In general, linear complex-field encoding techniques are proposed. For example, transmitter of a wireless communication system includes an encoder and a modulator. The encoder linearly encodes a data stream to produce an encoded data stream. The modulator to produce an output waveform in accordance with the encoded data stream for transmission through a wireless channel. The modulator generates the output waveform as a multicarrier waveform having a set of subcarriers, e.g., an Orthogonal Frequency Division Multiplexing (OFDM) waveform. The encoder linearly encodes the data stream so that the subcarriers carry different linear combinations of information symbols of the data stream.

Patent
   RE45230
Priority
Apr 22 2002
Filed
Apr 08 2013
Issued
Nov 04 2014
Expiry
Apr 21 2023

TERM.DISCL.
Assg.orig
Entity
Small
9
12
all paid
12. A wireless communication device comprising:
an encoder that applies a plurality of M matrices to linearly transform a stream of information bearing symbols selected from a constellation having a finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
a modulator to produce that produces an output waveform in accordance with the stream of precoded symbols for transmission through a wireless channel, where the matrices are identical and collectively have M*L redundant rows, where L represents an order of the wireless channel.
26. A method comprising:
applying a plurality of M matrices to linearly transform a stream of information bearing symbols selected from a constellation having a finite alphabet, wherein the M matrices linearly transform the stream of information bearing symbols to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
outputting a waveform in accordance with the stream of precoded symbols for transmission through a wireless channel, where the matrices are identical and have M*L redundant rows and, where L represents an order of the channel.
16. A method comprising:
performing error-control coding on an input stream to produce coded symbols;
interleaving the coded symbols to produce interleaved symbols;
mapping the interleaved symbols to produce a stream of information bearing symbols selected from a constellation having a finite alphabet;
applying a linear transformation to a the stream of information bearing symbols selected from a the constellation having a the finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
outputting a waveform in accordance with the stream of precoded symbols for transmission through a wireless channel.
7. A wireless communication device, comprising:
an encoder that applies a matrix to linearly transform blocks of K information bearing symbols selected from a constellation having a finite alphabet to produce blocks of N precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
a modulator that generates a multicarrier waveform having a set of subcarriers for transmission over a wireless channel, wherein N is the number of subcarriers and K is less than or equal to N, and wherein the size of the matrix is selected as a function of an order L of the wireless channel, and the number K of symbols per block is selected as a function of the channel order L.
0. 64. A method comprising:
applying a linear transformation to a stream of information bearing symbols selected from a constellation having a finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
outputting, via multiple antennas, waveforms in accordance with the stream of precoded symbols for transmission through a wireless channel,
wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix, wherein the first matrix is based on a fast Fourier transform (FFT) matrix, and wherein the second matrix is based on a diagonal matrix to phase-rotate each entry of a symbol vector.
0. 68. A device comprising:
an encoder that applies a linear transformation to a stream of information bearing symbols selected from a constellation having a finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
a modulator that produces output waveforms in accordance with the stream of precoded symbols for transmission through a wireless channel via multiple antennas,
wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix, wherein the first matrix is based on a fast Fourier transform (FFT) matrix, and wherein the second matrix is based on a diagonal matrix to phase-rotate each entry of a symbol vector.
27. A non-transitory computer-readable medium comprising instructions to cause a programmable processor to:
perform error-control coding on an input stream to produce coded symbols;
interleave the coded symbols to produce interleaved symbols;
map the interleaved symbols to produce a stream of information bearing symbols selected from a constellation having a finite alphabet;
apply a linear transformation to a stream of information bearing symbols selected from a the constellation having a the finite alphabet to produce a stream of precoded symbols that are complex numbers and are not restricted by the constellation of the information bearing symbols; and
output waveform in accordance with the stream of precoded symbols for transmission through a wireless channel.
0. 59. A wireless communication device comprising:
an encoder that applies a linear transformation to a stream of information bearing symbols selected from a constellation having a finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols;
a modulator that produces an output waveform in accordance with the stream of precoded symbols for transmission through a wireless channel; and
circuitry configured to (i) perform error-control coding on an input stream to produce coded bits, (ii) interleave the coded bits to produce blocks of interleaved bits, and (iii) map the blocks of interleaved bits to produce the stream of information bearing symbols selected from the constellation having the finite alphabet.
19. A method comprising:
performing error-control coding on an input stream to produce coded symbols;
interleaving the coded symbols to produce interleaved symbols;
mapping the interleaved symbols to produce blocks of K information bearing symbols selected from a constellation having a finite alphabet;
applying a matrix to linearly transform the blocks of K information bearing symbols that are selected from a the constellation having a the finite alphabet to produce blocks of N precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols, and
outputting a multicarrier waveform having a set of subcarriers in accordance with the stream of blocks of N precoded symbols for transmission through a wireless channel, where N is the number of subcarriers, and K is less than or equal to N.
1. A wireless communication device comprising:
a first encoder that encodes a data stream based on an error-control code to produce encoded symbols;
an interleaver that interleaves the encoded symbols to produce interleaved symbols;
a constellation mapper that maps the interleaved symbols to produce a stream of information bearing symbols selected from a constellation having a finite alphabet;
an a second encoder that applies a linear transformation to a the stream of information bearing symbols selected from a the constellation having a the finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
a modulator to produce an output waveform in accordance with the stream of precoded symbols for transmission through a wireless channel.
4. A wireless communication device comprising:
a first encoder that encodes a data stream based on an error-control code to produce encoded symbols;
an interleaver that interleaves the encoded symbols to produce interleaved symbols;
a constellation mapper that maps the interleaved symbols to produce blocks of K information bearing symbols selected from a constellation having a finite alphabet;
an a second encoder that applies a matrix to linearly transform the blocks of K information bearing symbols selected from a the constellation having a the finite alphabet to produce blocks of N precoded symbols that are complex numbers and that are not restricted to the constellation of the information bearing symbols; and
a modulator that generates a multicarrier waveform having a set of subcarriers, where N is the number of subcarriers of the multi-carrier multicarrier waveform and K is less than or equal to N.
13. A wireless communication device comprising:
a demodulator that receives a waveform carrying a encoded transmission and that produces a demodulated data stream, the encoded transmission including an encoded data stream, wherein the encoded data stream was produced by performing error-control coding on an input stream to produce coded symbols, interleaving the coded symbols to produce interleaved symbols, mapping the interleaved symbols to produce a stream of information bearing symbols selected from a constellation having a finite alphabet, and applying a linear transformation to a the stream of information bearing symbols selected from a the constellation having a the finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
a decoder that decodes the demodulated data stream to produce estimated data.
0. 49. A method comprising:
applying a linear transformation to a stream of information bearing symbols selected from a constellation having a finite alphabet to produce a stream of precoded symbols that are complex numbers and that are not restricted by the constellation of the information bearing symbols; and
outputting a waveform in accordance with the stream of precoded symbols for transmission through a wireless channel,
wherein the method further comprises:
performing error-control coding on an input data stream to produce coded bits;
interleaving the coded bits to produce interleaved bits;
mapping groups of the interleaved bits to produce the stream of information bearing symbols selected from the constellation having the finite alphabet, wherein a size of the constellation is larger than two, and wherein interleaving the coded bits comprises separating the coded bits so that neighboring coded bits are mapped to different information bearing symbols.
2. The wireless communication device of claim 1, wherein the modulator generates the output waveform as a multicarrier waveform having a set of subcarriers, and the second encoder encodes the stream of information bearing symbols so that the subcarriers carry different linear combinations of the information symbols.
3. The wireless communication device of claim 1, wherein the second encoder applies the linear transformation by applying a unitary matrix to the information bearing symbols.
5. The wireless communication device of claim 4, wherein the linear second encoder has a code rate r=K/N.
6. The wireless communication device of claim 4, wherein the linear second encoder applies a matrix of size N×K to the blocks of K information bearing symbols to produce the blocks of N precoded symbols.
8. The wireless communication device of claim 7, wherein K is selected so that K≧N−L.
9. The wireless communication device of claim 7, wherein K is selected so that K=N−L.
10. The wireless communication device of claim 6 7, wherein the linear encoder applies the matrix to perform a vector multiplication on the blocks of K information bearing symbols to produce the blocks of N precoded symbols, and applies each block of N precoded symbols across the N subcarriers.
11. The wireless communication device of claim 1, wherein the wireless communication device comprises one of a base station and a mobile device.
14. The wireless communication device of claim 13, wherein the decoder applies one of maximum-likelihood detection, zero-force (ZF) detection, minimum mean squared error (MMSE) detection, decision-directed detection, iterative detection, to decode the demodulated data stream.
15. The wireless communication device of claim 13, wherein the wireless communication device comprises one of a base station and a mobile device.
17. The method of claim 16, wherein outputting the waveform comprises:
outputting the output waveform as a multicarrier waveform having a set of subcarriers; and
encoding the stream of information bearing symbols so that the subcarriers carry different linear combinations of information symbols.
18. The method of claim 16, wherein applying the linear transformation to the stream of information bearing symbols comprises applying a unitary matrix to the stream of information bearing symbols.
20. The method of claim 19, wherein applying the linear transformation matrix comprises applying the linear transformation a matrix of size N×K to the blocks of K information bearing symbols to produce blocks of N precoded symbols at a code rate r=K/N.
21. The method of claim 19, wherein applying the linear transformation matrix comprises applying a unitary matrix of size N×K to the blocks of K information bearing symbols, wherein K is equal to N.
22. The method of claim 19, further comprising selecting the number of symbols per block K as a function of an order of the channel.
23. The method of claim 19, further comprising selecting the number of symbols per block K so that K≧N−L, wherein L represents an order of the channel.
24. The method of claim 19, further comprising selecting the number of symbols per block K so that K=N−L, where L represents an order of the channel.
25. The method of claim 19, wherein applying the linear transformation matrix comprises applying a matrix to perform a vector multiplication on the blocks of K information bearing symbols to produce the blocks of N precoded symbols.
28. The non-transitory computer-readable medium of claim 27, further comprising instructions to cause the programmable processor to:
output the output waveform as a multicarrier waveform having a set of subcarriers; and
encode the stream of information bearing symbols so that the subcarriers carry different linear combinations of information symbols.
0. 29. The wireless communication device of claim 1, wherein the output waveform comprises multiple output waveforms for transmission on multiple antennas, respectively.
0. 30. The wireless communication device of claim 29, wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix, wherein the first matrix is based on a fast Fourier transform (FFT) matrix, and wherein the second matrix is based on a diagonal matrix to phase-rotate each entry of a symbol vector.
0. 31. The wireless communication device of claim 30, wherein the number of the antennas is represented by Nt, wherein the first matrix is based on an Nt-point inverse version of the FFT matrix, wherein the linear transformation is based on:
Θ = F N t T diag ( 1 , α , , α N t - 1 ) , α := j 2 π / P
wherein FNtT represents the first matrix, and wherein
diag(1, α, . . . , αNt−1)
represents the second matrix, wherein P is an integer.
0. 32. The wireless communication device of claim 29, wherein the number of the antennas is represented by Nt, and wherein the linear transformation is based on a Vandermonde matrix of size Nt×Nt.
0. 33. The wireless communication device of claim 29, wherein the number of the antennas is represented by Nt, wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix,
wherein the first matrix is a matrix of size Nt×Nt, wherein each entry of the first matrix is based on a power of ej2π/Nt, each entry of a column of the first matrix being equal to one, and
wherein the second matrix is a diagonal matrix of size Nt×Nt having diagonal entries that are based respectively on different powers of ej2π/P including the zeroth power, wherein P is an integer.
0. 34. The wireless communication device of claim 33, wherein the multiple matrices include a third matrix, wherein the third matrix is a matrix of size Nt×Nt.
0. 35. The wireless communication device of claim 1, wherein the output waveform comprises an orthogonal frequency division multiplexing (OFDM) waveform.
0. 36. The method of claim 16, wherein interleaving the coded symbols to produce the interleaved symbols comprises writing the coded symbols into a matrix row-wise, and reading the encoded symbols from the matrix column-wise.
0. 37. The method of claim 16, wherein performing the error-control coding on the input stream comprises applying at least one of a turbo code and a convolutional code, wherein the constellation having the finite alphabet is a quadrature amplitude modulation (QAM) constellation, and wherein mapping the interleaved symbols comprises selecting symbols from the QAM constellation as the information bearing symbols.
0. 38. The method of claim 16, wherein applying the linear transformation comprises applying the linear transformation to the stream of information bearing symbols to produce the stream of precoded symbols such that the stream of precoded symbols differs, at least in part, from the stream of information bearing symbols, wherein the stream of precoded symbols includes complex numbers.
0. 39. The method of claim 16, wherein outputting the waveform comprises outputting multiple waveforms via multiple antennas, respectively.
0. 40. The method of claim 39, wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix, wherein the first matrix is based on a fast Fourier transform (FFT) matrix, and wherein the second matrix is based on a diagonal matrix to phase-rotate each entry of a symbol vector.
0. 41. The method of claim 40, wherein the number of the antennas is represented by Nt, wherein the first matrix is based on an Nt-point inverse version of the FFT matrix, wherein the linear transformation is based on:
Θ = F N t T diag ( 1 , α , , α N t - 1 ) , α := j 2 π / P
wherein FNtT represents the first matrix, and wherein
diag(1, α, . . . , αNt−1)
represents the second matrix, wherein P is an integer.
0. 42. The method of claim 39, wherein the number of the antennas is represented by Nt, and wherein the linear transformation is based on a Vandermonde matrix of size Nt×Nt.
0. 43. The method of claim 39, wherein the number of the antennas is represented by Nt, wherein the linear transformation is based on multiple matrices comprising a first matrix and a second matrix,
wherein the first matrix is a matrix of size Nt×Nt, wherein each entry of the first matrix is based on a power of ej2π/Nt, each entry of a column of the first matrix being equal to one, and
wherein the second matrix is a diagonal matrix of size Nt×Nt having diagonal entries that are based respectively on different powers of ej2π/P including the zeroth power, wherein P is an integer.
0. 44. The method of claim 43, wherein the multiple matrices include a third matrix, wherein the third matrix is a matrix of size Nt×Nt.
0. 45. The method of claim 16, wherein outputting the waveform comprises outputting an orthogonal frequency division multiplexing (OFDM) waveform.
0. 46. The method of claim 18, wherein applying the unitary matrix comprises applying a unitary matrix of size M×M in which all entries have equal norm of 1/√{square root over (M)}, where M is an integer greater than one.
0. 47. The method of claim 19, wherein outputting the multicarrier waveform comprises outputting multiple waveforms via multiple antennas, respectively.
0. 48. The method of claim 19, wherein outputting the multicarrier waveform comprises outputting an orthogonal frequency division multiplexing (OFDM) waveform.
0. 50. The method of claim 49, wherein interleaving the coded bits to produce the interleaved bits comprises writing the coded bits into a matrix row-wise, and reading the coded bits from the matrix column-wise.
0. 51. The method of claim 49, wherein interleaving the coded bits to produce the interleaved bits comprises positioning the coded bits to be mapped to different blocks of information bearing symbols.
0. 52. The method of claim 49, wherein performing the error-control coding on the input stream comprises applying a turbo code.
0. 53. The method of claim 49, wherein performing the error-control coding on the input stream comprises applying a convolutional code.
0. 54. The method of claim 49, wherein the constellation having the finite alphabet is based on quadrature amplitude modulation (QAM).
0. 55. The method of claim 49, wherein the constellation having the finite alphabet is based on quadrature phase shift keying (QPSK).
0. 56. The method of claim 49, wherein applying the linear transformation comprises applying a unitary matrix of size M×M in which all entries have equal norm of 1/√{square root over (M)}, where M is an integer greater than one.
0. 57. The method of claim 49, wherein outputting the waveform comprises outputting multiple waveforms via multiple antennas, respectively.
0. 58. The method of claim 49, wherein outputting the waveform comprises outputting an orthogonal frequency division multiplexing (OFDM) waveform.
0. 60. The wireless communication device of claim 59, wherein a size of the constellation is larger than two, and wherein the circuitry is configured to separate the coded bits so that neighboring coded bits are mapped to different information bearing symbols.
0. 61. The wireless communication device of claim 59, wherein the wireless communication device comprises one of a base station and a mobile device.
0. 62. The wireless communication device of claim 59, wherein the output waveform comprises multiple output waveforms for transmission on multiple antennas, respectively.
0. 63. The wireless communication device of claim 59, wherein the output waveform comprises an orthogonal frequency division multiplexing (OFDM) waveform.
0. 65. The method of claim 64, wherein the number of the antennas is represented by Nt, and wherein the first matrix is based on an Nt-point inverse version of the FFT matrix.
0. 66. The method of claim 65, wherein the linear transformation is based on:
Θ = F N t T diag ( 1 , α , , α N t - 1 ) , α := j 2 π / P
wherein FNtT represents the first matrix, and wherein
diag(1, α, . . . , αNt−1)
represents the second matrix, wherein P is an integer.
0. 67. The method of claim 64, wherein the method further comprises:
performing error-control coding on an input data stream to produce coded bits;
interleaving the coded bits to produce interleaved bits;
mapping groups of the interleaved bits to produce the stream of information bearing symbols selected from the constellation having the finite alphabet, wherein a size of the constellation is larger than two, and wherein interleaving the coded bits comprises separating the coded bits so that neighboring coded bits are mapped to different information bearing symbols.
0. 69. The device of claim 68, wherein the number of the antennas is represented by Nt, and wherein the first matrix is based on an Nt-point inverse version of the FFT matrix.
0. 70. The device of claim 68, wherein the linear transformation is based on:
Θ = F N t T diag ( 1 , α , , α N t - 1 ) , α := j 2 π / P
wherein FNtT represents the first matrix, and wherein
diag(1, α, . . . , αNt−1)
represents the second matrix, wherein P is an integer.
0. 71. The device of claim 68, further comprising:
circuitry configured to (i) perform error-control coding on an input stream to produce coded bits, (ii) interleave the coded bits to produce blocks of interleaved bits, and (iii) map the blocks of interleaved bits to produce the stream of information bearing symbols selected from the constellation having the finite alphabet.
0. 72. The device of claim 68, wherein the output waveforms comprises orthogonal frequency division multiplexing (OFDM) waveforms.
0. 73. The wireless communication device of claim 29, wherein the linear transformation is based on a Fourier transform.
0. 74. The wireless communication device of claim 34, wherein the third matrix is a unitary matrix.
0. 75. The method of claim 39, wherein the linear transformation is based on a Fourier transform.
0. 76. The method of claim 44, wherein the third matrix is a unitary matrix.
0. 77. The wireless communication device of claim 13, wherein the linear transformation is based on a Fourier transform.

This application claims priority from U.S. Provisional Application Ser. No. 60/374,886, filed Apr. 22, 2002, U.S. Provisional Application Ser. No. 60/374,935, filed Apr. 22, 2002, U.S. Provisional Application Ser. No. 60/374,934, filed Apr. 22, 2002, U.S. Provisional Application Ser. No. 60/374,981, filed Apr. 22, 2002, U.S. Provisional Application Ser. No. 60/374,933, filed Apr. 22, 2002, the entire contents of which are incorporated herein by reference.

This invention was made with [ETSI, “Broadband Radio Access Networks (BRAN); HIPERLAN Type 2 technical specification Part 1—physical layer,” DTS/BRAN030003-1, October 1999]
We want to design Θ so that a large diversity order can be guaranteed irrespective of the constellation that the entries of si are drawn from, with a small amount of introduced redundancy.

We can conceptually view Θ together with the OFDM modulation FH as a combined N×K encoder {tilde over (Θ)}:=FHΘ, which in a sense blends the single-carrier and multicarrier notions. Indeed, by selecting Θ, hence {tilde over (Θ)}, the system in FIG. 1 can describe various single and multicarrier systems, some of them are provided shortly as special cases of our LE-OFDM. The received vector {tilde over (x)} is related to the information symbol vector s through the matrix product {tilde over (H)}Θ.

We define the Hamming distance δ(u, u′) between two vectors u and u′ as the number of non-zero entries in the vector uc=u−u′ and the minimum Hamming distance of the set custom character as δmin (custom character):=min{δ(u, u′)|u, uεcustom character}. When there is no confusion, we will simply use δmin for brevity. The minimum Euclidean distance between vectors in custom character is denoted as dmin(custom character) or simply dmin.

Because such encoding operates in the complex field, it does not increase the dimensionality of the signal space. This is to the contrasted to the GF encoding: the codeword set of a GF (n, k) code, when viewed as a real/complex vector, in general has a higher dimensionality (n) than does the original uncoded block of symbols (k). Exceptions include the repetition code, for which the codeword set has the same dimensionality as that of the input.

Consider the binary (3, 2) block code generated by the matrix

[ 1 0 1 0 1 1 ] T ( 3 )
followed by BPSK constellation mapping (e.g., 0→−1 and 1→1). The codebook consists of 4 codewords
[−1 −1 −1]T, [1 −1 1]T, [−1 1 1]T, [1 1 −1]T.  (4)
These codewords span the R3×1 (or C3×1) space and therefore the codebook has dimension 3 in the real or complex field, as illustrated in FIG. 2.

In general, a (n, k) binary GF block code is capable of generating 2k codewords in an n-dimensional space Rn×1 or Cn×1. If we view the transmit signal design problem as packing spheres in the signal space (Shannon's point of view), an (n, k) GF block code followed by constellation mapping packs spheres in an n-dimensional space and thus has the potential to be better (larger sphere radius) than a k-dimensional packing. In our example above, if we normalize the codewords by a factor √{square root over (2/3)} so that the energy per bit Eb is one, the 4 codewords have mutual Euclidean distance √{square root over (8/3)}, larger than the minimum distance √{square root over (2)} of the uncoded BPSK signal set (±1,±1). This increase in minimum Euclidean distance leads to improved system performance in AWGN channels, at least for high signal to noise ratio (SNR). For fading channels, the minimum Hamming distance of the codebook dominates high SNR performance in the form of diversity gain (as will become clear later). The diversity gain achieved by the (3, 2) block code in the example is the minimum Hamming distance 2.

CF linear encoding on the other hand, does not increase signal dimension; i.e., we always have dim(U)≦dim(S). When Θ has full column rank K, dim(U)=dim(S), in which case the codewords span a K-dimensional subspace of the N-dimensional vector space CK×1. In terms of sphere packing, CF linear encoding does not yield a packing of dimension higher than K.

We have the following assertion about the minimum Euclidean distance.

Proposition 1 Suppose tr(ΘΘH)=K. If the entries of sεδ are drawn independently from a constellation custom character of minimum Euclidean distance of dmin (custom character) then the codewords in u:={Θs|sεδ} have minimum Euclidean distance no more than dmin(custom character).

Proof: Under the power constraint tr(ΘΘH)=K, at least one column of Θ will have norm no more than 1. Without loss of generality, suppose the first column has norm no more than 1. Consider sα=(α, 0, . . . , 0)T and sβ=(β, 0, . . . , 0)T, where α and β are two symbols from the constellation that are separated by dmin. The coded vectors uα=Θsα and uβ=Θsβ are then separated by a distance no more than dmin.

Due to Proposition 1, CF linear codes are not effective for improving performance for AWGN channels. But for fading channels, they may have an advantage over GF codes, because they are capable of producing codewords that have large Hamming distance.

The encoder

Θ = 4 15 [ 1 1 1 0.5 - 0.5 0.5 ] T , ( 5 )
operating on BPSK signal set δ={±1}2, produces 4 codewords of minimum Euclidean distance √{square root over (4/5)} and minimum Hamming distance 3. Compared with the GF code in Example 1, this real code has smaller Euclidean distance but larger Hamming distance. In addition, the CF coding scheme described herein differs from the GF block coding in that the entries of the LE output vector u usually belong to a larger, although still finite, alphabet set than do the entries of the input vector s
where N0/2 is the noise variance per dimension, y:=DHΘs, y′:=DHΘs′, and d(y,y′)=∥y−y′∥ is the Euclidean distance between y and y′.

Let us consider the N×(L+1) matrix V with entries [V]n,t=exp(−j2πnl/N), and use it to perform the N-point discrete Fourier transform Vh of h. Note that DH=diag(Vh); i.e., the diagonal entries of DH are those in vector Vh. Using the definitions e:=s−s′εδ|e, ue:=Θe, and De:=diag(ue), we can write y−y′=DHue=diag(Vh)ue. Furthermore, we can express the squared Euclidean distance d2(y,y′)=∥DHue2=∥DeVh∥2 as
d2(y,y′)=hHVHDeHDeVh:=hHAeh.  (7)

An upper bound to the average PEP can be obtained by averaging (6) with respect to the random channel h to obtain
where C is the search radius, a decoding parameter. Since R is upper triangular, in order to satisfy the inequality in (18), the last entry of s must satisfy |[R]K,K[s]K|<C, which reduces the search space if C is small. For one possible value of the last entry, possible candidates of the last-but-one entry are found and one candidate is taken. The process continues until a vector of s0 is found that satisfies (18). Then the search radius C is set equal to ∥QHx−Rs0∥ and a new search round is started. If no other vector is found inside the radius, then s0 is the ML solution. Otherwise, if s1 is found inside the sphere, the search radius is again reduced to ∥QHx−Rs1∥, and so on. If no s0 is ever found inside the initial sphere of radius C, then C is too small. In this case, either a decoding failure is declared or C is increased.

The complexity of the SD is polynomial in K
respectively, where (·)T denotes pseudo-inverse, ση2 is the variance of entries of noise η, and Rs is the autocorrelation matrix of s. Given the ZF and MMSE equalizers, they each require custom character(N×K) operations per K symbols. So per symbol, they require only custom character(N) operations. To obtain the ZF or MMSE equalizers, inversion of a N×N matrix is involved, which has complexity custom character(N3). However, the equalizers only needs to be recomputed when the channel changes.

The ML detection schemes in general have high complexity, while the linear detectors may have decreased performance. The class of decision-directed detectors lies between these categories, both in terms of complexity and in terms of performance.

Decision-directed detectors capitalize on the finite alphabet property that is almost always available in practice. In the equalization scenario, they are more commonly known as Decision Feedback Equalizers (DFE). In a single-user block formulation, the DFE has a structure as shown in FIG. 6, where the feed-forward filter is represented as a matrix W and the feedback filter is presented as B
W=URsΘHDHH(Rη+DHΘRsΘHDHH)−1, B=U−I,  (20)
where the R's denote autocorrelation matrices, (19) was obtained using Cholesky decomposition, and U is an upper triangular matrix with unit diagonal entries. Since the feed-forward and feedback filtering entails only matrix-vector multiplications, the complexity of such decision directed schemes is comparable to that of linear detectors. Because decision directed schemes capitalize on the finite-alphabet property of the information symbols, the performance is usually (much) better than linear detectors.

As an example, we list in the following table the approximate number of flops needed for different decoding schemes when K=14, L=2, N=16, and BPSK modulation is deployed; i.e., δ={±1}K.

TABLE 1
Decoding Scheme order of Flops/symbol
Exhaustive ML >2K = 214 = 16.384
Sphere Decoding ≈800 (empirical)
ZF/MMSE ≈N = 16
Decision-Directed ≈N = 16
Viterbi for ZP-only 2L = 22 = 4

Other possible decoding methods include iterative detectors, such as successive interference cancellation with iterative least squares (SIC-ILS) (T. Li and N. D. Sidiropoulos, “Blind digital signal separation using successive interference cancellation iterative least squares,” IEEE Transactions on Signal Processing, vol. 48, no. 11, pp. 3146-3152, November 2000, herein incorporated by reference), and multistage cancellations (S. Verdú, Multiuser Detection, Cambridge Press, 1998, herein incorporated by reference). These methods are similar to the illustrated DFE in the interference from symbols that are decided in a block is canceled before a decision on the current symbol is made. In SIC-ILS, least squares is used as the optimization criterion and at each step or iteration, the cost function (least-squares) will decrease or remain the same. In multistage cancellation, the MMSE criterion is often used such that MF is optimum after the interference is removed (supposing that the noise is white). The difference between a multistage cancellation scheme and the block DFE is that the DFE symbol decisions are made serially; and for each undecided symbol, only interference from symbols that have been decided is cancelled; while in multistage cancellation, all symbols are decided simultaneously and then their mutual interferences are removed in a parallel fashion.

As illustrated in FIG. 7, another embodiment may utilize for LEOFDM equalization an iterative “sum-product” decoding algorithm, which is also used in Turbo decoding (F. R. Kschischang, B. J. Frey, and H-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498-519, February 2001, herein incorporated by reference). In particular, the coded system is represented using a factor graph, which describes the interdependence of the encoder input, the encode output, and the noise-corrupted coded symbols.

As a simple example, suppose the encoder takes a block of 3 symbols s:=[s0, s1, s2]T as input and linearly encodes them by a 4×3 matrix Θ to produce the coded symbols u:=[u0, u1, u2, u3]. After passing through the channel (OFDM modulation/demodulation), we obtain the channel output xi=H(ej2πi/4)ui, i=0, 1, 2, 3. The factor graph for such a coded system is shown in FIG. 7, where the LE is represented by linear constraints between the LE input symbols s and the LE output symbols u.

When the number of carriers N is very large (e.g., 1,024), it is desirable to keep the decoding complexity manageable. To achieve this we can split the encoder into several smaller encoders. Specifically, we can choose Θ=PΘ′, where P is a permutation matrix that interleaves the subcarriers, and Θ′ is a block diagonal matrix: Θ′=diag(Θ0, Θ1, . . . , ΘM-1). This is a essentially a form of coding for interleaved OFDM, except that the coding is done in complex domain here. The matrices Θm, m=0, . . . , M−1 are of smaller size than Θ and all of them can even be chosen to be identical. With such designed Θ, decoding s from the noisy DHΘs is equivalent to decoding M coded sub-vectors of smaller sizes and therefore the overall decoding complexity can be reduced considerably. Such a decomposition is particularly important when a high complexity decoder such as the sphere decoder is to be deployed.

The price paid for low decoding complexity is a decrease in transmission rate. When such parallel encoding is used, we should make sure that each of the Θm matrices can guarantee full diversity, which requires Θm to have L redundant rows. The overall Θ will then have ML redundant rows, which corresponds to an M-fold increase of the redundancy of a full single encoder of size N×K. If a fixed constellation is used for entries in s, then square Θm's can be used, which does not lead to loss of efficiency (Z. Liu, Y. Xin, and G. B. Giannakis, “Linear Constellation Precoding for OFDM with Maximum Multipath Diversity and Coding Gains,” Proceedings of 35th Asilomar Conference on Signals, Systems & Computers, Pacific Grove, Calif., Nov. 4-7, 2001, pp. 1445-1449, herein incorporated by reference).

FIGS. 8-10 are graphs that illustrate exemplary results of simulations of the described techniques. In the illustrated results, we compare the proposed wireless communication techniques with existing coded OFDM systems that deploy existing GF block codes and convolutional codes. In all cases, BPSK constellation is used, and in Test Case 2 and 3, the binary encoded symbols are mapped to ±1's before OFDM modulation.

Test case 1 (Decoding of LE-OFDM): We first test the performance of different decoding algorithms. The LE-OFDM system has parameters K=14, N=16, L=2. The channel is i.i.d. Rayleigh and BER's for 200 random channel realizations according to As1) are averaged. FIG. 8 shows the performance of ZF, MMSE, DFE, and sphere decoding (ML) for LE-OFDM. We notice that at BER of 10−4 DFE performs about 2 dB better than the MMSE detectors, while at the same time it is only less than 1 dB inferior to the sphere decoder, which virtually achieves the ML decoding performance. The complexity of ZF, MMSE, DFE is all about N=16 flops per symbol, which is much less than the sphere decoding algorithm, which empirically needs about 800 flops per symbol in this case.
Test case 2 (Comparing LE-OFDM with BCH-coded OFDM): For demonstration and verification purposes, we first compare LE-OFDM with coded OFDM that relies on GF block coding. The channel is modeled as FIR with 5 i.i.d. Rayleigh distributed taps. In FIG. 9, we illustrate Bit Error Rate (BER) performance of CF coded OFDM with Vandermonde code of Theorem 4, and that of binary BCH-coded OFDM. The system parameters are K=26, N=31. The generating polynomial of the BCH code is g(D)=1+D2+D5. Since we can view this BCH as a rate 1 convolutional code with the same generator and with termination after 26 information symbols (i.e., the code ends at the all-zero state), we can use the Viterbi algorithm [J. Hagenauer, and P. Hoeher, “A Viterbi Algorithm with Soft-Decision Outputs and Its Applications,” in Proc. Of the IEEE 1989 Global Communications Conference, Dallas, Tex., pp. 1680-1686, November 1989, herein incorporated by reference] for soft-decision ML BCH decoding. For LE-OFDM, since the transmission is essentially a ZP-only single-carrier scheme, the Viterbi algorithm is also applicable for ML decoding.

Since the binary (26, 31) BCH code has minimum Hamming distance 3, it possesses a diversity order of 3, which is only half of the maximum possible (L+1=6) that LE-OFDM achieves with the same spectral efficiency. This explains the difference in their performance. We can see that when the optimum ML decoder is adopted by both receivers. LE-OFDM outperforms coded OFDM with BCH coding considerably. The slopes of the corresponding BER curves also confirm our theoretical results.

Test case 3 (Comparing LE-OFDM with convolutionally coded OFDM): In this test, we compare (see FIG. 10) our LE-OFDM system with convolutionally coded OFDM (with a rate ½ code punctured to rate ¾ followed by interleaving) that is deployed by the HiperLAN2 standard (ETSI, “Broadband Radio Access Networks (BRAN), herein incorporated by reference; HIPERLAN Type 2 technical specification Part 1—physical layer,” DTS/BRAN030003-1, October 1999, herein incorporated by reference) over the channels used in Test Case 2. The rate ½ mother code has its generator in octal form as (133, 171) and there are 64 states in its trellis. Every 3rd bit from the first branch and every second bit from the second branch of the mother code are punctured to obtain the rate ¾ code, which results in a code whose weight enumerating function is 8W5+31W6+160W7+ . . . . So the free distance is 5, which means that the achieved diversity is 5, less than the diversity order 6 achieved by LE-OFDM.

The parameters are K=36, N=48. We use two parallel truncated DCT encoders; that is, Θ=I2×2{circle around (x)}Θ0, where {circle around (x)} denotes Kronecker product, and Θ0 is a 24×18 encoder obtained by taking the first 18 columns of a 24×24 DCT matrix. With ML decoding, LE-OFDM performs about 2 dB better than convolutionally coded OFDM. From the ML performance curves in FIG. 10, LE-OFDM seems to achieve a larger coding advantage than the punctured convolutional code we used.

Surprisingly, even with linear MMSE equalization, the performance of LE-OFDM is better than coded OFDM for SNR values less than 12 dB. The complexity of ML decoding for LE-OFDM is quite high in the order of 1,000 flops per symbol. But the ZF and MMSE decoders have comparable or even lower complexity than the Viterbi decoder for the convolutional code.

The complexity of LE-OFDM can be dramatically reduced using the parallel encoding method with square encoders (Z. Liu, Y. Xin, and G. B. Giannakis, “Linear Constellation Precoding for OFDM with Maximum Multipath Diversity and Coding Gains,” Proceedings of 35th Asilomar Conference on Signals, Systems & Computers, Pacific Grove, Calif., Nov. 4-7, 2001, pp. 1445-1449, herein incorporated by reference). It is also possible to combine CF coding with conventional GF coding, in which case only small square encoders of size 2×2 or 4×4 are necessary to achieve near optimum performance (Z. Wang, S. Zhou, and G. B. Giannakis, “Joint coded-precoded OFDM with low-complexity turbo-decoding,” in Proc. of the European Wireless Conf., Florence, Italy, Feb. 25-28, 2002, pp. 648-654, herein incorporated by reference).

Various embodiments of the invention have been described. The described techniques can be embodied in a variety of receivers and transmitters including base stations, cell phones, laptop computers, handheld computing devices, personal digital assistants (PDA's), and the like. The devices may include a digital signal processor (DSP), field programmable gate array (FPGA), application specific integrated circuit (ASIC) or similar hardware, firmware and/or software for implementing the techniques. If implemented in software, a computer readable medium may store computer readable instructions, i.e., program code, that can be executed by a processor or DSP to carry out one of more of the techniques described above. For example, the computer readable medium may comprise random access memory (RAM), read-only memory (ROM), non-volatile random access memory (NVRAM), electrically erasable programmable read-only memory (EEPROM), flash memory, or the like. The computer readable medium may comprise computer readable instructions that when executed in a wireless communication device, cause the wireless communication device to carry out one or more of the techniques described herein. These and other embodiments are within the scope of the following claims.

Giannakis, Georgios B., Xin, Yan, Wang, Zhengdao

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