A Soft Decoding Scheme for Vector Quantization Over a ... - IEEE Xplore
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 10, OCTOBER 2005
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A Soft Decoding Scheme for Vector Quantization Over a CDMA Channel Ha H. Nguyen, Senior Member, IEEE
Abstract—The optimal decoding of vector quantization (VQ) over a code-division multiple-access (CDMA) channel is too complicated for systems with a medium-to-large number of users. This paper presents a low-complexity, suboptimal decoder for VQ over a CDMA channel. The proposed decoder is built from a soft-output multiuser detector, a soft bit estimator, and the optimal soft VQ decoding of an individual user. Simulation results obtained over both additive white Gaussian noise and flat Rayleigh fading channels show that with a lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the more complicated optimal decoder.
The soft-output MUDs considered include the jointly optimal MUD (OPT-MUD), the minimum mean-squared error MUD (MMSE-MUD), and the decorrelating MUD (DC-MUD) [9]. For each type of MUD, the soft-bit estimates are calculated and fed into the soft VQ decoders. Due to its lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the complicated optimal decoder.
Index Terms—Code-division multiple access (CDMA), combined source and channel coding, multiuser detection, soft decoding, vector quantization (VQ).
The system model considered in this letter is the same as users in a synchronous CDMA the one in [6]. There are system, where each user transmits its source vectors by means of VQ. The th user produces a -dimensional random vector . The vector is then encoded into an index , where for some integer . The bits per source transmission rate of the system is thus dimension. The th encoder is described by a partition
I. INTRODUCTION
T
HE source and channel coding of a communication system are often designed and implemented separately. This common practice is mainly due to the work by Shannon [1], where it was shown that such a separation can perform optimally. However, the positive coding theorems of information theory only show such separability in the limit of infinite codeword length, and hence, infinite delay [1], [2]. Furthermore, there exist channels for which the separation theorem is not valid, even asymptotically. One important class of such channels is the class of multiple-access channels [2]. It is, therefore, important to study the combined source-channel coding for this type of channel. The combined source-channel coding considered in this letter is restricted to block coding, where the code is defined by a robust vector quantization (VQ). In robust VQ, the channel imperfections are taken into account when assigning VQ codevectors to the transmission codewords. The codeword-assignment problem is generally referred to as the index-assignment (IA) problem [3]–[5]. The optimal soft decoding of such VQ over code-division multiple-access (CDMA) channels was recently considered in [6]–[8]. Such an optimal decoder is, however, too complicated for CDMA systems with a medium or large number of users. In this letter, a suboptimal approach to VQ decoding over CDMA channels is developed. The proposed decoder is built from a soft-output multiuser detector (MUD), a soft bit estimator, and the optimal soft VQ decoders of individual users.
Paper approved by X. Wang, the Editor for Modulation, Detection, and Equalization of the IEEE Communications Society. Manuscript received March 15, 2004; revised November 10, 2004; February 8, 2005; May 8, 2005. This work was supported by NSERC under a Discovery Grant. This paper was presented in part at the IASTED International Conference on Communications Systems and Applications (CSA), Banff, AB, Canada, July 2004. The author is with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857151
II. SYSTEM MODEL
of the Euclidean source space . Let
, such that if
, then . Also, de-
fine the th encoder centroid of user as . For a VQ with a mean squared error (MSE) distortion measure, are the optimal reconstruction vectors for a the centroids noiseless channel. For a noisy channel, the optimal hard-decision reconstruction vectors are formed as linear combinations of the centroids [18]. With binary phase-shift keying (BPSK) modulation, the is converted into a block of index bits, where . For simplicity, it is assumed that all users have the same block length . The bits of the th user’s indexes are transmitted over a synchronous CDMA channel by modulating the th user’s distinct signature waveform. Let be the bit duration, and , be the signature waveform of the th user whose energy is normalized to unity. Because the system is synchronous, it suffices to consider the transmission of a single index of every user. Two channel models, namely, the additive white Gaussian noise (AWGN) and the slow flat Rayleigh fading channels, are considered in this letter. For an AWGN channel, the received ) can baseband signal over the first index interval ( be expressed as (1) where is the received amplitude of the th users and W/Hz. real AWGN of power spectral density (PSD)
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Fig. 1. Structure of the proposed decoder.
For a slow flat Rayleigh fading channel, the parameters ’s in (1) are changed to ’s. These parameters are modeled as independent zero-mean complex Gaussian random variables with independent real and imaginary parts. Furthermore, the is zero-mean, complex white Gaussian noise of PSD noise per dimension. For simplicity, an AWGN channel shall be used in presenting the proposed decoding scheme. Application of the proposed decoder to a flat Rayleigh fading channel is straightforward. It is well known (see [9]) that the sufficient statistics for the channel model in (1) can be obtained by a bank of matched filters (sampled at the bit rate), and are given by (2) is the correlation matrix of the signature wavewhere forms with , is a Gaussian vector of zero mean and covariance matrix and independent of the transmitted . bits, and , the decoder needs Based on the sufficient statistics to make a decision on the transmitted source vectors of all users. Of course, different processing algorithms on yield different decoders. In the remainder of this section, two such decoders shall be reviewed, as they will be used as benchmark decoding schemes to compare with the decoder proposed in this letter. A. Jointly Optimal Multiuser-VQ Decoder
is the sample received data, where readily follows from the and the exact expression for CDMA channel model in (2) (see [6]). In [6], the implementation of the optimal decoder in (3) based on Hadamard matrix description of the VQs is presented. Such an optimal decoder is named the Hadamard-based multiuser decoder (HMD). The structure of HMD shows how to use the a priori and channel information in an optimal fashion to counteract channel noise and multiple-access interference (MAI). The total decoding complexity of HMD is about operations, which is clearly prohibitive for a CDMA system with a medium-to-large number of users. B. Suboptimal Decoder Based on Table Lookup An alternative decoding approach is based on a combination of separate multiuser detection and table-lookup (or hard) VQ decoding. The multiuser detection can be, for example, the OPT or the MMSE receiver [9]. Such a tandem approach first gives the hard decision for the transmitted vector of bits . For each user , the bits are then converted to the corresponding estimated index . The VQ decoder of the th user then finds and outputs the centroid for VQ decoding. If the OPT-MUD is used, then the complexity of the suboptimal hard decoder is about operations per user. On the other hand, the decoding operations if the MMSE-MUD complexity is about is employed. III. A SUBOPTIMAL SOFT DECODER
denote the vector Similar to [6], let . consisting of all users’ indexes having sample values . Also define Let and
Assume that the sources of the different users are statistically . independent, which implies that The jointly optimal decoder minimizes the distortion for each user . From estimation is the conditional mean theory, the optimal estimate of and is given by [6] (3)
The suboptimal decoder proposed in this letter is also based on separate multiuser detection and VQ decoding, as illustrated in Fig. 1. However, the key difference, compared with the decoder described in Section II-B, is that the soft bit estimates from the soft-output MUD are first generated and then fed into the individual soft VQ decoders. It should also be pointed out here that unlike the receiver in [10] for (channel) coded CDMA, there are no iterations between the soft-output MUD and the individual soft VQ decoders in Fig. 1. Although such iterations might also be implemented (similar to the single-user systems [11]) by appropriately modeling the VQ encoders with Markov sources, they are not considered here for two main reasons. First, the emphasis of the letter is on the low complexity of the receiver, which does not favor iterations. Second, as will be seen in Section IV, the performance of the proposed decoder can approach that of the op-
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timum decoder quite closely, which suggests that there might be only a little room for performance improvement with iterations. Furthermore, the iterations are only helpful for VQ encoders that have a large amount of redundancy.1 This is because VQ’s redundancy is essential to produce enhanced a priori probabilities of the binary bits for the soft-input soft-output MUDs through iterations. To see what are the soft bit estimates needed for individual soft VQ decoders in Fig. 1, it is appropriate to consider the optimal decoding of VQ over a single-user channel first.
ated from the soft-output MUD in Fig. 1. To this end, define the following soft bit estimate for a given MUD: MUD (8) where of the bit
MUD MUD
A. Optimal Soft VQ Decoder for a Single-User Channel If there is only the user signaling over an AWGN channel, the discrete channel output is simply (4) is a zero-mean Gaussian random variable with variwhere . The optimal decoder that minimizes the MSE comance putes the following conditional expectation [7]: (5) where
is the a posteriori log-likelihood ratio (LLR) at the output of the MUD, defined as
is the sample value of
. A detailed treatment of the above decoder based on a Hadamard matrix and the related Hadamard transform is given in [7]. Such a decoder is useful, since it provides a description of the optimal decoder in terms of the estimates of the individual bits of the transmitted index. The main operation of the Hadamard-based decoder can be summarized as follows. Let be the encoder matrix that satisfies , where is Sylvester-type Hadamard matrix the th column of an . Then (5) can be computed as [7]
(6)
where (7) denotes the Kronecker matrix product. The quantity is the MMSE soft estimate of the bit . For the channel model in (4), this soft bit estimate is computed as [7] .
Computation of the a posteriori LLR for each type of MUD is described below. For the optimal MUD, the a posteriori LLR is readily given in [10, eq. (28)]. Furthermore, it should be pointed out that with the use of the optimal MUD, the resulting decoder is the same as the user-separated HMD (US-HMD) in [6]. The complexity , which is still exponential of this decoder is about in the number of users . To further reduce the computational complexity of the US-HMD, an approximation to the US-HMD (called US-HMD) is presented in [6]. In essence, the US-HMD is obtained by reducing the number of terms in the summations needed to compute the LLR to be . For example, the sums nearest neighbors of , a hard can be limited to the decision on the transmitted value of obtained by applying a hard-decision DC-MUD. Such an approach, however, requires nearest neighbors for each of all determining and storing possible vectors of . The total decoding complexity of US-HMD is reduced to about . The choice of depends on the number of users , but generally, it has to grows to maintain a particular performance be increased as level [6]. Here we take a different and simpler approach to reduce the computational complexity of the US-HMD. Instead of the optimal MUD, a soft-output MMSE-MUD or a soft-output DC-MUD with a much lower complexity is applied. The soft bit estimates are then generated accordingly to be the inputs to the individual soft VQ decoders. denote a -vector of all zeros, except for the th eleLet ment, which is one. Then the output of the linear MMSE-MUD is2 [9], [10] for user that corresponds to bit
and
B. Proposed Suboptimal Soft Decoder Observe that the optimal Hadamard-based soft VQ decoder for a single-user channel can also be employed for an individual user in a CDMA channel, if the soft bit estimates can be gener1Note that because there is a much larger amount of redundancy typically introduced by channel coding, the use of interleavers/deinterleavers and iterative processing is very effective for channel-coded CDMA systems, as demonstrated in [10].
(9) It is shown in [12] that the distribution of the residual interference-plus-noise at the output of the linear MMSE-MUD is well approximated by a Gaussian distribution. Thus, it can be in (9) represents the output of the following assumed that as its input symbol: equivalent AWGN channel having (10) in (10) is the equivalent amplitude of the th The parameter user’s signal, and is a zero-mean Gaussian noise sample 2There
is a typo in [10, eq. (42)]. The term A should be changed to A
.
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Fig. 2. Performance comparison of different decoding schemes in a CDMA system with two users (the first user is illustrated). Sources are 256 256 monochrome images, and the channel is AWGN.
formance of different decoders is measured at different values , where of the channel signal-to-noise ratio (CSNR) is the energy per bit. The simulation results were obtained for the case where the two users’ amplitudes are equal, i.e., . Note that also shown in Fig. 2 is the performance of the Hadamard-based optimal decoder to serve as the upper bound. The advantage of the proposed decoder with soft-output MUD and soft VQ decoding over the table-lookup decoder can be clearly observed from Fig. 2 for each type of MUD, especially at the low-to-medium CSNR region. Such an advantage can be achieved with essentially the same decoder complexity (as a function of the number of users). Another observation is that there seems to be a little performance improvement by the use of MMSE-MUD over DC-MUD in this case. This is due to the fact that the level of MAI in a two-user system is typically small, and the two MMSE-MUD and DC-MUD perform fairly close. The advantage of using MMSE-MUD over DC-MUD becomes more evident when a system with a larger number of users (i.e., a higher level of MAI) is considered next. Here the system has four users. In order to make the results more general, different decoders are tested for “standard” synthetic data produced by a Gauss–Markov source. The Gauss–Markov source has been widely used in VQ research, because the statistical properties of the source can be easily adjusted [5]–[8], [13]–[18]. Specifically, the individual user’s source is modeled as a zero-mean, unit-variance, stationary, and first-order Gauss–Markov random process with correlation , coefficient . The source is described by is an independent and identically distributed where . The parameters of the Gaussian process with variance . The VQ was trained VQ used in the simulations are for the Gauss–Markov source having and a noiseless channel. The VQ codevectors are then given good IAs based on the LISA algorithm [13]. For this VQ, the entropy is 2.88 bits, and the signal-to-distortion ratio, which is the highest achievable value of the SNR, is 9.4 dB. Also, a simple channel model of four users and the following cross correlation matrix [6] is considered:
2
with variance . The expressions for shown in [10] to be
and
are
and . It is then simple to show that the a posteriori LLR delivered by the soft MMSE-MUD is given by . On the other hand, the soft output of the DC-MUD for the th user is , where is a Gaussian random variable with zero mean and variance . The a posteriori LLR provided by the soft DC-MUD is thus . Finally, the computational complexity of the proposed decoder based on either the soft-output MMSE-MUD or the soft, which is clearly lower output DC-MUD is about than that of the US-HMD. IV. NUMERICAL RESULTS AND COMPARISON Simulation results using real images are first presented to compare the proposed decoder with other decoders described in Section II. In designing the VQ codebook, 20 different 512 512 monochrome images, including “baboon,” “bridge,” “pepper,” and “f16” images, are used as the training data. The pixels of all images are represented by 8 bits. The codevector dimension and and , the number of codevectors are set to respectively. The compression ratio is thus bits/pixel. The IAs of the codevectors are based on the LISA algorithm [13]. A two-user system with a correlation coefficient of 0.7 between the two users’ signature waveforms is simulated over an AWGN channel. User 1 transmits the “Lena” image, while user 2 sends the “Zelda” image. Both images are not included in the training data. Fig. 2 presents the peak signal-to-noise ratio (PSNR) of the received image data for user 1, which is de. The perfined as PSNR
(11)
The performance of VQ decoders is measured in terms of the , versus the output SNR, SNR CSNR . All the simulations for this system were obtained with 120 000 source samples. Also, for simplicity, it is assumed that all users’ amplitudes are equal. Fig. 3 compares the performance of the proposed soft decoder with that of the table-lookup decoder and the optimal HMD over an AWGN channel. Observe that all the decoders can offer the highest achievable value of the SNR (which is 9.4 dB for the selected VQ) at a high CSNR region (more than 10 dB). The superiority of the proposed soft decoder over the table-lookup hard decoder can also be clearly seen. Moreover, there is a clear advantage of using a more complicated soft-output MUD in the proposed decoder at a medium range of the CSNR (about 0–8 dB) for the system under consideration. More specifically,
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Fig. 3. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.
Fig. 5. Performance comparison of the proposed decoding schemes with the US-HMD. The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.
a given type of MUD, the proposed soft decoder always outperforms the table-lookup decoder at no extra complexity. Due to the fading effects, it requires a much higher CSNR to achieve the same SNR level, compared with the case of an AWGN channel. Finally, Fig. 5 compares the performance of the proposed decoder with different types of soft-output MUD with that of the suboptimal US-HMD suggested in [6] over an AWGN channel. Note that with the four-user system under consideration, the complexity of the US-HMD is about the same as that of the proposed decoder (using either soft-output MMSE-MUD . The superiority of the proposed deor DC-MUD) if coder is thus clear from Fig. 5, where it is seen that US-HMD with produces the worst performance. A closer look at Fig. 5 reveals that the proposed decoder with soft-output MMSE-MUD still performs better than US-HMD, with at low CSNR and almost the same at high CSNR. Also observe that US-HMD with performs almost identically to the ). true US-HMD (i.e., Fig. 4. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is flat Rayleigh fading.
with the optimal MUD, the proposed decoder (which is the same as the US-HMD in [6] in this case) can perform very close to that of the optimal HMD. At a low CSNR region (less than 0 dB), the performance degradation due to a lower complexity soft-output MUD becomes smaller. In particular, it is interesting to point out that the performance of the proposed decoder using a soft-output MMSE-MUD closely approaches that of the US-HMD at a low CSNR region. Note that the former decoding scheme has a much lower computational complexity, compared with that of the latter one. Simulations of the four-user system (with the same average received power for all users) employing different decoders are also carried out over a flat Rayleigh fading channel. The results are reported in Fig. 4. Again, the general observation is that for
V. CONCLUSIONS A suboptimal approach to VQ decoding over CDMA channels has been presented. The proposed decoder is built from a soft-output MUD, a soft bit estimator, and the optimal soft VQ decoders of individual users. The soft-output MUDs can be the OPT-MUD, the MMSE-MUD, or the DC-MUD. It has been demonstrated that the proposed decoding scheme offers a great flexibility to trade performance for receiver complexity over both AWGN and flat Rayleigh fading channels. The extension of the technique to a multipath fading channel is also being investigated. ACKNOWLEDGMENT The author would like to thank the anonymous reviewers and the Editor for their comments and suggestions, which improved the presentation of this letter.
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REFERENCES [1] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, Jul./Oct. 1948. [2] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [3] N. Rydbeck and C. E. Sundberg, “Analysis of digital errors in nonlinear PCM systems,” IEEE Trans. Commun., vol. COM-24, no. 1, pp. 59–65, Jan. 1976. [4] K. A. Zeger and A. Gersho, “Zero redundancy channel coding in vector quantization,” Electron. Lett., vol. 23, pp. 654–655, Jun. 1987. [5] N. Farvardin, “A study of vector quantization for noisy channels,” IEEE Trans. Inf. Theory, vol. 36, no. 7, pp. 799–809, Jul. 1990. [6] M. Skoglund and T. Ottosson, “Soft multiuser decoding for vector quantization over a CDMA channel,” IEEE Trans. Commun., pp. 327–337, Mar. 1998. [7] M. Skoglund and P. Hedelin, “Hadamard-based soft decoding for vector quantization over noisy channels,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 515–532, Mar. 1999. [8] T. Ottosson and M. Skoglund, “Joint source-channel multiuser decoding for Rayleigh fading CDMA channels,” IEEE Trans. Commun., vol. 48, no. 1, pp. 13–16, Jan. 2000. [9] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [10] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999.
[11] N. Görtz, “On the iterative approximation of optimal joint sourcechannel decoding,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1662–1670, Sep. 2001. [12] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inf. Theory, vol. 43, no. 5, pp. 858–871, May 1997. [13] P. Knagenhjelm and E. Agrell, “The Hadamard transform—A tool for index assignment,” IEEE Trans. Inf. Theory, vol. 42, no. 7, pp. 1139–1151, Jul. 1996. [14] J. Han and H. Kim, “Joint optimization of VQ codebooks and QAM signal constellations for AWGN channels,” IEEE Trans. Commun., vol. 49, no. 5, pp. 816–825, May 2001. [15] V. Vaishampayan and N. Farvardin, “Joint design of block source codes and modulation signal sets,” IEEE Trans. Inf. Theory, vol. 38, no. 7, pp. 1230–1248, Jul. 1992. [16] F. H. Liu, P. Ho, and V. Cuperman, “Joint source and channel coding using a nonlinear receiver,” in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, 1993, pp. 1502–1507. [17] , “Sequential reconstruction of vector quantized signals transmitted over Rayleigh fading channels,” in Proc. IEEE Int. Conf. Commun., New Orleans, LA, 1994, pp. 23–27. [18] N. Farvardin and V. Vaishampayan, “On the performance and complexity of channel-optimized vector quantizers,” IEEE Trans. Inf. Theory, vol. 37, no. 11, pp. 155–159, Nov. 1991.
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A Soft Decoding Scheme for Vector Quantization Over a CDMA Channel Ha H. Nguyen, Senior Member, IEEE
Abstract—The optimal decoding of vector quantization (VQ) over a code-division multiple-access (CDMA) channel is too complicated for systems with a medium-to-large number of users. This paper presents a low-complexity, suboptimal decoder for VQ over a CDMA channel. The proposed decoder is built from a soft-output multiuser detector, a soft bit estimator, and the optimal soft VQ decoding of an individual user. Simulation results obtained over both additive white Gaussian noise and flat Rayleigh fading channels show that with a lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the more complicated optimal decoder.
The soft-output MUDs considered include the jointly optimal MUD (OPT-MUD), the minimum mean-squared error MUD (MMSE-MUD), and the decorrelating MUD (DC-MUD) [9]. For each type of MUD, the soft-bit estimates are calculated and fed into the soft VQ decoders. Due to its lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the complicated optimal decoder.
Index Terms—Code-division multiple access (CDMA), combined source and channel coding, multiuser detection, soft decoding, vector quantization (VQ).
The system model considered in this letter is the same as users in a synchronous CDMA the one in [6]. There are system, where each user transmits its source vectors by means of VQ. The th user produces a -dimensional random vector . The vector is then encoded into an index , where for some integer . The bits per source transmission rate of the system is thus dimension. The th encoder is described by a partition
I. INTRODUCTION
T
HE source and channel coding of a communication system are often designed and implemented separately. This common practice is mainly due to the work by Shannon [1], where it was shown that such a separation can perform optimally. However, the positive coding theorems of information theory only show such separability in the limit of infinite codeword length, and hence, infinite delay [1], [2]. Furthermore, there exist channels for which the separation theorem is not valid, even asymptotically. One important class of such channels is the class of multiple-access channels [2]. It is, therefore, important to study the combined source-channel coding for this type of channel. The combined source-channel coding considered in this letter is restricted to block coding, where the code is defined by a robust vector quantization (VQ). In robust VQ, the channel imperfections are taken into account when assigning VQ codevectors to the transmission codewords. The codeword-assignment problem is generally referred to as the index-assignment (IA) problem [3]–[5]. The optimal soft decoding of such VQ over code-division multiple-access (CDMA) channels was recently considered in [6]–[8]. Such an optimal decoder is, however, too complicated for CDMA systems with a medium or large number of users. In this letter, a suboptimal approach to VQ decoding over CDMA channels is developed. The proposed decoder is built from a soft-output multiuser detector (MUD), a soft bit estimator, and the optimal soft VQ decoders of individual users.
Paper approved by X. Wang, the Editor for Modulation, Detection, and Equalization of the IEEE Communications Society. Manuscript received March 15, 2004; revised November 10, 2004; February 8, 2005; May 8, 2005. This work was supported by NSERC under a Discovery Grant. This paper was presented in part at the IASTED International Conference on Communications Systems and Applications (CSA), Banff, AB, Canada, July 2004. The author is with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857151
II. SYSTEM MODEL
of the Euclidean source space . Let
, such that if
, then . Also, de-
fine the th encoder centroid of user as . For a VQ with a mean squared error (MSE) distortion measure, are the optimal reconstruction vectors for a the centroids noiseless channel. For a noisy channel, the optimal hard-decision reconstruction vectors are formed as linear combinations of the centroids [18]. With binary phase-shift keying (BPSK) modulation, the is converted into a block of index bits, where . For simplicity, it is assumed that all users have the same block length . The bits of the th user’s indexes are transmitted over a synchronous CDMA channel by modulating the th user’s distinct signature waveform. Let be the bit duration, and , be the signature waveform of the th user whose energy is normalized to unity. Because the system is synchronous, it suffices to consider the transmission of a single index of every user. Two channel models, namely, the additive white Gaussian noise (AWGN) and the slow flat Rayleigh fading channels, are considered in this letter. For an AWGN channel, the received ) can baseband signal over the first index interval ( be expressed as (1) where is the received amplitude of the th users and W/Hz. real AWGN of power spectral density (PSD)
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Fig. 1. Structure of the proposed decoder.
For a slow flat Rayleigh fading channel, the parameters ’s in (1) are changed to ’s. These parameters are modeled as independent zero-mean complex Gaussian random variables with independent real and imaginary parts. Furthermore, the is zero-mean, complex white Gaussian noise of PSD noise per dimension. For simplicity, an AWGN channel shall be used in presenting the proposed decoding scheme. Application of the proposed decoder to a flat Rayleigh fading channel is straightforward. It is well known (see [9]) that the sufficient statistics for the channel model in (1) can be obtained by a bank of matched filters (sampled at the bit rate), and are given by (2) is the correlation matrix of the signature wavewhere forms with , is a Gaussian vector of zero mean and covariance matrix and independent of the transmitted . bits, and , the decoder needs Based on the sufficient statistics to make a decision on the transmitted source vectors of all users. Of course, different processing algorithms on yield different decoders. In the remainder of this section, two such decoders shall be reviewed, as they will be used as benchmark decoding schemes to compare with the decoder proposed in this letter. A. Jointly Optimal Multiuser-VQ Decoder
is the sample received data, where readily follows from the and the exact expression for CDMA channel model in (2) (see [6]). In [6], the implementation of the optimal decoder in (3) based on Hadamard matrix description of the VQs is presented. Such an optimal decoder is named the Hadamard-based multiuser decoder (HMD). The structure of HMD shows how to use the a priori and channel information in an optimal fashion to counteract channel noise and multiple-access interference (MAI). The total decoding complexity of HMD is about operations, which is clearly prohibitive for a CDMA system with a medium-to-large number of users. B. Suboptimal Decoder Based on Table Lookup An alternative decoding approach is based on a combination of separate multiuser detection and table-lookup (or hard) VQ decoding. The multiuser detection can be, for example, the OPT or the MMSE receiver [9]. Such a tandem approach first gives the hard decision for the transmitted vector of bits . For each user , the bits are then converted to the corresponding estimated index . The VQ decoder of the th user then finds and outputs the centroid for VQ decoding. If the OPT-MUD is used, then the complexity of the suboptimal hard decoder is about operations per user. On the other hand, the decoding operations if the MMSE-MUD complexity is about is employed. III. A SUBOPTIMAL SOFT DECODER
denote the vector Similar to [6], let . consisting of all users’ indexes having sample values . Also define Let and
Assume that the sources of the different users are statistically . independent, which implies that The jointly optimal decoder minimizes the distortion for each user . From estimation is the conditional mean theory, the optimal estimate of and is given by [6] (3)
The suboptimal decoder proposed in this letter is also based on separate multiuser detection and VQ decoding, as illustrated in Fig. 1. However, the key difference, compared with the decoder described in Section II-B, is that the soft bit estimates from the soft-output MUD are first generated and then fed into the individual soft VQ decoders. It should also be pointed out here that unlike the receiver in [10] for (channel) coded CDMA, there are no iterations between the soft-output MUD and the individual soft VQ decoders in Fig. 1. Although such iterations might also be implemented (similar to the single-user systems [11]) by appropriately modeling the VQ encoders with Markov sources, they are not considered here for two main reasons. First, the emphasis of the letter is on the low complexity of the receiver, which does not favor iterations. Second, as will be seen in Section IV, the performance of the proposed decoder can approach that of the op-
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timum decoder quite closely, which suggests that there might be only a little room for performance improvement with iterations. Furthermore, the iterations are only helpful for VQ encoders that have a large amount of redundancy.1 This is because VQ’s redundancy is essential to produce enhanced a priori probabilities of the binary bits for the soft-input soft-output MUDs through iterations. To see what are the soft bit estimates needed for individual soft VQ decoders in Fig. 1, it is appropriate to consider the optimal decoding of VQ over a single-user channel first.
ated from the soft-output MUD in Fig. 1. To this end, define the following soft bit estimate for a given MUD: MUD (8) where of the bit
MUD MUD
A. Optimal Soft VQ Decoder for a Single-User Channel If there is only the user signaling over an AWGN channel, the discrete channel output is simply (4) is a zero-mean Gaussian random variable with variwhere . The optimal decoder that minimizes the MSE comance putes the following conditional expectation [7]: (5) where
is the a posteriori log-likelihood ratio (LLR) at the output of the MUD, defined as
is the sample value of
. A detailed treatment of the above decoder based on a Hadamard matrix and the related Hadamard transform is given in [7]. Such a decoder is useful, since it provides a description of the optimal decoder in terms of the estimates of the individual bits of the transmitted index. The main operation of the Hadamard-based decoder can be summarized as follows. Let be the encoder matrix that satisfies , where is Sylvester-type Hadamard matrix the th column of an . Then (5) can be computed as [7]
(6)
where (7) denotes the Kronecker matrix product. The quantity is the MMSE soft estimate of the bit . For the channel model in (4), this soft bit estimate is computed as [7] .
Computation of the a posteriori LLR for each type of MUD is described below. For the optimal MUD, the a posteriori LLR is readily given in [10, eq. (28)]. Furthermore, it should be pointed out that with the use of the optimal MUD, the resulting decoder is the same as the user-separated HMD (US-HMD) in [6]. The complexity , which is still exponential of this decoder is about in the number of users . To further reduce the computational complexity of the US-HMD, an approximation to the US-HMD (called US-HMD) is presented in [6]. In essence, the US-HMD is obtained by reducing the number of terms in the summations needed to compute the LLR to be . For example, the sums nearest neighbors of , a hard can be limited to the decision on the transmitted value of obtained by applying a hard-decision DC-MUD. Such an approach, however, requires nearest neighbors for each of all determining and storing possible vectors of . The total decoding complexity of US-HMD is reduced to about . The choice of depends on the number of users , but generally, it has to grows to maintain a particular performance be increased as level [6]. Here we take a different and simpler approach to reduce the computational complexity of the US-HMD. Instead of the optimal MUD, a soft-output MMSE-MUD or a soft-output DC-MUD with a much lower complexity is applied. The soft bit estimates are then generated accordingly to be the inputs to the individual soft VQ decoders. denote a -vector of all zeros, except for the th eleLet ment, which is one. Then the output of the linear MMSE-MUD is2 [9], [10] for user that corresponds to bit
and
B. Proposed Suboptimal Soft Decoder Observe that the optimal Hadamard-based soft VQ decoder for a single-user channel can also be employed for an individual user in a CDMA channel, if the soft bit estimates can be gener1Note that because there is a much larger amount of redundancy typically introduced by channel coding, the use of interleavers/deinterleavers and iterative processing is very effective for channel-coded CDMA systems, as demonstrated in [10].
(9) It is shown in [12] that the distribution of the residual interference-plus-noise at the output of the linear MMSE-MUD is well approximated by a Gaussian distribution. Thus, it can be in (9) represents the output of the following assumed that as its input symbol: equivalent AWGN channel having (10) in (10) is the equivalent amplitude of the th The parameter user’s signal, and is a zero-mean Gaussian noise sample 2There
is a typo in [10, eq. (42)]. The term A should be changed to A
.
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Fig. 2. Performance comparison of different decoding schemes in a CDMA system with two users (the first user is illustrated). Sources are 256 256 monochrome images, and the channel is AWGN.
formance of different decoders is measured at different values , where of the channel signal-to-noise ratio (CSNR) is the energy per bit. The simulation results were obtained for the case where the two users’ amplitudes are equal, i.e., . Note that also shown in Fig. 2 is the performance of the Hadamard-based optimal decoder to serve as the upper bound. The advantage of the proposed decoder with soft-output MUD and soft VQ decoding over the table-lookup decoder can be clearly observed from Fig. 2 for each type of MUD, especially at the low-to-medium CSNR region. Such an advantage can be achieved with essentially the same decoder complexity (as a function of the number of users). Another observation is that there seems to be a little performance improvement by the use of MMSE-MUD over DC-MUD in this case. This is due to the fact that the level of MAI in a two-user system is typically small, and the two MMSE-MUD and DC-MUD perform fairly close. The advantage of using MMSE-MUD over DC-MUD becomes more evident when a system with a larger number of users (i.e., a higher level of MAI) is considered next. Here the system has four users. In order to make the results more general, different decoders are tested for “standard” synthetic data produced by a Gauss–Markov source. The Gauss–Markov source has been widely used in VQ research, because the statistical properties of the source can be easily adjusted [5]–[8], [13]–[18]. Specifically, the individual user’s source is modeled as a zero-mean, unit-variance, stationary, and first-order Gauss–Markov random process with correlation , coefficient . The source is described by is an independent and identically distributed where . The parameters of the Gaussian process with variance . The VQ was trained VQ used in the simulations are for the Gauss–Markov source having and a noiseless channel. The VQ codevectors are then given good IAs based on the LISA algorithm [13]. For this VQ, the entropy is 2.88 bits, and the signal-to-distortion ratio, which is the highest achievable value of the SNR, is 9.4 dB. Also, a simple channel model of four users and the following cross correlation matrix [6] is considered:
2
with variance . The expressions for shown in [10] to be
and
are
and . It is then simple to show that the a posteriori LLR delivered by the soft MMSE-MUD is given by . On the other hand, the soft output of the DC-MUD for the th user is , where is a Gaussian random variable with zero mean and variance . The a posteriori LLR provided by the soft DC-MUD is thus . Finally, the computational complexity of the proposed decoder based on either the soft-output MMSE-MUD or the soft, which is clearly lower output DC-MUD is about than that of the US-HMD. IV. NUMERICAL RESULTS AND COMPARISON Simulation results using real images are first presented to compare the proposed decoder with other decoders described in Section II. In designing the VQ codebook, 20 different 512 512 monochrome images, including “baboon,” “bridge,” “pepper,” and “f16” images, are used as the training data. The pixels of all images are represented by 8 bits. The codevector dimension and and , the number of codevectors are set to respectively. The compression ratio is thus bits/pixel. The IAs of the codevectors are based on the LISA algorithm [13]. A two-user system with a correlation coefficient of 0.7 between the two users’ signature waveforms is simulated over an AWGN channel. User 1 transmits the “Lena” image, while user 2 sends the “Zelda” image. Both images are not included in the training data. Fig. 2 presents the peak signal-to-noise ratio (PSNR) of the received image data for user 1, which is de. The perfined as PSNR
(11)
The performance of VQ decoders is measured in terms of the , versus the output SNR, SNR CSNR . All the simulations for this system were obtained with 120 000 source samples. Also, for simplicity, it is assumed that all users’ amplitudes are equal. Fig. 3 compares the performance of the proposed soft decoder with that of the table-lookup decoder and the optimal HMD over an AWGN channel. Observe that all the decoders can offer the highest achievable value of the SNR (which is 9.4 dB for the selected VQ) at a high CSNR region (more than 10 dB). The superiority of the proposed soft decoder over the table-lookup hard decoder can also be clearly seen. Moreover, there is a clear advantage of using a more complicated soft-output MUD in the proposed decoder at a medium range of the CSNR (about 0–8 dB) for the system under consideration. More specifically,
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Fig. 3. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.
Fig. 5. Performance comparison of the proposed decoding schemes with the US-HMD. The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.
a given type of MUD, the proposed soft decoder always outperforms the table-lookup decoder at no extra complexity. Due to the fading effects, it requires a much higher CSNR to achieve the same SNR level, compared with the case of an AWGN channel. Finally, Fig. 5 compares the performance of the proposed decoder with different types of soft-output MUD with that of the suboptimal US-HMD suggested in [6] over an AWGN channel. Note that with the four-user system under consideration, the complexity of the US-HMD is about the same as that of the proposed decoder (using either soft-output MMSE-MUD . The superiority of the proposed deor DC-MUD) if coder is thus clear from Fig. 5, where it is seen that US-HMD with produces the worst performance. A closer look at Fig. 5 reveals that the proposed decoder with soft-output MMSE-MUD still performs better than US-HMD, with at low CSNR and almost the same at high CSNR. Also observe that US-HMD with performs almost identically to the ). true US-HMD (i.e., Fig. 4. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is flat Rayleigh fading.
with the optimal MUD, the proposed decoder (which is the same as the US-HMD in [6] in this case) can perform very close to that of the optimal HMD. At a low CSNR region (less than 0 dB), the performance degradation due to a lower complexity soft-output MUD becomes smaller. In particular, it is interesting to point out that the performance of the proposed decoder using a soft-output MMSE-MUD closely approaches that of the US-HMD at a low CSNR region. Note that the former decoding scheme has a much lower computational complexity, compared with that of the latter one. Simulations of the four-user system (with the same average received power for all users) employing different decoders are also carried out over a flat Rayleigh fading channel. The results are reported in Fig. 4. Again, the general observation is that for
V. CONCLUSIONS A suboptimal approach to VQ decoding over CDMA channels has been presented. The proposed decoder is built from a soft-output MUD, a soft bit estimator, and the optimal soft VQ decoders of individual users. The soft-output MUDs can be the OPT-MUD, the MMSE-MUD, or the DC-MUD. It has been demonstrated that the proposed decoding scheme offers a great flexibility to trade performance for receiver complexity over both AWGN and flat Rayleigh fading channels. The extension of the technique to a multipath fading channel is also being investigated. ACKNOWLEDGMENT The author would like to thank the anonymous reviewers and the Editor for their comments and suggestions, which improved the presentation of this letter.
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