4) M0-PSK modulates the time-frequency shifted signature by the j′-th
0
-ary symbol
and transmits the modulated signal. Consequently, the
(kd(j),D(j)))
-shifted signature is again distorted by being passed through the Main Channel (MC) with shifts
(kd,D).
The inventor designs a CCF between an estimated and received template and a received CE as follows. As shown in the middle block, connected to the Switch2-1 in
1′) decomposes an estimate of k
{circumflex over (k)}
as
2′) 2-D BPSK modulates a chip pulse by the
3′) time-frequency-shifts it by the
ĵ
shifts(kd(ĵ),D(ĵ)));
4′) and M0-PSK demodulates the resultant signal by the
ĵ′
-th
0
-ary symbol
The resultant signal is an estimated and received template.
Maximization of the real parts of N′ TD-CCFs and N FD-CCFs using their associated LFs, in array forms, is performed in terms of the cardinal numbers of the CCFs, as a function of the chip-level address (ρ′,ρ) and the data-level address
{right arrow over (p)}=(p,p′),
and a pair of estimated decoding integers of k:
(ĵ,ĵ′).
The receiver 1) chooses the
ĵ
-th 2-D PC from
2-D PCs; 2) cancels-out the phase of the data dp→ with address
{right arrow over (p)}=(p,p′)
by the PSK signal
3) maximizes the real parts of the 2 CCFs based the PUL algorithm, as shown in the lower block, connected to the Switch2-2, in
(kd,D)
and
ĵ,ĵ′,
and 4) obtains the MLE of k
k*=j*
0
+j*
,t.
This is realized by constructing a low
0
-ary PSK-modulation-based encoder-decoder system and combining the system, together with exploiting
ACs with non-commutative time-frequency shifts. This provides a high
-ary PSK modulation communication system. Namely, this system is capable of being used also for a synchronizer (or radar) for estimating parameters, in cooperation with decoder of k, from the output signal of one AC chosen, according to k, from
ACs with non-commutative shifts, each of which is connected to the MC with shifts
(kd,D).
Thus, the multiplexed system using non-commutative AC shifts may be a pradigm shift. Note that the computational complexity of
-ary PSK-demodulation is about
times larger than that of synchronizer (or radar), O(N+N′):
In accordance with an embodiment of the present invention, the inventor describes details of theoretical issues of communication systems and specific examples of an embodiment of the communication systems.
One of important problems to be solved in communication is to design a wireless communication system that can accommodate more traffic within a limited amount of radio spectrum[1]. Orthogonal Frequency Division Multiplex (OFDM) has been the dominant technology, as a TD- and FD-division multiplexed system to convey data of time Ts and bandwidth Fs (see
“Synchronisation” is the first procedure for communication through the channel with td, fD. However, phase distortion (PD)s due to time and frequency shift operator (TFSO) needed to the TD- and FD-division multiplex
e
j2πT
F
are followed by the PD of the channel
e
j2πt
f
.
Thus it is not easy to establish synchronisation. Moreover, there is no effective solution to the problem of estimating td, fD from an echo signal of radars.
The inventor first modulates TD-signal and its FT, i.e., FD-signal satisfying the time and frequency symmetrical property (TFSP) in terms of td, fD (see
τ,υ
e
iπτυ
τ,0·0,υ.
Secondly, the inventor defines arrays of TD- and FD-CCFs (see
As shown in [27]-[32], patent[1]-patent[6], the proposed estimation method firstly determines td, fD and the cardinal numbers of TD- and FD-CCFs maximizing the real parts of TD- and FD-CCFs; with no advance information about td, fD and secondly updates estimates of td, fD alternatively. Furthermore, it is shown that this method becomes a good solution to the radar problem.
{circumflex over (t)}
d
,{circumflex over (f)}
D,
and convergent values td, fD.
The problem of estimating td, fD from an echo signal is equivalent to that of determing two unknowns from PDs due to the NCP of TFSs. Hence, this belongs to the category of signal detection and estimation based on the Weyl-Heisenberg Group (WHG) theory. However, except Auslander & Tolimieri's remark[17], many radar researchers haven't obtained effective estimation methods with high precision yet because the NCP of TFSs was not taken into care in their estimation methods. On the contrary, the invention described in this disclosure is based on the inventor's belief that the NCP of TFSs only serves to strengthen the efficiency of communication systems including radars. The wavelet transform[8]. using the time and frequency shifted function, i.e., the Gabor elementary functions
g
m,m′(t)g(t−mTc)ei2πm′F′
discusses series representations of the form
and is primarily concerned with the coefficients am,m′. In wireless communication, the 5G, after 5G candidates: OFDM/OQAM, the filter bank multi-carrier (FBMC), and GFDM [9, 10, 12] have an interest in designing a multiplexed signal f(t) with information am,m′ to be transmitted and the elementary pulse g(t). Namely, the primary concern is the shape of g(t) that makes intersymbol interference (ISI) and interchannel interference (ICI) to zero and the orthogonality of gm,m′(t).
In wireless communication, synchronisation that's resistant to time and frequency offsets is necessary. However, there are a few attempts to estimate td, fD. Furthermore, most of communication engineers considers that the PD due to TFSs mτ0,m′ν0
e
iπmr
m′υ
can be negligible. However, the group theoretic property of the WHG tells us that the PD of the channel
e
iπt
f
is followed by the PD due to TFSs
e
iπmswm′υ
and the PD due to the multi-carrier technique
e
iπm′v
t
arises simultaneously. The mechanism of phase errors arising is not simple.
One must start by solving the following three themes behind the radar problem. Note that Linear FM Continuous Wave with a chirp signal, a compressed pulse modulated by pulse of short time, and its multi-carrier version are used as a transmit signal in conventional radars [19].
(theme1) The radar problem in range and velocity is essentially a problem of two unknowns td, fD; most of receivers is based on searching peaks of the magnitude of complex-valued ambiguity function (AF) of two variables, called the ambiguity surface or using the AF-characteristic of a chirp signal, It's a natural belief that when solving the problem relating to a given function of two unknowns, a way of dealing jointly with (independent) some other functions of the unknowns may be a better solving method.
(theme2) The radar problem is suffering from the PD
e
j2πt
f
due to the NCP of the TFSs. Such a similar situation due to the NCP of position and momentum operators in quantum mechanics is observed. A chirp pulse sequence of time interval Tp and frequency shift Fp generates a PD
e
j2πT
F
,
Furthermore, a communication system multiplexed by non-overlapped superposition of a signal on the TFP to covey data-symbols of time duration Ts and bandwidth Fs, through the channel with td, fD is causing PD
e
j2πt
f
D
followed by PD
e
j2πT
F
.
(theme3) Closely relating to the theme2, it is not easy to understand a mechanism whereby PD arises. That is, in communication systems and radars, two shift operators are usually defined as
S(a)f(t)=f(t+a),M(a)f(t)=ei2πatf(t), t∈
and give the NCP, represented as M(v)S(u)=e−i2πuvS(u)M(v) i.e., the NCP is manifested as the PD e−i2πuv, in other words, the phase term of the product of two shifts comes from the fact that u, v are exponentiated. This observation is the same as the mathematical basis for the introduction of the Heisenberg group in quantum mechanics, proposed by Weyl. Hence such PD is an important clue to solve the radar or synchronisation problem and is purely symbolic one. Careful consideration should be given to the exponential function, i.e., e−i2πuv. This leads to the symmetrical TFSO (see
Let s(t) denote a real-valued pulse, and let σ(t) denote the Hilbert transform of s(t). Then one gets a complex signal, referred to as an analytic signal, defined as ψ(t)=s(t)+iσ(t)[1]. A typical echo signal can be represented in the form
r
e(t;td,fD)=Aψ(t−td)eiΩ(t−t
where ψ(t) is the Complex Envelope (CE) of a pulse,
A,t
d,Ω,φ,Ω−Ωr=2πfD
denote its amplitude, time of arrival, carrier frequency, the phase of its carrier, and a change in its carrier frequency, called the Doppler shift of the reference carrier Ωr=2πfr. For brevity, let us assume Ωr=0 for a moment. Let
[⋅]
denote the Fourier Transform (FT) and
Ψ(f)=[ψ(t)]
denote the FT of ψ(t), then the FT of re(t; td, fD) is in the form
R
e(f;tdfD)=A[ψ(t−td)ei2πf
Ψ(f−fD)e−i2πft
A pair of re(t; td, fD) and Re(f; td, fD) is not symmetrical in td and fD because the product of unknowns td and fD appears only in the TD function (1).
But it can be represented in a slightly modified form[24, 26]
If a TD function x(t) and its FD function
X(f)=[x(t)]
are symmetrical in terms of td and fD, then this property is called the time and frequency symmetrical property (TFSP) (see
and get a trivial identity between the two operators
f
,−t
1=d,f
−1. (5)
Usually, the time shift operator S(−td)x(t)=x(t−td) and the modulation operator
M(fD)x(t)=ei2πf
are used. On the contrary, the shifts
in a TD signal x(t) and
in its FD signal X(f) in (4), hereafter called “half shifts”, seem to be a few slight modifications to usual time-frequency representations of signal:
S(−td)M(fD)x(t)=x(t−td)ei2πf
or
M(fD)S(−td)x(t)=x(t−td)ei2πf
But such half shifts are our intention to get effective representations of a radar signal and its received one so that their phase factors are fully traced in both the TD and FD as shown below.
Proposition 1: The TFSO (4) is identical to von Neumann's canonical commutative relations(usually abbreviated to CCRs) in quantum mechanics [4, 6] as referred to as the Stone-von Neumann theorem [5, 4, 25], defined as a two-parameter family of unitary operators S(a, b)=e−1/2 iabU(a)V(b) with its group-theoretic property, where U(a), V(b) are defined below. Tτ,ν and Tfν,−τ are referred to as von Neumann's TFSO s hereafter.
The Heisenberg commutation relation, referred to as the Heisenberg's uncertainty principle, is given as[4, 5]
[Q,P]=QP−PQ=ih. (6)
Proposition 2: Let us assume that Q and P can be exponentiated to one-parameter unitary groups U(a)=exp(iaQ), V(b)=exp(ibP), respectively (a, b real) and let us associate Q, P with TFSOs Tτ,0, T0,ν, respectively, then
Thus, the classical limit, i.e.,
h→0
limit[20] corresponds to
[r,0,
0,ν]=0, PD-free, i.e., τν∈
, (8)
where
h
s[⋅,⋅]
denote the reduced Planck's constant
and the quantum mechanics commutator.
The composition of TFSOs
gives
Eq. (7) is an example of the first equation in eq. (9).
The fact that the product of unknowns td and fD appear symmetrically in the exponent of PDs of TD- and FD-functions from their symmetry property is the more important property of the symmetrical TFSOs. Chip- and data-level addresses of the multiplexed signal appeared in the the exponent of PDs as well. The product of such exponents enables us to easily estimate parameters as discussed below. For example, in wireless communication[9, 10] an orthogonal frequency division multiplex (OFDM) signal
is the main subject, where the coefficients am,n take complex values representing an encoded transmit data, and x(t) is a prototype function. The OFDM signal is rewritten as
Hence the PD eiπnτ0mν0 does not alter the am,n except its sign as far as the condition τ0ν0=1 holds (cf. (8)). But if there are offsets such as τ′=τ0+ετ and ν′=ν0+εν, then the PD
inevitably arises followed by the PD
e
−iπt
f
through a doubly dispersive channel with td and fD. That is, we have to confront a phase-distorted signal
The conventional non-overlapping superposition of a signal in the TFP [1, 2, 9, 10, 12], defined as in (10) is causing an accumulation of the PD
e
iπ2t
mν
because of its group-theoretic property like (12). Hence the PDs directly weaken the output of the receiver and give a serious deterioration in “synchronisation” needed in digital communication systems[3]. On the contrary, the main concern is the design of the prototype function x(t) to reduce inter-symbol interference (ISI) and inter-channel interference (ICI). Such simple observation provides a starting point for this study.
To begin with, we have to study a brief review of Woodward's[15] statistical approach to the analysis and design of optimum radar systems, and Helstrom's[18] comprehensive study on theory and practice of signal detection and estimation.
It is an important observation that these approaches used only Abelian harmonic analysis as Auslander and Tolimieri's remark on Wilcox's study[16].
However, the fundamentals of radar theory reside in the following signal detection and parameter estimation theory.
When a radar signal appears in a receiver, its detection is made uncertain by the simultaneous presence of noise.
Consider the simplest signal-detection problem, that of deciding whether a signal s(t) of specified form has arrived at a definite time in the midst of Gaussian noise n(t). An input w(t) to the receiver is measured during an observation interval
0≤t≤T.
On the basis of this input an observer must choose one of two hypotheses, H0, “there is no signal,” i.e., w(t)=n(t), and H1, “the signal is present,” i.e., w(t)=s(t)+n(t). When wk=w(tk) is measured at time t=tk during the observation interval, the n samples wk are random variables having a joint probability density function (p.d.f.) pi(w) under hypotheses Hi, i=0, 1, and the observer's decision is best made on the basis of the likelihood ratio Λ(w)=p1(w)/p0(w), w=(w1, . . . , wn).
For a fixed decision level Λ0 the observer chooses hypothesis H0 if Λ(w)<Λ0; H1 if Λ(w)>Λ0.
A radar signal can be written simply as[15, 18]
s(t)=ψ(t)eiΩt,
where ψ(t) is its CE and Ω=2πfc the carrier frequency. If the spectrum of the signal s(t)
S(f)=[s(t)]=½[Ψ(f−fc)+Ψ*(−f−fc)], Ψ(f)=
[ψ(t)]
exhibits two narrow peaks, one near the frequency fc and the other near −fc, and if the widths of the bands are much smaller than Ω, the signal is termed narrowband (NB) or quasi-harmonic.
Assume that the input to the receiver
w(t)=ψw(t)eiΩt
ψw(t)
can be measured by a mixer.
In the presence of stationary NB white Gaussian noise with the auto-covariance function
ϕ(τ)={tilde over (ϕ)}(τ)eiΩτ,{tilde over (ϕ)}(τ)=N0δ(τ),
an optimum detector of the NB signal
s(t)=ψ(t)ei(Ωt+φ)
has the logarithm of its LF (LLF)[18, p. 106]
in which N0 is the unilateral spectral density of the white noise, g, being generated by passing the input ψw(t) through a filter[19] matched to the signal ψ(t) to be detected, and d2 are referred to as the statistic and the signal-to-noise ratio (SNR) of the LF Λ[ψw(t)].
The principle in hypothesis testing can be applied to choices among multiple hypotheses as follows. Suppose that a transmitter is sending a signal using one of M signals. The receiver is to decide which of these M signals is present during the observation interval (0,T). Namely, under hypothesis Hk, “a signal sk(t) was sent”, an input to the receiver is
w(t)=sk(t)+n(t),sk(t)=ψk(t)ei(2πf
where ψk(t) is the NB CE, fk the carrier,
φk
denotes the phase of sk(t), and n(t) random noise.
The receiver chooses one of the M hypotheses on the basis of measurements of its input w(t). Suppose that the receiver makes n measurements w1, . . . , wn of its input w(t). Let pk(w) be the joint p.d.f. of these data under hypothesis Hk and let ζk be the prior probability of that hypothesis. The likelihood ratio for detecting the k th signal in the presence of n(t) is defined by Λk(w)=pk(w)/p0(w), where p0(w) denotes the p.d.f. under a dummy hypothesis. For simplicity, assume that ζk=M−1, under the orthogonality of the signals sk(t)
∫0Ts*i(t)sjdt=Eiδij,
(Ei is the energy of the i th signal) then the receiver simply decides Hk if Λk(w)>Λj(w), for all k≠j.
Denote the unknown parameters of the signal by θ1, . . . , θm and represent them by a vector θ=(θ1, . . . , θm) in an m-dimensional parameter space, designated by Θ.
A radar echo can be represented in the form
s
echo(t;A,κ,td,fD)=Aeiκψ(t−td)ei2πf
where Aeiκ denotes the attenuation factor, A its amplitude, td its time of arrival, fc its carrier frequency, κ the phase of its carrier, and fD the Doppler shift of its reference carrier fr. Unknown parameters in the echo signal (15) are given as θ=(A, κ, td, fD). When
w(t)=ψw(t)ei2πf
and the noise is white with unilateral spectral density N0, the LLF[18, p. 251] is
Using a change of variables such as
Ae
iκ
=u+iv, u=
Ae
iκ
, v=Ae
iκ.
ℑ
denotes the imaginary part of the complex number following it. Thus one gets the maximum likelihood estimate (MLE) of A, κ containing θ′=(td, fD), given by [18, p. 251].
The MLEs of the remaining parameters θ′ are those that maximize[18, p. 251]
Hence one can concentrate one's efforts on estimating the θ′. For one of a closely spaced set of values of the Doppler shift
the MLE θ′ could be obtained by building a bank of parallel filters, each matched to a signal of the form
ψ(t)ei2π(f
where WD is the maximum range of the expected Doppler shift. It is, however, not easy to examine statistics by constructing a bank of NB filters in parallel. This fact leads one to decompose the 2-unknown-parameter problem into 2 single-unknown-parameter problems.
Let us rewrite (15) and its FT in the form
s
echo(t;A,κ,td,fD)=Aeiκ
t
0,f
S
echo(f;A,κ,td,fD)=Aeiκ
f
f
[ψ(t)],
where
t
,f
is a von Neumann's TFSO with td and fD where
ψ(t),φ0
denote the CE to be designed and its phase (its detail is omitted here), and
0,f
φ(t)
a passband signal shifted from the baseband CE by the reference frequency fr.
Using pulse code techniques with the TD-PC of period N and the FD-PC of period N′, one can want to make clear the exact location of (td, fD) in a 2-dimensional lattice
T
c
×F
c
in the TFP, where Tc=Ts/N, Fc=Fs/N′, Ts, and Fs denote a chip-pulse spacing, chip (sub-) carrier-spacing, signal (or data)-duration, and carrier-spacing, Divide the (td, fD)-parameter space Θ′ into a large number NN′ of small rectangular regions Δq→m,m′ with the data address
{right arrow over (q)}=(q,q′),
defined as
and denote by Hm,m′ the proposition “The parameter set θ′ lies in region
Δm,m′{right arrow over (q)}.”
However, NN′ such hypotheses Hm,m′ can be decomposed into N′ hypotheses for estimating fD in the TD signal secho(t; A, κ, td, fD) and N hypotheses for estimating td in the FD signal Secho(f; A, κ, td, fD) as discussed below.
Provided that the signals sk(t;θ′) are orthogonal, in hypothesis Hk associated with (14), for the CE ψk(t) and phase
φk,
consider the kth NB echo signal
s
k(t;θ′)=Aeiκψk(t−td)ei2π(f
If the noise is white and Gaussian with unilateral spectral density N0, then its TD-LLF [18, p. 129, p. 251] is
Let k=k0 be an integer satisfying
for a given decision level r0; Then the receiver decides the kth signal has been arrived; If all statistics gk lie below r0, the receiver decides that no signal was transmitted. This is referred to as an ML receiver. Hence a construction of orthogonal signals sk(t) is of crucial importance. Equation (18) suggests two ways of maximizing |gk(θ′)|: one is maximizing the integrand and the other is canceling out the phase factors eiκ of the carrier and
e
iφ
of the signal sk(t). The phase factor, however, is usually absorbed into
ψk(t)
by re-definition; moreover not gk(θ′) but |gk(θ′)| is simply evaluated. These strategies annihilate the phase information. Woodward[15] used a two-dimensional CCF, called the ambiguity function (AF), early defined by Ville[14], that plays a central role in the radar signal design. It is given by the definition
Non-commutative and group-theoretic properties of WHG-based time-frequency-shifted waveforms are manifested in the cisoidal factors in (9).
In addition, one can get:
Proposition 3 [Property3 of symmetric TFSOs]: For a TD signal z(t) and its FT
Z(f)=[z(t)],
the inner product (IP)s between the TD and FD time-frequency-shifted signals can be expressed as
where
<r(t),s(t)>t=∫−∞∞r(t)s*(t)dt
denotes the TD IP of r(t) and s(t), and the FD IP
<R(f),S(f)>f=∫−∞∞R(f)S*(f)df=<r(t),s(t)>t
of
R(f)=[r(t)]
and
S(f)=[s(t)].
Equation (19) shows that: i) both the real parts of the TD and FD IPs are maximized when t2=t1 and f2=f1, for which the maxima of the AF are attained; ii) if the left and right terms of the IP are thought of as the input signal ψw(t) and a signal ψ(t) to be detected, respectively of the statistic g in LLF (13), then ψ(t) may cancel-out the input signal's PD to enhance the statistic g. Proposition 3 emphasizes the importance of “phase” of a modulated signal like the “phasors” used in analysing alternating currents and voltages in electrical engineering[18, xv, p. 91], and tells us that two quantities td and fD always appear in a phase term. This is a major step toward (td, fD)-estimations in the TD and FD, based on the WHG theory, in distinct contrast to the conventional methods using matched filters[18], performed only in the TD. If one can design a signal so that its phase terms have been endowed with easy-traceability, one may utilize the both the PDs of ψw(t), ψ(t) of the statistic in the TD- and FD-LLFs.
Gabor[1] stressed the importance of analysis in the TFP and the utility of the Gaussian wave 21/4e−πt{circumflex over ( )}2 attaining the lower bound of the uncertainty relation of time and frequency.
He gave the time-frequency representation of a function f:
It is well known [9, p. 985] that the set of Gaussian functions gm,n(t) forms a basis of L2(R) having good properties regarding time and frequency localization but this basis is not orthonormal and the set of the functions is not even a frame[7]. See [7, 9] for a review on the double series representation (20). Most of communication engineers does not employ a Gaussian function primarily because it does not satisfy the Nyquist condition. In our (td, fD)-estimation problem, however, several favourable properties of Gaussians in the TFP play an important role.
TD-PC techniques, i.e., spreading spectrum (SS) techniques[3] can provide the simultaneous use of a wide frequency band, via code-division multiple access (CDMA) techniques, in which a signal to be transmitted s(t) is modulated by an independent pulse code c(t) so that its bandwidth is much greater than that of the message signal m(t), e.g., s(t)=m(t)c(t) and each user is assigned a pulse code such that the signals are orthogonal.
In order for the CE
ψ(t)
to satisfy the time and frequency symmetrical property (TFSP), its FT Ψ(f) should also be phase coded.
In place of continuous-time signals in
2(),
consider discrete-time signals in
2()
and assume that a TD signal s(t) is sampled with a sampling interval Δt, while a discrete-frequency signal is obtained by the L-point discrete Fourier transform (DFT). Hence the frequency gap between two adjacent frequency bins in the FD is Δf=1/(LΔt).
Let
(the truncation of fraction
and
(the truncation of fraction
be discrete variables of time t and frequency f. For the orthogonality of a chip pulse, let assume
and let
T
c
=MΔt,F
c
=M′Δf.
Thus one can define the L-point twiddle factor
Define the following 7 different kinds of discrete-time and discrete-frequency signals:
For a continuous chip-pulse g(t) having support [−LΔt/2,LΔt/2], i.e., duration LΔt, one can obtain its causal discrete-time LΔt-time-limited (TL) chip-pulse g[k] with delay[10] (D/2)Δt,D=L−1, L=(ΔtΔf)−1=MM′
and define its discrete-frequency LΔf-band-limited (BL) chip-pulse G[≃] having support [−LΔf/2,LΔf/2], i.e., bandwidth LΔf, by the DFT of g[k]
Introduce now a discrete-time TD-signature v[k;χ] and an FD-signature V[;χ] defined as
in terms of a TD-template of type-3
u
m′
(3)[k;X]
and an FD-template of type-4
U
m
(4)[;X′],
respectively defined by
where
a,b
d
g[k]
and
b,−a
J,d
G[],a,b,k,
∈
denote discrete analogues of von Neumann's TFSOs in the TD and FD in (4), respectively defined by
The TD-signature v[k;χ] contains N′ TD-templates of type-3
u
m′
(3)[k;X], 1≤m′≤N′,
while the FD-signature
V[;χ]
does N FD-templates of type-4
U
m
(4)[;X′], 1≤m≤N
so that the CCF between such a signature and its embedded template may have a large value via the phase coding. Note that the TD-template
u
m′
(3)[k;X]
has a rectangular support NMΔt×LΔf in the TFP, while the FD-template
U
m
(4)[;X′]
has a support LΔt×N′M′Δf. Substituting the compositions of TFSOs
0,m′M
d,m,M,0dg[k]
(resp.
0,−mM
f,d
m′M′,0
f,d
G′[]
) into (22) shows that the TD-signature and FD-signature
are perfectly symmetrical. Suppose that a radar TD-signal s[k;χ] with its CE
ψ[k;χ]
and the carrier
and its FT, FD-signal S[l;χ] have the form
Writing the CE and its DFT in the form
one can design the radar TD- and FD-signals
s[k;χ],S[;χ].
This is a 2-dimensional train of PP′ non-overlapped signatures
v[k;χ]
(resp.
V[;χ])
of duration Ts=NMΔt and carrier-spacing Fs=N′M′Δf, where
d
{right arrow over (q)}ϵ
is a data symbol on the lattice
T
s
×F
s
of the TFP, with address
{right arrow over (q)}=(q,q′).
Namely, a radar system needs PTs×P′Fs time-duration-bandwidth to search for targets whose delay td∈(0, PTs) and Doppler shift.
are not known in advance (simply set dq→=1), while a data communication system sends P·P′
-ary data
(e.g.,
Suppose now that such a signal s[k;χ] is transmitted through the channel with
modulation 0,l
0,−l
one can obtain its received TD signal, being demodulated at a mixer or a homodyne receiver
where
ψr[k]
is the signal component CE in the received signal, η[k] interference, and ξ[k] Gaussian noise. The FD expression, its DFT
R[;χ,A,κ,θ′,d]=
d[r[k;χ,A,κ,θ′,d]].
is omitted here. Its PD
may be absorbed into eiK by re-definition, but should be canceled-out at the correlation receiver as discussed below. Such received TD and FD signals provide the data w and its DFT W, observed in the mixer.
Independent and identically distributed (i.i.d.) TD- and FD-PCs give independent N′ TD-templates
u
m′
(3)[k;X], 1≤m′≤N′
U
m
(4)[;X′], 1≤m≤N
in the M-ary detection. Note that PCs have two functions: to randomise a signal and to generate several PDs caused by the NCP of TFSs; Fortunately, such a PD itself provides a good indication for parameter estimation in the sense that the transmitted signal is endowed with easy-traceability. This indicates the pros and cons of the use of PCs. In fact, the bandwidth of phase coded systems needs to be much greater than that of a classic radar system; The multi-carrier technique, i.e., the FD-PC multiples this bandwidth by the number of sub-carriers and so requires a super-wide-band signal.
Let us employ the M-ary detection strategy
using the TD-CE in (27)
ψ[k;χ]
(resp. FD-CE
Ψ[;χ])
to detect the radar signal in (26)
s[k;χ]
(resp.
S[;χ]).
Consider a strategy whereby the receiver chooses among the NN′ hypotheses Hm,m′ associated with (17). It suffices to find those values of the parameters θ′,d for which the LLF (or the real part of its associated CCF) in the TD (resp. the FD) is maximum. Consider first the problem of detecting the received TD-template CE
with its address
(ρ′,{right arrow over (p)}),{right arrow over (p)}=(p,p′)
of the lattice in the TFP
T
c
×F
c
,
where
ψρ′,{right arrow over (p)}(3)[k]=d{right arrow over (p)}pNM,p′N′M′d
0,ρ′M′dX′ρ′uρ′(3)[k;X], 0≤ρ′≤N′−1, (29)
is the TD-template CE of type 3 (cf. (22), (27)) with ρ′-th TD-template
X′
ρ′
u
ρ′
(3)[k;X]
{circumflex over (k)}
d
denotes an integer-valued estimate of kd,
μ
an integer-valued parameter for estimating
D.
This CE is embedded in the estimated and received CE
Ae
iκ
W
{circumflex over (k)}
l
{circumflex over (k)}
,l
dψ[k;χ]
of
ψ[k;χ]
in (27) (cf. ψr[k] in (28)), in which the relation
=
is used. Equation (29) shows that the CE contains a consequential phase, caused by several PDs as given below. Such a phase should be canceled-out in a CCF. Denote the CE complementary to
ψρ′,{right arrow over (p)}(3)[k]
by
ψη,ρ′,{right arrow over (p)}(3)[k], 0≤ρ′≤N′−1,
defined as
ψη,ρ′,{right arrow over (p)}(3)[k]=Σm′=0,m′≠ρ′N′−1ψm′,{right arrow over (p)}(3)[k],
ψm′,{right arrow over (p)}(3)[k]=d{right arrow over (p)}ρNM,p′N′M′d
0,m′M′dX′m′um′(3)[k;X], 0≤m′≤N′−1.
Equations (22) and (27) indicate that N′ TD-templates of type 3
u
ρ′
(3)[k;X], 0≤ρ′≤N′−1
are available and the receiver is to use N′ TD-template CEs of type 3
ψρ′,{right arrow over (p)}(3)[k], 0≤ρ′≤N′−1
and decide which of the N′ TD-LLFs is largest.
i) Discrete-time signal detection and Doppler-ML estimation problem in the TD: On the basis of measured values of the NT random variables w=(w[0], . . . , w[NT−1]), the receiver must choose between two hypotheses,
where
ψw[k]
is the NB CE of the observation data
at time k,
ψn[k]
the NB CE of white Gaussian noise
n[k]=ψD[k]
N
T
=└T/Δt┘>>1
the number of samples during the observation duration (0,T). Note that the signal component in hypothesis H1 is equal to
(N′)1/2Aeiκψ[k;χ].
Suppose that N′ TD-template CEs of type 3
ψm′,{right arrow over (p)}(3)[k], 0≤m′≤N′−1
of equal energies are quasi-orthogonal in the sense that
Then one can obtain:
Proposition 4: ii) Signal detection and Doppler-ML estimation problem in the TD:
On the basis of observed data w=(w[0], . . . , w[NT−1]) in the presence of white Gaussian noise with unilateral spectral density N0, the logarithm of the
(ρ′,{right arrow over (p)})
-th TD-LF for detecting and estimating the TD-template CE of type 3
Ae
iκ
ψρ′,{right arrow over (p)}(3)[k]
is given as[21]
where
Let
ρ′=ρ′0 and μ=
*μ
be integers for a given decision level r0 satisfying
where
g′
ρ′,{right arrow over (p)}(Â,{circumflex over (κ)},{circumflex over (k)}d;μ)
is a compensated variant of the statistic in (31)
g
ρ′,{right arrow over (p)}(A,κ,{circumflex over (k)}d;μ)
Âe
−i{circumflex over (κ)}
of Ae−iκ. Then the receiver decides the
(ρ′,{right arrow over (p)})
-th CE has been arrived at the address (p′,p→) of the lattice
T
c
×F
c
in the TFP and if all statistics
|g′ρ′,{right arrow over (p)}(kd,A,κ;μ)|/dρ′,{right arrow over (p)}
lie below r0, the receiver decides that no signal was transmitted. Thus
*μ
is an ML estimate of
μ
for a given
{circumflex over (k)}
d,
in which the use of the operator
needs the phase function
Next one moves on the detection and delay-estimation problem in the FD using the DFT of the measured w,
Let us consider the problem of detecting the received FD-template CE
Ae
iκ
Ψρ,{right arrow over (p)}(4)
with its address
(ρ,{right arrow over (p)})
of the lattice
T
c
×F
c
,
where
Ψρ,{right arrow over (p)}(4)[]=d{right arrow over (p)}
p′N′M′,−pNMf,d
0,−ρMf,dXρUρ(4)[
;X′], 0≤ρ≤N−1 (33)
is the FD-template CE of type 4 (cd. (22), (27)) with ρ-th FD-template of type 4
X
ρ
U
ρ
(4)[:X′]
This CE is embedded in the estimated and received FD-CE
Ae
iκ
Ψ[
;χ]
Ψ[;χ],
where
D
is an integer-valued estimate of
D,
kσ an integer-valued parameter for estimating kd, and the relation
=
is used. Denote the CE complementary to
Ψρ,{right arrow over (p)}(4)[]
by
Ψη,ρ,{right arrow over (p)}(4)[], 0≤ρ≤N−1,
Equations (22) and (27) suggest that N FD-templates of type 4
U
ρ
(4)[;X′], 0≤ρ≤N−1
are available and the receiver is to use N FD-template CEs of type 4
Ψρ,{right arrow over (p)}(4)[], 0≤ρ≤N−1
and decide which of the N LLFs is largest (
ii) Signal detection and delay-ML estimation problem in the FD:
On the basis of observed W=(W[0], . . . , W[NT−1]), the receiver must choose between two hypotheses in the FD,
where
W[]=
d[
ψw[k]
]
is the DFT of the observed data w[k] with the NB CE
ψw[k].
N[]=
d[
ψD[k]
]
the DFT of the noise n[k] with the NB CE
ψn[k].
N
B
=└B/Δf┘
the sample number of the bandwidth B, for simplicity NB=NT;
S
ρ,{right arrow over (p)}
(4)[]=
d[
−1,d[Ψρ,{right arrow over (p)}(4)[
]
]]
the
(ρ,{right arrow over (p)})
-th template signal spectrum with its FD-template CE of type 4 in (33)
Ψρ,{right arrow over (p)}(4)[],
and
E
ρ,{right arrow over (p)}
(4)[]=
d[
−1,d[Ψη,ρ,{right arrow over (p)}(4)[
]
]]
its complement spectrum with CE in (34)
Ψη,ρ,{right arrow over (p)}(4)[].
Note that the signal component in hypothesis H′1 is equal to
(N)1/2AeiκΨ[
;χ].
Suppose that N FD-template CEs of type 4 (cf. (22), (27))
Ψm,{right arrow over (p)}(4)[], 0≤m≤N−1
of equal energies are quasi-orthogonal in the sense that
Then one can obtain:
Proposition 5: On the basis of data
W=(W[0], . . . , W[NT−1]), W[]=
d[w[k]]
in the presence of white Gaussian noise with unilateral spectral density N0, the logarithm of the
(ρ,{right arrow over (p)})
-th FD-LF for detecting and estimating
S
ρ,{right arrow over (p)}
(4)[]
is given as
where
ρ=ρ0 and kσ=k*σ
be integers satisfying
for a given decision level r′0. Then the receiver decides the (ρ,p→)-th signal has been arrived at the address
(ρ,{right arrow over (p)})
of the lattice in the TFP
T
c
×F
c
and if all statistics
|G′ρ,{right arrow over (p)}(kσ;D,Â,{circumflex over (κ)})|/Dρ,{right arrow over (p)}
lie below r′0, the receiver decides that no signal was transmitted. Thus
k*
σ
is a ML estimate of kd for a given
D,
in which the use of the operator
∈k
needs the phase function
,
an FD analogue of the
,
so one can get (36) and (37).
Suppose that the observed input
w(t)=ψw(t)eiΩt
is NB and both the real and imaginary parts of the CE
ψw(t)
can be measured separately[18, 3].
In order to cancel out the phase factors of
Ae
iκ
ψρ′,{right arrow over (p)}(3)[k])
(resp.
Ae
w
Ψρ′,{right arrow over (p)}(4)[
])
and get a time and frequency symmetrical statistic,
instead of the statistic in (31)
g
ρ′,{right arrow over (q)}({circumflex over (k)}d,μ)
having
(resp. the statistic in (36)
G
ρ,{right arrow over (p)}(kσ,D)
having
,
one can use its associated CCF to be defined below (see
Lemma 1: Suppose that in two hypotheses H0 and H1 in (30) the CE
ψn[k]
is Gaussian. Then
<ψ0,ψρ′,{right arrow over (p)}(3)>d,k<<1,
where <⋅,⋅>d,k denotes the IP in the space
2()
of discrete-time TD functions. Thus one can define a CCF, called a type-3 correlator, between the received CE
ψ[k;χ]
(namely, the signal component of the received CE in (28)
ψr[k;χ]
with factor
Ae
iκ
),
instead of the input CE
ψw[k;χ]
to the receiver, and the complex impulse response of a NB filter matched to the estimated template CE with address
(ρ′,{right arrow over (p)})
ψρ′,{right arrow over (p)}(3)[k]|d
given as
in which the TD-template CE of type 3 in (29)
ψρ′,{right arrow over (p)}(3)[k]
and the codes X′ρ′,X are replaced by Y′ρ′,Y.
One can find after some labor that this CCF of type 3 is given as
Unfortunately, the AFs θgg(τ,υ) and ΘGG(υ,−τ) have many sidelobes in general. A Gaussian chip-pulse g(t), however, gives a radical solution to the estimation problems because it has its separable and exponentially decayed AF in terms of of τ and υ
where
For N,N′>>1, both the first and second arguments of θgg[⋅,⋅] should be relatively small, i.e., q→=p→ so that
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d)
has a large value; All of the terms with
{right arrow over (q)}≠{right arrow over (p)}
in (39) can be negligible. This property of Gaussian plays a central role in determining μ for maximizing
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d).
Evaluating threefold summation of PCs with the cisoidal factor leads us to define a summation of IDFT-type
symbolically denoted by a pair of the square bracket
m′,N′
−1,d[⋅]m′
and the round bracket (a)m′ with its convenient notation (a)m′=W−am′, and that of DFT-type
symbolically denoted by a pair of convenient notations
m,N
d[⋅]m′(b′)m=Wmb′.
Using the notations:
ν0[μ]=
μ−
D, τ0[{circumflex over (k)}d]={circumflex over (k)}d−kd,
one can obtain
Lemma 2: If the receiver of type-3 has its address
{right arrow over (p)}={right arrow over (q)}
and Y=X, Y′=X′, then its CCF becomes
Thus one can know necessary conditions for
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d)
to have a large value are
|ν0[μ]|<3sf and |τ0[{circumflex over (k)}d]|<3st.
Proof. Using the round bracket symbols in summations of IDFT-type and DFT-type, one can rearrange 5 components among 6 components in the second twiddle factor in (39)
and move its residue term W−1/2 ρ′M′τ0[k{circumflex over ( )}d] of the 6 components to the
first twiddle factor in (39). The use of the separability of the AF of the Gaussian proves (41).
Lemma 3: Suppose that in two hypotheses H′0 and H′1 in (35)
N[]
is Gaussian. Then the CCF
<N,Sρ,{right arrow over (p)}(4)<<1,
where the angular brackets <⋅,⋅ denotes the IP in the space
2()
of discrete-frequency FD functions.
Then one can define a CCF, called a type-4 correlator, between the received FD-CE
Ae
iκ
Ψ[
;χ],
i.e., the DFT of the signal component of the received CE in (28)
ψr[k]
with factor
Ae
iκ
,
instead of the input CE
ΨW[;χ]
to the receiver, and the FD complex impulse response of a NB filter matched to the FD estimated template CE in (33) with address
(ρ,{right arrow over (p)})
W
k
l
Ψρ,{right arrow over (p)}(4)[
]|d
given as
This CCF has the form
{right arrow over (q)}={right arrow over (p)}
so as to make
C
ρ,{right arrow over (p)}
(4)(kσ;D)
large; All of the terms in (44) with
{right arrow over (q)}≠{right arrow over (p)}
can be negligible. Then one can get the sum of the three exponents in the twiddle factor as above. Evaluating the threefold summation of PCs together with the cisoidal factor, one can have
Lemma 4: If the receiver of type-4 has its address
{right arrow over (p)}={right arrow over (q)}
and Y=X, Y′=X′, then its CCF becomes
Thus we know necessary conditions for
C
ρ,{right arrow over (p)}
(4)(kσ;D)
to have a large value are
|ν0[D]|<3Sf and |τ0[kσ]|<3st.
Proof. Similarly using the round bracket symbols in summations of IDFT-type and DFT-type, one can rearrange 5 components among 6 components in the second twiddle factor in (44)
and move its residue term W1/2 ρMν0[l{circumflex over ( )}D] of the 6 components to the first twiddle factor in (44). The use of the separability of the AF of the Gaussian proves (45).
Equations (41) and (45) show that correlator of type 3
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d)
and the one of type 4
C
ρ,{right arrow over (p)}
(4)(kσ:D)
are perfectly symmetrical in terms of
d
d and D.
Interchanging the TD-PC and FD-PC in the pair of type 3
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d)
and of type 4
C
ρ,{right arrow over (p)}
(4)(kσ;D),
called a complementary pair (CP)[27] gives other pair of type 1
c
ρ,{right arrow over (p)}
(1)(μ;{circumflex over (k)}d)
and of type 2
C
ρ′,{right arrow over (p)}
(2)(kσ;D),
called an original pair (OP)[24, 30], defined as follows.
[Original pair (OP) of TD and FD correlators]
Observe another decomposition of v[k;χ] and
C[;χ]
given as
whose TD-template of type-1
u
m
(1)[k;X′]
and FD-template of type-2
U
m′
(2)[;X]
are respectively defined by
ψρ,{right arrow over (p)}(1)[k]
be the (ρ,p→)-th estimated and received CE having the TD-template
X
ρ
u
ρ
(1)[k,X′]
with address
(ρ,{right arrow over (p)})
of the lattice
T
c
×F
c
defined by the TD-template CE of type 1
ψρ,{right arrow over (p)}(1)[k]=d{right arrow over (p)}ρNM,p′N′M′d
ρM,NdXρuρ(1)[k,X′]. (49)
Then one can get its associated CCF, called a type-1 correlator
and one can have
Let Y=X, Y′=X′ and discard all the terms
d
{right arrow over (q)}
except the term satisfying
{right arrow over (q)}={right arrow over (p)}.
Then one can have
in which using the round bracket symbols in summations of IDFT-type and DFT-type, one can rearrange 5 components among 6 components in the second twiddle factor in (51)
and move its residue term W1/2 ρMτ0[k{circumflex over ( )}d] of the 6 components to the first twiddle factor in (51). The use of the separability of the AF of the Gaussian gives (52).
Secondly let
Ψρ,{right arrow over (p)}(2)[]
be the (ρ,p→)th estimated and received CE having the FD-template
X′
ρ′
U
ρ′
(2)[,X]
with address
(ρ,{right arrow over (p)})
of the lattice
T
c
×F
c
,
defined by the FD-template CD of type 2
Ψρ,{right arrow over (p)}(2)[]=d{right arrow over (p)}
p′N′M′,−pNMf,d
0,−pMf,dX′ρ′Uρ′(2)[
,X]. (54)
Then one can get its associated FD correlator, called a type-2 correlator
and have
Let Y=X, Y′=X′ and discard all the terms
d
{right arrow over (q)}
except the term satisfying
{right arrow over (q)}={right arrow over (p)}.
Then one can have
in which using the round bracket symbols in summations of IDFT-type and DFT-type, one can rearrange 5 components among 6 components in the second twiddle factor in (56)
and move its residue term W−1/2 ρ′M′τ0[k∧d] of the 6 components to the
first twiddle factor in (56). The use of the separability of the AF of the Gaussian gives (57).
If one can get precise estimates
{circumflex over (k)}
d and D
that make all
|ν0[μ]|,|τ0[{circumflex over (k)}d]|
|ν0[D]|,|τ0[kσ]|
in Lemma 4 are within chip-pulse bandwidth and duration, then the two CCFs
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d) and Cρ,{right arrow over (p)}(4)(kσ;
D)
will filter out the interference
<ψη,ρ′,{right arrow over (p)}(3)[k],ψρ′,{right arrow over (p)}(3)[k]>d,k and <Ψη,ρ,{right arrow over (p)}(4)[],Ψρ,{right arrow over (p)}(4)[
]>d,l,
respectively contained in
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d) and Cρ,{right arrow over (p)}(4)(kσ:
D)
and recover
d
{right arrow over (p)},
in place of using conventional sharp filters. This is a radical solution to the digital signal processing for communication.
The simple way to update estimates
{circumflex over (k)}
d and D
in both pairs of correlators is the following algorithm, called a Phase-Updating Loop (PUL), different from a conventional “Phase-Locked Loop” for synchronisation in communication systems:
[PUL Algorithm with Updating Rule for Attenuation Factor-MLE]:
{cρ′,{right arrow over (p)}(3)(μ,{circumflex over (k)}d,s)}ρ′=0N′−1
and
{Cρ,{right arrow over (p)}(4)(kσ,D,s)}ρ=0N−1
be the CP of the type-3 and type-4 correlator arrays, and let
{cρ,{right arrow over (p)}(1)(μ,{circumflex over (k)}d,s)}ρ=0N−1
and
{Cρ′,{right arrow over (p)}(2)(kσ,D,s)}ρ′=0N′−1
be the OP of the type-1 and type-2 correlator arrays. For simplicity set
d
{right arrow over (p)}=1
until the PUL algorithm is terminated. Let
be the s-th MLE of the attenuation factor
with the s-th digitized estimate
{circumflex over (θ)}′s,d=({circumflex over (k)}d,s,D,s+1) or ({circumflex over (k)}d,s+1,
D,s)
of the parameter θ′=(td, fD) that is defined in (16), where
ψ*(t−td)e−i2πf
are replaced by
(ψρ′,{right arrow over (p)}(3)[k])*,
ψ[k],
respectively, given as
Then one can update an integer pair
({circumflex over (k)}d,s,D,s+1)(resp. ({circumflex over (k)}d,s+1,
D,s))
as {circumflex over ( )}D,s+1=
*μ, if s is even (resp. odd), and k{circumflex over ( )}d,s+1=k*σ, if s is odd (resp. even),
where
and the initial value
θ′0,d=({circumflex over (k)}d,0D,1) or ({circumflex over (k)}d,1,
D,0)
is chosen arbitrarily, e.g.,
θ′0,d=(0,0).
*μ and k*σ
to be candidates of
D,s+1 and {circumflex over (k)}d,s+1.
The (s+1)-th step estimation procedure is terminated if, for the chip pulse g[k] (or G[l]) duration LΔt and bandwidth LΔf,
Then such estimates are ML estimates and both of the two correlators become ML receivers. This algorithm can be written in the form of Youla's restoration[22] of a signal as given in Theorem below.
The recovery or restoration of a signal that has been distorted is one of the most important problems in signal processing. Youla[22] gave the answers to the restoration problems. Adroit use[23] of some of the mathematical machinery, introduced by Youla[22] will lead us to show that the convergence of the PUL algorithm hinges on how well the von Neumann's APT[21] does work.
Consider the Hilbert space
,
consisting of all 2(Z) space of square-summable continuous and discrete-time (or discrete-frequency) functions with an IP, defined by
(or
and the norm
∥f∥=√{square root over (<f,f>d,k)}
or
∥F∥=√{square root over (<F,)}
Let ε be any closed linear manifold (CLM) in the Hilbert space
.
The projection theorem[22] tells us that if ε′ and ε″ are orthogonal sub-spaces of
,
then every f∈ε possesses a unique decomposition f=g+h, g∈ε′, h∈ε″, where g, h are projections of f onto ε′ and ε″, respectively, denoted by g=Pf, h=Qf; P denotes the associated projection operator (PO) projecting onto ε′, Q=I−P its associated PO projecting onto ε″, and I the identity operator.
Let ε1 (resp. ε3) be the set of all
f∈
2()
that are LΔt-TL (resp. Ts-TL) signals. On the contrary, denote the set of all
f∈
2()
that are LΔf-BL signals by ε2 and that of Fs-BL signals by ε4. Each of
εi, 1≤i≤4
Let Pi be a PO projecting onto
εi, 1≤i≤4.
and Qi=I−Pi the PO projecting onto the orthogonal complement of εi, written as
⊥εi, 1≤i≤4,
each of which is defined below. The CCF plays a role of a PO in the sense that given any two signals r and s, the signal r∈ε has a unique decomposition of the form
where
s
⊥
is orthogonal to s. The coefficient
is regarded as a PO projecting onto ε′.
The CP of type-3, type-4 correlators and the OP of type-1, type-2 correlators provide also POs written as:
where
ψρ,{right arrow over (ρ)}(1)⊥,Ψρ′,{right arrow over (p)}(2)⊥,ψρ′,{right arrow over (p)}(3)⊥,Ψρ,{right arrow over (p)}(4)⊥
are orthogonal complements of
ψρ,{right arrow over (ρ)}(1),Ψρ′,{right arrow over (p)}(2),ψρ′,{right arrow over (p)}(3),Ψρ,{right arrow over (p)}(4).
The Alternative Projection Theorem (APT) [21, p. 55, theorem 13.7] (
tells us that:
If E and F are projections on CLMs
ε,
in a Hilbert space, respectively, then the sequence of operators E, FE, EFE, FEFE, . . . has a limit G, the sequence F, EF, FEF, . . . has the same limit G, and G is a projection on
ε.
(The condition EF=FE need not hold.)
With use of APT, the following result is obtained:
Let us consider the 4 POs in the TD and FD that contain (s+1)-step estimates
(ρ′,*,*μ)(resp.(ρ*,
*μ)),
determined by the argmax-operations in terms of (ρ′,μ) (resp. (ρ,
μ)) with the s-step estimate
{circumflex over (k)}
ds
in (60) (resp. (61)), and (ρ*,k*σ) (resp. (ρ″*,k*σ)), determined by the argmax-operations in terms of (ρ,kσ) (resp. (ρ′,kσ)) with the s-step estimate
Ds
in (60) (resp. (61)). Write these POs symbolically as
and
Q
k
=I−P
k, 1≤k≤4.
Then the PUL algorithm converges.
[Proof]: Consider the recursion formula for the CP only because that for the OP is exactly the same except an exchange between the TD-PC X and FD-PC X′, i.e., between suffixes (3, 4) and (1, 2). Two different orderings of the time-truncation operator P3 and the band-limited operator P4 provide two different iteration equations;
First consider an algorithm for the recovery of
ψ
as follows: If
ψ∈ε4,
then
ψ=−1,dP4
dψ
and
g
1
=P
3
ψ=P
3
−1,d
P
4
d
ψ=ψ−Q
3
−1,d
P
4
dψ,
ψ
satisfies the operator equation
A
1
ψ=g
1
, A
1
=I−Q
3
−1,d
P
4
d.
The recursion, defined by
ψ=g1+Q3−1,dP4
dψ
enables us to have the iterative equation in the TD[22, 23]
ψi=(I−Q3−1,dP4
d)ψ+Q3
−1,dP4
dψi−1,
or ψi−ψ=(Q3−1,dP4
d)i(ψ0−ψ), if (ψ0−ψ)∈ε4. (65)
becomes the projection of
(ψ0−ψ)
onto a CLM εc=⊥ε3∩ε4. The εc contains only the trivial function[22, p. 699], [23, p. 637]; i.e.,
This is one of Youla's[22] main results. Hence the operator
P
3
−1,d
P
4
d
singles out the rectangle of chip-pulse duration LΔt and chip bandwidth LΔf with chip-level and data-level addresses
((ρ,ρ′),{right arrow over (p)}),
i.e., the intersection of the rectangular support of its associated CE
ψρ′,{right arrow over (p)}(3)
and that of the CE
Ψρ,{right arrow over (p)}(4),
i.e.,
[((p−1)N+ρ−Δa′)Tc,((p−1)N+ρ+Δb′)Tc]×[((p′−1)N′+ρ′−Δa)Fc,((p′−1N′+ρ′+Δb)Fc]
of the lattice
T
c
×F
c
and filters out the rest in the TFP, where Δa′, Δb′, Δa, and Δb are integers satisfying Δa′+Δb′=M′ and Δa+Δb=M. Such an operator is referred to as a phase-space (or time-frequency) localization operator[7]. Thus the CE ψ[k] is restored so that the parameters
k
d and D
are estimated within LΔt and LΔf. It should be noted that unlike usual rectangles of strictly TL (resp. BL) operator[23], the time-truncation operator P3 (resp. the band-limited operator P4) is defined by using a template, i.e., a non-overlapped superposition of N coded TD Gaussian chip-pulses g[k] (resp. N′ coded FD Gaussian chip-pulses
G()
with no guard interval.
Conversely, in the FD, if ψ∈ε3, then
Ψ=dP3
−1,dΨ
and
g
2
=P
4
Ψ=P
4
d
P
3
−1,d
Ψ=Ψ−Q
4
d
P
3
−1,dΨ.
That is, ψ satisfies the operator equation [23]
A
2
Ψ=g
2
, A
2
=I−Q
4
d
P
3
−1,d.
This gives the iterative equation
Ψi=(I−Q4dP3
−1,d)Ψ+Q4
dP3
−1,dΨi−1,
or Ψi−Ψ=(QidP3
−1,d)i(Ψ0−Ψ), if (Ψ0−Ψ)∈ε3. (66)
becomes the projection of (ψ0-ψ) onto a CLM
ε′c=⊥ε4∩ε3.
This CLM contains only the trivial function. Thus limi→∞ψi=ψ.
The FD CE
Ψ[]
is restored so that the parameters
k
d and D
are estimated within LΔt and LΔf. Similarly, the operator
P
4
d
P
3
−1,d,
i.e., another localization operator singles out the rectangle of LΔt×LΔf with chip-level and data-level addresses
((ρ,ρ′),{right arrow over (p)})
i.e., the intersection of the rectangular support of the CE
ψρ′,{right arrow over (p)}(3)
and that of the CE
Ψρ,{right arrow over (p)}(4),
[((p−1)N+ρ−Δa′)Tc,((p−1)N+ρ+Δb′)Tc]×[((p′−1)N′+ρ′−Δa)Fc,((p′−1)N′+ρ′+Δb)Fc]
of the lattice
T
c
×F
c
and filters out the rest in the TFP. [End of proof]
Consider a multi-carrier filter bank for generating the TD-signature v[k;χ] and FD-signature
V[;χ],
defined in (25). Using a repetition of the TD-PC Xm of period N to get an infinite code sequence at the time: −∞, . . . , −Tc, 0, Tc, . . . , ∞ and that of the FD-PC X′m′ of period N′ to get an infinite code sequence at the frequency: −∞, . . . , −Fc, 0, Fc, . . . , ∞, one can get: Proposition 6: Write v[k;χ] and
V[;χ]
in (25) respectively as
where
f
m,m′
TD,[k],fm,m′FD[]
are modulated filters, defined by
For an input x[k] and output y[n], Vaidyanathan[13, p 117] defined the input-output relation in the TD using three types of multi-rate filters with filter coefficient h[⋅]: Mf-fold decimation filter:
Lf-fold interpolation filter:
and Mf/Lf-fold decimation filter:
[Proof]: The TD-signature υ[k;χ] in (25) (resp. the FD-signature
V[;χ]
) looks like the expression of a signal obtained at the output of a synthesis filter bank (SFB)[10, 12] with N′ (resp. N) sub-bands and with an expansion factor equal to M (resp. M′) and being phase-coded by the FD-PC X′m′ (resp. the TD-PC Xm) on each sub-band. Indeed, if the
X
m
,m∈(resp. X′m,m′∈
)
is the input signal and the
f
m,m′
TD[k](resp. fm,m′FD[])
is the filter on the (m,m′)th sub-band of this SFB, then the output
υ[k;χ](resp. V[;χ])
can be written as (67). [End of proof]
Together with the symbols lcm[M,N′]=M0N′=MN′0, lcm[M′,N]=M′0 N=M′N0 [13],[10],[11] using M0N′,M′0 N polyphase component
one can get polyphase filters (Vaidyanathan's[13, p 121] Type 1 polyphase)
For lm=lcm[M,N′], gd=gcd[M,N′], there exist N′0,M0 satisfying lm=MN′0=N′M0. On the other hand, by the identity lm·gd=M·N′, M=gd·M0,N′=gdN′0 hold.
Thus one can get the SFBs of 2-D modulated v[k] and V[] by TD-, FD-PC as shown in
u
m′
TD[k], 0≤m′≤N′−1.
V[]
in (25), (67), containing m-th FD template
U
m
FD[], 0≤m≤N−1.
While eq. (27) provides the SFBs for generating
ψ[k;χ]
and
Ψ[;χ].
Proposition 7: The CEs ψ[k;χ] and
Ψ[;χ].
can be written respectively as
where
k
q,q′
TD[k],kq,q′FD[]
are modulated filters of their associated SFBs, defined by
[Proof]: Equation (71) indicates that if
{dq,q′}q=0P−1
(resp.
{dq,q′}q′=0P′−1)
denotes the input signal, and
k
q,q′
TD[k]
(resp.
k
q,q′
FD[])
is the filter on the (q,q′)th sub-band of its associated SFB, then the
TD-CE:ψ[k;χ]
(resp.
FD-CE:Ψ[;χ])
is the output of its SFB with PP′ sub-bands and with an expansion factor equal to NM (resp. N′M′) on each sub-band. [end of proof]
Thus one can obtain the SFB for generating a transmit signal, that contains data symbols as shown in
one can obtain polyphase filters (Vaidyanathan's[13, p 121] Type 1 polyphase)
where P′0,P0 denote integers satisfying P′0MN=lcm[P′,MN], P0MN=lcm[P,MN]. [end of proof]
ψ[k]
with input, complex-valued data
{dp,p′}p,p′=1P,P′,
Ψ[]
with input, complex-valued data
{dp,p′}p,p′=1P,P′.
Besides the attenuation factors Aeiκ, the PD due to non-commutativity of modulation/demodulation by the carrier lc operations, accompanied by the delay k{circumflex over ( )}d,
arises. The received signal in (28)
r[k;χ,A,κ,θt,d]
(resp. its FT
R[;χ,A,κ,θ′,d]),
(simply denoted by
r[k],R[])
contains those PDs.
Define a type-3 (resp. type-4) CCF between the received signal r[k](resp. R[l] in (28) and the estimated TD-template CE of type 3 in (29)
[k]|d
(resp. FD-CE in (33)
Ψρ,{right arrow over (p)}(4)[
]d
respectively by
where the phase factor
(resp.
)
is designed to cancel out the one
in the signal component of r[k] in (28), (38) (cf.
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d))
(resp. that of
R[])
(cf. in (28), (43)
C
ρ,{right arrow over (p)}
(4)(kσ;D)).
Then one can get a TD (resp. FD) analysis filter bank (AFB)[10, 12]:
Proposition 8: The CCFs of type-3 and of type-4 can be rewritten, suited to AFB realizations, respectively as
where D=L−1 and
h
ρ′,{right arrow over (p)}
(3)[k],hρ,{right arrow over (p)}(4)[]
are modulated filters of TD and FD AFBs
-ary symbols
are transmitted,
one may evaluate the output at the address
{right arrow over (p)}=(p,p′)
in the correlator array of p-band, p′-duration to estimate
{d{right arrow over (p)}}p,p′=1P,P′.)
[Proof]: If the phase-modulated received TD signal
(resp. the FD one
denotes the input signal of the
(ρ′,{right arrow over (p)})-th
(resp.
(ρ,{right arrow over (p)})-th)
-sub-band of the AFB with P′ (resp. P) sub-bands with a decimation factor equal to NM (resp. N′M′) on each sub-band, given by (75) (resp. (76)), then two facts:
i) one is the TD symmetry property[10] (cf. (21))
g(−t)=g(t), i.e., g[k]=g[D−k], uρ′(3)[k;X]=uρ′(3)[D−k;X]
and ii) the other is the property of the FD signal
G[−]=W−D
G[
], Up(4)[−
;X′]=W−D
Uρ(4)[
;X′],
that the TD symmetry entails, prove that its associated filter
h
ρ′,{right arrow over (p)}
(3)[k]
(resp.
h
ρ,{right arrow over (p)}
(4)[])
is defined by (77). [end of proof]
Note that two kinds of filters given in patent[2] contain neither the PD nor its canceling-out terms
,
.
If AFB filter in (77) and P′0MN, P0M′N′ polyphase components:
then one can obtain N′,N polyphase filters (Vaidyanathan's[13] Type 2 polyphase)
Thus one can get the AFBs as shown in μ;kd),C(4)ρ,p→(kσ;
D), respectively;
The output bipolar symbol of the TD-AFB (resp. the FD-AFB) is given as the sign of
(resp. that of
where
Âe
i{circumflex over (κ)}
is the MLE of the attenuation factor Aeiκ with the MLE
{circumflex over (θ)}′,d=({circumflex over (k)}d,D)
(cf. (59)). Such a pair of SFBs and AFBs symmetrical in the TD and FD is referred to as a “twinned-FBMC”.
{dp,p′}p=1P, 1≤p′≤P′.
{dp,p′}p′=1P′, 1≤p≤P,
In which the TD-correlator
C
{right arrow over (p)},s′
TD(μ;{circumflex over (k)}d)
C
{right arrow over (p)},s′
FD(kσ;D)
in
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d)
and
C
ρ,{right arrow over (p)}
(4)(kσ;D).
Note that if the PUL algorithm is implemented in the interface between the TD- and FD-AFBs, as shown in
-ary symbol d{right arrow over (p)}∈
for data communication on the basis of complex-valued CCFs
and
All the filters, defined by (68), (72), and (77) in this FBMC can be realized by the polyphase filters (called Vaidyanathan's type 1 or type 2 polyphase[13]). But such subjects are beyond our present interest and so are omitted here.
In the radar detection technique as a typical example of communication exploiting the non-commutative property (NCP) of time and frequency shift (TFS)s, a pair of TFSO of type 3 with td, fD
t
,f
and TFSO of type 4
{circumflex over (t)}
,{circumflex over (f)}
with estimated parameters
{circumflex over (t)}
d
,{circumflex over (f)}
D
at the receiver, has shown that a priori half shifts of td, fD play an important role in the estimation of those parameters. Other examples of communication systems exploiting the NCP of TFSs are given as follows.
Our discussion has been restricted to a single-target problem for simplicity. Of course, a simple way for detecting multiple targets is to adequately use the decision level r0 of the statistic for estimating Doppler and the r′0 of the one for delay, respectively in (32), (37) and to enumerate several targets as functions of r0.r′0.
Consider another approach to multiple target problems. Let {(kd,j, D,j)}NPathj=1 be Npath pairs of delay and Doppler. Divide the target space Θ′=[0,T)×[0,F) into 4 regions:
1=[0,T/2)×[0,F/2),2=[0,T/2)×[F/2)×[F/2,F),
3=[T/2,T)×[0,F/2),4=[T/2,T)×[F/2,F)
and assign TD- and FD-PCs:
χ={X,X′},={Y,Y′},
={Z,Z′},
={W,W′}
to each region. Suppose that the 2-D PCs
χ,,
,
have their chip address sets, defined as
and their associated signature waveforms, defined by
where χ(1), χ(2), χ(3), χ(4) correspond to
χ,,
,
.
Then the CE of the radar signal has the multiple targets form
If this CE is transmitted through Npath doubly dispersive channels with delay, Doppler, and attenuation factor, respectively denoted by
{(td,i,fD,iAieiπ
then the signal component of its received signal is given by
While, the receiver uses the 2-D PC
χ(or )
for the TD-template CE of type 3 and FD-template CE of type 4
ψρ′,{right arrow over (ρ)}(3)[k] and Ψρ,{right arrow over (ρ)}(4)[]
in (29), (33) and restricts the range of controlling
μ and kσ,
respectively of the CCR of type 3 in (38) and of the CCR of type 4 in (43):
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d) and Cρ,{right arrow over (p)}(4)(kσ;
D)
to
for R2, then it gets the MLEs for the target subspaces R1,R2. Similarly the receiver gets MLEs for other regions R3,R4. This technique is based on the philosophy behind the CDMA and so is referred to as the code-division multiple target (CDMT)s.
Numerical simulations with N=N′=64, Npath=4, SNR ≥5 dB showed that 3,4 targets are successfully detected with probability 80% by using the PUL algorithm.
As shown in the proof of the convergence of PUL algorithm, TD-CCF of type 3 and FD-CCF of type 4
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d) and Cρ,{right arrow over (p)}(4)(kσ;
D),
respectively using the orthogonal projection operators P3 onto the Ts-time limited (TL) TD-space and P4 onto Fs-band limited (BL) FD-space tell us that a pair of TFSOs of type 3 and of type 1:
(,
)
(or that of the frequency dual of TFSO of type 3 and the TFSO of type 2:
(,
))
is an inherent operator due to the NCP of delay and Doppler, and thus plays an essential role in radar and multiplexed communication systems because such a pair contains either a pair of two unknowns
(D,kd)
or that of two control-parameters
(μ,kσ)
for getting a pair of two MLEs
(D,{circumflex over (k)}d).
On the contrary, the modulation and demodulation TFSOs
,
and the data-level and chip-level TFSOs
pNM,p′N′M′
d
ρM,0
d,ρ′M′,0f,d,
with data-level and chip-level addresses
((ρ,ρ′),{right arrow over (p)})
are independent of (D,kd).
While, the TD- and FD-CCF pair in (41), (45)
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d),Cρ,{right arrow over (p)}(4)(kσ;
D),
with address parameters
((ρ,ρ′),{right arrow over (p)}),
contains the data
d
{right arrow over (p)}
together with several PDs.
This suggests that if the artificial parameters
{kd,i,D,i}i=1N
(for simplicity setting
A
i
e
iκ
=1)
are embedded in
d
{right arrow over (p)},
then such parameters are available for a multiple target problem. For example, consider communication systems based on the NCP of delay and Doppler using Npath pairs of TD-TFSOs and TD-CCFs and Npath pairs of FD-TFSOs and FD-CCFs, defined as
with Npath pairs of parameters
{(kd,i,D,i)}i=1N
Such systems use Npath pairs of TD-, FD-templates embedded in transmitted TD- and FD-signatures and PUL algorithm for estimating
(kd,i,D,i),
where the PUL algorithm was called the “active” PUL[25, 30] in the sense that updating is done at the transmitter; but the adjective active was something of a misnomer because updating is applicable to the receiver only. The above MLE is based on identifying Npath pairs of known shifts
{(kd,i,D,i)}i=1N
from signals with embedded those shifts. Namely, it is relatively different from the single-target problem. Hence the above PUL is an ordinary PUL.
The CDMT technique can be applied to
-ary
phase-shift-keying (PSK) communication with data
Using Npath 2-D PCs and two twiddle factors
,
, 0≤
,
≤
−1
placed in the front of N′ TD-CCF and N FD-CCF arrays
c
ρ′,{right arrow over (p)}
(3)(μ;{circumflex over (k)}d) and Cρ,{right arrow over (p)}(4)(kσ;
D),
called a “phase tuned layer (PTL)” and replacing the
operations for MLE in the PUL by
by augmenting two PTL variables
,
.
one can get a PTL that plays a role of replacing
d
{right arrow over (p)}
of the pair of CCFs in (41), (45) by
d
{right arrow over (p)}
·
,d
{right arrow over (p)}·
with the result that phase errors in the real parts of the CCFs have Gaussian distribution with mean 0.
The inventor can get good numerical simulation results with N=N′=16,
=8,16
and SNR 30 dB. However, when
=16,
sidelobes arise the left- and right-hand sides of the main lobe in the distribution, which means decoding errors may occur. Hence only use of simple PD-cancellation methods limit to setting
=8.
Thus, in order to try M-ary communication with
≥32,
one can divide the delay-Doppler space Θ′ and the signal TFP S into
delay-Doppler sub-spaces and sub-TFPs and assign
to each sub-TFP. Moreover, let us suppose that artificial shifts
(kd,i,D,i)
are around the center of i-th delay-Doppler sub-space and use the PTLs of W8=exp(−i2π/8) and WM=exp(−i2π/M), placed in the front of TD- and FD-CCFs, namely,
This results in the following replacement of the PTL for
d
{right arrow over (p)}
and arg max operation
Hence one should require
CCFs in total. Therefore, one can obtain numerical decoding simulation results for 128-PSK, 256-PSK using 16, 32 2-D PCs. This M-ary PSK communication realized a new class of multiplexing communication, called “delay-Doppler space division multiplex (dD-SDM)”[35, 34]. However, the phase resolution of this system remains within WM=exp(−i2π/M). So, it is consequently unable to realize a high
-ary PSK communication. A new technique to realize a high M-ary PSK communication is proposed as follows.
The inventor can give communication systems with modulation and demodulation of high
-PSK in cooperating with establishing synchronisation, that are capable of being used also as a synchroniser for communication and or radar. First of all, one can provide explanations of
-ary PSK encoder and decoder as follows.
-ary PSK communication.
There are two different kinds of states: one is synchronizer/radar state (when the data dq→ is on the upward sides of the switches 1-1,1-2) and another is
-ary PSK communication state (when the data dq→ is connected to the down-ward of the switches 1-1,1-2). Each state is controlled by the switches. When
d
{right arrow over (q)}
is
-ary PSK, i.e., the data communication state, the switches are on downward.
The leftmost part of the transmmitter is the input
d
{right arrow over (q)}.
When the transmitter is in the synchronizer/radar state, it performs four procedures to a chip waveform:
while when the transmitter is in the data communication state, it performs six procedures to a chip waveform:
Next, in the block of the k-encoder, consisted of the j′-th AC chosen from
parallel ACs, where the symbol
denotes the set of the ACs and the dot . simply its one.
In accordance with an embodiment of the present invention, the encoder-switches 1-3, 1-4 choose the j-th AC in the block of the k-encoder, according to j, and the transmitter phase-modulates the output of the encoder by the j′-th PSK symbol
which is determined by an encoding of k with integers j,j′, defined as
When the switch 1-2 is on upward and downward sides, respectively
TD-CEψ[k] and FD-CEΨ[]
in (27) and
TD-CEψAC[k] and FD-CEΨAC[]
as defined below in (87) respectively, might be passed through the switch. Furthermore, in accordance with an embodiment of the present invention, the transmitter (or the encoder) modulates those CEs by a carrier
c,
transmits the resultant passband (PB) signal through the main channel (MC) with shifts
(kd,D),
and demodulates the noisy PB signal, contaminated by noise, by the carrier
c,
resulting a received signal.
The explanation of the block-diagram shown in
If a high
-PSK modulated symbol
is transmitted and is contaminated by both phase noise and additive noise, then a signal
should be resolved. However, its resolution is unrealizable. To solve it, in a k-encoder (the right block in the lower part of
so as to transmit a lower
0
-PSK modulated symbol
While to transmit the encoded integer
the transmitter relies heavily on the joint estimation method of delay and Doppler with high precision, proposed by this disclosure. Namely, the transmitter firstly divides the (td, fD)-parameter space Θ′ and the signal time-frequency plane (TFP) S equally into
sub-parameter spaces Θ(i),′ and sub-TFP S(i); secondly assigns 2-D PC χ(i) to each sub-TFPs; thirdly 2-DBPSK modulates a chip waveform by
2-D PCs; fourthly combines those BPSK-modulated signals by
-multiplexing; fifthly assumes a situation that the resultant multiplexed signal is transmitted through the j-th channel, called the j-th artificial channel (AC), chosen from
parallel ACs, according to j, so that those multiplexed signal is time-frequency shifted by about the center of the sub-parameter space Θ(j),′; sixthly PSK-modulates the output of the j-th AC by the j′-th
0
-PSK symbol
to get a transmit CE. The transmitter modulates this CE by the carrier
c
and inputs the resultant PB signal to the MC with shifts
(kd,D).
External noise is added to the output of the MC. The transmitter demodulates the noisy PB signal by the carrier
c.
The resultant BB signal yields a received CE.
Next, the explanation of the block-diagram as shown in
-ary PSK communication.
When the receiver is in the synchronizer/radar state (i.e., the switch 2-1 is on the upward side), it performs two procedures to a chip waveform:
while when the receiver is in the data communication state (i.e., the switch 2-1 is on the downward side), it firstly decodes k in the k-decoder, denoted by k{circumflex over ( )} and defined as
and secondly performs two procedures a chip waveform:
so as to select the j{circumflex over ( )}-th AC from
parallel ACs, where the symbol
denotes the set of the ACs and the dot . simply its one.
In accordance with an embodiment of the present invention, when the switch 2-1 is on downward side, the decoder chooses the
ĵ
-th AC in the block of the k-decoder, according to j{circumflex over ( )}, and the receiver phase-modulates the output of the AC by the j{circumflex over ( )}′-th PSK symbol
to get a template CE. The receiver calculates the CCR between the received CE and this template CE, in the COR-block of the lower part of the diagram of
The symbol COR means correlation. The big block in the right-hand-side of the COR indicates the block of maximum likeklihood estimating
(kd,D),
and Aeiκ, recovering
d
{right arrow over (q)},
and decoding k{circumflex over ( )} for the selected chip-level addresses (ρ′,ρ): When the receiver is in the synchroniser/radar state (i.e., the switch 2-2 is on the upward side), it performs
While when the receiver is in the data communication state (i.e., the switch 2-2 is on the downward side), it performs
The explanation of the block-diagram shown in
Next, consider a situation that a received CE is 2-DBPSK demodulated by a 2-D PC phase modulated chip waveform. When the receiver is in the the data communication state (i.e., the switch 2-1 is on downward), it 2-D BPSK modulates a chip waveform by the estimated
ĵ
χ(ĵ)
and inputs the 2-DBPSK modulated signal to the
ĵ
-th AC, chosen from
parallel ACs and
0
-PSK demodulates the output signal of the AC by the symbol
.
In the final step in the MLE block, the real parts of TD-CCFs and FD-CCFs, normalized by the PC and the attenuation factor
[cρ′,{right arrow over (p)},ĵ,ĵ′AC,(3)(
μ;{circumflex over (k)}d,j,j′)/(X′ρ′Aeiκ)],
[Cρ′,{right arrow over (p)},ĵ,ĵ′AC,(4),(kσ;
D,j,j′)/(XρAeiκ)]
are maximized in terms of 4 variables
(ρ′,{right arrow over (p)},ĵ,ĵ′),(ρ′,{right arrow over (p)},ĵ,ĵ′)
(kd,D).
If the PUL algorithm converges, then the receiver gets convergence
(k*d,*D).
Thus the convergence is common in two states. When the receiver is in the synchroniser/radar state (i.e., the switch 2-2 is on the upward side), it recovers
d
{right arrow over (q)},
while, when the receiver is in the data communication state (i.e., the switch 2-2 is on the downward side), it decodes k in the form
k*=j*
0
+j*
,′
by using MLEs for
(ĵ,ĵ′).
Note that even when the receiver is in the usual synchroniser/radar state (i.e., the switch 2-2 is on the upward side), it is capable of low
0
-PSK modulating/demodulating to convey data
d
{right arrow over (q)}.
e.g.,
0=8.
When the transmitter is in the synchroniser/radar state, the main lobe locates at
(kd,D)
and it is discriminated from side lobes, where the double-wave symbol denotes background noise.
Three parallel ACs are connected in series in the front of the MC in order to encode k using the
-PSK modulated signal. The MC still remains at
(kd,D).
However, when the ACs are added to the MC and the three series channels, MC+AC0,MC+AC1,MC+AC2 locate at
(kd+kd(0),D+
D(0)),(kd+kd(1),
D+
D(1)),(kd+kd(2),
D+
D(2)),
respectively.
The time shifts and frequency shifts are simply additive in quantitative terms, However, several new PDs simultaneously arise and are accompanied by group-theoretical property. In order to obtain the distribution as shown in
different values, respectively.
In
(kd(0),D(0)),(kd(1)
D(1)),(kd(2),
D(2)),(kd(3),
D(3)),
is attached to it, associated as a third variable of the TFP besides the T and F coordinate axes.
As shown in
(kd,D)
having its fundamental unit plane
[0,Ts)×[0,Fs).
That is, if a target exists around the TFP with data-level address
{right arrow over (p)}=(p,p′)
i.e., around the neighborhood of sub-TFP [(p−1)Ts·pTs)×[(p′−1)Fs,p′Fs) then the target should be discriminated within the data-level address
{right arrow over (p)}=(p,p′), 1≤p≤P−1, 1≤p′≤P′−1
Hence the argmax operation on the two real parts of the CCFs in
{right arrow over (p)}
regardless of the use of data communication.
As discussed above, in the transmitter and receiver, respectively in
The systems connected to the upward sides of the switches 1-1, 1-2, 2-1, 2-2 in
In accordance with an embodiment of this disclosure, in the transmitter and receiver system, in order to transmit
-ary PSK
modulated symbol
and to embed efficiently “information” k into its transmit signal, for an integer
0, 1≤0≤
,
the transmitter prepares
pairs of 2-DBPSK codes, i.e., pairs of TD-PCs of period N
X
(j)
={X
m
(j)}
and FD-PCs of period N′
X
(j),
′={X
m
(j),t}
and divides the fundamental unit plane Θ′=[0,Ts)×[−Fs/2, Fs/2) of the delay and Doppler-parameter space
Θmax=[0,Tmax)×[−Fmax/2,Fmax/2)
and the signal's TFP S=[0,Ts)×[0,Fs) equally into
sub-parameter spaces and sub-TFPs, respectively denoted as
where Ts=NMΔt, Fs=N′M′Δf denote the time duration and band-width of the symbol
d
{right arrow over (q)}
,{right arrow over (q)}=(q,q′),
Tmax=PTs, Fmax=P′Fs denote the maxima of delay and Doppler to be detected with integers P, P′, and the sampling intervals Δt, Δf.
Next the transmitter decomposes information k given as
and transmits an integer pair (j, j′), i.e., an integer part
and its fraction fraction part
in place of k. To transmit the encoded integer pair (j, j′), the transmitter 1) 2-D BPSK-modulates a chip waveform of time duration Tc=MΔt and bandwidth Fc=MΔf by 2-D PC X(j); 2) gets a 2-D phase-modulated signal, called a TD-signature and its Fourier Transform (FT), called an FD-signature; 3) time-frequency shifts by data-level shift (Ts, Fs) and combines these time-frequency shifted signatures in a non-overlapping form to obtain a
signal. Next the transmitter assumes that the multiplexed signal may transmit through an artificial channel (AC) with delay-Doppler shifts
(kd(j),D(j)),
located at around the center of the sub-parameter space Θ(j),′. To simulate such a situation, the transmitter makes two TFSOs of type 4
,
act on the multiplexed TD- and FD-signals, respectively. Thirdly, the transmitter PSK-modulates the resulting time-frequency shifted TD-signal by
0-ary
symbol
(i.e., multiplies the PSK-modulated signal by the symbol
d
{right arrow over (q)},
), and combines these (P/P′) signals in non-overlapping form on the TFP to generate TD-CE and its DFT, FD-CE (cf.
0
-ary PSK communication, in place of (27), given by
where
υ[k;χ(i)],V[;χ(i)]
denote TD- and FD-signatures (cf. (25)). defined by
where
denotes the i-th sub-set of the chip address set {(m,m′)}N−1,N′−1m=0,m′=0 of S, associated with the i-th sub-TFP S(i) as shown in
The TD- and FD-CEs in (87) are designed to convey the encoded integers (j,j′) of k to receivers through the MC. However, M-ary communication leads us to set the periods of the TD-PCs and FD-PCs to be nearly
multiples of those for synchronization of communication or target detection to keep estimation precision (a situation that the switches (1-1,1-2,2-1,2-2) are on the downward and upward sides in
Eq. (87) tells us that TD- and FD-signatures
υ[k;χ(i)],V[;χ(i)]
are firstly 2-D BPSK-modulated by 2-D PC X(i) and secondly the resultant phase modulated signatures are combined in non-overlapped form such as a
signal and thirdly such a doubly-multiplexed signal is PSK-modulated by
0-ary
symbol
and fourthly the PSK-modulated signal is transmitted through the j-th AC with shifts
(kd(j),D(j)),
and fifthly is PSK-modulated by
0-ary
data symbol
d′
{right arrow over (q)}·
The final signal is the TD-CE
where
d′
{right arrow over (q)}
with data-level address q→=(q, q′) has the value of 1.
At the receiver, an estimated k, denoted by k{circumflex over ( )}
{circumflex over (k)}, 0≤{circumflex over (k)}≤−1
is decomposed with a decoded pair of integers
The receiver first makes the estimated TD- and FD-TFSOs of type 4
and
representing the j{circumflex over ( )}-th-AC with shifts
(kd(ĵ),D(ĵ))
act on such TD- and FD-signatures, respectively and secondly PSK-demodulates those time-frequency shifted TD- and FD-signatures by
0-ary
to cancel out the PSK-modulation phase (cf.
The CCFs of type 3 and of type 4 exploiting AC shifts, with encoded integer pair (j,j′) and its augmented pair for argmax operations
(ĵ,ĵ′),
are concretely given by respectively, in place of (38) and (43)
each of which is called a correlator of type-3 with AC shifts and a correlator of type-4 with AC shifts, respectively. In the transmitter-signal-part (cf. (87)) of the two CCFs above, (i.e., the input part of statistic g (13)) the TFSO of type 4 (or its frequency dual)
or
is inserted between the product of chip- and data-level shifting TFSO of type 4 (or its frequency dual) at the transmitter (cf. υ[k;X(i)], V[;X(i)] in (88))
qNM,q′N′M′
d·m,M,m′M′d or
q′N′M′,−qNMf,d·
m′M′,−mMf,d
and TFSO of type 3, representing the MC with shifts
(kd,D),
While, in the receiver-signal-part of the two CCFs (89), (90) (i.e., the signal-to-be-detected part of statistic g(13)), for
-ary PSK modulation, there are
different pairs of AC shifts
(kd(j),D(j))
Then the associated estimated TFSOs of type 3 representing those ACs
(or their frequency dual ones
)
are defined. These TFSOs are inserted between the estimated TFSO of type 1
(or the estimated TFSO of type 2
{circumflex over (l)}
,−k
f,d)
for estimating the MC shifts
(kd,D)
and the product of chip- and data-level shifting TFSOs of type 4 or its frequency dual at the receiver
pNM,p′N′M′
d·n,M,ρ′M′d or
p′N′M′,−pNMf,d·
n′M′,−ρMf,d
The transmitter/receiver system shown in
-ary PSK communication is a typical example of multiplexing communication systems utilizing non-commutative delay and Doppler shifts of the MC and AC parameter spaces of co-dimension 2, where the fundamental unit plane Θ′ of the delay and Dopler parameter space of the MC is exclusively divided, denoted as Θ(i),′. So, it is obvious that the term “delay-Doppler space division multiplexing (dD-SDM)” [30], patent[6] is used to refer such a multiplexed system. In this system, in place of transmitting
-ary data k, the transmitter sends a signal containing encoded integers
The integer j is recovered by using the PD due to the delay shift
k
d
(ĵ)
and Doppler shift
D
(ĵ),
located at around the center of Θ(j∧),′ (cf.
-multiplexing, thirdly inputs it to the
ĵ
-th AC, and fourthly
0
-PSK demodulates the output signal of the AC by the symbol
different values of
ĵ,
the transmitter requires
parallel ACs and the set of those associated 2-D PCs
While, at the receiver to detect the template from a fourfold-multiplexed chip waveform, the receiver selects an estimated
ĵ
-th AC and constructs maximum-likelihood estimates of
(kd,D).
-ary PSK communication at the transmitter is realized by
-multiplexed low
0
-ary PSK modulations based on using non-commutative AC shifts and 2-D PCs.
Such a multiplexed system, called “delay-Doppler space division multiplexing (dD-SDM)” is a system by dividing the signal space S=[0,Ts)×[0,Fs) into several sub-TFPs and assigning a 2-D PC to each sub TFP for 2-D BPSK modulation. The use of 2-D PC results in adding a new axis perpendicular to the TFP as a third axis (see
The references[30] and patent[6] regarded a multiple targets problem with
{(kd,i,D,i)}i=1N
as an application of Code-Division Multiple Target (CDMT)s and gave an target-detection method using Npath 2-D PCs and TFSOs with shifts
{(kd,i,D,i)}i=1N
together with the phase-cancellations by the product of two twiddle factors
However, this method forces us to resolve small phase quantity
Thus when
≥16,
decoding errors arise due to phase noise and phase distortions. Furthermore, in the references above, without using both the TFSO of type 3
representing MC shifts
(kd,D)
and its estimated TFSO of type 4
,
-ary communication were done. Moreover, this approach was not suited to multiple targets problems of a doubly delay-Doppler shifted signal and has no estimation algorithm with guaranteed proof.
To solve this, in accordance with an embodiment of the present invention, the inventor defines two new CCFs exploiting the NCP of AC shifts
c
ρ′,{right arrow over (p)},ĵ,ĵ′
AC,(3)(μ;{circumflex over (k)}d,j,j′),Cρ,{right arrow over (p)},ĵ,ĵ′AC,(4)(kσ;
D,j,j′)
respectively defined by (89), (90), in which several PDs are estimated and canceled out, precisely and rigorously. It is not easy to recover
d′
{right arrow over (q)}·
embedded in TD-CE in (87)
by
0-ary PSK
The detail of cumbersome calculations of (89), (90) are omitted here. However, the two CCFs for high
-ary PSK demodulation are required by using
and replacing
(kd,D)⇒(kd+kd(j),
D+
D(j)),d{right arrow over (p)}
d′{right arrow over (p)}
at the transmitter and
({circumflex over (k)}d,D)⇒({circumflex over (k)}d+kd(ĵ),
D+
D(ĵ)),d{right arrow over (q)}⇒d′{right arrow over (q)}
,(N,N′)⇒(N1,N′1)
at the receiver. This replacement is accompanied by the occurrence of several new PDs. Hence one should replace 2 variables for MLE operation of 2 variables
argmaxρ′{right arrow over (p)}
(or
argmaxρ,{right arrow over (p)})
by the one of 4 variables
argmaxρ′{right arrow over (p)},ĵ,ĵ′
(or
argmaxρ,{right arrow over (p)},ĵ,ĵ′),
where
ρ′∈IF(ĵ)
(or
ρ∈IT(ĵ)).
One can obtain generalized versions of (39), (44):
where
{tilde over (τ)}0[kσ,ĵ]=kσ−kd+kd(ĵ)−kd(j),{tilde over (ν)}0[μ,ĵ]=
μ
D+
D(j)−
D(j),
I
T
(ĵ)
,I
F
(ĵ)
denote the sets of chip TD- and FD-addresses n,n′, associated with the j{circumflex over ( )}-th sub-TFP
S
(ĵ),
and the product of the address sets
I
(ĵ)
=I
T
(ĵ)
×I
F
(ĵ).
The ambiguity function (AF)s: θgg[τ,υ] and ΘGG[τ,υ] respectively in (91) and (92) with Gaussian functions g,G are separable and decay exponentially in terms of τ,υ. This well-known property and the disjoint chip address sets
I
(ĵ)
,I
T
(ĵ)
,I
F
(ĵ)
and the fact that
(kd(j),D(j))
is at around the center of Θ(j) tell us that among the first and second arguments of the AF all of the terms relating to (p−q)MN1; (p′−q′)M′N1′ with p′≠q′, p≠q and
(m,m′)∈I(i),n,ρ∈IT(ĵ),n′,ρ′∈IF(ĵ)
with i≠j{circumflex over ( )} can be neglected. Thus, the terms with p′=q′, p=q and
i=ĵ
remain and thus only the term equal to
{right arrow over (p)}
is selected from the summation
and
ĵ=j
is identified via the functions τ˜0[kσ,j{circumflex over ( )}],ν˜0[μ,j{circumflex over ( )}′] with kσkd,
μ≃
D.
This is the reason why the transmitter divides exclusively Θ, 2-D BPSK modulates a chip waveform by 2-D PCs χ(i), and combines these modulated signatures in non-overlapped form to realize
-multiplexing based on
parallel ACs that are intended to introduce non-commutative shifts. While
ĵ′=j′
is identified by the PD cancellation term
.
The two CCFs above use independent 2-D PCs
and PDs due to non-commutative AC shifts
(kD(ĵ),D(ĵ))
as functions of an estimated integer
for an encoded integer
of an integer k.
The TD- and FD-CCFs provide us with a decoding method using PUL-based MLEs. This method can guarantee the phase resolution at least
and hence is resistant to phase noise.
It is natural to precisely estimate delay and Doppler for radar and establish synchronization of communication systems before encoding and decoding
-ary PSK data. Since one should confront a realistic situation that a transmitted signal through the MC with shifts
(kd,D)
is processed at the receiver, we need two CCFs with AC shifts in (91) and in (92). Such receivers when the switches are on upward in
-ary PSK communication, and work as a radar equipped with data transmission. This system is an amplitude-shift-keying(ASK)-free
-ary PSK communication and provides a secret wireless communication system whose secret keys are 2-D PCs
The main reason that receivers employ ASK-free modulation is twofold: (1) adaptive and dynamic estimation of the attenuation factor Aeiκ; (2) a simplification of a decoder. Estimating and Canceling out PDs enable us to realize high
-ary PSK communication as an efficient use of radio resources. Note that the use of
2-DPCs implies
-user multiplexed communication.
Daughman [33, 34] gave a 2-D version of Gabor elementary functions
g
m,m′(t)g(t−mTc)ei2πm′F′
and pointed out that 2-D Gabor representation(to be defined below) is useful for image analysis and segmentation. Before discussing Daughman study, one should consult the Heisenberg group theory reviewed by Howe [5].
t,τ,ν∈
n
define two shift operators
T
τ
f(t)f(t−τ),Mνf(t)
ei2πν·tf(t). (93)
where ν·t denotes the inner product. For
z∈
introduce the scalar operator
S
z
x(t)zx(t).
{MνTτSz:ν,τ∈n,z∈
}
is a group of unitary operators on
L
2().
H=
n×n×
,
called the (reduced) Heisenberg group of degree n and define on it a law of combination
(ν1,1,z1)(ν2,
2,z2)=(ν1+ν2,
1+
2,e−2πiν
Define a map ρ, i.e., a faithful unitary representation of H and an automorphism r of H by the rule
Then it is immediate to verify that the FT, denoted as {circumflex over ( )} intertwines the two representations of H
ρ and ρor
with each other. That is, [5]
{circumflex over ( )}ρ(h)=ρ(r(h)){circumflex over ( )}. (96)
Daughman[35] pointed out that the model of the 2-D receptive field profiles encountered experimentally in cortical simple cells, which captures their salient tuning properties of spatial localization, orientation selectivity, spatial frequency selectivity, and quadrature phase relationship, is the parameterized family of “2-D Gabor filters”. In terms of a space-domain (SD) impulse response function (or a complex Gabor function) g(x) with SD variables
x=(x,y)
and its associated 2-D FT, i.e., spatial frequency-domain (SFD) function
ĝ(u)=2[g(x)]
with SFD variables
u=(u,υ),
the general functional form of 2-D Gabor filter family is specified in [35, 36]
where P denotes the phase of the sinusoid such that the dc component is zero, and
(x−x0)r,(y−y0)r
stand for a clockwise rotation of the operation by θ such that
The SD function g(x), SFD function
ĝ(u)
are perfectly symmetrical and have their shifts
x
0=(x0,y0), u0=(u0,v0)
at which 2-D Gaussian envelope has the peak value. An important property of the family of 2-D Gabor filters is their achievement of the theoretical lower bound of joint uncertainty in the two conjoint domains of x and u. Defining uncertainty in each of the four variables by the normalized second moments
√{square root over (σx2)},√{square root over (σy2)},√{square root over (σu2)},√{square root over (σv2)}
about the principle axes, a fundamental uncertainty principle exists[35, 36]:
√{square root over (σx2)}√{square root over (σu2)}≥¼π, √{square root over (σy2)}√{square root over (σv2)}≥¼π (99)
and the lower bound of the inequality is achieved by the family of 2-D Gabor filters (97). where
u
0=(u0,v0),{right arrow over (α)}=(α,β)
are modulation, scaling parameters of Gaussian.
He proposed a simple neural network (NN) architecture for finding optimal coefficient values in arbitrary 2-D signal transforms which in general might be neither complete nor orthogonal as follows.
Consider some discrete 2-D signal I[x], say, an image supported on [256×256] pixels in x, which one wants to analyze or compress by representing it as a set of expansion coefficients {ai} on some set of 2-D elementary functions
{g[x;{right arrow over (θ)}i]}, {right arrow over (θ)}i=(x0iu0i,{right arrow over (α)}i),x0i=(x0i,y0i), u0i=(u0i,v0i),αi=({right arrow over (α)}i,βi).
The attempt to represent I[x] either exactly or in some optimal sense by projecting it onto a chosen set of vectors {g[x;θ→i]} is reduced to finding projection coefficients {ai} such that the resultant vector H[x]
minimizes the squared norm of the difference-vector E=∥I[x]−H[x]∥2. If a given image I[x] is regarded as a vector in an n-D vector space (e.g. n=65 536), then the optimal coefficients
{ai}i=1n
is determined by a system of n simultaneous equations in n unknowns; but it is impractical to solve this huge system. So he proposed NN architectures with connection strength
{g[x;{right arrow over (θ)}i]}
and input image I[x]. 2-D Gabor representations for image analysis and compression, edge detections, feature detectors etc. have been proposed. [37, 38, 39]
In this disclosure, the inventor points out that the 2-D Gabor expansion (100) can be regarded as a conjoint 2-D spatial/spatial frequency (S/SF) representation using non-commutative shifts (x0i,u0i) in terms of 2-D Gabor functions
{g[x;{right arrow over (θ)}i]}
with shifts (x0i,u0i): According to an important precedent for 1-D signal processings exploiting the non-commutativity of time-frequency shifts, presented in the previous sections in this disclosure, it is natural to consider SD representation (100) and its associated SFD representation, symmetrically. To do it, let 2-D FT of I[x] and H[x]
Î[u]=(2)[I[x]], Ĥ[u]=
(2)[H[x]].
Then the conjoint spatial/spatial frequency representation, in place of (100), is defined as
and let ε,ε′ be two subspaces of the Hilbert space. Suppose that
I[x]∈ε,Î[u]∈ε′.
The squared norms of errors
∥I[x]−H[x]∥2, ∥I[u]−H[u]∥2
are minimized by the use of two orthogonal projection operator (PO)s projecting onto εi, ε′i (defined below), respectively
{Pg(x;{right arrow over (θ)}
The projection theorem [27] tells us that
I[x],Î[u]
possess their unique decomposition
where g⊥(x;θ→i), g{circumflex over ( )}⊥ (u; θ→i) are orthogonal complements of g(x; θ→i), g{circumflex over ( )}(u; θ→i), respectively. Therefore it is important to choose a set of templates
{g(x;{right arrow over (θ)}2),ĝ(u;{right arrow over (θ)}i)}i=1n,
For example, let εi be the set of all f∈2(Z) that are (L1Δx×L2Δy) space-limited (SL) signals and let ε′i be that of (L1Δu×L2Δv) spatial frequency-limited (SFL) signals. Suppose that
g(x;{right arrow over (θ)}i)∈εi and ĝ(u,{right arrow over (θ)}i)∈ε′i.
P
g(x;{right arrow over (θ)}
)
,P
ĝ(u;{right arrow over (θ)}
)
are POs projecting onto εi,ε′i; respectively. Apply the von Neumann's APT to the two POs and we use 2 2×2-D localization operators (LOs) defined by
P
g(x;{right arrow over (θ)}
)
−1,(2),d
P
ĝ(u;{right arrow over (θ)}
)
(2),d
,P
ĝ(u,{right arrow over (θ)}
)
(2),d
P
g(x;{right arrow over (θ)}
)
−1,(2),d (103)
then one can get optimized expansion coefficients given as
a
i
˜P
g(x;{right arrow over (θ)}
)
I(x),a′i˜Pĝ(u;{right arrow over (θ)}
Such 2-D LOs select a limited 4-D cube
[L1Δx×L2Δy]×[L′1Δu×L′2Δv]⊂4
around the peak address (x0i,u0i) in the 4-D spatial/spatial frequency plane and filter out the rest. where
g
⊥(x;{right arrow over (θ)}i)∈εi⊥ and ĝ⊥(u;{right arrow over (θ)}i)∈εi⊥,t.
εi⊥,εi⊥,′
are orthogonal complements of
εi,ε′i.
Contrary to Howe's shift operators (93), for the SD- and SFD-signals
z(x),{circumflex over (z)}(u)
the inventor introduces “half-shifts” for the symmetrical property
e
−πu
·x
,e
iπx
·u
,x
0=(x0,y0), u0=(u0,v0)∈2
and uses 2×2-D von Neumann's symmetrical spatial-spatial frequency shift operator (SSFSO)s with those half-shifts, defined by
for the 2-D Gabor function with peak-address x0=(x0, y0), u0=(u0, v0)∈R2 in 2×2-D SD-SFD space
g(x;{right arrow over (θ)}i)
ĝ(u;{right arrow over (θ)}i)=(2)[g(x;{right arrow over (θ)}i)].
Thus, the inventor can rewrite the Gabor functions as
Consider the DFT of the above signals. Let (Δx,Δy) be the sampling intervals of 2-D space variables
x=(x,y)
with Cartesian lattice dimensions (L1,L2). Let
z[{right arrow over (k)}]
be a discrete SD function whose discrete space variable is
Then the sampling interval of 2-D spatial frequencies
u=(u,v)
appropriate to the L1, L2 spatial lattice are given by
Δu=1/(L1Δx),Δv=1/(L2Δy)
and its discrete spatial frequency variable is given by
Using 2 twiddle factors
(2),d[⋅], and 2-D IDFT −1,(2),d[⋅],
one can obtain the relationships between an SFD function
Z[]
and an SD function
z[{right arrow over (k)}]
Thus the SD function
z[{right arrow over (k)}]
(or the SFD function
Z[])
has support
L
1
Δx×L
2
Δy
(or
L
1
Δu×L
2
Δv).
If the spacing of the peak-address x0i (or u0i) on the SD space (or SFD space) is
(MxΔx,MyΔy) or (MuΔu,MvΔv)
and the normalization condition
M
x
Δx×M
u
Δu=M
y
Δy×M
v
Δv=1
is imposed, then
L
1
=M
x
M
u
, L
2
=M
y
M
v.
If discrete space/spatial frequency shifts of
g(x;{right arrow over (θ)}i),ĝ(u;{right arrow over (θ)}i)
are given as
then 2×2-D von Neumann's discrete SSFSOs become
These 2×2-D symmetrical SSFSOs contain 2-D half shifts
respectively, as phase terms of SD and SFD signals
z[{right arrow over (k)}],Z[].
Thus the SFSOs (108) may be useful for image analysis, feature extraction, and data compression because
(ai,bi) i.e., (x0i,u0i)
are efficiently extracted through the inner product of the projection coefficients in (102). Note that image analysis which involves a temporal change is considered to be a subject of a signal processing exploiting three-dimensional NCP.
Embodiments, mathematical formulae and figures, described in a written description of the invention, and of the manner and process of making and using it are merely just ones of examples of the process and/or method of the invention. The inventor can honestly say that some of embodiments are realized examples of communication systems exploiting the NCP of TFSs, that Gabor prospected deliberately in the 1952 paper.[2]
The non-overlapping superposition of signals on the TFP necessarily entails PDs due to the NCP of TFSs. Most of researchers in the field of communication, however, has little or no understanding of TFSs with the NCP. Under such a situation, communication systems exploiting the NCP of TFSs are expected to provide a new class of digital multiplexed communication systems.
Communication systems exploiting the NCP of TFSs, given in this disclosure, can be restated from the following five different points of view.
(Viewpoint 1)
The symmetrical time and frequency shift operator (TFSO), defined and introduced in the references (patent[1] and [26, 27, 30]) has the properties: (i) its “half shifts”, i.e., TD- and FD-PDs that are embedded into TD- and FD-signatures makes clear shift's important role in parameter estimation; (ii) the PD due to modulation and demodulation by the carrier lc, accompanied by the delay kd
always arises; (iii) the fact that any PDs being embedded into a signal can be compactly expressed in terms of powers of the twiddle factor
makes estimations of several PDs and canceling out the PDs easy.
(Viewpoint 2)
The TD- and FD-PC modulation, i.e., 2-D binary phase-shift-keying (BPSK)) is a well-known modulation technique. However if one can regard it as an example that TFSOs are available, then one can see that TD- and FD-templates are embedded in a wide-band signal modulated by 2D-PCs.
(Viewpoint 3)
The non-overlapped superposition of signals on the TFP is a conventional method as an efficient use of radio resources. However, this superposition needs the use of TFSO that causes a data-level PD. Fortunately, such a PD plays an important role in parameter estimation.
(Viewpoint 4)
Contrary to conventional use of maximum likelihood (ML) TD-functionals only, the ML TD- and FD-functionals define arrays of TD- and FD-CCFs, respectively for detecting TD- and FD-templates. TD- and FD-CCFs define orthogonal projection operator (PO)s P3, P4 (or P1, P2), respectively projecting onto time-limited (TL) and band-limited (BL) spaces as subspaces of the Hilbert space. These POs provide a frame on which a deeper understanding of the APT can be built.
(Viewpoint 5)
Contrary to popular belief that a non-Nyquist Gaussian is useless,
The Alternative POs Representing a Conjunction of the Two POs
P
8
−1,d
P
4
d
(or
P
4
d
P
3
−1,d)
becomes a localization operator in cooperation with the use of Gaussian having properties: self-dual of the FT, separable and exponentially decaying Gaussian's AF in terms of delay and Doppler. This leads the fact that arrays of TD- and FD-CCFs yield excellent receivers.
The inventor makes all of the above viewpoints clear after publications in the patents and non-patent references. This disclosure features its own novelty. All features of novelty are prescribed as follows
(Point 1 of Novelty)
Multiplexed communication systems exploiting the NCP of the TFSs on the TFP are provided. Also algorithms for evaluating PDs due to the TFSs and for canceling out their PDs are given based on symmetrical TFSOs in (4), (24). Furthermore, such an operation pair of embedding- and canceling-out-PDs is shown to be of crucial importance.
(Point 2 of Novelty)
Doppler- and delay-estimate likelihood TD- and FD-functionals, respectively enable us to get ML estimators of parameters and realize canceling-out-PDs by their associated MLE correlators.
(Point 3 of Novelty)
It is shown that the classical BPSK modulation is causing PDs due to the NCP of its TFSs, but fortunately these PDs provide a clue about parameter estimation and signal reconstruction.
(Point 4 of Novelty)
Both the non-overlapped superposition of signals on the TFP and modulation (or demodulation) for a baseband (or passband) signal, accompanied by a delay, as a conventional technique in communication, induce PDs due to the NCP of their TFSs. Fortunately, embedding-and-canceling-out-PD techniques play an important role in detecting a data-level signal.
(Point 5 of Novelty)
All of the above PDs are canceled out by TD- and FD-CCFs. These CCFs induce TD- and FD-POs projecting onto the TL- and BL-signal spaces, respectively as subspaces of the Hilbert space. Applying the von Neumann's alternative projection theorem (APT) to the two subspaces defines the combined operator of the two POs, called a localization operator (LO) on the TFP.
(Point 6 of Novelty)
Youla's signal restoration method using the APT enables us to prove that the LO guarantees the convergence of the alternative parameter-updating algorithm, called PUL. This implies that the LO, as a function of data- and chip-level addresses, becomes a new kind of filters in place of a conventional sharp filter in the DSP and works well for a radar with no limitation on the maxima of delay and Doppler.
(Point 7 of Novelty)
A 2-D signal, i.e., image is usually described by conjoint spatial (S)/spectral frequency (SF) representations, 2-D Gabor representations in particular. However, many researchers using conventional techniques are unaware that 2-D S and SF shift (SSFS)s, implicitly involved in 2-D Gabor functions induce unexpectedly PDs due to their NCP like 1-D TFSs. The inventor treats an S domain (SD) signal and an SF domain (SFD) signals symmetrically and introduces conjoint S/SF representations of dimension 2×2 in terms of 2-D Gabor functions and their 2-D FTs, defined as (101), in place of the problem of representing image in terms of 2-D Gabor functions (100).
Such a new symmetrical conjoint problem of representing image (101) can be solved by the SD and SFD decomposition (102) using orthogonal projection operator (PO)s projecting onto SD- and SFD-limited spaces as subspaces of the Hilbert space, respectively defined as
P
g(x;{right arrow over (θ)}
)
,P
ĝ(u;{right arrow over (θ)}
).
2-D Gabor functions are defined using symmetrical space and spatial frequency shift operator (SSFSO)s (105) (or (108)). Furthermore, the combined 2×2-D localization operator (LO)s (103) of the two POs that the von Neumann's APT can be applied provide the basics of 2-D signal processing theory exploiting the NCP of SSFSs for purposes such as image analysis, feature extraction, and data compression.
An example as an embodiment of transmitter-receiver systems based on the above-described theoretical aspects is given together with cited figures.
According to an embodiment of the present invention,
(The Transmitter Apparatus 100)
The transmitter apparatus 100 is an apparatus that executes a program imposed on a transmitter as described in the above description.
As shown in
(The Acquisition Part of Transmitting Data 101)
The acquisition part of transmitting data 101 makes acquisition of data to be transmitted; The transmitting data is, e.g., acoustic data, image data, and may be text data such as any digitized data.
If the transmitter-receiver system with the number 1 is used for radar, then the transmitting data may be a radar pulse wave.
(The Generator Part of a Transmitting Signal 102)
The generator part of a transmitting signal 102 performs signal processing transmitting data obtained by the acquisition part of transmitting data 101 and generates a transmitting signal.
The generator part of a transmitting signal 102 comprises SFBs as shown in
As above mentioned, an example of processing for generators of a signal to be transmitted is the modulation by TD- and FD-phase code (PC)s of periods N,N′, i.e., 2-D BPSK modulation.
Examples of a transmitting signal generated by the generator part of a signal to be transmitted 102 are e.g., as mentioned above
The generator part of a signal to be transmitted with the number 102 may be an apparatus that executes a program imposed on a transmitter as described in subsection <7.1 transmitters for generating signature and radar signal>. Or the generator part of a signal to be transmitted with the number 102 may be an apparatus that executes a program imposed on a transmitter apparatus as described in subsections <8.1 multiple target detection using CDMT> and <8.2 multiple target detection using artificial delay-Doppler>.
(Transmitter Part 103)
The transmitter part 103 transmits a transmitter signal generated by the generator part of a signal to be transmitted with the number 102.
(Receiver Apparatus 200)
The receiver apparatus 200 is an apparatus that executes a program imposed on a receiver as described in the above description.
As shown in
(Receiver Part 201)
The receiver part 201 receives a transmitter signal, transmitted by the transmitter apparatus 100. Examples of a received signal that the receiver part 201 receives are e.g., as mentioned above
received TD-, FD-signals:r[k;χ],R[;χ],
(Shift Estimation and Received-Data Extraction Parts 202)
The shift estimation and received-data extraction parts 202, (simply called the estimation part) shifts a received signal by the receiver part 201 by estimated shifts and extract the received data.
The shift estimation and received-data extraction parts 202 comprises at least one of AFBs shown in
The shift estimation and received-data extraction parts 202 is an apparatus that executes a program as described in subsection <7.2 AFB; receivers and encoder design>. Moreover, the generator part of a signal to be transmitted 102 is an apparatus that executes a program as described in subsection <8.2 High M-PSK communication based on exploiting non-commutative time-frequency shifts>.
Thus, the shift estimation and received-data extraction parts 202 is a method for receiving a signal and executes a program according to the estimation step for estimating a time shift and a frequency shift from a received signal (cf.
The generator part of a signal to be transmitted 102 executes a program according to the shift steps for time-frequency shifting the signal to be transmitted (cf.
The non-commutative shift parameter space of co-dimension 2 is a 3-dimensionalized space version of the 2-D time-frequency plane with the time- and frequency-coordinates by augmenting non-commutative time-frequency shifts as a third coordinate (cf.
As above mentioned, a pair of unknown delay and Doppler
(kd,D)
is itself the pair of non-commutative shifts on the parameter space of co-dimension 2 like the pair of AC shifts. The parameter space is related to the shift operations acting on a signal on the time-frequency plane (TFP); Such a “shift plane” should be discriminated against the TFP (
Several comments to the patent[6]-patent[8] are listed as follows.
a) The phase distortion (PD)
(or its discretized version
that the modulation and demodulation by the carrier fc accompanied by the delay td (or kd) are causing was neglected in the above patents. While in this disclosure, this PD and the PD eiκ of the attenuation factor Aeiκ are jointly and co-operatively canceled out. The reason that the co-operative treatment of PDs is needed is as follows; Firstly, each PD is not independent of each other; Secondly, the PD is propagated due to its group-theoretic property; Thus it is not an easy task to cancel out these PDs precisely. Consequently, the process in updating estimates of time and frequency shifts should be undertaken in co-operation with the phase cancellations (i.e., simultaneous and parallel processing is necessary: e.g., (59)).
b) Time and frequency shifts in this disclosure are referred to as variables of the parameter space of co-dimension 2, associated with two variables: time and frequency on the signal's TFP, respectively. The non-commutative shift operations induce the PDs. While, time and frequency offsets in communication are deviations from the pulse's period and the carrier, respectively.
(1) the estimated time and frequency offsets in patents[7, 8] and time and frequency offsets in OFDM-based communication system in patents[7] are deviations from the period of a chip pulse and the carrier, respectively and correspond to the time shift td and the frequency shift fD in this disclosure, respectively. Namely, the above both of two patents can be regarded as conventional acquisition and tracking methods for synchronisation because these are unaware of the non-commutativity of td, fD.
(2) To avoid confusion between the intended shifts and offsets, the inventor uses the term: not the offsets but “half-shifts of the exponent of the exponential function” (in the sense that the conventional shifts are halved). One of main features in this disclosure is to design a transmitting signal so that its phase function always contains half shifts beforehand and to cancel out the PDs based on the group-theoretic property of the NCP of TFSs.
c) The patent[8] discussed the preamble of the received signal as a template signal only in TD but was unaware of signals in FD and gave a conventional channel estimation method, being unaware of the NCP.
(1) In order to give a convergence proof of the phase updating loop (PUL) algorithm, proposed in the reference patent[6], the inventor 1) defines subspaces of the Hilbert space εi, 1≤i≤4 to which the template belongs; 2) shows that each of N′ TD-CCFs, CCFs of type 3 (or CCFs of type 1) yields a projection operator P3 (or P1) projecting onto NMΔt- (or LΔt-)time limited (TL) Hilbert subspace ε3 (or ε1) and that each of N FD-CCFs, CCFs of type 4 (or CCFs of type 2) gives a projection operator P4 (or P2) projecting onto N′M′Δf- (or LΔf-) band limited (BL) Hilbert subspace ε4 (or ε2); 3) proves that the PUL algorithm using the localization operator (LO), i.e., the von Neumann's alternative projection Theorem (APT) operator P3F−1,dP4Fd (or P4FdP3F−1,d) for 2-D template matching on the TFP converges on the intersection of the TL Hilbert subspace and the BL Hilbert subspace (F−1,d,Fd denote IDFT and DFT); furthermore, 4) shows that the interaction is empty by the Youla's theorem and that the PUL can give estimates of td, fD with precision within time duration LΔt and bandwidth LΔf of a Gaussian chip waveform and with computational complexity of but O[N+N′] of not O[N×N′], where N,N′ denote the numbers of TD- and FD-CCFs (L=(ΔtΔf)−1=MM′).
(2) The MMSE algorithm for estimating the channel characteristics, proposed by the patent[8] is based on Bayes's rule and 2-parameter estimation method. Thus this algorithm has computational complexity of O[N×N′]. While this disclosure uses a large number of likelihood functional (LF)s that comprises N′ TD-LFs for estimating Doppler and N FD-LFs for estimating delay. Each LF is based on Bayes's rule. One TD-LF among TD-LFs and FD-LF among FD-LFs in TL-TD and BL-FD Hilbert subspaces, respectively are chosen alternatively and in the sense of maximum likelihood estimate (MLE). Thus the alternative algorithm with computational complexity of O[N+N′] is radically different from conventional methods.
The shift estimation and received-data extraction parts 202 executes a program e.g., a program according to the estimation step for estimating a time shift and a frequency shift from a received signal, as described in (84) and its related description. The estimation step uses the TFSO of type-1 with estimated time shift and frequency shift
and the TFSO of type-2 with time shift and estimated frequency shift
and estimates a time shift and a frequency shift that are embedded in the received signal.
The estimation step mentioned above uses 4 different types of TFSOs: a TFSO of type-1 with estimated time shift, frequency shift, and a phase term representing a half shift of the estimated time shift
a TFSO of type-2 with time shift, estimated frequency shift, and a phase term representing a half shift of the estimated frequency shift
a TFSO of type-3 with an observable time shift to be estimated, an observable frequency shift to be estimated, and a phase term representing a half shift of the time shift to be estimated
(or a frequency dual of the TFSO of type 3
),
and a TFSO of type-4 representing the estimated time shift, the estimated frequency shift, and the phase term due to the half shift of the estimated time shift
(or a frequency dual of the TFSO of type 4
),
and estimates a time shift and a frequency shift that are embedded in the received signal.
The phase function of a TD-signal to be transmitted contains the phase term representing half shifts of one or more time shift-parameters; The phase function of an FD-signal to be transmitted contains the phase term representing half shifts of one or more frequency shift-parameters.
The receiver apparatus 200 mentioned above can realize a receiving apparatus with high efficiency. For example, when the receiver 200 is used for a radar receiver, it becomes a high rate receiving apparatus.
As described in (84) and its related description, in the estimation step above in the program that the shift estimation and received-data extraction parts 202 executes, the receiver uses the CCF of type-1 (C.f., MF309, (38)) that is represented by using the TFSO of type-1 mentioned above
c
ρ′,{right arrow over (p)}
(3)(u,i;{circumflex over (k)}d,i)
and the CCF of type-2 (C.f., MF336, (43)) that is represented by using the TFSO of type-2 mentioned above
C
ρ,{right arrow over (p)}
(4)(kσ,i;D,i),
refers to these CCFs, and estimates a time shift and a frequency shift that are embedded in the received signal as an example.
As mentioned above, the above CCF of type-1 can be represented by (c.f., MF309, (38))
and the above CCF of type-2 can be represented by (c.f., MF336, (43))
In the estimation step above in the program, the receiver apparatus 1) uses the CCF of type-1 with AC shifts that is represented by using the TFSO of type-1, of type-3, and of type-4, mentioned above, (cf. MF715, (89)).
and the above CCF of type-2 with AC shifts that is represented by using the TFSO of type-2, of type-3, and of type-4, mentioned above, (cf. MF715, (90)).
2) refers to these CCFs, and 3) estimates a time shift and a frequency shift that are embedded in the received signal as an example.
As described above, the TD-CCF of type-1 with AC shifts (see MF715, (89)) and the FD-CCF of type-2 with AC shifts (see MF715, (90)) contain the PD due to the NCP of modulation and demodulation by the carrier fc (or lc), accompanied by the delay td (or kd)
e
i2πt
f
or its discretized version
and the PD due to the phase term eiκ of the attenuation factor Aeiκ.
As described in MF400, (cf. (60), (61)) and in its related description, the shift estimation and received-data extraction parts 202 updates a pair of the estimated delay and estimated Doppler
({circumflex over (k)}d,s,D,s+1)
or
({circumflex over (k)}d,s+1,D,s)
by using the recursion formula for the CP (the AC shift version of (60), MF400, but the AC shift version of the formula for the OP (61) is omitted)
and sets the pair of the MLEs
(*μ,k*σ)
to the the pair of two estimates
({circumflex over (k)}d,s+1,D,s+1).
Furthermore, executions in the shift estimation and received-data extraction parts 202 are
1) to refer to the TD-CCF of type-1 with AC shifts (cf. MF715, MF898, (89))
c
ρ′,{right arrow over (p)}
AC,(3)(μ;{circumflex over (k)}d,j,j′)
whose real part is maximized by varying μ; 2) to determine the MLE
*μ
and 3) to set it to
D,s+1.
Other executions in the parts 202 are 1) to refer to the FD-CCF of type-2 with AC shifts (cf. MF715, MF899, (90))
C
ρ,{right arrow over (ρ)}
AC,(4)(kσ;D,j,j′)
whose real part is maximized by varying kσ; 2) to determine the MLE
k*
σ
and 3) to set it to
{circumflex over (k)}
d,s+1.
Thus the execution in the shift estimation and received-data extraction parts 202 is to alternatively update estimates of a time shift and a frequency shift.
The above estimation step in the shift estimation and received-data extraction parts 202 estimates the time shift and the frequency shift that are embedded in the signal, with use of MLEs of frequency shift for detecting N′ TD-templates and MLEs of time shift for detecting N FD-templates.
Furthermore, in the above estimation step in the shift estimation and received-data extraction parts 202, when referring to the TD-CCF of type-1 with AC shifts (cf. MF715, MF898, (89))
c
ρ′,{right arrow over (p)}
AC,(3)(μ;{circumflex over (k)}d,j,j′)
and the FD-CCF of type-2 with AC shifts (cf. MF715, MF899, (90))
C
ρ,{right arrow over (p)}
AC,(4)(kσ;D,j,j′),
the receiver apparatus firstly replaces the rhs of the first equation of (60)
c
ρ′,{right arrow over (p)}
(3)
and the rhs of the second equation of (60)
C
ρ,{right arrow over (p)}
(4)
by the associated CCFs with AC shifts and secondly applies the argmax operation to those CCFs with AC shifts, and thirdly augments variables of the argmax operation by a pair of two estimated integers(cf. MF904, (109))
(ĵ,ĵ′).
The receiving apparatus in the above estimation step in the shift estimation and received-data extraction parts 202 is a receiver part of receiving a signal
TD-CEψAC[k]/FD-CEΨAC[]
that is the output signal of the j-th AC with delay and Doppler shifts
(kd(j),D(j))
having an integer j, one of two encoded integers by an encoder of k for
-ary communication.
In accordance with an embodiment of the present invention, as discussed above, the receiving apparatus that is an apparatus of receiving a signal comprises an estimation part for estimating a time shift and a frequency shift that are embedded in the signal, with reference to a non-commutative shift parameter space of co-dimension 2.
Furthermore, in the MLE, the transmitting TD- and FD-CE signals (87) (or (27)) (according to the state that the switches 1-1,1-2 in the transmitter of
-ary communication are obtained as follows; The TD- and FD-signatures first are multiplexed using 2-D BPSK modulation by the j-th PC chosen among the set of independent 2-D PCs
according to the encoded integer pair of k
and secondly, are passed to the AC channel with its associated time and frequency shifts
(kd(j),D(j))
or equivalently the transmitter makes the associated TFSOs
(or
)
act on the multiplexed TD- or FD-signature.
In accordance with an embodiment of the present invention, as discussed above, the transmitting method that is a method for transmitting a signal to be transmitted comprises a shift step for time-frequency shifting the signal to be transmitted, with reference to a non-commutative shift parameter space of co-dimension 2.
Furthermore, in the above MLE step, when using the CCFs in (89), (90) that are cross-correlations between the received signal
TD-CEψAC[k]/FD-CEΨAC[],
being passed through the AC channel with its associated time and frequency shifts
(kd(j),D(j)),
characterized by the encoder of k for
-ary communication and an estimated received template (the first and second terms of the rhss of (89), (90) indicate a transmitting signal and an estimated received template, respectively), one should notice the fact that the symbol Σ in (87)-(90) implies the multiplexing, being equivalent to the non-overlapped superposition of a signal, and that such multiplexing can be simulated by using several kinds of TFSOs. In the generation process of a transmitting signal in
χ(i)=({Xm(i)},{X′m′(i)})
respectively of periods
The transmitting TD- and FD-CE are obtained by time-frequency shifting the resultant TD- and FD-signatures by data level-shifts and multiplexing those signatures. If the superposition of signals is simply done without the use of time and frequency shifts, then the detection of templates is difficult; One should pay the price for the PDs due to shift operations: It is, however, obvious that fortunately, the PDs are valuable clues in detecting templates and thus the TFSOs are essential tools for precise evaluations of PDs and canceling out the PDs. Furthermore, the rhss of (89), (90) contain the PD due to modulation and demodulation by the carrier c, accompanied by the delay kd
and the compensation
0-ary
for canceling out the PD due to the phase modulation by the encoded integer j′ of k
As shown above, the shift step generates the signal to be transmitted by PSK-modulating a chip pulse wave in the TD by the TD-PC of period N and PSK-modulating the TD-PSK modulated TD-pulse wave by the FD-PC of period N′ to generate a multicarrier version of the TD-PSK modulated TD-pulse wave that is a transmitting signal.
In accordance with an embodiment of the present invention, in the transmitting apparatus in accordance with an embodiment of the present invention, the transmitter, when the switches 1-1,1-2 are on downward in
(kd(j),D(j)),
having the encoded integer j of k
The output signal of the AC gives
TD-CEψAC[k]/FD-CEΨAC[].
In accordance with an embodiment of the present invention, in the receiver apparatus in accordance with an embodiment of the present invention, the receiver, when the switches 2-1,2-2 switch on downward in
(ρ′,{right arrow over (p)},ĵ,ĵ′)(or (ρ′,{right arrow over (p)},ĵ,ĵ′)).
If the PUL algorithm for updating MLEs
(k*σ,*μ)
converges, then the data symbol
d
{right arrow over (q)}
can be recovered and the decoding of k can be done.
In accordance with an embodiment of the present invention, the generalization of several apparatuses with applications to images in subsection 8.3 is a natural 2-D extension of 1-D signal, exploiting the half shifts due to the NCP of the space-spatial frequency shift (SSFS)s (see (102)-(108)).
(An Extended Version of an Embodiment for Images)
In accordance with an embodiment of the present invention, this disclosure gives a method for receiving an image that comprises an estimation step for estimating a space shift and a spatial frequency shift that are embedded in the received image signal, with reference to a parameter space of co-dimension 2, wherein each of the space shift and the spatial frequency shift has dimension ≥2.
(An Extended Version of an Embodiment for Images)
In accordance with an embodiment of the present invention, the receiver in the above estimation step refers to the 2×2-D symmetrical SSFS operator (SSFSO)s
a
,b
(2),d
and its frequency dual
b
,−a
(2),f,d
and estimates space shifts and spatial frequency shifts.
(An Extended Version of an Embodiment for Images)
In accordance with an embodiment of the present invention, this disclosure gives a method for transmitting an image signal, comprising a shift step for space-spatial frequency shifting the image signal to be transmitted, with reference to a parameter space, wherein each of the space shift and the spatial frequency shift has dimension ≥2.
(An Extended Version of an Embodiment for Images)
In accordance with an embodiment of the present invention, the transmitter in the above estimation step refers to the 2×2-D symmetrical SSFS operator (SSFSO)s
a
,
(2),d
and its frequency dual
b
,−a
(2),f,d
and space- and spatial frequency-shifts an image.
(The Received-Data Output Part 203)
The received-data output part 203 outputs the extracted data by the shift estimation and received-data extraction parts 202
<The Signal Flow-Chart Between a Transmitter and a Receiver>
(S101)
In the step S101, the data acquisition part 101 gets data to be transmuted. Concrete procedures by the part 101 have been described above.
(S102)
In the step S102, the signal generation part 102 generates a signal to be transmitted. Concrete procedures by the part 102 have been described above.
(S103)
In the step S103, the transmitter part 103 transmits the signal to be transmuted. Concrete procedures by the part 103 have been described above.
(S201)
In the step, the receive part 201 receives a transmitted signal by the transmitter part S103. Concrete procedures by the part 201 have been described above.
(S202)
In the step S202, the shift estimation and received-data extraction parts 202 estimates shifts and extracts received data, simultaneously. Concrete procedures in the shift estimation and received-data output part 202 are as described above.
(S203)
In the step S203, the received-data output parts 203 outputs the received data extracted by the shift estimation and received-data extraction parts 202. Concrete procedures in the received-data output part 203 are as described above.
The example 1 of embodiments of communication systems that is based on the theoretical basis as explained above is explained, together with cited figures.
<Transmitter Apparatus 100a >
The transmitter apparatus 100a is an apparatus that executes a program imposed on a transmitter as described in the above description.
As shown in
(The Transmitting Signal Generation Part 102a)
the transmitting signal generation part 102a comprises the generator part of a transmitting signal 102 relating to the example 1 of embodiments of transmitter-receiver system and the embedding shift part 110 as well.
As discussed in (84) just below, the embedding shift part 110 generates a transmitting signal into which a set of Npath delay and Doppler parameters
{(kd,iD,i)}i=1N
are embedded.
The generator part of a transmitting signal 102a time-frequency shifts a signal to be transmitted by using one time shift (or multiple time shifts) and one frequency shift (or multiple frequency shifts)
{(kd,iD,i)}i=1N
<Receiver Apparatus 200a>
The receiver apparatus 200a is an apparatus that executes a program imposed on a receiver as described in the above description.
As shown in
(The Shift Estimation and Received-Data Extraction Parts 202a)
The shift estimation and received-data extraction parts 202a comprises the parts similar as parts that the shift estimation and received-data extraction parts 202 comprises, relating to the example 1 of embodiments of transmitter-receiver system as an example.
<The Signal Flow-Hart Between a Transmitter and a Receiver>
(S102a)
In the step S102a, the generator part of a transmitting signal 102a with the embedding-shift part 110 generates a transmitting signal. Concrete procedures by the generator part of a transmitting signal S102a have been described above.
(S202a)
In the S202a, the shift estimation and received-data extraction parts 202a estimates shifts and extracts the received data, simultaneously. Concrete procedures in the shift estimation and received-data extraction parts 202a are as described above.
The transmitter-receiver system that is explained by referring to
As described in subsection <8.3 signal processing exploiting multi-dimensional non-commutative spatial-spatial frequency shifts>, the received apparatus 200 executes the estimation step for estimating SSFSs that are embedded in the received images by referring to the parameter space, i.e., SSFS parameter space of co-dimension ≥2 as an example.
As described in subsection <8.3 signal processing exploiting multi-dimensional non-commutative spatial-spatial frequency shifts>, the above estimation step, the receiver refers to the 2×2-D symmetrical spatial shift and spatial frequency shift (SSFS) operator (SSFSO)s representing the half-shift-PDs due to the NCP of the SSFSs
a
,b
(2),d
and its frequency dual
b
,−a
(2),f,d
and estimates SSFSs.
Similarly, in accordance with an embodiment of the present invention, the transmitter, i.e., the method for transmitting an image refers to the SSFS parameter space of co-dimension ≥2 and comprises 2 different SSFSOs that represent spatial-spatial frequency shifting an image to be transmitted and the PD due to the half-shift of the spatial frequency shift (or the PD due to the half-shift of the spatial shift).
In the above shift step, the transmitter refers to the symmetrical SSFSOs
a
,b
(2),d
and its frequency dual
b
,−a
(2),f,d
and estimates SSFSs.
[Example of Embodiments Using Software]
The control-block in the transmitter apparatuses 100, 100a and the receiver apparatuses 200,200a (in particular, the generator parts of a transmitting signal 102, 102a and the shift estimation and received-data extraction parts 202,202a) may be realized by logical circuits implemented by integrated circuit (IC chip)s or by software.
In the case of the implementation by software, the transmitter apparatuses 100, 100a and the receiver apparatuses 200,200a are equipped with a computer that executes programs for each of several functions; e.g., this computer comprises multiple processors and computer-readable memories. The computer executes the programs that the processors read from the memories so that it can perform the aim of the present invention. non-transitory memory, e.g, read-only-memory (ROM), tape, disc, card, IC memory, and programmable logical circuits, etc are used for the memories. The computer may be equipped with random access memory (RAM) performing the above programs. The above programs may be installed via any transmission media (such as communication network, broadcast, radio wave) and supplied by the above computer. In accordance with an embodiment of the present invention, the above program that may be embodied in an embedded data signal into a carrier wave by electronic transmission and may be realized.
Program products by computer that realize several functions of the transmitter apparatuses 100, 100a and the receiver apparatuses 200,200a include one of embodiments of the present invention. The above program product by computer loads programs that are provided by any transmission media by using at least one computer and makes the computer to execute at least one program. Thus, at least one processor associated with the computer executes at least one program. Thus each of functions of the transmitter apparatuses 100, 100a and the receiver apparatuses 200,200 can be realized. The program product by computer makes at least one of program-installed computers to execute steps in transmitting process(method of transmission) and receiving process(method of receiving).
In accordance with an embodiment of the present invention, the above mentioned embodiments have no limitation. The embodiments are modified within the claims. Comprising several methods or techniques given in different embodiments leads to an embodiment of the present patent and includes one of methods of the present patent. Furthermore, adequate comprising several methods or techniques given in different embodiments may provide a new technical characteristic feature.
The present patent is ideally suited to wireless transmitter-receiver communication systems and to radar systems.
| Number | Date | Country | Kind |
|---|---|---|---|
| PCT/JP2018/001735 | Jan 2018 | JP | national |
This Nonprovisional application is a continuation of PCT International Application No. PCT/JP2018/024592 filed in Japan on Jun. 28, 2018, which claims priority under 35 U.S.C. § 119 on PCT International Application No. PCT/JP2018/001735 filed in Japan on Jan. 22, 2018, the entire contents of which are hereby incorporated by reference.
| Number | Date | Country | |
|---|---|---|---|
| Parent | PCT/JP2018/024592 | Jun 2018 | US |
| Child | 16369053 | US |
