USRE45230E1 - Wireless communication system having linear encoder - Google Patents
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- USRE45230E1 USRE45230E1 US13/858,734 US201313858734A USRE45230E US RE45230 E1 USRE45230 E1 US RE45230E1 US 201313858734 A US201313858734 A US 201313858734A US RE45230 E USRE45230 E US RE45230E
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
- H04L27/26—Systems using multi-frequency codes
- H04L27/2601—Multicarrier modulation systems
- H04L27/2602—Signal structure
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
- H04L1/0041—Arrangements at the transmitter end
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/02—Arrangements for detecting or preventing errors in the information received by diversity reception
- H04L1/04—Arrangements for detecting or preventing errors in the information received by diversity reception using frequency diversity
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
- H04L2025/0335—Arrangements for removing intersymbol interference characterised by the type of transmission
- H04L2025/03375—Passband transmission
- H04L2025/03414—Multicarrier
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
- H04L2025/03433—Arrangements for removing intersymbol interference characterised by equaliser structure
- H04L2025/03439—Fixed structures
- H04L2025/03445—Time domain
- H04L2025/03471—Tapped delay lines
- H04L2025/03484—Tapped delay lines time-recursive
- H04L2025/0349—Tapped delay lines time-recursive as a feedback filter
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L5/00—Arrangements affording multiple use of the transmission path
- H04L5/0001—Arrangements for dividing the transmission path
- H04L5/0003—Two-dimensional division
- H04L5/0005—Time-frequency
- H04L5/0007—Time-frequency the frequencies being orthogonal, e.g. OFDM(A), DMT
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L5/00—Arrangements affording multiple use of the transmission path
- H04L5/003—Arrangements for allocating sub-channels of the transmission path
- H04L5/0044—Arrangements for allocating sub-channels of the transmission path allocation of payload
Definitions
- the invention relates to communication systems and, more particularly, transmitters and receivers for use in wireless communication systems.
- a channel that couples a transmitter to a receiver is often time-varying due to relative transmitter-receiver motion and multipath propagation. Such a time-variation is commonly referred to as fading, and may severely impair system performance.
- ISI intersymbol interference
- IFFT Inverse Fast Fourier Transform
- OFDM Orthogonal Frequency Division Multiplexing
- a cyclic prefix (CP) of length greater than or equal to the channel order is inserted per block at the transmitter and discarded at the receiver.
- the CP also converts linear convolution into cyclic convolution and thus facilitates diagonalization of an associated channel matrix (Z. Wang and G. B. Giannakis, “Wireless multicarrier communications: where Fourier meets Shannon,” IEEE Signal Processing Magazine, vol. 47, no. 3, pp. 29-48, May 2000, herein incorporated by reference).
- OFDM transfers the multipath diversity to the frequency domain in the form of (usually correlated) fading frequency response.
- Each OFDM subchannel has its gain being expressed as a linear combination of the dispersive channel taps.
- Error-control codes are usually invoked before the IFFT processing to deal with the frequency-selective fading. These include convolutional codes, Trellis Coded Modulation (TCM) or coset codes, Turbo-codes, and block codes (e.g., Reed-Solomon or BCH).
- TCM Trellis Coded Modulation
- Turbo-codes e.g., Reed-Solomon or BCH.
- block codes e.g., Reed-Solomon or BCH.
- Such coded OFDM schemes often incur high complexity and/or large decoding delay (Y. H. Jeong, K. N. Oh, and J. H. Park, “Performance evaluation of trellis-coded OFDM for digital audio broadcasting,” in Proc. of the IEEE Region 10 Conf, 1999, vol. 1, pp. 569-572, herein incorporated by reference).
- Some of these schemes also require Channel State Information (CSI) at the transmitter (A. Ruiz, J. M.
- CSI Channel
- Cioffi, and S. Kasturia “Discrete multiple tone modulation with coset coding for the spectrally shaped channel,” IEEE Transactions on Communications, vol. 40, no. 6, pp. 1012-1029, June 1992, herein incorporated by reference; H. R. Sadjadpour, “Application of Turbo codes for discrete multi-tone modulation schemes,” in Proc. of Intl. Conf. on Com., Vancouver, Canada, 1999, vol. 2, pp. 1022-1027, herein incorporated by reference), which may be unrealistic or too costly to acquire in wireless applications where the channel is rapidly changing.
- each IFFT-processed block can be zero padded (ZP) by at least as many zeros as the channel order (B. Muquet, Z. Wang, G. B. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixed or zero padded multicarrier transmissions?” IEEE Transactions on Communications, August 2000 (to appear), herein incorporated by reference; Z. Wang and G. B. Giannakis, “Wireless multicarrier communications: where Fourier meets Shannon,” IEEE Signal Processing Magazine, vol. 47, no. 3, pp. 29-48, May 2000, herein incorporated by reference).
- the CF block code described herein can also be viewed as a form of real-number or analog codes (W. Henkel, Zur Decodierung algebraischer Blockcodes über precisen Alphabeten, Ph.D. thesis, VDI Fort suits-Berichte, Erasmus 10, Nr. 109, VDI-Verlag, Düsseldorf, 1989, herein incorporated by reference; T. G. Marshall Jr., “Coding of real-number sequences for error correction: A digital signal processing problem,” IEEE Journal on Selected Areas in Communications, vol. 2, no. 2, pp. 381-392, March 1984 herein incorporated by reference; J. K. Wolf, “Redundancy, the discrete Fourier transform, and impulse noise cancellation,” IEEE Transactions on Communications, vol. 31, no. 3, pp. 458-461, March 1983, herein incorporated by reference).
- Linear Encoder (LE),” 0 and the corresponding encoding process is called “linear encoding,” also abbreviated as LE when no confusions arise.
- the resulting CF coded OFDM will be called LE-OFDM.
- the linear encoder is designed so that maximum diversity order can be guaranteed without an essential decrease in transmission rate.
- the described LE can be designed to guarantee maximum diversity order irrespective of the information symbol constellation with minimum redundancy.
- the described LE codes are maximum distance separable (MDS) in the real or complex field, which generalizes the well-known MDS concept for Galois field (GF) codes (F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977, herein incorporated by reference).
- MDS maximum distance separable
- GF Galois field
- a wireless communication device comprises an encoder that linearly encodes a data stream to produce an encoded data stream, and a modulator to produce an output waveform in accordance with the encoded data stream for transmission through a wireless channel.
- a wireless communication device comprises a demodulator that receives a waveform carrying a linearly encoded transmission and produces a demodulated data stream, and a decoder that applies decodes the demodulated data and produce estimated data.
- a method comprises linearly encoded a data stream with to produce an encoded data stream, and outputting a waveform in accordance with the data stream for transmission through a wireless channel.
- a computer-readable medium comprises instructions to cause a programmable processor to linearly encode a data stream with to produce an encoded data stream, and output a waveform in accordance with the data stream for transmission through a wireless channel.
- FIG. 1 is a block diagram illustrating an exemplary wireless communication system in which a transmitter and receiver implement linear precoding techniques.
- FIGS. 2A , 2 B illustrate uncoded and GF-coded BPSK signals.
- FIG. 3 illustrates an example format of a transmission block for CP-only transmissions by the transmitter of FIG. 1 .
- FIG. 4 illustrates an example format of a transmission block for ZP-only transmissions by the transmitter of FIG. 1 .
- FIG. 5 illustrates sphere decoding applied in one embodiment of the receiver of FIG. 1 .
- FIG. 6 illustrates an example portion of the receiver of FIG. 1 .
- FIG. 7 is factor graph representing an example linear encoding process.
- FIGS. 8-10 are graphs that illustrate exemplary results of simulations of the described techniques.
- FIG. 1 is a block diagram illustrating a telecommunication system 2 in which transmitter 4 communicates data to receiver 6 through wireless channel 8 .
- Transmitter 4 transmits data to receiver 6 using one of a number of conventional multi-carrier transmission formats including Orthogonal Frequency Division Multiplexing (OFDM).
- OFDM has been adopted by many standards including digital audio and video broadcasting (DAB, DVB) in Europe and high-speed digital subscriber lines (DSL) in the United States.
- OFDM has also been proposed for local area mobile wireless broadband standards including IEEE802.11a, MMAC and HIPERLAN/2. [ETSI, “Broadband Radio Access Networks (BRAN); HIPERLAN Type 2 technical specification Part 1—physical layer,” DTS/BRAN030003-1, October 1999].
- system 2 represents an LE-OFDM system having N subchannels.
- the techniques described herein robustify multi-carrier wireless transmissions, e.g., OFDM, against random frequency-selective fading by introducing memory into the transmission with complex field (CF) encoding across the subcarriers.
- transmitter 4 utilizes different linear combinations of the information symbols on the subcarriers.
- the techniques described herein may be applied to uplink and/or downlink transmissions, i.e., transmissions from a base station to a mobile device and vice versa.
- transmitters 4 and receivers 6 may be any device configured to communicate using a multi-user wireless transmission including a cellular distribution station, a hub for a wireless local area network, a cellular phone, a laptop or handheld computing device, a personal digital assistant (PDA), and the like.
- a cellular distribution station a hub for a wireless local area network
- a cellular phone a laptop or handheld computing device
- PDA personal digital assistant
- transmitter 4 includes linear encoder 10 and an OFDM modulator 12 .
- AWGN additive white Gaussian noise
- the set ⁇ is always finite. But we allow it to be infinite in our performance analysis.
- the encoder ⁇ considered here does not depend on the OFDM symbol index i. Time-varying encoder may be useful for certain purposes (e.g., power loading), but they will not be pursued here. Hence, from now on, we will drop our OFDM symbol index i for brevity.
- the matrix ⁇ can be naturally viewed as the generating matrix of a complex field block code.
- the set is a subset of the C N ⁇ 1 vector space. More specifically, is a subset of the K dimensional subspace spanned by the columns of ⁇ .
- 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.
- 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 R 3 ⁇ 1 (or C 3 ⁇ 1 ) space and therefore the codebook has dimension 3 in the real or complex field, as illustrated in FIG. 2 .
- a (n, k) binary GF block code is capable of generating 2 k codewords in an n-dimensional space R n ⁇ 1 or C n ⁇ 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.
- 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).
- SNR signal to noise ratio
- 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).
- CF linear encoding does not yield a packing of dimension higher than K.
- 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 cyclic prefix in this case consists of L zeros, which, together with the L zeros from the encoding process, result in 2L consecutive zeros between two consecutive uncoded information blocks of length K. But only L zeros are needed in order to separate the information blocks. CP is therefore not necessary because the L zeros created by ⁇ already separate successive blocks.
- ZP-only transmission is essentially a simple single-carrier block scheme.
- viewing it as a special case of the LE-OFDM design will allow us to apply the results about LE-OFDM and gain insights into its performance. It turns out that this special case is indeed very special: it achieves the best high-SNR performance among the LE-OFDM class.
- the PEP can be approximated using the Chernoff bound as: P(s ⁇ s′
- An upper bound to the average PEP can be obtained by averaging (6) with respect to the random channel h to obtain (V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744-765, March 1998, herein incorporated by reference):
- the diversity order G d,e determines the slope of the averaged (w.r.t. the random channel) PEP (between s and s′) as a function of the SNR at high SNR (N 0 ⁇ 0).
- G c,e determines the shift of this PEP curve in SNR relative to a benchmark error rate curve of (1 ⁇ 4N 0 ) ⁇ r e .
- Diversity order herein to mean the asymptotic slope of the error probability versus SNR curve in a log-log scale.
- “diversity” refers to “channel diversity,” i.e., roughly the degree of freedom of a given channel.
- three conditions may be satisfied: i) Transmitter 4 is well-designed so that the information symbols are encoded with sufficient redundancy (enough diversification); ii) Channel 8 is capable of providing enough degrees of freedom; iii) Receiver 4 is well designed so as to sufficiently exploit the redundancy introduced at the transmitter.
- G d Since the diversity order G d determines how fast the symbol error probability drops as SNR increases, G d is to be optimized first.
- Theorem 1 (Maximum Achievable Diversity Order): For a transmitted codeword set with minimum Hamming distance ⁇ min , over i.i.d. FIR Rayleigh fading channels of order L, the diversity order is min( ⁇ min , L+1). Thus, the Maximum Achievable Diversity Order (MADO) of LE-OFDM transmissions is L+1 and in order to achieve MADO, we need ⁇ min ⁇ L+1.
- MADO Maximum Achievable Diversity Order
- Theorem 1 is intuitively reasonable because the FIR Rayleigh fading channel offers us L+1 independent fading [S. G. Wilson, Digital Modulation and Coding, Prentice-Hall, Inc, 1996, herein incorporated by reference] taps, which is the maximum possible number of independent replicas of the transmitted signal in the serial transmission mode. In order to achieve the MADO, any two codewords in should be different by no less than L+1 entries.
- the results in Theorem 1 can also be applied to GF-coded/interleaved OFDM systems provided that channel coding or interleaving is applied only within an OFDM symbol and not across successive OFDM symbols.
- the diversity is again the minimum of the minimum Hamming distance of the code and L+1. To see this, it suffices to view as the codeword set of GF-coded blocks.
- Theorem 2 (Symbol Detectability MADO): Under the channel conditions of Theorem 1, the maximum diversity order is achieved if and only if symbol detectability is achieved; i.e., ⁇ D H ⁇ c ⁇ 2 ⁇ 0, ⁇ e ⁇ e and ⁇ h ⁇ 0.
- the result in Theorem 2 is somewhat surprising: it asserts the equivalence of a deterministic property of the code, namely symbol detectability in the absence of noise, with a statistical property, the diversity order. It can be explained though, by realizing that in random channels, the performance is mostly affected by the worst channels, despite their small realization probability. By guaranteeing detectability for any, and therefore the worst, channels, we are essentially improving the ensemble performance.
- the symbol detectability condition in Theorem 2 should be checked against all pairs s and s′, which is usually not an easy task, especially when the underlying constellations are large and/or when the size K of s is large. But it is possible to identify sufficient conditions on ⁇ that guarantee symbol detectability and that are relatively easy to check.
- One such condition is provided by the following theorem.
- Vandermonde encoders in i) satisfy the conditions of Theorem 3. Any K rows of the matrix ⁇ ( ⁇ ) form a square Vandermonde matrix with distinct rows. Such a Vandermonde matrix is known to have a determinant different from 0. Therefore, any K rows of ⁇ ( ⁇ ) are linearly independent, which satisfies the conditions in Theorem 3.
- ⁇ 1 [ cos ⁇ ( 1 2 ⁇ ⁇ 0 ) cos ⁇ ( 3 2 ⁇ ⁇ 0 ) ⁇ cos ⁇ ( 2 ⁇ ⁇ K - 1 2 ⁇ ⁇ 0 ) cos ⁇ ( 1 2 ⁇ ⁇ 1 ) cos ⁇ ( 3 2 ⁇ ⁇ 1 ) ⁇ cos ⁇ ( 2 ⁇ ⁇ K - 1 2 ⁇ ⁇ 1 ) cos ⁇ ( 3 2 ⁇ ⁇ 1 ) ⁇ cos ⁇ ( 2 ⁇ ⁇ K - 1 ) ⁇ ⁇ ⁇ cos ⁇ ( 1 2 ⁇ ⁇ K - 1 ) cos ⁇ ( 3 2 ⁇ ⁇ K - 1 ) ⁇ cos ⁇ ( 2 ⁇ ⁇ K - 1 2 ⁇ ⁇ K - 1 ) ] ( 11 )
- z n cos ⁇ ( 1 2 ⁇ ⁇ n ) .
- Theorem 5 (MADO of Correlated Rayleigh Channels): Let the channel h be zero-mean complex Gaussian with correlation matrix R h . The maximum achievable diversity order equals the rank of R h , which is achieved by any encoder that achieves MADO with i.i.d. Rayleigh channels. If R h is full rank and MADO is achieved, then the coding advantage is different from the coding advantage in the i.i.d. case only by a constant
- R h [ U 1 U 2 ] ⁇ [ ⁇ 1 0 0 ⁇ 2 ] ⁇ [ U 1 H U 2 H ] . ( 12 ) where U 1 is (L+1) ⁇ r h , U 2 is (L+1) ⁇ (L+1 ⁇ r h ), A 1 is r h ⁇ r h full rank diagonal, and A 2 is an (L+1 ⁇ r h ) ⁇ (L+1 ⁇ r h ) all-zero matrix.
- a ⁇ e ⁇ 1 1 2 ⁇ U 1 H ⁇ A e ⁇ U 1 ⁇ ⁇ 1 1 2 is an r h ⁇ r h matrix.
- a ⁇ e ⁇ 1 1 2 ⁇ U 1 H ⁇ B e H ⁇ B e ⁇ U 1 ⁇ ⁇ 1 1 2 is the Gram matrix of
- Theorem 5 asserts that the rank(R h ) is the MADO for LE-OFDM systems as well as for coded OFDM systems that do not code or interleave across OFDM symbols. Also, MADO-achieving transmissions through i.i.d. channels can achieve the MADO for correlated channels as well.
- ZP-only transmission is one of the coding advantage maximizers (Z. Wang, X. Ma, and G. B. Giannakis, “OFDM or single-carrier zero-padded block transmissions?” IEEE Transactions on Communications, August 2001 (accepted), herein incorporated by reference; “Optimality of Single-Carrier Zero-Padded Block Transmissions,” in Proc. of Wireless Comm. and Networking Conf., pp. 660-664, 2002, Orlando, Fla., herein incorporated by reference).
- Theorem 6 (ZP-only: maximum coding advantage): Suppose the entries of s(i) are drawn independently from a finite constellation with minimum distance of d min ( ). Then the maximum coding advantage of an LE-OFDM for i.i.d. Rayleigh fading channels under as1) is G c,max ⁇ L d min 2 ( ). The maximum coding advantage is achieved by ZP-only transmissions with any K.
- GF MDS codes F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North Holland, 1977, herein incorporated by reference.
- Examples of GF MDS codes include single-parity-check coding, repetition coding, generalized RS coding, extended RS coding, doubly extended RS coding, algebraic-geometry codes constructed using an elliptic curve.
- Reed-Solomon codes are the least restrictive among them in terms of the number of elements in the field, they are constrained on the code length and the alphabet size.
- Our linear encoders ⁇ operate over the complex field with no restriction on the input symbol alphabet or the coded symbol alphabet.
- a generator ⁇ for an MDS code is called systematic if it is in the form [I K , P] T where P is a K ⁇ (N ⁇ K) matrix.
- Theorem 8 (Systematic MDS code): A code generated by [I, P] T is MDS if and only if every square submatrix of P is nonsingular.
- Any square submatrix of a Cauchy matrix is nonsingular (R. M. Roth and G. Seroussi, “On generator matrices of MDS codes,” IEEE Transactions on Information Theory, vol. 31, no. 6, pp. 826-830, November 1985, herein incorporated by reference).
- ⁇ is a finite set, e.g., a finite constellation carved from (possibly sealed and shifted) Z K .
- This includes BPSK, QPSK, and QAM [X. Giraud, E. Boutillon, and J. C. Belfiore, “Algebraic tools to build modulation schemes for fading channels,” IEEE Transactions on Information Theory, vol. 43, pp. 938-952, May 1997, herein incorporated by reference; Z. Liu; Y. Xin, and G. B. Giannakis, “Unitary precoded OFDM with maximum multipath diversity and coding gains,” in Proc.
- LE-OFDM requires ML decoding.
- the minimum-distance detection rule becomes ML and can be formulated as follows:
- ML decoding of LE transmissions belongs to a general class of lattice decoding problems, as the matrix product D H ⁇ in (2) gives rise to a discrete subgroup (lattice) of the C N space under the vector addition operation.
- finding the optimum estimate in (16) requires searching over
- the search radius C is set equal to ⁇ Q H x ⁇ Rs 0 ⁇ and a new search round is started. If no other vector is found inside the radius, then s 0 is the ML solution. Otherwise, if s 1 is found inside the sphere, the search radius is again reduced to ⁇ Q H x ⁇ Rs 1 ⁇ , and so on. If no s 0 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 (U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,” Math. Comput., vol. 44, pp. 463-471, April 1985, herein incorporated by reference), which is better than exponential but still too high for practical purposes. Indeed, it is not suitable for codes of block size greater than, say, 16. When the block size is small, the sphere decoder can be considered as an option to achieve the ML performance at manageable complexity.
- Viterbi decoding can be used at a complexity of (Q L ) per symbol, where Q is the constellation size of the symbols in s (G. D. Jr. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Transactions on Information Theory, vol. 18, pp. 363-378, May 1972, herein incorporated by reference).
- Zero-forcing (ZF) and MMSE detectors (equalizers) offer low-complexity alternatives.
- the ZF and MMSE equalizers based on the input-output relationship (2) can be written as (A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Linear precoding for estimation and equalization of frequency-selective channels,” in Signal Processing Advances in Wireless and Mobile Communications, G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Eds. 2001, vol.
- ( ⁇ ) T denotes pseudo-inverse
- ⁇ ⁇ 2 is the variance of entries of noise ⁇
- R s is the autocorrelation matrix of s.
- 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.
- DFE Decision Feedback Equalizers
- B is usually chosen to be a strictly upper or lower triangular matrix with zero diagonal entries.
- the feedback loop is represented as a matrix, the operations happen in a serial fashion: the estimated symbols are fed back serially as their decisions are formed one by one.
- the matrices W and B can be designed according to ZF or MMSE criteria (N. Al-Dhahir and A. H. Sayed, “The finite-length multi-input multi-output MMSE-DFE,” IEEE Transactions on Signal Processing, vol. 48, no. 10, pp. 2921-2936, October 2000, herein incorporated by reference; A. Stamoulis, G. B. Giannakis, and A.
- 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. Verd ⁇ , 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.
- SP-ILS successive interference cancellation with iterative least squares
- 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.
- 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.
- 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).
- 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.
- M ⁇ 1 are of smaller size than ⁇ and all of them can even be chosen to be identical.
- decoding s from the noisy D H ⁇ s 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.
- FIGS. 8-10 are graphs that illustrate exemplary results of simulations of the described techniques.
- 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 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.
- 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.
- Test case 3 Comparing LE-OFDM with convolutionally coded OFDM:
- 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
- the rate 1 ⁇ 2 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 3 ⁇ 4 code, which results in a code whose weight enumerating function is 8W 5 +31W 6 +160W 7 + . . . . So the free distance is 5, which means that the achieved diversity is 5, less than the diversity order 6 achieved by LE-OFDM.
- ⁇ 0 is a 24 ⁇ 18 encoder obtained by taking the first 18 columns of a 24 ⁇ 24 DCT matrix.
- 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.
- 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.
- LE-OFDM 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.
- 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.
- DSP digital signal processor
- FPGA field programmable gate array
- ASIC application specific integrated circuit
- 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.
- 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.
- RAM random access memory
- ROM read-only memory
- NVRAM non-volatile random access memory
- EEPROM electrically erasable programmable read-only memory
- 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.
Abstract
Description
We assume the channel to be random FIR, consisting of no more than L+1 taps. The blocks within the dotted box represent a conventional uncoded OFDM system.
with H(jω) denoting the channel frequency response at ω; and ηi=Fηi standing for the FFT-processed additive white Gaussian noise (AWGN).
x=F{tilde over (x)}=F({tilde over (H)}FHΘs+ñ)=DHΘs+η. (2)
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.
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
operating on BPSK signal set δ={±1}2, produces 4 codewords of minimum Euclidean distance √{square root over (4/5)} and
- As 1) The channel h:=[h(0), h(1), . . . , h(L)]T has independent and identically distributed (i.i.d.) zero-mean complex Gaussian taps (Rayleigh fading). The corresponding correlation matrix of h is Rh:=E[hhH]=αL 2IL+1, where the constant αL:=1/(L+1).
P(s→s′|h)≦exp(−d2(y,y′)/4N0), (6)
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′.
d2(y,y′)=hHVHDe HDeVh:=hHAeh. (7)
where λe,0, λe,1, . . . , λe,L are the non-increasing eigen-values of the matrix Ae=VHDe HDeV.
We call re the diversity order, denoted as Gd,e, and (IIt=0 r
and the equality is achieved when δmin≧L+1.
satisfies
Using Chebyshev polynomials of the first kind
each entry
of Θ1 is a polynomial T2m+1(zn) of order 2m+1 of some
The determinant det(Θ1) is therefore a polynomial in z0, . . . , zK-1 of order Σn=1 K(2n−1)=K2. It is easy to see that when zn=0, or when zm=±zn, m≠n, Θ1 has an all-zero row, or, two rows that are either the same or the negative of each other. Therefore, zn, zm−zn, and zm+zn are all factors of det(Θ1). So, g(z0, z1, . . . , zK-1):=IInznIIm>n(zm 2−zn 2) is also a factor of det(Θ1). But g(z0, z1, . . . , zK-1) is of order K+K(K−1)=K2, which means that it is different from det (Θ1) by at most a constant. Using the leading
where U1 is (L+1)×rh, U2 is (L+1)×(L+1−rh), A1 is rh×rh full rank diagonal, and A2 is an (L+1−rh)×(L+1−rh) all-zero matrix. Define
and {tilde over (h)}:=[{tilde over (h)}1 T{tilde over (h)}2 T]T, where
is defined by
Since h2 has an autocorrelation matrix R{tilde over (h)}
Since
the entries of {tilde over (h)}1, which are jointly Gaussian, are i.i.d.
where
is an rh×rh matrix.
where {tilde over (λ)}e,l, l=1, . . . , rh, are the eigen-values of Ãe.
is the Gram matrix of
and thus Ae has rank equal to
the MADO for this correlated channel.
SD starts its search by looking only at vectors s such that
∥QHx−Rs∥<C, (18)
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.
Gzf=(DHΘ)T and Gmmse=RsΘHDH H(ση 2IN+DHΘRsΘHDH H)−1,
respectively, where (·)T denotes pseudo-inverse, ση 2 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 (N×K) operations per K symbols. So per symbol, they require only (N) operations. To obtain the ZF or MMSE equalizers, inversion of a N×N matrix is involved, which has complexity (N3). However, the equalizers only needs to be recomputed when the channel changes.
Rs −1+ΘHDH HRη −1DHΘ=UHΛU, (19)
W=URsΘHDH H(Rη+DHΘRsΘHDH H)−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.
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 | ||
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
Claims (77)
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