Patent Grant
7720168
1. Field of the Invention
The invention described herein is related to encoding data streams in a wireless communications network. More specifically, the invention is directed to encoding data in multiple-channel, multiple-frequency communication systems so as to provide maximum diversity in space-frequency or in space-time-frequency.
2. Description of the Prior Art
Multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) communication systems, referred to as MIMO-OFDM have exhibited great potential in broadband communications. By employing multiple transmit and multiple receive antennas, many of the adverse effects typically encountered in wireless transmission may be reduced. OFDM modulation may mitigate frequency selective transmission by establishing a plurality of parallel flat-fading channels, thereby providing high spectral efficiency and eliminating the need for complex equalization technology.
Two MIMO-OFDM coding techniques are prevalent in the current art: space-frequency (SF) coding within an OFDM block exploits spatial and frequency diversities, and space-time-frequency (STF) coding, where coding is applied over multiple OFDM blocks to additionally exploit temporal diversities.
Previous SF coding systems have fallen short of achieving full diversity in both space and frequency, usually achieving full diversity in one realm at the expense of full diversity in the other. The same is true for STF codes—full diversity across all three of space, time and frequency has proven elusive. Thus, there is an apparent need for SF and STF coding which can guarantee full-diversity concurrently across all applicable domains.
In one aspect of the present invention, a method is provided for conveying an encoded data stream in a wireless communication system in a manner that achieves maximum space-frequency or space-time-frequency diversity. A transmitter is provided with a plurality of transmit antennas and means for simultaneously transmitting through each antenna a corresponding symbol over a corresponding one of a plurality of frequencies. A receiver is provided with a plurality of receive antennas and means for recovering the data stream from the symbols transmitted over the plurality of frequencies. The region of space between each transmit antenna and each receive antenna defines a corresponding transmission channel. By way of the present invention, a first codeword is formed from symbols corresponding to data in a data stream. A transform is applied to the first codeword to produce a second codeword. The transform may operate on the symbols of the first codeword themselves or may simply arrange the first codeword into the second codeword in an appropriate manner. The second codeword includes a plurality of column vectors equal in number to the number of transmit antennas and a plurality of rows equal in number to an integer multiple of the plurality of frequencies. The second codeword is transmitted such that the symbols in each column thereof are transmitted via a corresponding one of the transmit antennas and symbols in each row thereof are transmitted over a corresponding frequency. The second codewords are received at the receiver and the data stream is recovered therefrom.
In another aspect of the invention, an apparatus for conveying data in a broadband wireless communication system is provided. The apparatus includes a transmitter having a plurality of transmit antennas and a plurality of orthogonal frequency modulators. The frequency modulators are operable to modulate a transmission symbol onto each of a corresponding plurality of frequencies. The apparatus further includes an encoder for encoding a data stream into a codeword, where the codeword is characterized by a matrix of transmission symbols having a plurality of columns equal in number to the number of transmit antennas and a plurality of rows equal in number to an integer multiple of the number of frequencies. The encoder is coupled to the transmitter such that a transmission symbol in a row of the codeword and at a column of the codeword and is transmitted via a transmit antenna corresponding to the column of the codeword over a frequency corresponding to the row of the codeword. The encoder is further operable to establish a set of symbols in which a minimum product distance is maximized, where the established set encompasses the transmission symbols of the codeword. The apparatus also includes a receiver having a plurality of receive antennas and a plurality of orthogonal frequency demodulators for demodulating each of the corresponding frequencies. A decoder is coupled to the receiver for decoding the transmitted codeword and recovering therefrom the data stream.
In yet another aspect of the present invention, an apparatus is provided having a transmitter with a plurality of transmit antennas and a plurality of orthogonal frequency modulators for modulating a transmission symbol onto each of a corresponding plurality of frequencies and a receiver having a plurality of receive antennas and a plurality of orthogonal frequency demodulators for demodulating each of the plurality of frequencies. The apparatus includes an encoder for encoding a data stream into a first codeword and a mapper for mapping the first codeword onto a second codeword. The second codeword is characterized by a matrix of transmission symbols having a plurality of columns equal in number to the number of transmit antennas and a plurality of rows equal in number to an integer multiple of the plurality of frequencies. The encoder is coupled to the transmitter such that a transmission symbol at a column and row of the second codeword is transmitted, respectively, via a transmit antenna corresponding to the column and over a frequency corresponding to the row. A decoder coupled to the receiver decodes the transmitted codeword and recovers the data stream therefrom.
1. Transmission Channel Model
To illustrate certain aspects of the present invention, it is believed beneficial to characterize the transmission channels by way of a model. Referring to
where τl is the delay and αi,jk(l) is the complex amplitude of the l-th path between transmit antenna i and receive antenna j. The αi,jk(l)'s are modeled as zero-mean, complex Gaussian random variables with variances E|αi,jk(l)|2, =δl2, where the operator E is that of probabilistic expectation. The power of the signal in the L paths are normalized such that Σl=0L−1 δl2=1. From (1), the frequency response of the channel is given by
where j=√{square root over (−1)} is the imaginary unit.
To achieve diversity across space, time and frequency, space-time-frequency (STF) coding across Mt transmit antennas, N OFDM subcarriers and K consecutive OFDM blocks is first considered. The input bit stream 135 is divided into b bit long segments, and each segment is mapped onto a codeword. In the STF case, an STF codeword can be expressed as a KN×Mt matrix
C=[C1T C2T . . . CKT]T, (3)
where the channel symbol matrix Ck is given by
and cik(n) is the channel symbol transmitted over the n-th sub-carrier by transmit antenna i in the k-th OFDM block. The STF code is assumed to satisfy the energy constraint E∥C∥F2=KNMt, where ∥C∥F is the Frobenius norm of C i.e.,:
During the k-th OFDM block period, the transmitter, at each OFDM Tx processors 115, applies an N-point IFFT to each column of the matrix Ck. After appending a cyclic prefix, the OFDM symbol corresponding to the i-th (i=1, 2, . . . , Mt) column of Ck is transmitted by transmit antenna i. In accordance with the conventions of the art, all of the Mt OFDM symbols are transmitted simultaneously from different transmit antennas.
At the receiver, after matched filtering, removing the cyclic prefix, and applying FFT at OFDM Rx processor 125, the received signal at the n-th subcarrier at receive antenna j in the k-th OFDM block is given by
is the channel frequency response at the n-th subcarrier between transmit antenna i and receive antenna j, Δf=1/T is the subcarrier separation in the frequency domain, and T is the OFDM symbol period. It is assumed that the channel state information Hi,jk (n) is known at the receiver, but not at the transmitter. In (6), zjk(n) denotes the additive white complex Gaussian noise with zero mean and unit variance in the n-th sub-carrier at receive antenna j in the k-th OFDM block. The factor √{square root over (ρ/Mt)} in (6) ensures that ρ is the average signal to noise ratio (SNR) at each receive antenna, independently of the number of transmit antennas.
In certain implementations of the present invention, temporal variability in the transmission channel is ignored. As such, the channel impulse response from transmit antenna i to receive antenna j is simplified and is modeled as
where, still, τl is the delay of the l-th path, and αi,j(l) is the complex amplitude of the l-th path, where each αi,j(l) is modeled as a zero-mean, complex Gaussian random variables with variance E|αi,j(l)|2=δl2. The powers of the L paths are once again normalized such that Σl=0L-1 δl2=1. From (7), the frequency response of the temporally invariant channel is given by
It is assumed that the MIMO channel is spatially uncorrelated, i.e. the channel taps αi,j(l) are independent for different indices (i, j).
In the space-frequency (SF) diversity case, each b bit long segment of the input bit stream 135 is mapped onto an SF codeword, which can be represented by an N×Mt matrix
where ci(n) denotes the channel symbol transmitted over the n-th subcarrier by transmit antenna i. The SF code is assumed to satisfy the energy constraint E∥C∥f2=NMt. The OFDM transmitter applies an N-point IFFT via OFDM Tx processor 115 to each column of the matrix C, and after appending the cyclic prefix, the OFDM symbol corresponding to the i-th (i=1,2, . . . , Mt) column of C is transmitted by transmit antenna i.
At the receiver, after matched filtering, removing the cyclic prefix, and applying FFT via OFDM Rx processor 125, the received signal at the n-th subcarrier at receive antenna j is given by
is the channel frequency response at the n-th subcarrier between transmit antenna i and receive antenna j, Δf=1/T is the subcarrier separation in the frequency domain, and T is the OFDM symbol period. Again, it is assumed that the channel state information Hi,j(n) is known at the receiver, but not at the transmitter. In (11), zj(n) denotes the additive complex Gaussian noise with zero mean and unit variance at the n-th subcarrier at receive antenna j. The noise samples zj(n) are assumed to be uncorrelated for different j's and n's.
2. Code Design Criteria
The performance criteria for coded MIMO-OFDM systems will now be derived based on the STF channel model to determine the maximum achievable diversity order for such systems. Using the notation ci((k−1)N+n)cik(n), Hi,j((k−1)N+n)
Hi,jk(n), yj((k−1)N+n)
yjk(n), and zj((k−1)N+n)
zjk(n) for 1≦k≦K, 0≦n≦N−1 1≦i≦Mt and 1≦j≦Mr, the received signal in (6) can be expressed as
for m=0, 1, . . . KN−1. The received signal may be rewritten in vector form as
where D is a KNMr×KNMtMr, matrix constructed from the STF codeword C in (3) as follows:
D=IM
where {circle around (×)} denotes the tensor product, IM, is the identity matrix of size Mr×Mr, and
Di=diag{ci(0), ci(1), . . . , ci(KN−1)} (16)
for any i=1, 2, . . . , Mt. The channel vector H of size KNMtMr×1 is formatted as
H=[Hl,1T . . . HM
where
Hi,j=[Hi,j(0) Hi,j(1) . . . Hi,j(KN−1)]T (18)
The received signal vector Y of size KNMr×1 is given by
Y=[y1(0) . . . y1(KN−1) y2(0) . . . y2(KN−1) . . . yM
and the noise vector Z has the same form as Y, i.e.,
Z=[z1(0) . . . z1(KN−1) Z2(0) . . . z2(KN−1) . . . zM
Suppose that D and {tilde over (D)} are two matrices constructed from two different codewords C and {tilde over (C)}, respectively. Then, the pair-wise error probability between D and {tilde over (D)} can be upper bounded as
where r is the rank of (D−{tilde over (D)})R(D−{tilde over (D)})Ψ, γ1, γ2, . . . , γr are the non-zero eigenvalues of (D−{tilde over (D)})R(D−{tilde over (D)})Ψ, and R=E {HHΨ} is the correlation matrix of H. The symbol Ψ is used herein to denote the complex conjugate transpose of a matrix, i.e., (A*)T=(AT)*=AΨ is the transpose of the complex conjugate of the matrix A. Based on the upper bound on the pair-wise error probability in (21), two general code performance criteria can be proposed as follows: Diversity (rank) criterion where the minimum rank of (D−{tilde over (D)})R(D−{tilde over (D)})Ψ over all pairs of different codewords C and {tilde over (C)} should be as large as possible, or product criterion; where the minimum value of the product Πi=1rγi over all pairs of different codewords C and {tilde over (C)} should be maximized.
The channel frequency response vector between transmit antenna i and receive antenna j for the time invariant case can be similarly denoted by
Hi,j=[Hi,j(0) Hi,j(1) . . . Hi,j( N−1)]T. (22)
Using the notation ω=e−j2πΔf, Hi,j can be decomposed as:
Hi,j=W·Ai,j, (23)
where
which is related to the delay distribution, and
Ai,j=[αi,j(0)αi,j(1) . . . αi,j(L−1)]T,
which is related to the power distribution of the channel impulse response. In general, W is not a unitary matrix. If all of the L delay paths fall at the sampling instances of the receiver, W is part of the DFT-matrix, which is unitary. From (23), the correlation matrix of the channel frequency response vector between transmit antenna I and receive antenna j can be calculated as:
The third equality follows from the assumption that the path gains αi,j(l) are independent for different paths and different pairs of transmit and receive antennas. Note that the correlation matrix R is independent of the transmit and receive antenna indices i and j.
For two distinct SF codewords C and {tilde over (C)}, the notation
Δ=(C−{tilde over (C)})(C−{tilde over (C)})Ψ (25)
is used. The pair-wise error probability between C and {tilde over (C)} can be upper bounded as
where ν is the rank of Δ∘R, γ1, γ2, . . . γν are the non-zero eigenvalues of Δ∘R, and ∘ denotes the Hadamard product. Recall that if A={ai,j} and B={bi,j} are matrices of size m×n, then the Hadamard product
Based on the upper bound (26), the two SF code design criteria are similar to those above. Diversity (rank) criteria may be used, where the minimum rank of Δ∘R over all pairs of distinct codewords C and {tilde over (C)} should be as large as possible, and product criteria may be used, where the minimum value of the product Πi=1νλi over all pairs of distinct codewords C and {tilde over (C)} is maximized.
3. Full-Rate and Full-Diversity SF Code Design
If the minimum rank of Δ∘R is ν0 for any pair of distinct codewords C and {tilde over (C)}, the SF code is said to achieve a diversity order of ν0Mr. According to a rank inequality on Hadamard products, it is known that rank(Δ∘R)≦rank(Δ)rank(R). Since the rank of Δ is at most Mt, the rank of R is at most L, and the rank of Δ∘R is at most N, the maximum achievable diversity (or full diversity) is at most min {LMtMr, NMr}. Clearly, in order to achieve a diversity order of ν0Mr, the number of non-zero rows of C and {tilde over (C)} cannot be less than ν0 for any pair of distinct SF codewords C and {tilde over (C)}. If a SF code achieves full diversity, the diversity product, which is the normalized coding advantage, is given by:
where λ1, λ2, . . . λν are the non-zero eigenvalues of Δ∘R for any pair of distinct SF codewords C and {tilde over (C)}.
Certain embodiments of the present invention provide a systematic method to obtain full-rate SF codes achieving full diversity in space and frequency. Specifically, a class of SF codes can be designed through the present invention that can achieve a diversity order of ΓMtMr for any fixed integer Γ, (1≦Γ≦L).
In accordance with certain embodiments of the invention, each SF codeword C is a concatenation of some matrices Gp:
C=[G1T G2T . . . GpT 0N−PΓM
where P=└N/(ΓMt)┘, and each matrix Gp, p=1, 2, . . . , P, is of size ΓMt by Mt. The zero padding in (28) is used if the number of subcarriers N is not an integer multiple of ΓMt. Each matrix Gp, (1≦p≦P), has the same structure given by
G=√{square root over (M)}tdiag(X1, X2, . . . , XM
where diag(X1, X2, . . . , XM
Sufficient conditions for the SF codes described above to achieve a diversity order of ΓMtMr will now be derived. As noted above, to determine the diversity criterion, the rank of Δ∘R must be known, where Δ is defined in (25) and R is the correlation matrix defined in (24). Suppose that C and {tilde over (C)} are two distinct SF codewords which are constructed from G1, G2, . . . , Gp and {tilde over (G)}1, {tilde over (G)}2, {tilde over (G)}p, respectively. Then, there exists at least one index p0, (1≦p0≦P) such that GP
From (24), we know that the correlation matrix R{ri,j}1≦i,j≦N is a Toeplitz matrix. The entries of R are given by
Under the assumption that Gp={tilde over (G)}p for any p≠p0, the non-zero eigenvalues of Δ∘R are the same as those of [(Gp
Note that Q is independent of the index p0, i.e., it is independent of the position of Gp
where diag(X−{tilde over (X)})diag ((x1−{tilde over (x)}1), (x2−{tilde over (x)}2), . . . , (xΓM
In the above derivation, the second equality follows from the identities [IM
where Q0={qi,j}, 1≦i,j≦Γ and qi,j is specified in (31). Similar to the correlation matrix R in (24), Q0 can also be expressed as
Clearly, with τ0<τ1< . . . <τL−1, Q0 is nonsingular. Therefore, from (34), it can be observed that if Πk=1ΓM
The assumption of Gp={tilde over (G)}p for any p≠p0 is also sufficient to calculate the diversity product defined in (27). If the rank of Δ∘R is ΓMt and Gp≠{tilde over (G)}p for some p≠p0, the product of the non-zero eigenvalues of Δ∘R should be not less than that with the assumption of Gp={tilde over (G)}p for any p≠p0. Specifically, the diversity product can be calculated as
is termed the “intrinsic” diversity product of the SF code and is independent of the power delay profile of the channel.
Thus, in accordance with fundamental aspects of the invention, the following holds: For any SF code constructed by (28) and (29), if Πk=1ΓM
It is to be noted that |det(Q0| depends only on the power delay profile of the channel, and the intrinsic diversity product ζin depends only on min X≠{tilde over (X)}(Πk=1ΓM
In a first exemplary design approach, the signal points X=[x1 x2 . . . xK] are transformed over a K-dimensional signal set. Specifically, assume that Ω is a set of signal points, such as a constellation known to those in the art (QAM, PAM, etc). For any signal vector S=[s1 s2 . . . sK]εΩK, let
X=SMK, (38)
where MK is K×K matrix. For a given signal constellation Ω, the transform MK should be optimized such that the minimum product distance of the set of X vectors is as large as possible. Both Hadamard transforms and Vandermonde matrices may be utilized for constructing MK, however the transforms MK based on Vandermonde matrices result in larger minimum product distance than those based on Hadamard transforms.
To illustrate these aspects of the invention, an exemplary transform MK based on Vandermonde matrices will be presently described. Recall that a Vandermonde matrix with variables θ1, θ2, . . . θK is a K×K matrix
In a first exemplary class of transforms, K=2s, (s≧1) and the optimum transform MK for a signal constellation from Z[j]{a +jb: both a and b are integers, j=√{square root over (−1)}} is given by
where θ1, θ2, θK are the roots of the polynomial θK−j over field Q[j]{c+dj: both c and d are rational numbers}, and they can be determined as
In another exemplary class, K=3·2s, (s≧0), and the optimum transform MK for a signal constellation Ω from Z[ω]{a+bω: both a and b are integers, ω=(−1+j)√{square root over (3)}/2} is given by
where θ1, θ2, . . . θK are the roots of the polynomial θK+ω over field Q[ω]{c+dω: both c and d are rational numbers}, and they are specified as
The signal constellations Ω from Z[j] are of practical interest. In the case where K=2s, (s≧1), optimum transforms (40) and (41) have been described in the prior art literature. Other transforms MK (not optimum) are known where K is not a power of two. If K=2s·3t (s≧1,t≧1), for example, a class of transforms MK for signal constellations Ω from Z[j] has been given as
Additionally, optimum transforms MK are known where K is not a power of two. For example, if K is not a power of two, but K=φ(J) for some J with J≢0(mod4), where φ(•) is the Euler function (φ(J) denotes the number of integers m, (1≦m≦J) such that m is relatively prime to J, i.e., gcd(m,J)=1), the optimum transform MK for a signal constellation Ω from Z[j] can be expressed as
For example, when K is 6, 10, 12, 18, the corresponding J is 7, 11, 13, 19, respectively. In these cases, θk=ej2kπ/J
In case of some odd K, for example K=5, 7, 9, and so on, it has been shown that:
where θ1, θ2, . . . , θK are the roots of polynomial θK−(1+j) over the field Q[j], and can be calculated as
In (46), the factor γ is equal to √{square root over ((21/K−1)/K)} for the energy normalization. Note that although the transforms given in (44) and (47) are not optimum, they do provide large minimum product distance.
In another approach to designing the signal set X, the structure of the diagonal space-time block codes is exploited. Suppose that the spectral efficiency of the SF code is r bits/s/Hz. The set of L0=2rK variables may then be directly designed under the energy constraint E∥X∥F2=K. For example, if diagonal space-time block codes may be constructed as follows:
Cl=diag(eju
where
and u1, u2, . . . , uK ε{0, 1, . . . , L0−1}. The parameters u1, u2, . . . , uK are to be optimized such that the metric
is maximized. A set of variables X=[x1 x2 . . . xK] may be constructed as follows: For any l=0, 1, . . . , L0−1, let
xk=eju
Then, the minimum product distance of the set of the resulting signal vectors X may determined by the metric in (49). The optimum parameters u=[u1 u2 . . . uK] can be obtained via computer search. For example,
In the previous discussion, a class of SF codes with full rate and full diversity was obtained under the assumption that the transmitter has no a priori knowledge about the channel. If channel information were available to the transmitter, aspects of the invention may bring into effect improved performance of the SF codes. For example, the correlation between adjacent subcarriers may be reduced by random interleaving of the transmitted symbols. Additionally, if the power delay profile of the channel is available to the transmitter, further improvement can be achieved through a data permutation (or interleaving) method that explicitly takes the power delay profile into account.
4. Maximizing the Coding Advantage of SF Codes by Permutation
Suppose that the path delays τ0, τ1 τL−1 and signal powers θ02, θ12, θL−12 are available to the transmitter. With this knowledge, the present invention provides an optimum permutation (or interleaving) method for the SF codes defined by (28) and (29) such that the resulting coding advantage is maximized. By permuting the rows of a SF codeword C, an interleaved codeword σ(c) is obtained.
An exemplary system for carrying out the permutations to be presently described is illustrated in
The receive antennas 320 are coupled to channel monitor 360, which performs analysis on the incoming transmission to extract characteristic data from each transmission channel 350. The channel characteristics may include symbol delay information as well as an indication of received power. The characteristic information is provided to interleaver 370 at the transmitter through feedback path 365. The interleaver 370 permutes the codeword as prescribed by the channel characteristics, as will be described in paragraphs that follow. It should be noted that while the exemplary channel characteristic acquisition mechanism of
As stated previously, for two distinct SF codewords C and {tilde over (C)} constructed from G1, G2, . . . , Gp and {tilde over (G)}1, {tilde over (G)}2, . . . , {tilde over (G)}p respectively, there exists at least one index p0(1≦p0≦P) such that Gp
Suppose that Gp
where q=(m−1)Γ+i and r=(m−1)Γ+j. Note that the non-zero eigenvalues of [σ(C−{tilde over (C)})(C−{tilde over (C)})Ψ]∘R are determined by the matrices Am, m=1, 2, . . . , Mt. The product of the non-zero eigenvalues of [σ(C−{tilde over (C)})σ(C−{tilde over (C)})Ψ]∘R. λ1, λ2, . . . , λΓM
From (51), for each m=1, 2, . . . , Mt, the Γ×Γ matrix Am can be decomposed as follows:
As a consequence, the determinant of Am is given by
Substituting (55) into (52), the expression for the product of the non-zero eigenvalues of [σ(C−{tilde over (C)})σ(C−{tilde over (C)})Ψ]∘R takes the form
Therefore, the diversity product of the permuted SF code can be calculated as
where ζin is the intrinsic diversity product defined in (37), and ζex is the “extrinsic” diversity product, is defined by
The extrinsic diversity product ζex depends only on the permutation and the power delay profile of the channel. The permutation does not affect the intrinsic diversity product ζin.
From (54), for each m=1, 2, . . . , Mt, Wm can be written as
Wm=Vm·diag(ωn
where
Thus, det(WmΛWmΨ)=det(VmΛVmΨ). It is to be observed that the determinant of WmΛWmΨ depends only on the relative positions of the permuted rows with respect to the position n(m−1)Γ+1, not on their absolute positions.
Thus, another fundamental aspect of the present invention is revealed. For any subcarrier permutation, the diversity product of the resulting SF code is
ζ=ζin·ζex (61)
where ζin and ζex are the “intrinsic” and “extrinsic” diversity products defined in (37) and (58), respectively. Moreover, the extrinsic diversity product ζex is upper bounded as: (i) ζex≦1; and more precisely, (ii) if the power profile δ0, δ1, . . . , δL−1 is sorted in a non-increasing order as: δl
where equality holds when Γ=L. As a consequence,
It is to be observed from the discussion above that the extrinsic diversity product ζex depends on the power delay profile in two ways. First, it depends on the power distribution through the square root of the geometric average of the largest Γ path powers, i.e., (Πi=1Γδl
By carefully choosing the applied permutation method, the extrinsic diversity product ζex may be increased so as to improve the overall performance of the SF code. Toward this end, certain embodiments of the invention may employ any of the following exemplary permutation strategies.
In certain embodiments of the invention, an integer n(0≦n≦N−1) is decomposed into
n=e1Γ+e0, (64)
where 0≦e0≦Γ−1,
and └x┘ denotes the largest integer not greater than x. For a fixed integer μ, (μ≧1), e1 in (64) is itself decomposed into
e1=ν1μ+ν0, (65)
where 0≦ν0≦μ−1 and
The rows of the N×Mt SF codeword constructed in accordance with (28) and (29) are permuted in such a way that the n-th (0≦n≦N−1) row of C is moved to the σ(n)-th row, where
σ(n)=ν1μΓ+e0μ+ν0, (66)
where e0,ν0,ν1, are defined in (64) and (65). The integer μ is referred to as the separation factor, and should be chosen such that σ(n)≦N for any 0≦n≦N−1, or equivalently,
Moreover, in order to guarantee that the mapping (66) is one-to-one over the set {0, 1, . . . , N−1} (i.e., it defines a permutation), μ must be a factor of N. The role of the permutation specified in (66) is to separate two neighboring rows of C by μ subcarriers.
The following result characterizes the extrinsic diversity product of the SF code that is permuted with the above described method. For the permutation specified in (66) with a separation factor μ, the extrinsic diversity product of the permuted SF code is
Moreover, if Γ=L, the extrinsic diversity product ζex can be calculated as
The permutation (66) is determined by the separation factor μ. Among the objectives of the invention is to find a separation factor μop that maximizes the extrinsic diversity product ζex:
If Γ=L, the optimum separation factor μop can be expressed as
which is independent of the signal power along individual paths. The optimum separation factor can be easily found via, for example, a low complexity computer search. However, in certain cases, closed form solutions can also be obtained. For example, if Γ=L=2, the extrinsic diversity product ζex is
To illustrate, consider an exemplary embodiment of the invention having N=128 subcarriers, and a total bandwidth of BW=1 MHz. Then, the OFDM block duration is T=128 μs without the cyclic prefix. If τ1−τ0=5 μs, then μop=64 and ζex √{square root over (2δ0δ1)}. If τl−τ0=20 μs then μop=16 and ζex=√{square root over (2δ0δ1)}. In general, if τl−τ0=2ab μs, where a is an non-negative integer and b is an odd integer, μop=128/2a+1, and the extrinsic diversity product is ζex √{square root over (2δ0δ1)}, which achieves the upper bound (58) of the diversity product (61).
As a further example of a closed form determination of the optimum separation factor, assume that τ1−τ0=lN0T/N, l=1, 2, . . . , L−1 and N is an integer multiple of LN0, where N0 is a constant and not necessarily an integer. If Γ=L or δ02=δ12= . . . =δL−12=1/L, the optimum separation factor is
and the corresponding extrinsic diversity product is ζex=√{square root over (L)}(Πl=0L-1 δl)l/L. In particular, in the case where δ02=δ12= . . . =δL−12=1/L, ζex=1. In both cases, the upper bound of the extrinsic diversity products is achieved. Note that if τl=lT/N for l=0, 1, . . . , L−1, Γ=L and N is an integer multiple of L, the permutation with the optimum separation factor μop=N/L is similar to a known prior art optimum subcarrier grouping method, which, however, is not optimal for arbitrary power delay profiles.
As an illustrative example of the aspects of the invention described above, the optimum separation factors for two commonly used multipath fading models will now be determined. The COST 207 6-ray power delay profiles for typical urban (TU) and hilly terrain (HT) environment are described in Tables 1 and 2, respectively. Two different bandwidths are considered: a) BW=1 MHz, and b) BW=4 MHz. In the exemplary embodiment, the OFDM has N=128 subcarriers. The plots of the extrinsic diversity product ζex as the function of the separation factor μ for the TU and HT channel models are shown in
Referring to the Γ=2 case, there is shown in
Referring to
To demonstrate aspects of the invention, consider a MIMO-OFDM system having Mt=2 transmit antennas, Mr=1 receive antenna and N=128 subcarriers. Full-rate full-diversity SF codes were constructed according to (28) and (29) with Γ=2, yielding the code block structure
The symbols x1, x2, x3, x4 were obtained as
where s1, s2, s3, s4 were chosen from BPSK constellation (siε{1,−1}) or QPSK constellation (siε{±1,±j}), V(•) is the Vandermonde matrix defined in (39), and θ=ejπ/8. This code targets a frequency diversity order of Γ=2, thus it achieves full diversity only if the number of delay paths is L≦2.
The SF codes of the present invention were implemented with three permutation schemes: no permutation, random permutation, and the optimum permutation method of the present invention. The random permutation was generated by the known Takeshita-Constello method, which is given by
The performance of the illustrative systems are presented herein in terms of average bit error rate (BER), and are shown in the Figures as functions of the average signal-to-noise ratio (SNR). In all of the illustrative results, the curves with squares, pluses and stars show the performance of the proposed full-rate full-diversity SF codes without permutation, with the random permutation (76) and with the inventive optimum permutation, respectively.
To compare the performance of the full-rate full-diversity SF coding schemes of the invention using different permutation schemes, the code (74) was implemented with the channel symbols S1, s2, s3, s4 chosen from BPSK constellation. The symbol rate of this code is 1, and its spectral efficiency is 1 bit/s/Hz, ignoring the cyclic prefix.
An illustrative system utilized a simple 2-ray, equal-power delay profile, with a delay τ μs between the two rays was respectively implemented in accordance with two cases: a) τ=5 μs, and b) τ=20 μs with OFDM bandwidth BW=1 MHz. From the BER curves, shown in
The code (74) was also implemented with the TU and HT channel models in accordance with two situations: a) BW=1 MHz, and b) BW=4 MHz.
The SF code of the present invention is a clear improvement over systems of the prior art in that full-rate transmission may be achieved with full-diversity in the space-frequency domain. However, certain implementations will require a shift in operating principles of existing systems, which may not be desirable. New codes in compliance with the present invention would replace old codes of existing systems, which, in certain cases, may be cost prohibitive. To provide certain ones of the advantageous features of the present invention, while, at the same time, utilizing some existing infrastructure, SF codes may be mapped, in accordance with aspects of the present invention, from existing space-time (ST) codes.
5. Space-Frequency (SF) Codes from Space-Time (ST) Codes via Mapping
Among the beneficial features of the invention is a simple repetition mapping, from which full-diversity SF codes can be constructed from any ST (block or trellis) code designed for quasistatic flat Rayleigh fading channels.
As shown in
Fl: (g1 g2 . . . gM
where 1l×1, is an all one matrix of size l×1. The resulting l×Mt matrix is actually a representation of the vector (g1 g2 . . . gM
In fact, SF code C is obtained by repeating each row of G l times and adding some zeros. The zero padding used here ensures that the space-frequency code C has size N×Mt. Typically, the size of the zero padding is small, and it can be used to drive the trellis encoder to the zero state. According to the present invention, if the employed ST code G has full diversity for quasistatic flat fading channels, the space-frequency code constructed by (78) will achieve a diversity of at least lMtMr.
As shown in
To show this to be true, it is known that in typical MIMO-OFDM systems, the number of subcarriers N is greater than LMt, and, thus, the proof for the lMt≦N case may be shown for a given mapping Fl, 1≦≦L. If lMt>N, the proof follows similar lines, but will be omitted for brevity. Assume that k is the largest integer such that klMt≦N.
For two distinct codewords C and {tilde over (C)} of size N×Mt, there are two corresponding codewords G and {tilde over (G)} of size kMt×Mt such that
Since the ST code achieves full diversity for quasistatic flat fading channels, (G−{tilde over (G)}) is of full rank for two distinct G and {tilde over (G)}, i.e., the rank of (G−{tilde over (G)}) is Mt.
Using the performance criteria described above, the objective of the proof is to show that the matrix Δ∘R has a rank of at least lMt. From (25) and (80), it is observed
Thus, in Δ∘R, all entries are zero, except for a klMt×klMt submatrix, which will be denoted as (Δ∘R)klM
On the other hand, from (24), it can be verified that the entries of the correlation matrix R={ri,j}1≦i,j≦N can be expressed as
Therefore, from (80)-(82), it can be ascertained that
where P={pi,j}1≦i,j≦klM
The last equality in (83) follows from the identities [IkMt{circle around (×)}11×l]Ψ and (A1{circle around (×)}B1)(A2{circle around (×)}B2)(A3{circle around (×)}B3)=(A1A2A3){circle around (×)}(B1B2B3).
The klMt×klMt matrix P is further partitioned into l×l submatrices as follows:
where each submatrix Pm,n, 1≦m,n≦kMt is of size l×l. Denoting the entries of Pm,n as pm,n (i,j), 1≦i,j≦l, it is observed that
As a consequence, each submatrix Pm,n, 1≦m,n≦kMt, can be expressed as
Pm,n=Wmdiag{δ02, δ12, . . . δL−12}WnΨ (87)
where Wm is specified as
for m=1, 2, . . . , kMt. The matrix (88) can be further decomposed as
Wm=Wldiag {w(m−1)/τ
for 1≦m≦kMt, where
The matrix consisting of the first l columns if W1 may be denoted by W0. In so doing, W0 is an l×l Vandermonde matrix in l variables wτ
Since Δf is the inverse of the OFDM block period T and the maximum path delay is less than T, Δf (τj−τi)<1 for any 0≦i<j≦l−1. Thus, W0 is of full rank, as is W1. It follows that for any m=1, 2, . . . , kMt, the rank of Wm is l.
Returning to (83), the rank of Δ∘R can be investigated. For convenience, the following notation is used:
and Λ=diag{δ02, δ12, δL−12,}. Then, substituting (85) and (87) into (83), it is observed that
Since the rank of (G−{tilde over (G)}) is Mt, there are Mt linearly independent rows in (G−{tilde over (G)}). Suppose that the fi-th, 1≦f1<f2< . . . <fM≦kMt, rows of (G−{tilde over (G)}) are linearly independent of each other. Then, the matrix
is a submatrix of (G−{tilde over (G)})(G−{tilde over (G)}), and the rank of A is Mt. Using the notation
from (93), it is clear that Q0{A{circle around (×)}Λ} Q0Ψ is a lMt×lMt submatrix of (Δ∘R)klM
Since the rank of A is Mt and the rank of Λ is L, according to a rank equality on tensor products
rank(A{circle around (×)}Λ)=rank(A)rank(Λ)=MtL (96)
so the matrix A{circle around (×)}Λ is of full rank. It is known that for any m=1, 2, . . . , kMt, the rank of Wm is l, so the rank of Q0 is lMt. Therefore, the rank of Q0{A{circle around (×)}Λ} Q0Ψ is lMt, which is what was intended to be shown.
The SF code obtained from a space-time block of square size via the mapping Fl (1≦l≦L) will achieve a diversity of lMtMr exactly, which follows from the discussion above. Since the maximum achievable diversity is upper bounded by min {LMtMr, NMr}, the following fundamental aspect of the invention is revealed: The SF code obtained from a full diversity ST code via the mapping Fl defined in (79) achieves the maximum achievable diversity min {LMtMr, NMr}.
The symbol rate of the resulting SF codes via the mapping Fl is 1/l times that of the corresponding ST codes. However, the SF codes of the present invention achieve full space-frequency diversity. For example, for a system with two transmit antennas, eight subcarriers, and a two-ray delay profile, the symbol rate of the full-diversity SF codes introduced above is ½, whereas the symbol rate in prior art systems is only ¼. In certain practical applications, this effect can be compensated for by expanding the constellation size, maintaining the same spectral efficiency. Furthermore, from a system performance point of view, there is a tradeoff between the diversity order and the coding rate. The present invention offers a flexible choice in the diversity order.
The coding advantage of the resulting SF codes will now be characterized in terms of the coding advantage of the underlying ST codes by defining and evaluating the diversity product for SF codes. The effect of the delay distribution and the power distribution on the performance of the inventive SF codes will also be analyzed.
The diversity product, which is the normalized coding advantage of a full diversity ST code, has been defined for quasistatic flat-fading channels
where β1, β2, . . . , βM are the nonzero eigenvalues of (G−{tilde over (G)})(G−{tilde over (G)})H for any pair of distinct ST codewords G and {tilde over (G)}.
Based on the upper bound on the pair-wise error probability, the diversity product of a full-diversity SF code can be defined as
where λ1, λ2, λLM
The diversity product of the full-diversity SF code described above is bounded by that of the corresponding ST codes as follows:
√{square root over (ηL)}ΦζST≦ζSF,R≦√{square root over (ηl)}ΦζST (99)
where Φ=(Πl=0L-1 δl)l/L, and ηl and ηL are the largest and smallest eigenvalues, respectively, of the matrix H, which is defined as
and the entries of H are given by
To show that this is so, we use the notation developed above by replacing the repetition factor l as L, since the full diversity is achieved by using the mapping Fl. For any n×n non-negative definite matrix A, we denote its eigenvalues as eig1(A)≧eig2(A)≧ . . . ≧eign(A).
For two distinct codewords C and {tilde over (C)}, there are two corresponding ST codewords G and {tilde over (G)}, such that the relationship of C−{tilde over (C)} and G−{tilde over (G)} in (80) and (81) holds. The rank of Δ∘R is exactly LMt, which means that Δ∘R has totally LMt eigenvalues, which are the same as the nonzero eigenvalues of (Δ∘R)klMt×klMt. Thus,
where eigkLMt (QQΨ)≦θi≦eig1(QQΨ) for i=1, 2, . . . LMt. In (102), the second equality follows from (93), and the last equality follows by Ostrowski's theorem. Since, Q=diag{W1, W2, . . . , WkM
QQΨ=diag{W1W1Ψ, W2W2Ψ, Wkm
As a requirement of Ostrowski's theorem, the matrix Q should be nonsingular, which is guaranteed by the fact that each matrix Wm is of full rank for any m=1, 2, . . . , kMt. Furthermore, from (89), it is observed that for any 1≦m≦kMt
WmWmΨ=W1DDΨW1Ψ=W1W1ψ
where D=diag {W(m−1)Lτ
Since the set of LMt nonzero eigenvalues of [(G−{tilde over (G)})(G−{tilde over (G)})Ψ]{circle around (×)}Λ can be expressed as
{eigi((G−{tilde over (G)})(G−{tilde over (G)}))·eigj(Λ):1≦i≦Mt, 1≦j≦L} (103)
substituting (103) into (102), we arrive at
Since ηL≦θi≦ηl for any i=1, 2, . . . ,LMt, the results shown in (99) are proven.
From the above, it is apparent that the larger the coding advantage of the ST code, the larger the coding advantage of the resulting SF code, suggesting that to maximize the performance of the SF codes, the best known ST codes existing in the art should be utilized. Moreover, the coding advantage of the resulting SF code depends on the power delay profile. First, it depends on the power distribution through the square root of the geometric average of the path powers, i.e., Φ=(Πl=0L-1 δl2)l/L. Since the sum of the powers of the paths is unity, it is apparent that the best performance is expected from the uniform power distribution case, i.e. δl2=l/L. Secondly, the entries of the matrix H defined in (100) are functions of path delays, and therefore the coding advantage also depends on the delay distribution of the paths. For example, in case of a two-ray delay profile, (i.e., L=2), the matrix H has two eigenvalues:
Typically, the ration of (τl−τ0)/T is less than ½ and therefore cos π(τl−τ0)/T is non-negative. Thus, the smaller the separation of the two rays, the smaller the eigenvalues η2. If the two rays are very close compared with the duration T of one OFDM symbol, the lower bound in (99) approaches zero.
In an illustrative example, the signal coding through mapping of the present was implemented via SF codes from ST block code and ST trellis codes. The SF block codes were obtained form orthogonal ST block codes for two and four transmit antennas. For the case of two transmit antennas, 2×2 orthogonal ST block codes according to the known Alamouti's structure was used, which is given by
The SF block code for four transmit antennas was obtained from the 4×4 orthogonal design
In both cases, the entries xi were taken from BPSK or QPSK constellations. Note that the 2×2 orthogonal design could carry one channel symbol per subcarrier, while the 4×4 block code had a symbol rate of only ¾. For SF trellis codes, the two-antenna, four-state, QPSK space-time code, and the three-antenna, 16-state, QPSK space-time code were utilized.
The SF block codes and trellis were implemented with different power delay profiles: a two-ray equal power delay profile and COST207 typical urban (TU) six-ray power delay profile. The subcarrier path gains were generated according to the system model characterized by (12), independently for different transmit and receive antennas. The Figures illustrate the results in the form of bit-error rate (BER) curves as a function of the average SNR per bit, where the inventive full-diversity SF block codes are compared with the SF codes using ST codes directly at the same spectral density.
In a first example of the present invention, a simple two-ray, equal power delay profile of τμs between the two rays is assumed. Two cases were implemented: i) τ=5 μs and ii) τ=20 μs. In the illustrative example, the communication system had N=128 subcarriers, and the total bandwidth was BW=1 MHz. Thus, the OFDM block duration was T=128 μs without the cyclic prefix. The length of the cyclic prefix was set to 20 Us in all cases. The MIMO-OFDM systems had one receive antenna. In all of the illustrative curves, the dashed lines correspond to the τ=5 μS case and the solid lines correspond to the 20 μs case. The curves with squares show the performance of the full-diversity SF codes obtained by repeating each row of the ST codes two times. The curves with the circles show the results for the case of using ST codes directly as SF codes (i.e., no repetition).
The performance of the SF trellis codes for two and three transmit antennas are depicted in
Certain observations can be made from the illustrative examples. It is apparent that by repeating each row of the space-time code matrix, codes may be constructed whose error performance curve is steeper than that of the codes without repetition, i.e., the obtained codes have higher diversity order. However, the actual performance of the code depends heavily on the underlying channel model. In all cases, both the absolute performance and the performance improvement obtained by repetition are considerably better in case of the longer delay profile (i.e., τ=20 μs), and the performance of the obtained full-diversity SF codes degrade significantly in case of the delay profile with τ=5 μs. These phenomena can be explained as follows.
The delay distribution of the channel has a significant effect on the SF code performance. If the delays of the paths are large with respect to one OFDM block period, there will be fast variations in the spectrum of the channel impulse response; therefore, the probability of simultaneous deep fades in adjacent sub-channels will be smaller. This observation is in accordance with previous discussions, it should be expected that better BER performance occurs when transmitting data over channels with larger path delays. On the other hand, if the two delay paths are very close, the channel may cause performance degradation.
A second implementation was performed using a more realistic channel model. The SF codes simulated in the previous subsection were tested over the COST207 Typical Urban (TU) six-ray channel model. The power delay profile of the channel is shown in
The performance of the SF block codes from the 2×2 orthogonal design with and without repetition are shown in
The performance of the inventive full-rate full-diversity SF codes was also compared with that of full-diversity SF codes, such as the mapped SF codes described herein. The full-rate code (74) was implemented with symbols s1, s2, s3, s4 chosen from a QPSK constellation. The symbol rate of the code is 1, and the spectral efficiency is 2 bits/s/Hz, ignoring the cyclic prefix. The full-diversity SF code for comparison is a repetition of the Alamouti scheme two times as follows:
where the channel symbols x1 and x2 were chosen from 16-QAM in order to maintain the same spectral efficiency. In
First, a 2-ray, equal-power profile, with a) τ=5 μs, and b) τ=20 μs was implemented. The total bandwidth was BW=1 MHz. From the BER curve of the τ=5 μs case, depicted in
The two SF codes were also implemented using the TU and HT channel models. Two situation were considered: a) BW=1 MHz, and b) BW=4 MHz.
The codes described above have among their beneficial features full space-frequency diversity, which is a clear improvement over the prior art. The concepts of the invention described above may be extended to achieve maximum diversity in all three of space, frequency and time, as is described below.
6. Space-Time-Frequency (STF) Codes
In the description of the channel model above, it was shown in (21) that
where ν is the rank of Δ∘R, and λ1, λ2, . . . , λν are the non-zero eigenvalues of Δ∘R. The minimum value of the product Πi=lνλi over all pairs of distinct signals C and {tilde over (C)} is termed as coding advantage, denoted by
If the minimum rank of Δ∘R is ν for any pair of distinct STF codewords C and {tilde over (C)}, it is said that the STF code achieves a diversity order of νMr. For a fixed number of OFDM blocks K, number of transmit antennas Mt and correlation matrices RT and RF, the maximum achievable diversity or full diversity is defined as the maximum diversity order that can be achieved by STF codes of size KN×Mr.
According to the rank inequalities on Hadamard products and tensor products,
rank(Δ∘R)≦rank(Δ)rank(RT)rank(RF).
Since the rank of Δ is at most Mt and the rank of RF is at most L, then
rank(Δ∘R)≦min{LMtrank(RT),KN}.
Thus, the maximum achievable diversity is at most min {LMtMrrank (RT)×KNMr}. Previous work has not discussed whether this upper bound can be achieved or not. However, through the present invention, this upper bound can indeed be achieved. It can also be observed that if the channel stays constant over multiple OFDM blocks (rank (RT)=1), the maximum achievable diversity is only min {LMtMr,KNMt}. In this case, STF coding cannot provide additional diversity advantage compared to the SF coding approach described above.
Note that the proposed analytical framework includes ST and SF codes as special cases. If only one sub-carrier (N=1) is considered, and one delay path (L=1), the channel becomes a single-carrier, time-correlated, flat fading MIMO channel. The correlation matrix R simplifies to R=RT, and the code design problem reduces to that of ST code design. In the case of coding over a single OFDM block (K=1), the correlation matrix R becomes R=RF, and the code design problem simplifies to that of SF codes, such as those described above involving mapping.
Two exemplary STF code design methods to achieve the maximum achievable diversity order min {LMtMrrank(RT),KNMr} will now be described. Without loss of generality, it is assumed that the number of sub-carriers, N, is not less than LMt so the maximum achievable diversity order is LMtMrrank(RT).
Suppose that CSF is a full-diversity SF code of size N×Mt, such as the SF codes described above. A full-diversity STF code, CSTF, may be constructed by repeating CSF K times over K OFDM blocks as follows:
CSTF=1k×1{circle around (×)}CSF, (109)
where 1k×l is an all one matrix of size k×1. Let
ΔSTF=(CSTF−{tilde over (C)}STF)(CSTF−{tilde over (C)}STF)Ψ
and
ΔSF=(CSF−{tilde over (C)}SF)(CSF−{tilde over (C)}SF)ψ
Then,
ΔSTF=[1k×l{circle around (×)}(CSF−{tilde over (C)}SF)][1l×k{circle around (×)}(CSF−{tilde over (C)}SF)]1k×k{circle around (×)}ΔSF.
Thus,
Since the SF code CSF achieves full diversity in each OFDM block, the rank of ΔSF∘RF is LMt. Therefore, the rank of ΔSTF ∘R is LMt rank(RT), so CSTF in (109) is guaranteed to achieve a diversity order of LMtMTrank(RT).
It is to be noted that the maximum achievable diversity depends on the rank of the temporal correlation matrix RT. If the fading channels are constant during K OFDM blocks, i.e., rank(RT)=1, the maximum achievable diversity order for STF codes (coding across several OFDM blocks) is the same as that for SF codes (coding within one OFDM block). Moreover, if the channel changes independently in time, i.e. RT=IK, the repetition structure of STF code CSTF in (109) is sufficient, but not necessary to achieve the full diversity. In this case,
Δ∘R=diag(Δ1∘RF, Δ2∘RF, . . . , ΔK∘RF),
where Δk=(Ck−{tilde over (C)}k)(Ck−{tilde over (C)}k)Ψ for 1≦k≦K. Thus, in this case, the necessary and sufficient condition to achieve full diversity KLMtMT is that each matrix Δk∘RF be of rank LMt over all pairs of distinct codewords simultaneously for all 1≦k≦K.
The repetition-coded STF code design of the present invention ensures full diversity at the price of symbol rate decrease by a factor of 1/K OFDM blocks compared to the symbol rate of the underlying SF code. The advantage of this approach is that any full-diversity SF code (block or trellis) can be used to design full-diversity STF codes.
In another aspect of the invention, a class of STF codes that can achieve a diversity order of ΓMtMTrank(RT) for any fixed integer Γ(1≦Γ≦L) by extending the full-rate full-diversity SF code construction method proposed for one OFDM block (K=1 case).
Consider a STF code structure consisting of STF codewords C of size KN×Mt:
C=[C1T C2T . . . CKT]T (110)
where
Ck=[Gk,1T Gk,2T . . . Gk,PT 0N−PΓM
for k=1, 2, . . . , K. In (111), P=└N/(ΓMt)┘, and each matrix Gk,p(1≦k≦K,1≦p≦P) is of size ΓMt×Mt. The zero padding in (111) is used if the number of subcarriers N is not an integer multiple of ΓMt. For each p(1≦p≦P), the code matrices G1,p, G2,p, . . . , GK,p are designed jointly, but the design of G1,p, G2,p, . . . GK,p is independent for different p values. For a fixed p(1≦p≦P), let
Gk,p=√{square root over (M)}tdiag(Xk,1Xk,2Xk,M
where diag(Xk,1, Xk,2, . . . Xk,M
The symbol rate of the proposed scheme is PΓMt/N, ignoring the cyclic prefix. If N is a multiple of ΓMt, the symbol rate is 1. If not, the rate is less than 1, but since N is typically much greater than ΓMt, the symbol rate is very close to 1. Full rate is defined as one channel symbol per subcarrier per OFDM block period, so the inventive method can either achieve full symbol rate, or it can perform very close to it. Note that this scheme includes the code design method described above as a special case when K=1.
A sufficient condition will now be shown for the STF codes described above to achieve a diversity order of ΓMtMrrank(RT). For simplicity, we use the notation X=[xl,1 . . . xl,ΓM
(Mtδmin)ΓM
where
Furthermore, if the temporal correlation matrix RT is of full rank, i.e., rank (RT)=K, the coding advantage is
ζSTF=δMtKΓM
where
From the above, it should be observed that with the code structure specified in (110)-(112), it is not difficult to achieve the diversity order of ΓMtMTrank(RT). The remaining problem is to design a set of complex symbol vectors, X=[x1,1 . . . xl,ΓMt . . . xK,1 . . . XK,Γ,Mt], such that the coding advantage ζSTF is as large as possible. One approach is to maximize δmin and δmax in (113) according to the lower and upper bounds of the coding advantage. Another approach is to maximize δ in (120). The latter case will be illustrated and offers advantages over the former approach. First, the coding advantage ζSTF in (119) is determined by δ in closed form, although this closed form only holds with the assumption that the temporal correlation matrix RT is of full rank. Second, the problem of designing X to maximize δ is related to the problem of constructing signal constellations for Rayleigh fading channels, which has been well solved in the prior art. In the literature, δis called the minimum product distance of the set of symbols X.
To implement X in order to maximize the minimum product distance δ, denote L=KTMt and assume that Ω is a constellation such as QAM, PAM, and so on. The set of complex symbol vectors is obtained by applying a transform over a L-dimensional signal set ΩL. Specifically,
where S=[s1 s2 . . . sL]εΩK is a vector of arbitrary channel symbols to be transmitted, and V(θ1, θ2, . . . , θL) is a Vandermonde matrix with variables θ1, θ2, θL:
The optimum θ1,,1≦l≦L, have been specified for different L and Ω. For example, if Ω is a QAM constellation, and L=2s(S≧1), the optimum θ1's are given by
In case of L=2s·3t(s≧1,t≧1), a class of θ1's are given as
These relationships are known in the field of coding through various signal constellations. The STF code design discussed in this subsection achieves full symbol rate, which is much larger than that of the repetition coding approach. However, the maximum-likelihood decoding complexity of this approach is high. Its complexity increases exponentially with the number of OFDM blocks, K, while the decoding complexity of the repetition-coded STF codes increases only linearly with K. Fortunately, known sphere decoding methods can be used to reduce the complexity.
The inventive STF code design methods have been implemented for different fading channel models, and their performance compared. The OFDM modulation had N=128 sub-carriers, and the total bandwidth was 1 MHz. Thus, the OFDM block duration was 128 μs for all cases. As in previous examples, the average bit error rate (BER) curves are illustrated as functions of the average signal-to-noise radio (SNR).
Both a block code and a trellis code were implemented in the illustrative example. The simulated communication system had MT=1 receive antenna as well as a two-ray, equal power delay profile (L=2), with a delay of 20 μs between the two rays. Each ray was modeled as a zero mean complex Gaussian random variable with variance 0.5.
The full-diversity STF block codes were obtained by repeating a full-diversity SF block code via (109) across K=1, 2, 3, 4 OFDM blocks. The full-diversity SF block code for Mt=2 transmit antennas was constructed from the Alamouti scheme with QPSK modulation via mapping described above. The spectral efficiency of the resulting STF code is 1, 0.5, 0.33, 0.25 bit/s/Hz (omitting the cyclic prefix) for K=1, 2, 3, 4, respectively. The full-diversity STF block code was simulated without temporal correlation (RT was an identity matrix). From
The simulated full-diversity STF trellis code was obtained from a full-diversity SF trellis code via (109) with K=1, 2, 3, 4, respectively. The used full-diversity SF trellis code for Mt=3 transmit antennas was constructed by applying the repetition mapping above to the prior art 16-state, QPSK ST trellis code. Since the modulation was the same in all four cases, the spectral efficiency of the resulting STF codes were 1, 0.5, 0.33, 0.25 bit/s/Hz (omitting the cyclic prefix) for K=1, 2, 3, 4, respectively. Similarly to the previous case, it was assumed that the channel changes independently from OFDM block to OFDM block. The obtained BER curves can be observed in
In a second illustrative demonstration, we used a more realistic Typical Urban (TU) channel model, which is shown in
αi,jk(l)=εαi,jk−1(l)+ηi,jk(l),0≦1≦L−1 (125)
where the constant ε(0≦ε≦1) determines the amount the temporal correlation, and ηi,jk(l) is a zero-mean, complex Gaussian random variable with variance δl√{square root over (1−ε2)}. If ε=0, there is no temporal correlation (independent fading), while if ε=1, the channel stays constant over multiple OFDM blocks. We considered three temporal correlation scenarios: ε=0,ε=0.8 and ε=0.95.
The simulated full-rate STF codes were constructed by (110)-(112) for Mt=2 transmit antennas with Γ=2. The set of complex symbol vectors X were obtained via (121) by applying Vandermonde transforms over a signal set Ω4K for K=1, 2, 3, 4. The Vandermonde transforms were determined for different K values according to (123) and (124). The constellation Ω was chosen to be BPSK. Thus, the spectral efficiency the resulting STF codes were 1 bit/s/Hz (omitting the cyclic prefix), which is independent of the number of jointly encoded OFDM blocks, K.
The performances of the full-rate STF codes are depicted in
Referring to
Once the second codeword has been formed, certain embodiments of the invention will ascertain whether channel characteristic information is available to the transmitter, as indicated at decision block 1835. If the decision is in the affirmative, certain embodiments of the invention will perform permutations on the second codeword, such as those permutations described above, as indicated at block 1840. If the channel information is not available, the permutations are not performed and flow is transferred immediately to block 1850. The second codeword, either permuted or not, is then transmitted in accordance with the invention, as indicated at block 1850. The method then exits at block 1850.
The descriptions above are intended to illustrate possible implementations of the present invention and are not restrictive. Many variations, modifications and alternatives will become apparent to the skilled artisan upon review of this disclosure. For example, components equivalent to those shown and described may be substituted therefor, elements and methods individually described may be combined, and elements described as discrete may be distributed across many components. The scope of the invention should therefore be determined with reference to the appended claims, along with their full range of equivalents.
The invention described herein is based on Provisional Patent Application Ser. No. 60/574,468, filed on 26 May 2004.
The invention described herein was developed through research funded under federal contract. As such, the United States Government has certain rights thereto.
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| 60574468 | May 2004 | US |
