Field of the invention
The present disclosure relates to channel estimation in wireless networks. More specifically, the present disclosure relates to a system and method for disintegrated channel estimation in wireless networks.
Related Art
In the field of wireless telecommunications, providing communications channels that are reliable and of high quality of service are paramount concerns. Cooperative communication has become one of the most essential aspects of recent mobile communication. A relay network represents one form of cooperative communication and has attracted a large amount of research in recent years. A space-time coding technique supported by relays has, in the past, been employed to achieve spatial diversity. The achievable diversity obtained using cooperative networks was studied previously. It has been found that the relay nodes alter the received signals according to the space-time block code (STBC) arrangement and then send the resultant signals to the destination node. The diversity achieved using the distributed STBC (D-STBC) with relays can be achieved by accurately estimating multiple timing errors and multiple frequency offsets as well as acquiring precise knowledge of the channel state information (CSI). Additionally, previous studies have successfully addressed the problems caused by timing errors and frequency offsets.
To acquire accurate knowledge of CSI, several channel estimation techniques have been investigated previously. These techniques are often categorized as (1) cascaded channel estimation; and (2) disintegrated channel estimation. However, cascaded channel estimation is ineffective for several applications such as optimal relay matrix design and beamforming. In existing relay beamforming techniques, relay gains were adjusted to suppress interference. Concerning disintegrated channel estimation, two methods are usually adopted to obtain disintegrated CSI at the destination node. In the first method, the source-relay channels are estimated at the relay nodes first by means of a conventional single-hop channel estimation, and the estimates are then quantized, compressed and sent toward the destination node via dedicated channels. Thus, the relay nodes are required to conduct source-relay channel estimation and compression with high complexity, and additional time slots or subchannels are required to deliver the source-relay estimates, decreasing both the spectral and energy efficiencies. In the second method, both the source-relay and the relay-destination channels are estimated at the destination node. A method utilizing relay pilots to conduct disintegrated channel estimation was studied previously, in which several pilot subcarriers are preserved for sending the relay pilots. It was found that the destination node can estimate the source-relay-destination and the relay-destination channels by exploiting the source and the relay pilots, respectively. The orthogonality requirement of the relay pilots was addressed in a previous study. In addition, the orthogonality was exploited to generate training sequences that can be applied in an amplify-and-forward (AF) relay network. Because prior efforts only studied a disintegrated channel estimation in a single-relay network, generalizations and improvements of the disintegrated channel estimation techniques in multiple cooperative relay network are strongly desired.
Accordingly, what would be desirable is a system and method for disintegrated channel estimation in wireless networks, which addresses the foregoing shortcoming of existing systems.
The present disclosure relates to a system and method for disintegrated channel estimation in wireless networks. The system provides a disintegrated channel estimation technique required to accomplish the spatial diversity supported by cooperative relays. The system includes a filter-and-forward (FF) relaying method with superimposed training sequences for separately estimating the backhaul and the access channels. To reduce inter-relay interference, a generalized filtering technique is provided. Unlike the interference suppression method commonly used in conventional FF relay networks, the generalized filtering matrix provided herein multiplexes the superimposed training sequences from different relays to the destination by time-division multiplexing (TDM), frequency-division multiplexing (FDM) and code-division multiplexing (CDM) methods. The Bayesian Cramér-Rao lower bounds (BCRBs) are derived as the estimation performance benchmark. The mean square errors (MSEs) of the disintegrated channel estimation are also derived. Finally, the improvements offered by the present system are verified by comprehensive computer simulations in conjunction with calculations of the BCRBs and the MSEs.
The foregoing features of the invention will be apparent from the following Detailed Description, taken in connection with the accompanying drawings, in which:
The present disclosure relates to a system and method for disintegrated channel estimation in wireless networks, as disclosed herein in detail in connection with
A generic relaying signal model is disclosed herein to effectively represent the AF and filter-and-forward (FF) relaying variants. The system can oeprate with multiple relays arranged in a parallel manner to accomplish D-STBC. Orthogonal training sequences are generated to assist the destination node in estimating all disintegrated channels. Multi-hop relaying can also be performed by utilizing the superimposed training methods. Rather than choosing the relay gain to minimize the inter-link interference between the source-relay and the source-destination links, the sytem of the present disclosure applies a generalized filtering matrix as a kernel to multiplex the relay pilot sequences prior to the relay-destination transmission. The generalized filtering method of the present disclosure can coordinate the relay pilot sequences by exploiting the time-division multiplexing (TDM), the frequency-division multiplexing (FDM), or the code-division multiplexing (CDM) methods. The system can effectively overcome inter-relay interference (IRI), thereby reducing both the mean square errors (MSEs) of the disintegrated channel estimation and the symbol-error rate (SER) obtained using the STBC in the relay network. The channel estimation technique of the present disclosure can be effectively applied in a scalable FF relay network, especially when a relay selection mechanism is applied or a relay occasionally joins/leaves the currently operating network without changing the bi-phase protocol.
The protocol of the present disclosure can be briefly described as follows. The mobile unit can detect several relays as candidates to support the cooperative communication and then select the required relays to establish STBC. When a relay occasionally joins (or leaves) the currently operating network, the mobile unit can stack it in (or pop one up from) the candidate list without informing the base station because the protocol and the frame structure are unchanged.
Some mathematical notations frequently used herein are summarized as follows. Matrices (uppercase) and vectors (lowercase) are in boldface. Cn×m in and Cm represent the complex matrices of size n×m and complex vectors of size m, respectively. The superscripts (•)*, (•)T, (•)H and (•)−1 represent the complex conjugate, the vector (or matrix) transpose, the Hermitian and the inverse operator, respectively. Φ(•) denotes a circulant function that converts a vector in Cn into a circulant matrix in Cn×n; it aids in representing the circular convolution of the signal and the channel impulse response (CIR) vectors. Θ(•) denotes a diagonal matrix generating function that allocates the parameter vector into the diagonal terms of a square matrix, Θ−1(•) denotes an inverse matrix of the diagonal matrix. FN represents the N-point discrete Fourier transform (DFT) matrix with (p, q)-entry expressed as
where p, q=0,1, . . . , N−1 and
Frequency domain (FD) signals are represented as ˜(•), e.g., {tilde over (x)}=FNx. Additionally, it is noted that the training slot length employed herein is the duration of an OFDM symbol, not a slot that contains 7 OFDM symbols, as defined in the LTE specification. The terms “slot” and “subslot” are employed to avoid the confusion that might arise from the use of the terms “symbol” and “subsymbol.”
Reference is now made to , M relay nodes, and one destination node denoted as
. The kth relay node is denoted as
k, where k=1,2, . . . , M. The direct link is absent or is omitted in
-
signal-to-noise ratio (SNR) is very low such that the
-
link can be neglected. A quasi-static multipath channel that can usually be represented as a tapped-delay-line (TDL) model is considered here. The channel tap-weighting coefficients are assumed to be constant within a time slot but to vary between slots. The zero-padded CIR vectors are used to give the transmitted signal and CIRs the same length N. The
-
k CIR vector is denoted as hk=[{hacek over (h)}kT01×(N−L
k-
CIR vector is defined as gk=[{hacek over (g)}kT01×(N−L
-
k-
CIR vector is represented as vk=[{hacek over (v)}kT01×(N−L
Referring to of
1 . . .
k of
1 . . .
k receive and filter the pilot signal. In step 16, the relay nodes
1 . . .
k process the pilot signal and generate an orthogonal pilot signal. Next, in step 18., the relay nodes
1 . . .
k superimpose the orthogonal pilot signal onto the filtered pilot signal. In step 20, the relay nodes
1 . . .
k transmit the superimposed and filtered pilot signals to the destination node
of
receives the superimposed and filtered pilot signals, and estimates the disintegrated channel information using the superimposed and filtered pilot signals. In step 24, the relay nodes
1 . . .
k of
. Finally, in step 26, the destination node
receives the spce-time coded data signals from the relay nodes
1 . . .
k and decodes the space-time coded data signals using the estimated disintegrated channel information.
The processing steps discussed above in connection with
The basic transmission protocol of a relay network is often composed of two phases. In the first phase, the relay nodes receive the signal transmitted from the source node. The received signal on the kth relay can be written as:
y
k=√{square root over (Ps)}Φ(hk)uS+n′k, (1)
where us is the training signal transmitted from , Ps denotes the transmitted power of the training signal, Φ(hk) denotes the circulant CIR matrix of the
-
k link, and n′k is an additive white Gaussian noise (AWGN) introduced at
k. In the second phase, any individual relay
k sends a signal to node
with or without further signal processing, depending on the relaying strategy. The generic relaying model considered herein can be represented in the form of a matrix linear equation, and the received signal at
can be expressed as:
y
D=Σk=1M[Φ(gk)(Wkyk+uk)]+nD, (2)
where Φ(gk) is the circulant CIR matrix of the k-
link and nD denotes the AWGN at
. A general filtering matrix Wk and a superimposed signal vector uk are jointly employed in (2) to consider various relaying strategies.
In the information-bearing transmission, D-STBC are employed to achieve the spatial diversity. To utilize the D-STBC technique in relay networks, the symbols on the relay nodes should be altered in accordance with its cooperative pattern. For example, considering a 2 -relay network, the relay should be capable of storing two symbols to conduct the rearrangement. In the first phase, k receives the signal from
on the mth symbol block,
y
k,m=Φ(hk)xm+nk,m, (3)
where k and m denote the relay index and the stored symbol index, respectively, and xm, is the mth data block transmitted from . The coding matrix can be expressed as:
If a set of data signals transmitted from is [x1Tx2T]T, then by using equation (3) and the STBC arrangement in equation (4) and taking the complex conjugate on the second symbol of the received signal, the signal received in
can be expressed as:
Because the relay nodes herein are assumed to be unable to acquire -
k CSI through channel estimation, the filtering matrix Wk in the data transmission duration is thus chosen as an identity matrix with a constant gain αkI, which can be considered as a fixed-gain AF relaying scenario.
To detect the STBC signal, the matrix in the first term of equation (5) should be evaluated first. However, the off-diagonal terms containing Φ(g2)Φ*(h2) and Φ(h2)Φ*(g2) cannot be obtained directly from equation (2). Therefore, the STBC technique unavoidably requires disintegrated channel estimation. To keep the relay complexity low, a relay does not estimate the -
k channel, and
is the only node to conduct channel estimation.
A technique that effectively maintains the currently existing protocols, spectral efficiency, and network scaleability in multi-relay networks is highly desired, and its development is one goal of the present disclosure. From equation (2), if the filtering matrix Wk is substituted by αkI, the signal model of the AF relaying strategy assisted from the superimposed training sequences can be obtained as:
y
D=√{square root over (Ps)}Σk=1M[αkΦ(gk)Φ(hk)]uS+Σk=1M√{square root over (Pk)}Φ(gk)uk+Σk=1MαkΦ(gk)nk+nD, (6)
where αk denotes the gain of k. The individual
k-
CIR vector gk can be estimated at
via the corresponding relay pilot uk. The source pilot us can be employed to estimate the compound
-
-
channel, v=Σk=1M [αkΦ(gk)Φ(hk)], as in the first term of equation (6). In a parallel multi-relay scenario, individual
-
k CIR vectors, hk, k=1,2, . . . , M, are mixed with each other and cannot be separately estimated; thus, IRI is unavoidable. To generalize the application of multi-relay networks, a novel filtering strategy to address IRI in conjunction with a superimposed training method is proposed.
The system of the present disclosure can be considered a FF relaying strategy with superimposed training sequences (FF/SITS). By inserting the received signal on the kth relay as written in equation (1) into (2), the received signal at can be rewritten as yD=√{square root over (Ps)}Σk=1M Φ(gk)WkΦ(hk)uS+Σκ=1M √{square root over (Pκ)}Φ(gκ+Σκ=1M Φ(gκ)nκ+nD, where nκ=Wκn′κ. If Wk is chosen to be a circulant matrix with relay gain αk, yD can be reformulated as:
y
D=√{square root over (Ps)}Σk=1M αkΦ(gk)Φ(hk)uS,k+Σκ=1M √{square root over (Pκ)}Φ(gκ)uκ+ΣK=1M ακΦ(gκ)nκ+nD, (7)
Hereafter, αk is set to 1 in order to focus on the work of the filter Wk, uS,k=WkuS according to the relation: WkΦ(hk)uS=Φ(hk)WkuS. In equation (7), the first term is the desired term for conducting the -
-
channel estimation by means of uS,k; the second term is the desired term for conducting the
κ-
estimation by means of uκ; the third term denotes the relay noises experiencing
κ-
links; and the fourth term represents the destination noise. In the filtering process, all the pilots must be chosen such that the following orthogonality requirements are satisfied:
where δ[•] represents the discrete-time Kronecker delta function, k, κ ∈ {1,2, . . . , M} , AL=[IL0L×(N−L)]L×N is a segmenting matrix that consists of an identity matrix and a zero matrix, L denotes the maximum delay spread or may simply be set as the CP length to consider the worst condition, and e1=[101×(N−1)]T denotes the N×1 elementary vector. In equation (8), the first orthogonal requirement is adopted to separate the channel estimations between gk and gκ; the second orthogonal requirement is adopted to separate the channel estimations between vk and vκ; and the third orthogonal requirement is adopted to separate the channel estimations between the links of -
k-
and
κ-
. The orthogonality requirements hold if the degrees of freedom of the CSI are assumed to be lower than the CP length. Wk is utilized to effectively multiplex the CSIs on different links in either the time domain (TD) or the FD to maintain the orthogonality. To obtain complete knowledge of the disintegrated channels, the maximum number of relays under the coordination of the proposed filtering method cannot exceed N/2L. The third orthogonality requirement is also met. Three primary multiplexing methods, TDM, FDM and CDM, are applied here to the parallel multi-relay scenario to satisfy the first and second orthogonality requirements.
The TDM approach equally partitions a single training time slot[1] into M subslots, within any of which a relay node κ sends its own relay pilot uκ. To achieve the filter function described in equation (7), a filter arranges
-
k channel information to a segmented orthogonal pilot ūS,k. Therefore, the kth relay filtering matrix can be formulated as:
WkDT,kHΦ(ūS,k)AN
k=1,2, . . . , M , (9)
where “” represents “defined as”; uS,k=FN
D
T,k
=[T
10N
Tj=IN-
k channel, and WFDLS can be replaced by any
-
k channel estimation, e.g., a TD LS channel estimation: WTDLS=(ΦH(uS)Φ(uS))−1ΦH(uS). The FD LS channel estimate is then segmented to Nr by AN
-
k is then placed in the kth subslot. As an example, the pilot structure employed in the scenario in which 2 relays are arranged in parallel is depicted in
k, Wk in equation (9) can be pre-calcul ated and pre-stored.
At destination node (see
where AL,N-
k-
can be conducted by:
Using the above channel estimates, the -
k CIR vector hk can be estimated using the LS criterion, i.e.:
ĥ
k=(ΦH(ĝk)Φ(ĝk))−1ΦH(ĝk){circumflex over (v)}k,
k=1,2, . . . , M. (13)
As disclosed above, the channel estimates of different -
k links are allocated to the interleaved subslots for an TDM approach. In effect, the FDM approach interleaves the channel estimates of different
-
k links onto different subcarriers. Therefore, the filtering matrix on the kth relay node can be formulated as:
WkFNH{tilde over (D)}F,kFN, k=1,2, . . . , M , (14)
where {tilde over (D)}F,k denotes the subcarrier sift matrix that is a diagonal matrix consisting 1's or 0's, i.e.,
mod denotes the modulo operator. The -
k CSI are projected to different dimensions by filtering the signal sent from
. A commonly used allocation method is to load individual relay pilots onto equally spaced subcarriers. An example of the subcarrier arrangement in a 2-parallel-relay network is shown in
At destination node (see
k D CIR vector gk can be estimated by:
The compound channel of the link -
-
can be estimated as:
where the vector AB=[1LT0N/M−LT1LT0N/M−LT . . . 1LT0N/M−LT]1×NT. To individually estimate the cascaded channel of -
k-
, k=1,2, . . . , M , {circumflex over (v)} should first be transformed into the FD. The
-
k-
CSI can be extracted from the interleaved subcarriers by means of {tilde over (D)}F,k, and the FD estimates on the interleaved subcarriers are then interpolated by {tilde over (L)}k. If TD channel estimation is required, FNH is applied. Therefore,
-
k-
can be estimated by:
{circumflex over (v)}k=ALFNH{tilde over (L)}k{tilde over (D)}F,kFN{circumflex over (v)},
k=1,2, . . . , M. (18)
where {tilde over (L)}k represents an interpolation operator in the FD. Because the signal after {tilde over (D)}F,k is considered as expended channel samples in the FD, the signal would be interpolated in the TD after the inverse DFT (IDFT) operation. Utilizing the channel estimates ĝk and {circumflex over (v)}k, hk can be estimated as in equation (13).
In the CDM approach, the original source pilot must be replaced by the orthogonal source pilots. This process is similar to the TDM approach without the segmenting and reallocating steps, where the replaced pilots must be designed to be orthogonal to other pilots. The filtering matrix can thus be reformulated as:
W
k
Φ(uS,k)FNHΘ−1(ũS)FN, k=1,2, . . . , M. (19)
k-
link can be achieved by:
The cascaded -
k-
CIR vector vk can be estimated with:
Utilizing the channel estimates ĝk and {circumflex over (v)}k, hk can be estimated as in equation (13). It can be proved that the filtering matrices written in equations (9), (14) and (19) are in the form of a circulant matrix or the equivalent FN−1Θ(v)FN [39] and can therefore be considered a TDL filter.
The BCRB of disintegrated channel estimation in single relay scenario has previously been derived. However, multiple relay and filtering matrix are considered into the signal model, and the property of the proposed filtering matrix should be taken into consideration. To extend the derivation from single to multiple relay, a general form of exponential integral function εn(•) has been introduced. As a result, the BCRB is included as a spacial case of our general BCRB derivation. The Fisher information matrix (FIM) can be expressed as:
where the parameter z consists of all channel pairs of -
k and
k-
links, i.e.:
z=[z1Tz2T . . . zMT]T, (23)
and each channel pair zk contains both gk and hk, i.e., zk=[gkThkT]T. The FIM can be shown to be a block diagonal matrix, i.e., J=Θ([J1J2 . . . JM]). Each diagonal submatrix of J can be expressed as:
where the elements of Jk can be derived as:
J
k
(11)=αk2ρh
J
k
(22)=αk2c2NPsI+Rh
J
k
(12)
=J
k
(21)=0. (25)
In equation (25), ρh
where ρg
and has the iterative property
The BCRBs of gk and hk can be obtained from the inverses of the diagonal submatrices of the FIM, i.e.:
BCRB
g
=tr{J
k
(11)−1},
BCRB
h
=tr{J
k
(22)−1},
k=1,2, . . . , M. (27)
Based on equations (9), (14), and (19), the MSEs of the proposed disintegrated channel estimation approaches are obtained.
The MSEs of the k-
channel estimations can be expressed as equations 28-30, as follows:
where
are the SNRs at the kth relay and the destination, respectively.
The MSEs of the -
k-
channel estimations can be obtained from those of the
k-
channel estimations, i.e.:
MSET;v=MSET;g,
MSE
F;v
=M×MSE
F;g,
MSEC;v=MSEC;g. (31)
The general MSE expressions of the k-
channel estimations can be obtained as:
where σĝ
Computer simulations were conducted to verify the analysis results derived herein and to reconfirm the improvements obtained using the system of the present disclosure. The system studied herein follows the LTE specifications, in which the DFT size is 256 for a 3 MHz bandwidth transmission, the CP length is 18, the sampling rate is 3.84 MHz, the sample duration is 260 ns, and the CP duration is 4.69 μs. In the following simulations, all links, including -
k and
k-
, k=1,2, . . . , M , are simulated with independent TDL models, in which the tap spacing is commonly set as the sample duration. The TDL parameters are obtained from the reduced variability models in Table A1-19: the International Telecommunication Union (ITU-R) channel model for rural macrocell (RMa) test environments of a non-line of sight (NLoS) scenario. The RMa NLoS channel has a mean delay spread of 21.37 ns, root-mean-square (rms) delay spread of 36.68 ns, and maximum excess delay time of 220 ns, which roughly corresponds to a 2-tap TDL channel model in the studied system. Each tap is simulated as an independent complex Gaussian random variable. To avoid the inter-carrier interference (ICI) effects caused by the random frequency modulation resulting from the relative movement, the simulations are all conducted in the context of block fading. Although the channels are considered to be time-varying, all the tap-weighting coefficients remain unchanged within an OFDM symbol duration and are updated every transmission slot. The quadrature-phase-shift keying (QPSK) mapping with Gray coding is employed. Four parallel relays are simulated and cooperated in ½ code rate STBC. The SNR at
k is set to 20 dB to represent moderately noisy conditions.
k-
and the
-
k channel estimations, respectively, obtained using the proposed approaches with 4 parallel relays. For simplicity, xDM is used to represent TDM, FDM and CDM approaches. The simulation results indicate that the NMSEs of the
k-
and
-
k channel estimations obtained using the xDM approaches are close to those obtained through the statistical derivations described in equations 28-32. The NMSEs of the
k-
channel estimation obtained using the FDM and CDM are lower than that obtained using the TDM, in which case
k-
channel estimation. The IRI obtained is reduced by a factor of 1/M. The simulation with
The -
k channel estimation can be achieved by the deconvolution of the
-
k-
and
k-
CIRs in the TD. Meanwhile, the deconvolution process in the TD can be conducted by dividing the estimates of
-
k-
by the estimates of
k-
in the FD. Because the TD deconvolution or the FD division process inevitably increases the estimation error, the NMSEs of the
-
k channel estimations are usually higher than those of
k-
channel estimates. Meanwhile, the FDM spends the advantage earned from its
-
k channel estimation and
-
k-
channel estimation, as the FD channel reconstruction employed in the FDM results in a non-negligible interpolation error.
The LS channel estimation is employed in the -
k channel estimation to allow a fair comparison. Prior studies have shown that a 5 bit/dim quantization process can provide sufficient accuracy for the CF strategy. Without loss of generality and for the simplicity of channel coding, CFs with the 6 bit/dim scheme were simulated. Considering both the in-phase and quadrature-phase components of the CIRs, 12 bit/tap was chosen for the simulations. Furthermore, twice and four times the channel standard deviation ρg
-
k channel estimation after the channel reconstruction conducted in the destination for the CF strategy are much higher than those obtained using the proposed techniques.
-
k channel estimations due to the relay noises, quantization errors are unavoidable in the quantizer at the relay nodes and error also occurs within the signal detection conducted at the destination due to the non-negligible destination noise. An error correction code can help reduce the third impairment but cannot effectively reduce the first two. The SER obtained using the CF relaying strategy can be reduced to those of xDM approaches by increasing
-
links are perfectly demodulated and decoded in relays. Then, the relays cooperatively establish D-STBC in the destination.
In summary, the three approaches disclosed herein can successfully perform the disintegrated channel estimation. From the figures, it can be seen that the BCRBs can be considered effective performance benchmarks because the NMSEs obtained using the proposed approaches are close to the derived BCRBs. This finding indicates that the proposed channel estimation approaches can effectively coordinate IRI to achieve low NMSE and thus low SER, as shown in
Because the complexities of achieving the STBCs based on the FF or CF relaying strategies are similar, the complexity comparison may focus on these strategies and the channel estimation processes. The proposed channel estimation approaches are all based on a similar relay filtering process Wk. Because an FFT operation requires N log2N multiplications and a diagonal matrix requires N multiplications, Wk requires 2(N log2N+Nr log2 Nr)+N+Nr multiplications and 2(N log2N+Nr log2 Nr)+N accumulations for the TDM, 2N log2N multiplications and 2N log2N+N accumulations for the FDM, and 2N log2 N+N multiplications and 2N log2N+N accumulations for the CDM. The filtering process Wk of xDM approaches can be pre-calculated and pre-stored on the kth relay after us is assigned because all the other components in Wk are constant. In each training cycle, an xDM strategy requires N2 multiplications and N2+N accumulations, in which N accumulations result from superimposing pilots. Considering the channel estimation at , TDM requires 2Nr log2 N+Nr multiplications, and both CDM and FDM require 2N log2 N+N multiplications. For a disintegrated channel estimation in a CF relaying network, the FD LS channel estimation requires at least 2N log2 N+N multiplications and 2Nlog2N accumulations per training cycle. Although a CF relay seems to have lower complexity than xDM approaches on a relay node, the complexity of the quantization, compression and forwarding of the source-relay CIR to the destination node has not been taken into account. The xDM strategy employs the original pilot or subcarriers employed by the conventional AF and DF relaying strategies, keeping the protocol unchanged, and therefore has higher relay network scalability and feasibility. However, CF needs to forward the source-relay CIR estimates to the destination; therefore, extra bandwidth is needed. The extra bandwidth is significant; meanwhile, the complexities of quantization, compression and remodulation of the source-relay CIR estimates are also high, even if the protocol can be adjusted accordingly (and/or in a timely manner).
Additional graphs illustrating simulated performance tests in connection with the system of the present disclosure are illustrated in
The MSEs of LS CE on R_k-D are shown in
The MSEs of S-R_k-D CE are shown in
A generic FF relaying strategy with multiplexed superimposed training sequences has been disclosed herein. The generalized filtering approaches effectively multiplex the superimposed training sequences to overcome IRI. Moreover, distinguished channel estimation techniques based on the generalized filtering approaches have been systematically derived, statistically analyzed and compared with the BCRBs.
The following additional equations and derivations are provided for reference.
The FIM in equation (22) can be reformulated as equation (33):
By considering the parameter vector z defined in (23), a matrix F can be defined as
from the signal model in (7), the likelihood function can be expressed as equation (36):
The mean and covariance matrix of yD are obtained as:
To derive a closed form from equation (36), the expectations in equation (38) must be evaluated:
From equation (20), by assuming Ps=Pk, the LS channel estimation of the ĝk can be evaluated using equations (7) and (8), i.e.,
The second term of the above equation represents the estimation error of ĝk and can be rewritten as:
Considering the generic form of the FF/CDM-SITS filtering matrix on the kth relay, the following should be taken into account:
WkWkH=IN, (1)
E{n′kn′kH}=σk2IN, (2)
E{Φ(gk)ΦgkH}=M(gk)=ρg
Φ(uk)HΦ(uk)=NIN, (4)
E{nDnDH}=σD2IN, (5)
ALALH=IN
The covariance matrix of êĝ
The MSE can be evaluated with MSEC;g
The channel estimation error of -
-
can be expressed as:
Therefore, the covariance matrix of e{circumflex over (v)}
The MSE can thus be evaluated as MSEC;v=tr{M(e{circumflex over (v)}
The MSE of the -
k channel estimation is derived in the FD for simplicity. The estimates of gk and vk can be expressed as:
{tilde over (ĝ)}
k
={tilde over (g)}
k
+e
{tilde over (ŷ)}
and {tilde over ({circumflex over (v)})}k={tilde over (v)}k+e{tilde over ({circumflex over (v)})}
where e{tilde over (ĝ)}
The division in the above equation is an element-wise division. The -
k channel can be derived from the
-
k-
and
k-
channel estimations as:
{tilde over (ĥ)}
k
=h
k
+F
N
H[Θ−1({tilde over (g)}k+e{tilde over (ĝ)}
Therefore, the channel estimation error of {tilde over (ĥ)}k is as written in equation (32), and the covariance matrix is derived as:
where Jensen's inequality is applied. Therefore, the MSE of the -
k channel estimation can be evaluated with MSEC;h
-
k channel estimations in the proposed TDM and FDM approaches can be derived by substituting the error variances of
-
k-
and
k-
channel estimation into equation (46).
Based on the signal model in equation (7), the relay matrix of the proposed TDM approach in equation (9) and the k-
channel estimation in equation (11), the channel estimation error can be evaluated as:
According to the following:
the covariance matrix is derived as:
The MSE can be evaluated as MSET;g=tr{M(eĝ
From the k-
channel estimation in equation (12), the channel estimation error can be evaluated as:
and the covariance matrix can be written as:
The MSE can be evaluated as MSET;v=tr{M(e{circumflex over (v)}
Based on the signal model in equation (7), the relay matrix of the proposed FDM approach in equation (14) and the k-
channel estimation in equation (16), the channel estimation error can be evaluated as:
According to the following facts,
the covariance matrix of eĝ
The MSE can then be evaluated as MSEF;g
From the k-
channel estimation in equation (17), the channel estimation error can be evaluated as:
Let A=ALFNH{tilde over (L)}k{tilde over (D)}F,kFNAB, then AAH=MIL. Assuming σk2=σκ2 and ρg
The MSE can be evaluated as MSEF;v=tr{M(e{circumflex over (v)}
Having thus described the invention in detail, it is to be understood that the foregoing description is not intended to limit the scope of the present invention. What is intended to be protected by Letters Patent is set forth in the following claims.
This application claims priority to U.S. Provisional Patent Application No. 62/277,704 filed on Jan. 12, 2016, the entire disclosure of the application hereby expressly incorporated by reference.
This invention was made with government support under Grant No. CNS-1456793 and Grant No. ECCS-1343210 awarded by the National Science Foundation. The government has certain rights in the invention.
Number | Date | Country | |
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62277704 | Jan 2016 | US |