An Importance Sampling Technique for a Symbol

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In [22, Theorem 2], Sadowsky and Bucklew give conditions under which the density (15) yields an asymptotically efficient. IS simulation. The necessary condition ...
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 7, JULY 2000

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An Importance Sampling Technique for a Symbol-by-Symbol TCM Decoding/Equalization Algorithm KyeongJin Kim and Ronald A. Iltis, Senior Member, IEEE

Abstract—An importance sampling simulation technique is used to estimate the bit-error rate (BER) of a symbol-by-symbol trelliscoded modulation (TCM) decoder/equalizer proposed for digital microwave radio applications. To evaluate the performance at a BER smaller than 10 6 , we investigate a randomized bias technique previously developed by the authors. To properly select the bias vectors, the asymptotic (infinite signal-to-noise ratio) decision boundary is first determined. The simulation technique is then extended to the one-shot symbol-by-symbol TCM decoding/equalization algorithm which is equivalent to the recursive symbol-bysymbol detection algorithm of Abend and Fritchman. By using this novel importance sampling technique, we can speed up the simulation and efficiently evaluate the BER performance of the TCM decoder/equalizer. Index Terms—Demodulation, trellis-coded modulation.

Monte

Carlo

methods,

I. INTRODUCTION

W

HEN digital data is to be transmitted across a digital microwave radio (DMR) channel, a very low bit-error rate (BER) must be maintained. Due to the bandwidth limitation, trellis-coded modulation (TCM) [1] has been proposed for DMR applications [2], [3]. The advantage of TCM is that bandwidth expansion is not required, and coding gains are readily obtained. The DMR channel is usually characterized by a slowly time-varying, frequency-selective multipath fading channel model due to Rummler [4], resulting in intersymbol interference (ISI). Although this DMR channel is much easier to equalize than rapidly time-varying mobile channels [5], TCM schemes encounter a substantial degradation in coding gain with the DMR channel impairments [3]. To fully benefit from the TCM coding gain, it is thus important to properly equalize the channel. Most previous work in joint equalization and decoding for TCM has focused on variations of maximum-likelihood sequence estimation (MLSE) [5], [6]. Interleaving plays an important role by breaking up burst errors caused by fading. Although interleaving improves BER performance, it intro-

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received September 26, 1998; revised Sepember 30, 1999. This work was supported in part by the University of California MICRO Program and P-COM Corporation. This paper was presented in part at the CISS Conference, Princeton, New Jersey, 1998. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA USA 93106 (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)06155-9.

duces delay and is difficult to combine with a joint equalization and decoding algorithm [5]. An alternative symbol-by-symbol detection (SBSD) equalizer for TCM was considered in [7] for the mobile channel. However, due to the need for interleaving, this structure could not perform joint decoding/equalization. This paper investigates a different approach to joint equalization/decoding using an extension of the Abend and Fritchman SBSD [8] to TCM coded channels. Unfortunately, it is difficult to evaluate the performance of the TCM receivers at low BER’s. Although an upper bound for the BER performance of SBSD TCM decoding/equalization algorithm has recently been obtained in [9], there is no SBSD analog to the transfer function bound used for evaluation of MLSE decoding [10], [11]. Monte–Carlo (MC) simulation is often used to estimate the SBSD BER performance. However, evaluation at a BER smaller than 10 requires a large number of simulation trials. To speed up the simulation, we extend an importance sampling (IS) technique originally developed in [12] for uncoded equalization to the joint TCM decoding/equalization application. The remainder of the paper is organized as follows. In Section II, the channel and signal models are described. The SBSD TCM decoding/equalization algorithm is described in Section III. The IS technique to evaluate the BER performance is described in Section IV with simulation results provided in Section V, and conclusions are summarized in Section VI. II. SIGNAL AND CHANNEL MODELS We will use the following notation to describe the various SBSD algorithms. : DMR ISI channel length and length • of the composite channel formed by the convolutional encoder with constraint length and DMR ISI channel, respectively. : complex-valued DMR ISI channel • coefficients. : equally probable independent binary • data sequence. : dibit data sequence. • : cumulative dibit data sequence. • : dibit data • subsequence of length . : cumulative th dibit • data sequence. : th cumulative sequence in the set • .

0090–6778/00$10.00 © 2000 IEEE

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where is the symbol rate and is a bandwidth-efficient (e.g., raised cosine) pulse shape. The DMR channel is usually characterized by the Rummler model [4], with frequency response in low-pass equivalent form (3) where is the flat fade parameter, is the relative notch depth parameter, is the frequency measured from the center of the channel to the notch, and is the delay difference in the channel. The receiver is assumed matched to the transmitted pulse. The in the time domain, sampled at received noiseless signal , is then given by instants

Fig. 1. TCM digital microwave radio system with SBSD.



th cumulative sequence in the set . : cumulative received • sample vector. : cumulative received • sample sequence. : complex-valued symbols at the output of • the trellis encoder as a function of the length- dibit data subsequences. : DMR ISI channel • output due to the transmitted signal. : cu• mulative DMR ISI channel output vector. : cumulative • symbol sequence. : th subsequence in the set • . : th subsequence in the set • . : circular white • Gaussian noise vector. The system model for this problem is given in Fig. 1. The seis transmitted across the quence of information dibits are input into the DMR channel. Referring to Fig. 1, the convolutional encoder with constraint length . The output bits . In genare mapped into the complex-valued symbols are nonlinear functions of the inputs. eral, the symbols For example, the modulated symbols for the constraint length , rate , 8-ary PSK code, in [1], are given by

(4)

:

(1) where denotes modulo-2 addition. The transmitted signal is given by (2)

. FurtherWe note that (4) assumes an ideal sinc pulse for more, a noise-whitening filter is not required for an ideal sinc pulse. With the equivalent discrete-time channel model, it can be shown that the received samples in the presence of noise are given by (5) is circular white Gaussian noise with correlawhere . The equivalent DMR tion function ISI channel coefficients are (6) Note that with the assumption of a small roll-off factor, can be approximated as an independent noise without using the noise-whitening filter. Due to the unrealistic assumption of a small roll-off factor, it is seen that the channel is noncausal. coHowever, it is easily seen that only a small number of efficients are significant. For example, using a typical value for ns, and a symbol the DMR channel for the path delay, MHz, we can see that , rate, , , and . In general, for the equivalent discrete-time DMR channel, it is anticipated that a maximum of three or four coefficients will be significant for symbol rates lower than 100 MHz. Since the DMR channel is slowly time varying, the equalization/decoding algorithm can assume that the DMR channel is known a priori. The overall goal of the work presented here is to find an effective joint equalization and decoding strategy for determining from received samples and to evaluate the the data bits performance at a BER smaller than 10 . III. JOINT SYMBOL-BY-SYMBOL TCM DECODING/EQUALIZATION ALGORITHM While the MLSE computes the optimum sequence decisions on the cumulative data sequence , the SBSD computes an opbased on the timum decision on binary symbol

KIM AND ILTIS: SYMBOL-BY-SYMBOL TCM DECODING/EQUALIZATION ALGORITHM

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cumulative data sequence . The key to the development of the is circular Gaussian when it is conditioned SBSD is that . The conditional probability density on the subsequence is given by function of

the error probability, , indepenare generated using the true dent sample vectors in conventional MC simulations. From the redensity is expressed as follows: ceived samples, the estimator

(7)

(11)

denotes a circular Gaussian density with where and variance . The optimum symbol-by-symbol demean cision is given by

(8) A recursion for the data subsequence ployed following [13] as

where

is the indicator function and

. The relative error defined by [15] grows without bound as . Thus, MC estimation is often impractical for BER’s less than 10 . as the weighting function in IS. Here, Define is the true density for the received sample and is the simulation density. The IS estimator is given by (12)

is em-

(9) is the cardinality of the coded signal constellation and means that the first dibits in equal the last dibits in . This recursion is key to the final form of the SBSD given by

where

(10) where is a normalization constant. Although the SBSD is more complex than the better-known Viterbi algorithm, it offers some advantages when extended to blind equalization applications [13], [14]. Unfortunately, a direct analysis of the BER performance of the SBSD seems to be intractable. However, with the method of [12], we show that a randomized bias vector IS technique can accurately estimate the BER performance for the SBSD. IV. IMPORTANCE SAMPLING USING RANDOMIZED BIAS VECTORS A. Review of Large Deviations Theory Applied to Importance Sampling IS is an MC technique where data is generated using a simulation distribution different from the true probability distribuis tion. In the sequel, it is assumed that a decision on bit , denoted on the th to be made using the cumulative vector . To form an unbiased estimate of simulation run by

is a sample vector generated according to the simwhere for the th run. It is easily shown that ulation density , that is, the IS estimator is unbiased. If is chosen as (13) will have zero variance [17]. Equation (13) indicates then has support confined to . Since and that the ideal are unknown, (13) is not practical. However, the structure of the optimum simulation density (13) does provide guidance for the design of good IS methods [22]. Many IS methods have been proposed corresponding to different choices of the simulation density. Linearly shifted densities are used in [18] and [19]. However, the linear shift is only is defined by a single hyperplane, or halfoptimum when space. As discussed in [12], the decision boundary for the equalization problem is a nonconvex union of partial half-planes, and hence the linear shift method performs poorly in such applications. An IS simulation method using a recursive adaptation technique [20] to update the simulation density has also been proposed. However, this method cannot guarantee the convergence of simulation density parameters for our SBSD structure. An alternative technique based on large deviation theory (LDT) [21] is developed in [22]. Under certain geometric conditions on , this approach yields an asymptotically efficient simulation density. Specifically, in [22], an IS simulation is said to decreases at be asymptotically efficient if the variance of least at an exponential rate with increasing signal-to-noise ratio (SNR) The randomized bias technique based on LDT has also been proposed for finding the optimum bias vectors for an uncoded equalization problem [12]. In this paper, we will extend this randomized bias technique to the TCM equalization/decoding problem using the one-shot SBSD algorithm. We first review a simple simulation density developed using LDT in [22], and then introduce the randomized bias technique as an extension of LDT results. When is characterized by a yields single half-plane, the following simulation density an asymptotically efficient IS method [22]. The density is given

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here in slightly modified form to accommodate complex vectors . (14) , , is a lower semicontinuous where log-moment generating function or cumulant defined by , and is the unique , where is a unique dominating point. solution to A dominating point is characterized as follows for the additive Gaussian noise case, using the results of [22] and [12]. As, where is deterministic (e.g., a signal sume that vector) and is zero-mean, circular Gaussian noise. Define the . Then, is a dominating point if: 1) is the vector , to , and 2) the closest point on the error region boundary, half space defined by

completely covers . Using this definition of a dominating point, it is readily seen that the simulation density (14) is asymptotically efficient if the error region is characterized by a single half-space. As discussed below, the error region for TCM decoding/equalization is, in general, a union of Voronoi cells, and thus, a more complex simulation density must be chosen to obtain an importance sampling gain. To find an appropriate simulation density, we use the following results of Sadowsky and Bucklew [22]. Definition 1 (Sadowsky and Bucklew): A point is defined to be a minimum rate point of if: 1) ; 2) , where is a well-defined is related to the cumulant through convex rate function and the Legendre–Fenchel transform, , such that . In applications where a single dominating point does not exist, Sadowsky and Bucklew [22] proposed the following exponentially twisted sum density with vectors chosen according to the probabilities .

tical simulation method, as first suggested in [23], we can estimate the error rate using the density of conditioned on transmission of a specific data sequence . An error then occurs if , since under . Denote as the received vector generated according to the simulation density, with transmitted, on the th simulation run. Assume that the condiis to be estimated. By choosing tional error rate for a given as an IS weighting function, the conditional estimator is rewritten as follows:

(16) where denotes the simulation run, denotes a possible data and denotes the true consequence, and ditional density and the simulation density assuming a specific was transmitted. Note that the outer sum in (16) sequence possible dibits through , and the ranges over the . two possible values for the remaining bit A randomized bias technique for the TCM equalization/decoding problem using the one-shot SBSD [12], [24] is developed as follows. First, we review the SBSD structure and resulting decision boundaries following the analysis of [12]. The vector channel output can be represented as

.. . .. .

.. .

(17)

Using the defined vector quantities, (17) can now be rewritten as

(15)

(18)

has the unique solution . where In [22, Theorem 2], Sadowsky and Bucklew give conditions under which the density (15) yields an asymptotically efficient IS simulation. The necessary condition for efficiency is that the includes all minimum rate points. When is set conditionally circular Gaussian, it is shown in [12] that the sufficient condition for efficiency of [22, Theorem 2] holds when the are chosen as follows. First, at least one point must points , and must equal . Second, the half-spaces be on formed by the hyperplanes orthogonal to the vectors must completely cover the error region , where is the conditional mean of .

and is a circular white where Gaussian noise vector. When a symbol takes on one of the possible dibits with equal probability, and under the circular white , the one-shot detector to Gaussian noise assumption for at time using the received vector is decode defined by

B. Conditional Importance Sampling Technique for TCM

Note that the one-shot algorithm (19) is exactly equivalent to the Abend and Fritchman recursive algorithm (10). Following [12], we can obtain the following sufficient conditions for a point

is the most It has been shown that choosing a suitable important aspect in applying the IS technique. To obtain a prac-

(19)

KIM AND ILTIS: SYMBOL-BY-SYMBOL TCM DECODING/EQUALIZATION ALGORITHM

to lie on the asymptotic decision boundary, as

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TABLE I ALGORITHM FOR DETERMINATION OF BIAS VECTORS

.

(20) and are candidate received vectors such where and , respectively. Since there are, in that possible input symbols to the TCM encoder, general, , from the definition, equals . the number of in (19) The asymptotic decision region for deciding , is readily seen to correspond to a union of Vorononi cells con( , since as , only one taining the vectors closest to dominates term corresponding to the point the sum of exponentials. The decision boundary (20) is similarly defined by portions of half-spaces whose boundary points satisfy the three rules given. Note that the decision region is not necessarily convex for SBSD, since the decision boundary is a union of Voronoi cells and not a single cell in general. (For an example of such a nonconvex region, see [12].) For the case of TCM with ISI, only a subset of the possible determine the decision boundaries, correpairs sponding to the Voronoi neighbors [26]. In some cases, the point will satisfy the three rules in (20), and . In this case, the pair constihence lie on tutes a Gabriel neighbor as discussed in [26]. As shown in [26], all Voronoi neighbors are Gabriel neighbors for the case of linear binary block codes. However, even in uncoded systems with ISI, it is easy to find example boundaries where some Voronoi neighbors are not Gabriel neighbors. In the IS technique used here, we will first find all pairs which are Gabriel neighbors and lie on , and to each rethen construct bias vectors from the transmitted . Although, sulting boundary point in general, the resulting simulation density will not be asymptotically efficient in the sense of [22], the simulation results presented demonstrate a large IS gain, which suggests that the bias vector technique is constructing a large enough set of halfspaces, such that is well covered. Following the form of [12], the algorithm for determination of Gabriel neighbors and bias vector computation is described in Table I. in a ranIt is now shown that adding the bias vectors to according to the domized manner is equivalent to generating exponentially twisted sum density (1). To show this, first derive the log-moment generating function, under the assumption that is a sequence of conditionally circular Gaussian vectors, with variance decreasing inversely with . (21) Then straightforward calculations show that (22) The Legendre–Fenchel transform then becomes (23)

To find the twisted sum density (15), the points satisfying , where must be de. termined. Using (22), it is seen that Substituting for into (15) yields, after some algebra (24) The interpretation of (24) is that a bias vector equal to is added to , with probability . In general, we cannot show that the randomized bias simulation technique is asymptotically efficient, since not all Voronoi neighbors are Gabriel neighbors, and the resulting half-spaces do not completely cover . However, the following theorem shows that for the case of zero ISI (TCM decoding only), the randomized bias vector IS method satisfies the necessary, but not sufficient conditions for asymptotic efficiency. Theorem 1: Consider a TCM symbol-by-symbol de, using the sequence . tector (decoder) for bit , where Then, the set of points is the set of minimum rate points defined by Sadowsky and Bucklew [22], s.t. and . Hence, the set of bias vectors satisfies the necessary (but not sufficient) conditions for asymptotic efficiency of the IS simulation. The proof of this Theorem is given in Appendix A. A somewhat more restrictive version of Theorem 1 holds for ISI channels as follows. Theorem 2: Consider the joint TCM decoder/equalizer for using the ISI channels, where a decision is made on bit be the set of points at minimum Eusequence . Let from . Define the set of midpoints clidean distance . If all such lie on , as satisfies then the set of bias vectors the necessary conditions for asymptotic efficiency of the IS simulation. The proof is given in Appendix B. Although the necessary conditions for asymptotic efficiency do not always hold for ISI channels, we have found in simula, there are numerous Gabriel neightions that for each , and hence numerous bias vectors. The resulting bors half-spaces are conjectured to cover a large portion of , since the simulated IS gains discussed below increase exponentially with SNR.

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Fig. 2.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 7, JULY 2000

Encoder structure for 8-PSK TCM.

Fig. 4. Channel frequency response of

Fig. 3.

Channel frequency response of

H.

H.

The performance improvement resulting from the IS estimate [12] can be estimated as follows: (25) Fig. 5. Error-rate performance of the one-shot SBSD TCM decoding/equalization algorithm over a perfect channel (no ISI) with AWGN.

where (26) Note that (25) is the approximate ratio of the number of IS runs to the conventional MC runs required to attain a given estimation . error variance, V. SIMULATION RESULTS Two types of simulation results using the proposed IS technique are presented, those obtained for an uncoded 4-PSK SBSD decoder/equalizer with or without ISI and those obtained for SBSD TCM decoder/equalizer with or without ISI. The selected TCM encoder in our simulation is shown in Fig. 2. The randomized IS technique is applied to two complex channels with the transfer functions

The channel, as seen in Fig. 3, has infinite nulls on the unit circle, which makes it difficult to equalize with a linear equalizer channel, plotted in Fig. 4, has , [10], and the MHz, ns, and the notch depth, dB. are The number of received samples for the decision on . In all cases, a sufficient number of vectors are generated at each SNR to have at least an estimated 10%

Fig. 6. Error-rate performance of the one-shot SBSD TCM decoding/equalization algorithm over a perfect channel (no ISI) with AWGN depending on .

k

relative precision. The bias vectors were selected with uniform probability in the simulations. Furthermore, since BER’s with and are not symmetric, we take the avrespect to erage value of the corresponding conditional BER’s. Fig. 5 shows the comparison of the performance of the IS SBSD TCM decoder/equalizer and uncoded 4-PSK system, when there is only additive Gaussian noise. For this case, we obtain the coding gain of approximately 2.2 dB at a BER of 10 and 2.6 dB at a BER of 10 . For the same conditions and a perfect channel (no ISI), Figs. 6 and 7 show the BER performance of the SBSD TCM decoder/equalizer for different values of and the resulting IS gain defined by (25), respectively. Fig. 6, as expected, shows that the BER performance improves as the number of received samples increases.

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Fig. 7. IS gain 0 for the one-shot SBSD TCM decoding/equalization algorithm over a perfect channel (no ISI) with AWGN.

Fig. 9. IS gain 0 for the one-shot SBSD TCM decoding/equalization algorithm over channel with ISI and AWGN.

Fig. 8. Error-rate performance of decoding/equalization algorithm over depending on .

Fig. 10. Error-rate performance of the one-shot SBSD TCM decoding/equalization algorithm over DMR channel with ISI and AWGN depending on .

k

H

the one-shot SBSD TCM channel with ISI and AWGN

Fig. 7 shows that the estimated IS gain defined by (25) grows exponentially with increasing SNR. In practice, we have found that such an exponentially growing, monotonic simulated gain is obtained when the IS error rate agrees closely with the conventional MC rate, that is, when errors are not undercounted. Fig. 8 shows the BER performance of the IS simulated SBSD TCM decoder/equalizer compared with an uncoded is included. The SBSD 4-PSK system when the ISI channel TCM decoder/equalizer is robust for this particular ISI channel, and has a coding gain of approximately 1.9 and 2.3 dB at a BER of 10 and 10 , respectively. Due to the severe ISI, we experience a relative degradation in the coding gain. This figure, however, shows that as we did in the perfect channel, we can improve the BER performance by increasing . For the same ISI channel, Fig. 9 shows the estimated IS gain which grows exponentially with increasing SNR. As in the case of the perfect channel, this is seen as a sign that the simulation is accurate. Fig. 10 shows the BER performance of the IS-simulated SBSD TCM decoder/equalizer and uncoded 4-PSK system is included. The SBSD TCM when the DMR ISI channel decoder/equalizer is also robust for this DMR ISI channel, and has a coding gain of approximately 1.3 and 1.6 dB at a BER of 10 and 10 , respectively. Fig. 11 is the corresponding estimated IS gain which again grows exponentially with increasing SNR. Table II compares the run time required to generate 1000 error events for IS and the conventional MC techniques on a 400-MHz

H

k

Fig. 11. IS gain 0 for the one-shot SBSD TCM decoding/equalization algorithm over DMR channel with ISI and AWGN.

Pentium II computer. The run time is almost constant for the perfect channel (no ISI) with respect to SNR. This indicates that the IS method is able to maintain a simulation-error rate of about one-half, and that the method is nearly asymptotically efficient. and , the run time does increase However, for channels with increasing SNR. This indicates that the error region is not sufficiently covered by the half-spaces defined by the bias vectors, and that the simulation-error rate is thus less than one-half. This result is expected given Theorem 2, which shows that the necessary conditions for asymptotic efficiency are not always satisfied for a channel with ISI. However, there is still a significant savings in run time at high SNR for the ISI channels over conventional MC simulation.

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TABLE II COMPARISON OF IS AND MC TECHNIQUES OVER DIFFERENT CHANNELS DEPENDING ON

k

of is less than 1 (which implies that there is no gain using the IS technique if the number of bias vectors is too small). 2) As SNR increases, the IS gain converges to that obtained using the full set of bias vectors. VI. CONCLUSIONS

Fig. 12. Error-rate performance of decoding/equalization algorithm over depending on and .

k

n

H

the one-shot SBSD TCM channel with ISI and AWGN

In this paper, we considered a symbol-by-symbol TCM equalization/decoding algorithm for DMR applications. To speed up the simulation, a novel importance sampling technique was presented based on the randomized bias technique. The bias vectors were found by first determining the asymptotic decision boundary for the one-shot symbol-by-symbol TCM decoding/equalization algorithm. For a channel without ISI, it was shown that the resulting randomized bias technique satisfied the necessary (but not sufficient) conditions for asymptotic efficiency, as defined in [22]. The simulated IS gain was found to increase exponentially and monotonically with SNR, which provides an empirical indication that the IS simulation results are accurate, and that the bias vectors chosen cover a significant portion of the error region with half-planes. APPENDIX A PROOF OF THEOREM 1

Fig. 13. IS gain 0 for the one-shot SBSD TCM decoding/equalization algorithm over channel with ISI and AWGN depending on and .

H

k

n

We also considered speeding up the simulation by using only of the total set of bias vectors which pass the tests in Table I. The simulation results in Figs. 12 and 13 apply the rechannel. Fig. 12 corresponds duced set of bias vectors to the to the channel and conditions in Fig. 8. By choosing a smaller with the same , the BER is lower which revalue for sults from the fact that errors are undercounted. However, the BER asymptotically approaches the result in Fig. 8, as increases. Fig. 13 shows that the estimated IS gain grows exponentially with increasing SNR. Looking at this figure, we can find the following trends. 1) At lower SNR, for a small value

In Theorem 1, it is claimed that the set of points are minimum rate points, where is in the set . To prove this, it is suffi, cient to show that all such points lie on the boundary . since then lies on . Then, since Assume that for all for a trellis code, with zero ISI, the conditions (20) reduce to the following. First, note that condition (3) is always satisfied for this choice of . Conditions (1) and (2) become

(A.1)

KIM AND ILTIS: SYMBOL-BY-SYMBOL TCM DECODING/EQUALIZATION ALGORITHM

The above inequalities are now shown to be always satisfied is at minimum for the trellis codes in [1]. By construction, from . (Note that refers to the Euclidean distance minimum distance to over all , although there may, in genthat are closer to than . In particular, eral, be vectors can be less than .) For the minimum free distance will, in the trellis codes in [1], it can be shown that multiple from . For example, if corgeneral, be at distance represponds to an uncoded bit equaling 1, then multiple resenting parallel transitions will be at the same minimum free distance from . Thus, for trellis codes, the following inequalities are evident:

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and from (20) (B.5) Hence, the triangle inequality yields (B.6) or (B.7) cannot be a minimum rate which contradicts (B.1). Hence, , which proves Theorem 2. point, since REFERENCES

(A.2) The second inequality in (A.2) arises from the symmetry of the codes in [1], and can be verified by examination of the trellises. fixed, pick one of the at minimum distance That is, for from . Noting that the encoder is initialized at time 0 to a that given state, say state 0, it is impossible to find another to . is closer than The conditions in (A.2) now reduce to the following inequalfor all . ities since

(A.3) However, substitution of (A.3) into (A.1) yields strict , and inequalities, since in (A.1), . Thus, the points are on the decision boundary. Furthermore, a simple geometric argument shows are the closest points on to , and hence that these over , thus completing minimize the proof. APPENDIX B PROOF OF THEOREM 2 From [22], we require that all minimum rates points must be to satisfy the contained in the set necessary conditions for asymptotic efficiency. That is, the set must satisfy . Note that , if is the closest point on to . The proof is by contradicwhich is closer tion. Assume that there exists a point then . Then to (B.1) Consider the triangle inequality, and let vectors defining in (20).

be the

(B.2) (B.3) However, recall from the conditions of Theorem 2 that (B.4)

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Kyeongjin Kim received the B.S. degree in electronics from Inha University, in 1987, and the M.S. degree in electrical engineering from the Korea Advanced Institute for Science and Technology (KAIST), in 1991. Since 1996, he has been working toward the Ph.D. degree in the Department of Electrical Computer Engineering at the University of California, Santa Barbara. From 1991 to 1995, he was a Research Engineer with the Video Research and Development Center at Daewoo Electronics Company, Ltd., Korea. His research is focused on detection algorithms for TCM and CDMA.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 7, JULY 2000

Ronald A. Iltis (S’83–M’84–SM’91) received the B.A. degree in biophysics from the Johns Hopkins University, Baltimore, MD, in 1978, the M.Sc. degree in engineering from Brown University, Providence, RI, in 1980, and the Ph.D. degree in electrical engineering from the University of California, San Diego, in 1984. Since 1984, he has been on the faculty at the University of California, Santa Barbara, where he is currently a Professor in the Department of Electrical and Computer Engineering. He has also served as a consultant to government and private industry in the areas of adaptive arrays, neural networks, and spread-spectrum communications, blind equalization, multisensor/multitarget tracking, and neural networks. Dr. Iltis has served as an an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS. He received the Fred W. Ellersick Award for Best Paper at the IEEE MILCOM Conference in 1990.