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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001

Joint Optimization of VQ Codebooks and QAM Signal Constellations for AWGN Channels Jong-Ki Han and Hyung-Myung Kim

Abstract—A joint design scheme has been proposed to optimize the source encoder and the modulation signal constellation based on the minimization of the end-to-end distortion including both the quantization error and channel distortion. The proposed scheme first optimizes the vector quantization (VQ) codebook for a fixed modulation signal set, and then the modulation signals for the fixed VQ codebook. These two steps are iteratively repeated until they reach a local optimum solution. It has been shown that the performance of the proposed system can be enhanced by employing a new efficient mapping scheme between codevectors and modulation signals. Simulation results show that a jointly optimized system based on the proposed algorithms outperforms the conventional system based on a conventional quadrature amplitude modulation signal set and the VQ codebook designed for a noiseless channel. Index Terms—Channel-optimized VQ (COVQ), joint optimization, QAM.

I. INTRODUCTION

I

N MODERN practical communication systems, the source coder is designed, in most cases, separately from the channel coder both physically and conceptually. However, this design approach might be inefficient for applications involving the transmission of video and images. This fact is generally acknowledged by researchers [1]–[5], but efficient methods of combining source and channel coders into a joint system are still under investigation. In this paper, we introduce a technique for combining vector quantization (VQ) and quadrature amplitude modulation (QAM) signaling. Our results are important, since VQ is widely used for signal compression [6] and QAM is a very common modulation method for high-speed data transmission [7]–[10]. A useful technique to combat channel noise is to design a VQ codebook optimized for noisy channels. The resulting encoder is often referred to as “channel-optimized vector quantization” (COVQ) and is optimized for a given set of crossover probabilities of the codevectors [11]–[14]. The design of the QAM signal set has been considered in [7]–[10]. Foschini [7] considered equally probable signal points and an equal cost for error events, and optimized the signal constellation under the criterion of minimum symbol-error probPaper approved by K. Illgner, the Editor for Speech, Image, Video, and Signal Processing of the IEEE Communications Society. Manuscript received November 3, 1999; revised May 26, 2000 and October 5, 2000. J.-K. Han is with the Digital Media System Laboratory, Corporate Research and Development Center, Samsung Electronics, Kyung-Ki-Do 442-742, South Korea (e-mail: [email protected]). H.-M. Kim is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(01)04081-8.

ability. Sundberg et al. [8] derived the weighted QAM constellations for 8-bit pulse-coded modulation, where the positions of the QAM constellation points are modified into the weighted square QAM constellations. In [7] and [8], the modulation signals were modified to reduce the symbol-error probability without considering codevectors. Liu et al. [9] optimized the QAM signals for general codebook by considering the relative effect of each codevector to the overall distortion. While the systems described in [7] and [9] had been optimized for additive white Gaussian noise (AWGN) channel, Webb [10] proposed the star QAM constellation for Rayleigh fading channels to reduce the symbol-error probability. While existing studies have considered those two problems (COVQ codebook optimization and QAM signals optimization) individually, in this work we propose a design scheme that jointly optimizes the COVQ encoder and QAM signal to minimize end-to-end distortion. The design of the QAM signal constellation is matched to minimize the expected distortion of the COVQ. Likewise, the COVQ codebook is optimized for the codevector crossover probabilities determined by the QAM signals. A similar study was conducted by Liu [9], who has also studied the problem of the joint design of the source coder and modulation signal. In optimizing the modulation signal constellations, Liu used Lagrange multiplier to produce the objective cost function to be minimized, and a conventional gradient search algorithm is directly applied. In this paper, we have optimized the QAM signal constellations by partitioning the modulation signal space with an energy constraint. Additionally, an efficient mapping scheme between codevectors and modulation signals has been proposed to enhance the performance of the proposed system. We compare the performance of the jointly optimized system based on the proposed scheme with those of subsystems optimized individually: COVQ optimized for a given square QAM signal, and new QAM signals optimized for a VQ encoder based on a noiseless channel. Further distortion reduction can be achieved when the mapping scheme between codevectors and modulation signals is taking into account before the proposed joint optimization algorithm is applied for the COVQ codebook and QAM signals. The algorithms proposed in this paper are appropriate for QAM-based systems operating over noisy channels, such as are found in telephone and cable modems. When the channel bandwidth is limited [16], it makes sense to give different levels of protection to the data according to their importance. The data from video source encoders of HDTV are already classified into two priority classes; for example, the data presenting the low-frequency components of the video signal carrying more

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Fig. 1. A communication system consisting of VQ encoder over noisy channel.

important information than the high-frequency components. In this case, while the more important data are transmitted after applying the channel coding technique, the less important data can be transmitted without error protection using the communication system proposed in this paper. The paper is organized into seven sections as follows. In Section II, a general communication system model is described and the design problem considered in this paper is formulated. In Section III, the optimization method for QAM signal constellation for the fixed VQ codebook is proposed. In Section IV, a COVQ is derived for the optimized QAM signals. An overall iterative algorithm for the system design is presented in Section V. In Sections V-A and V-B, two efficient mapping algorithms (MAs) between codevectors and modulation signals are proposed. In Section VI, the simulation results are presented for the optimized systems. These results are compared to the performance of the conventional communication systems. Section VII contains the summary and conclusions.

II. SYSTEM MODEL It is assumed that the only channel effect is Gaussian noise and that all other transmission impairments (linear distortion, phase jitter, frequency offset, etc.) have been eliminated [7]–[10]. The system model considered is represented in Fig. 1, which shows a communication system using VQ encoder over a noisy channel, where denotes the AWGN of the channel. A -dimensional source vector is to be encoded by means of a vector quantizer. If belongs to a region among the disjoint of the space , where partition is the size of VQ codebook, the vector quantizer assigns a drawn from a finite reproduction alcodevector . The codevector phabet (codebook) can be directly mapped into the modulation signal as considered in [9] and [4]. The set forms the QAM signal constellation represented in Fig. 2, where denotes the number of signals. , are assumed to The modulated signals , be transmitted over the noisy channel that is exposed to AWGN. is displaced to the received signal The transmitted signal due to AWGN. As shown in Fig. 3, hard-deinto a modulacision detector maps the received signal if belongs to a region among the disjoint tion signal of the two-dimensional repartition . ceived signal space The performance of the above system is measured in terms of the mean squared error (MSE) between the source vector and its reconstructed vector . Let the transition probabilities

Fig. 2. Square QAM constellation.

Fig. 3. The operation of the hard-decision detector the signal space .

R

7 and partitioning 9 of

, denote the probability that the is received, given that is transmitted. Then, codevector the overall distortion due to both the quantizer and the channel error is given by

(1)

is the squared difference between and , where . In (1), includes the distortions due to the i.e., quantization error of VQ encoder and the channel noise. Since ’s are directly mapped into ’s, the transition probability is . Therefore, (1) can be rewritten as

(2)

For the fixed dimension and codebook size , we wish to . The joint optimization design and to minimize of signal constellation and the VQ encoder is performed as follows. First, given a fixed VQ codebook , we minimize by adjusting the transition probabilities the through optimization of modulation signal set . Then, for the , a vector quantizer adjusted transition probabilities will be optimized so that can be minimized.

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III. OPTIMIZATION OF QAM SIGNAL CONSTELLATION

TABLE I OPTIMIZATION OF QAM SIGNAL CONSTELLATION

In this section, for a fixed VQ codebook is minimized by the optimal construction of modulation signal constellation, or equivalently, by the optimal partition of the modulation signal space. , are When the modulated signals transmitted over the AWGN channel, the transition probability in (2) is given by (3) Substituting (3) into (2) gives

signal (4)

is in a region When a particular amount of distortion introduced by

, i.e., becomes

is chosen as the conditional expectation of , i.e.,

, the

(5)

(6)

, given

(10) denotes the conditional mean where and the signal squared distance between the received signal belongs to the region . Upon the complevector when tion of (8) and (10), the average energy of tends to be changed unless the energy constraint is incorporated. Thus, the signals are normalized as follows:

where (11) (7) Thus, for each to minimize

, a region number . The is such that

should be chosen optimum partition

(8) Another interpretation of (8) can be given as follows. The VQ codebook and partition are assumed to be known and fixed. Recall that the objective is to minimize the overall distortion is a nonnegative quantity, to mingiven by (4). Since it suffices to minimize imize (9) , it suffices to obtain a that Therefore, to minimize minimizes (9) for every value of . The optimization can be achieved by obtaining (8). On the other hand, the well-known estimation-theoretic reis minimized when the sults imply that for a fixed

and are the modulation signal calculated in (10) where and the initial signal, respectively. Equations (8), (10), and (11) result in and that satisfy the necessary and sufficient conditions for optimality for a fixed and . The optimization algorithm for QAM signals is described in Table I. A successive application of (8), (10), and (11) results in a sequence of signal set and the partition pairs for which ’s form a noninthe corresponding channel distortions creasing sequence of nonnegative numbers, hence it converges. Despite the fact that (8) and (10) are both necessary and sufficient conditions when and are fixed, respectively, the final solution obtained by the iterative algorithm need not satisfy the sufficient conditions for the system’s optimality. That is, the final solution obtained by this algorithm is only a locally optimum solution. We will refer the final - pair as the optimal QAM (O-QAM) signal constellation for AWGN channel. Since codevectors are considered in the optimization of QAM signal set, when two codevectors are similar, the corresponding modulation signals become close in the modulation signal space. IV. CHANNEL-OPTIMIZED VQ (COVQ) In this section, we consider the problem of minimizing by optimizing when modulation signal ’s are fixed.

HAN AND KIM: JOINT OPTIMIZATION OF VQ CODEBOOKS AND QAM SIGNAL CONSTELLATIONS

From (4), when a particular belongs to a partition amount of distortion introduced by becomes

, the

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TABLE II OVERALL OPTIMIZATION ALGORITHM OF VQ ENCODER AND MODULATION SIGNAL CONSTELLATION

(12) Thus, for each to minimize

, a partition number should be chosen . The optimum partition is found as

(13) represents the number of training source vectors. where Then, for the fixed , the optimum codebook is obtained as

(14) Note that (13) and (14) are the modified representation of is equivalent to optimal COVQ [11] where . V. AN ITERATIVE JOINT OPTIMIZATION FOR

AND

In the previous two sections, the optimization procedures of the modulation signal constellation and the COVQ codebook are described, individually. In this section, the strategy for jointly optimizing these two subsystems will be discussed. The goal of our joint system design is to minimize the overall distortion in (2). Rewriting (4) using (5), (12), and (15) provides (16)

(17)

(18) In O-QAM, the modulation signal constellation is obtained , which minimizes by minimizing with fixed , and in COVQ, the VQ codebook is designed by , which minimizes with minimizing fixed .

The joint design algorithm is summarized in Table II. In Step 3, a new QAM signal set has been designed for a given VQ codebook. The iterative design process then goes to Step 4 for the new QAM signal set obtained in to minimize Step 3. For a fixed and , updating the modulation signal set and corresponding space partition according to Step 3 . For a fixed does not increase the overall distortion and , constructing VQ codebook and according to Step 4 , either. Therefore, it is easy to does not increase the see that the overall distortion is a nonincreasing function of iteration index, and the joint optimization algorithm for the design of and converges. Since a local optimal solution is guaranteed by this design procedure, it is important to choose the proper initial values for the VQ codebook, the modulation signal set, and the mapping and signals . Both the convergence between codevectors rate and the performance of the final system can be improved substantially if the initial state of the design process is set properly. We found empirically that a squared QAM signal set and a VQ codebook made by LBG algorithm [17] for a noiseless channel are appropriate for the initialization in most cases. A. Heuristic Initial Mapping Between Codevectors and Modulation Signals In this section, we introduce a heuristic mapping scheme beand . tween For the construction of VQ codebook , the splitting algorithm is used as in [17]. In labeling modulation signals , we first classify the entire ’s into two sets, and , where a horizontal center line of the entire signal set is used as a threshold, and we call and , respectively. Then, the two the resulted sets as sets are partitioned into , where a vertical center , is used as a threshold for each parline of each subset, , tition, and the generated sets are represented as . If we continue in this manner, the number of sets is and sets which are doubled each time, we will end up with . Each set has one signal , labeled as i.e., each signal can be indexed with a -bit binary vector. and signal having the same -bit Finally, codevector binary index are mapped for an initial assignment of Overall Optimization Algorithm in Table II.

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B. Proposed Initial Mapping Between Codevectors and Modulation Signals

(27)

Here, we propose an efficient mapping scheme between the and modulation signals for a good initial codevectors mapping. For a sufficiently high channel-to-noise ratio (CSNR), the only significant type of error occurs when the received signal is perturbed to a-decision region adjacent to . In this case, and in (3) the transition probability between two signals can be approximated as [7]

(28) (29) the number of elements in

(30)

(31) (19) for a high CSNR, Thus, in the calculation of the summation will be significant only over the part of decision which is close to the boundary with . region (20) where

shares a boundary with . Equation (2) becomes

and

if

shares a boundary with otherwise.

Suppose that the terms and

(32)

, are appearing in (24) for the signal computed and stored initially. Then, using those values in (32), can be computed for . Choose and such that (33)

(21) (22) where (23) and , and let Consider two particular signals denote the reduction in due to the exchange of codevecand . The Appendix shows that tors assigned to can be expressed as

(24) where (25) (26)

and , indicate the identification where two numbers, numbers of two signals whose codevectors are chosen to shares a be switched. For the values of satisfying that or , the values of boundary with and in (32) have to be updated. Then using those updated indicating new two values, a new pair of numbers signals whose codevectors are to be switched can be selected. The MA based on this iterative procedure is summarized in Table III. The procedure consists of six steps as follows. In Step 1, initial mapping of codevectors is made by a heuristic mapping scheme described in Section V-A. In Step 2, the , and for values of are calculated. In Step 3, we select a switch which results in among all the possible switches, where maximum , may be positive or negative value. and with the maximum are selected The as a candidate to be permuted. The switching yields the greatest . But, if no selection results in the positive decrease in , then the algorithm halts in a locally maximum , optimal state in Step 6. Since the values ( ) which are pre-calculated in Step 2 are used, Step 3 and can be executed very simply. In Step 4, if the maximum , then the selected codevectors and are and for signals switched. Step 5 updates the values of whose region shares a boundary with or . The and are values of switched, respectively. Steps 3–5 are repeated until no further is achieved. Hence, a successive improvement in application of Steps 3–5 in Table III results in a sequence of the codevector mappings for which the corresponding form a strictly decreasing sequence of positive numbers, where a mapping of codevectors is different from any one that the MA had previously generated. The switching of two codevectors does not exclude the finding of the globally optimal mapping, but the algorithm converges to a locally optimal solution due to

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TABLE III MAPPING ALGORITHM TO ASSIGN THE CODEVECTORS INTO THE MODULATION SIGNALS

the restriction that only two codevectors are switched in each iteration. VI. SIMULATION RESULTS Computer simulation using real images were performed to evaluate the performance of the proposed algorithm. In designing the COVQ codebook and O-QAM signals, and assigning codevectors to modulation signals with MA, 19 different 512 512 monochromatic images including “baboom,” “barbara,” “toy,” “bridge,” “f16,” and “pepper” images are used as the training data. The pixels of all images are represented by 8 bit. The codevector dimension and the number of codevectors and , respectively, and the compresare set to sion ratio is 0.375 bits/pixel. The simulation was performed for the LENA image which is not in the training image set, and the results are presented in terms of peak signal-to-noise ratio of the received image data. The performance is measured at different values of the CSNRs whose values vary with the value , where CSNR is defined as of (34) and

Fig. 4. The performance comparison between the proposed system (with MA described in Table III) and the conventional system (without MA).

other systems in reducing the overall distortion. It can be observed that the proposed algorithm gives significant performance gain at low CSNRs. However, the gain of the proposed system becomes smaller at higher CSNRs. The reason is that is very small at high CSNR, i.e., the conventional systems become insensitive to the channel noise.

(35) B. Results for Optimization of QAM (O-QAM) which have been used as a measure of the system performance in [9] and [14]. A. Results for VQ Codevector Assignment to Modulation Signal Sets (MA) In Fig. 4, the performance of the MA proposed in Section V-B is compared with those of the conventional system (without MA) and the heuristic mapping scheme described in Section V-A. The codevectors are assigned to the signals by dB. This figure shows that applying the MA with the system using the proposed mapping scheme outperforms

In this section, numerical results on the performance of the O-QAM system are given and compared them with those of other QAM systems. In the simulation, ’s are generated by , where uniform sampling of signal space . By making the number of sampled ’s very , we let the interval between the ’s belarge come small and effectively cover the continuous signal space. The resulting O-QAM signal constellation is different from the square QAM signal constellation in Fig. 2. Foschini’s [7] constellation had been obtained by minimizing symbol-error probability for equally probable signal points. The constellation

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Fig. 5. The performance comparison of various optimized QAM systems over AWGN channels.

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Fig. 7. The performance comparison of various optimized VQ codebooks.

Fig. 8. The performance comparison of various optimized communication systems.

C. Results for COVQ

Fig. 6. The first three iterations of the optimization procedure of modulation signals to reduce the overall distortion D(C ; S ), where CSNR = 20 dB is assumed. The signals are optimized in 37 iterations.

proposed by Liu [9] has been obtained by minimizing the averaged distortion due to AWGN for the general codebook. In obtaining the results presented in Fig. 5, the modulation signals are assumed to be transmitted over AWGN channels. O-QAM signals have been optimized by applying (8) and (10) at dB. This figure shows that O-QAM and Liu’s systems significantly outperform Foschini’s and the square QAM systems. This is due to the fact that O-QAM and Liu’s systems are optimized for the general codebook, while Foschini’s is designed for equiprobable signals. Fig. 6 represents the procedure dB. As of optimization of the modulation signals at modulation signals are optimized, signals assigned to different codevectors which are far in source data space are also located far from each other in modulation signal space and cause large contributions to the overall distortion when transmission errors do occur. From these results, it can be assured that the O-QAM is very effective in reducing the overall distortion.

The VQ codebook has been generated and the result is shown in Fig. 7. In applying COVQ, due to computational problems in integrating functions over irregular regions of two-dimensional , a training sequence approach is used for Euclidean space the computation of the transition probability in (3) and its assodB),” “COVQ ( ciated centroid. “COVQ ( dB),” and “COVQ ( dB)” represent the cases dB, that the COVQ codebooks are optimized for dB, and dB, respectively. According to Fig. 7 the performances of COVQs are excellent at the range of CSNR where the COVQ codebooks are optimized. It can be seen from the results in Fig. 7 that the codebook for the case dB is not suitable for the channel condition with dB. with D. Results for Joint Optimization Algorithm Fig. 8 shows the results of applying the overall optimization algorithm described in Table II. Various conventional systems are presented for comparison. The separately optimized conventional system considered in the comparison is either uniformly spaced square type QAM constellation or the system whose VQ codebook is optimized for noiseless channel. Conventional systems (O-QAM/normal VQ and Liu QAM/normal VQ) were

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(a)

(b)

(c)

(d)

N

Fig. 9. The Lena images are encoded with VQ codebook ( = 256) and are transmitted over noisy channels. Images of (a) and (c) are reconstructed from the received data in the conventional system (VQ+QAM). Those of (b) and (d) are results for the proposed system (COVQ + O-QAM). CSNRs of the noisy channels are 30.26 dB in (a) and (b), 28.50 dB in (c) and (d), respectively.

designed by using the O-QAM algorithm and Liu QAM [9], respectively, with a VQ codebook optimized for noiseless channel. “Square QAM/COVQ” represents the system obtained by applying COVQ algorithm with the square type QAM. The system with “Joint Opt” which represents the proposed system outperforms other systems for the wide range of CSNRs. The gains of the designed system are mainly due to the optimal construction of both COVQ and the signal constellation. The without any performance of the source encoder transmission errors is 28.9338 dB. By comparing two conventional systems (Foschini QAM/Normal VQ and square QAM/Normal VQ), it can be observed that it is important to reduce the overall distortion by considering the joint optimization of the VQ codevectors, especially in the low CSNR region, where the performances of the systems designed for equally probable signal points are poor. The performance of “joint opt. ( R-MA)” which uses random initial mapping is slightly better than that of square QAM/COVQ. It means that O-QAM of the “joint opt. ( R-MA)” is not sufficiently effective since MA is absent, i.e., codevectors and modulation signals are randomly mapped. It is interesting to note that the performance of the jointly designed system is enhanced if H-MA (heuristic MA) or MA (proposed MA) are applied before joint optimization algorithm is applied

since they provide a good choice of initial mapping between codevectors and modulation signals. Fig. 9 shows the subjective effects of the proposed algorithm. The 512 512 Lena image has been encoded by VQ . The images reconstructed at the noisy codebook dB are shown in Figs. 9(a) and channel of (b). Additionally, Figs. 9(c) and (d) are those for the channel dB. It can be seen from the figure that a with perceptually significant improvement can be obtained by using the proposed algorithm [Figs. 9(b) and (d)] compared with a conventional system [Figs. 9(a) and (c)] which consists of uniformly-spaced, square QAM and VQ codebook optimized for noiseless channel. Experimental results on the real image indicate that the proposed system significantly outperforms the conventional system, especially at low channel signal-to-noise ratio. E. Performance of Joint Optimization Algorithm for Gauss–Markov Source In order to make the results more general, the algorithms are tested for standard synthetic data produced by a Gauss–Markov source and the results are given in Fig. 10. The first-order Gauss–Markov processes are of the form (36)

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where

and

denote integers in

, and (A2) (A3) (A4) (A5)

and ) mapped into and are When the codevectors ( exchanged, the distortion in (A1) after the switch becomes Fig. 10. The performance comparison of various optimized communication systems for first-order Gauss–Markov source data ( = 0:9).

where is a zero-mean, unit-variance, Gaussian white noise process. In this test, the performances are represented in terms of signal-to-noise ratio (SNR) of the received data, defined as (37)

(A6) where, when

does not share a boundary with (A7)

and are the signal and noise variances, rewhere spectively. According to Fig. 10, it can be observed that the proposed system outperforms the conventional systems for the Gauss–Markov sources.

(A8) (A9)

VII. CONCLUDING REMARKS We have described an iterative procedure to optimize a VQ source encoder and a QAM signal set jointly. The proposed scheme optimizes a VQ-based communication system by minimizing end-to-end distortion including both quantization error and channel distortion. Although the computational complexity of the design is quite high, it can be done off-line, and does not impact system operation. Our joint design scheme outperforms several conventional systems optimized individually. Additionally, the initial mapping scheme between codevectors and modulation signals enhances the robustness of the proposed system. It is possible to further enhance the performance, by introducing advanced initial mapping technique between VQ codevectors and QAM signals.

(A10) and since with

and

when

shares a boundary

(A11)

(A12)

APPENDIX DERIVATION OF In this appendix, particular signals rewritten as

and

in (24) is derived. Consider two . Then, in (21) can be (A13)

(A14) (A1)

HAN AND KIM: JOINT OPTIMIZATION OF VQ CODEBOOKS AND QAM SIGNAL CONSTELLATIONS

Let denote the reduction in and change of codevectors assigned to

due to the ex, i.e., (A15)

Since (A6) into (A15) gives

from (19), substituting (A1) and

(A16) where (A17) (A18) (A19)

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[4] M. Polley, S. Wee, and W. Schreiber, “Hybrid channel coding for multiresolution HDTV terrestrial broadcasting,” in Proc. IEEE Int. Conf. Image Processing, Nov. 1994, pp. 243–247. [5] V. A. Vaishampayan and N. Farvardin, “Joint design of block source codes and modulation signal sets,” IEEE Trans. Inform. Theory, vol. 38, pp. 1230–1248, July 1992. [6] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, MA: Kluwer, 1992. [7] G. J. Foschini, R. D. Gitlin, and S. B. Weinstein, “Optimization of twodimensional signal constellations in the presence of Gaussian noise,” IEEE Trans. Commun., vol. COM-22, pp. 28–38, Jan. 1974. [8] C. E. W. Sundberg, W. C. Wong, and R. Steele, “Logarithmic PCM weighted QAM transmission over Gaussian and Rayleigh fading channels,” in Proc. Inst. Elect. Eng., vol. 134, Oct. 1987, pp. 557–570. [9] F.-H. Liu, P. Ho, and V. Cuperman, “Joint source and channel coding using a nonlinear receiver,” in Proc. ICC’93, 1993, pp. 1502–1507. [10] W. T. Webb and L. Hanzo, Modern Quadrature Amplitude Modulation. New York: IEEE Press/Pentech, 1994. [11] N. Farvardin and V. Vaishampayan, “On the performance and complexity of channel-optimized vector quantizers,” IEEE Trans. Inform. Theory, vol. 37, pp. 155–160, Jan. 1991. [12] N. Phamdo, N. Farvardin, and T. Moriya, “A unified approach to treestructured and multistage vector quantization for noisy channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 835–850, May 1993. [13] D. Miller and K. Rose, “Combined source-channel vector quantization using deterministic annealing,” IEEE Trans. Commun., vol. 42, pp. 347–356, Feb./Mar./Apr. 1994. [14] K.-P. Ho and J. M. Kahn, “Combined source-channel coding using channel-optimized quantizer and multicarrier modulation,” in Proc. 1996 IEEE Int. Conf. Communications, June 1996, pp. 1323–1327. [15] H. Stark, F. B. Tuteur, and J. B. Anderson, Modern Electrical Communications. Englewood Cliffs, NJ: Prentice-Hall, 1988. [16] W. S. Lee, M. R. Pickering, M. R. Frater, and J. F. Arnold, “Error resilience in video and multiplexing layers for very low bit-rate video coding systems,” IEEE J. Select. Areas Commun., vol. 15, pp. 1764–1774, Dec. 1997. [17] Y. Linde, A. Buzo, and R. M. Gray, “An algorithm for vector quantizer design,” IEEE Trans. Commun., vol. COM-28, pp. 84–95, Jan. 1980.

(A20) (A21) the number of elements in

if

Jong-Ki Han was born in Seoul, Korea, on September 5, 1968. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1992, 1994, and 1999, respectively. He is currently working as a Member of Technical Staff at the Corporate Research and Development Center, Samsung Electronics Company, Suwon, South Korea. His research interests include image compression, transcoding, and VLSI signal

(A22)

shares a boundary with otherwise. (A23) processing.

ACKNOWLEDGMENT The authors would like to express their sincere thanks to the reviewers for their constructive and valuable comments. REFERENCES [1] K. Ramchandran, A. Ortega, K. M. Uz, and M. Vetterli, “Multiresolution broadcast for digital HDTV using joint source-channel coding,” IEEE J. Select. Areas Commun., vol. 11, pp. 6–23, Jan. 1993. [2] N. Tanabe and N. Farvardin, “Subband image coding using entropy-coded quantization over noisy channels,” IEEE J. Select. Areas Commun., vol. 10, pp. 926–943, June 1992. [3] K. Fazel and M. Ruf, “Combined multilevel coding and multiresolution modulation,” in Proc. IEEE Int. Conf. Communications, Geneva, Switzerland, May 1993, pp. 1081–1085.

Hyung-Myung Kim received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, in 1974, and the M.S. and Ph.D. degrees in electrical engineering from the University of Pittsburgh, Pittsburgh, PA, in 1982 and 1985, respectively. He is now a Professor at the Department of Electrical Engineering and Computer Science, the Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea. His research interests include digital signal/image processing, digital transmission of voice, data and image and multidimensional system theory. Dr. Kim was the Treasurer of the IEEE Taejon Section in 1992. He has been an editorial board member of Multidimensional Systems and Signal Processing since 1990.