Decision-feedback MAP receiver for time-selective ... - IEEE Xplore
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Decision-Feedback MAP Receiver for Time-Selective Fading CDMA Channels Tan F. Wong, Member, IEEE, Qian Zhang, and James S. Lehnert, Fellow, IEEE
Abstract—A decision-feedback maximum a posteriori (MAP) receiver is proposed for code-division multiple-access channels with time-selective fading. The receiver consists of a sequence-matched filter and a MAP demodulator. Output samples (more than one per symbol) from the matched filter are fed into the MAP demodulator. The MAP demodulator exploits the channel memory by delaying the decision and using a sequence of observations. This receiver also rejects multiple-access interference and estimates channel fading coefficients implicitly to give good demodulation decisions. Moreover, computer simulations are performed to evaluate the bit-error rate performance of the receiver under various channel conditions. Index Terms—Code-division multiaccess, fading channels, feedback communication, linear prediction, MAP estimation.
I. INTRODUCTION
T
IME-SELECTIVE fading due to Doppler spread is a major difficulty in wireless mobile communication systems [1]. Because code-division multiple-access (CDMA) systems tend to have larger symbol durations than time-division multiple-access systems, the effect of time-selective fading can be more severe in CDMA systems. For example, consider an IS-95 CDMA system [2], [3] operating in the personal communication systems (PCS) band at a carrier frequency around 2 GHz. For a mobile user moving at 60 mi/h, the Doppler spread is about 200 Hz. According to Jakes model [1] (see Section II), this corresponds to a coherence time of 2 ms. For a symbol rate at 9600 symbols/s, fading coefficients separated by 20 symbols are almost uncorrelated. For this example, it is no longer reasonable to approximate the channel as time invariant. Hence, the detection and interference cancellation schemes based on the time-invariant channel assumption may not be applicable. Demodulation schemes based on the methods of maximumlikelihood sequence estimation (MLSE) and maximum a posPaper approved by U. Mitra, the Editor for Spread Spectrum/Equalization of the IEEE Communications Society. Manuscript received April 8, 1998; revised June 15, 1999 and September 22, 1999. This work was supported by the Defense Advanced Research Projects Agency under Contract DAAB07-97-CD016. This paper was presented in part at the IEEE Military Communications Conference (MILCOM’98), Boston, MA, October 1998. T. F. Wong was with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA. He is now with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130 USA (e-mail: [email protected]). Q. Zhang was with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA. He is now with Globespan Semiconductor, Inc., Red Bank, NJ 07701 USA (e-mail: [email protected]). J. S. Lehnert is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)04019-8.
teriori (MAP) estimation [4]–[10] have been proposed for narrowband channels with intersymbol interference. It is shown in [9] that MAP algorithms outperform MLSE algorithms because MAP algorithms can give soft-outputs [9], which are useful for outer decoders. The applicability of the MAP technique for narrowband time-selective fading channels has also been demonstrated in [11]. By performing the demodulation process based on a sequence of observations, these techniques try to make use of the channel memory in order to lower the error probability. Although extending the MAP technique to CDMA systems seems natural, the presence of multiple-access interference (MAI) in CDMA systems presents two major difficulties, which hamper the application of the MAP demodulation technique. First, the MAI has a complex distribution function [12], which makes the development of the MAP demodulator difficult. Second, in near–far situations, the received power of the MAI can be stronger than that of the desired signal. In this case, the MAP method will not give good performance unless some kind of MAI cancellation is performed. In this paper, we develop an efficient way to apply the MAP technique to direct-sequence CDMA systems for time-selective flat fading channels. We assume that users in the system communicate asynchronously. For synchronous CDMA systems, the situation reduces to the narrowband case by employing orthogonal sequences. Hence, our focus is on asynchronous CDMA systems. Details of the system configuration and the fading model are given in Section II. We consider the use of long spreading sequences [13], [14] in this paper. As a result, most MAI rejection techniques developed for short spreading sequences are not applicable [15], [16]. Assuming no other system resources, such as receiving antenna arrays, are available, we employ the simple decentralized receiver considered in [15] to achieve MAI cancellation as a practical alternative. The receiver contains a sequence-matched filter whose impulse response is matched to the signature waveform of the desired user. Output samples from the matched filter are fed into a delayed-decision forward-recursive MAP demodulator similar to the ones described in [4], [9], and [11]. Although the output of the matched filter does not provide a sufficient statistic for the detection of the transmitted symbols, processing that is limited to observing the matched filter output greatly simplifies the complexity of the receiver and provides a certain degree of immunity to MAI when long sequences are used [15]. Based on the results in [17] and [18], we note that the MAI components in these output samples are asymptotically Gaussian distributed when the number of chips approaches infinity. Since is designed to be per symbol large in most practical CDMA systems, it is reasonable for us to assume that the matched filter output samples are Gaussian
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random variables. This Gaussian approximation simplifies the development of the MAP receiver. The receiving structure and the MAP demodulator are developed in Section III and IV, respectively. The complexity of the conventional MAP demodulator in [4] grows exponentially with the number of observations with our channel model. To reduce complexity, we employ the decision feedback technique, and approximate the autocorrelation function of the fading process by that of a finite-order moving average process. Based on these two complexity reduction techniques, a recursive algorithm whose complexity depends exponentially on the decision delay is developed in Section IV. The resulting algorithm performs two essential operations implicitly, namely, MAI rejection by noise-whitening [15] and linear prediction of the fading coefficients of the desired user signal. The recursive algorithm requires an estimation of the instantaneous covariance matrices of the interference components in the matched filter output samples. In Section V, we describe a method based on the delayed sampling technique in [16] to accomplish this estimation. In Section VI, we conduct Monte Carlo simulations to evaluate the bit-error rate (BER) performance of this decision-feedback MAP receiver with the time-selective fading channel model described in Section II for both single-user and multiple-user systems. Finally, we draw conclusions from the development of this receiver in Section VII. II. SYSTEM MODEL In this section, we describe the DS-CDMA system and the channel model. simultaneous users in the We assume that there are , generates a sequence of system. The th user, for . We assume that the data symbols are indata symbols dependent random variables taking values from a finite alphabet such that . We note that both binary phase-shift ) and quadrature phase-shift keying (BPSK) ( ) satisfy this assumpkeying (QPSK) ( tion. Generalizations to other signaling schemes, such as -ary phase-shift keying (MPSK) and quadrature amplitude modulation, are straightforward. Moreover, the data symbols are not assumed to be identically distributed to allow the possible use of pilot-assisted modulation schemes. The th user is provided a randomly generated signature segiven by quence (1) are independent random variables taking The elements or with values from either is used to generate the equal probabilities. The sequence spectrally spread signal given by (2) is the chip duration and is the common chip where is time-limwaveform for all signals. The chip waveform
ited to ,1 and is normalized so that . , where is the The symbol duration is given by number of chips per symbol. Moreover, we assume that the signature sequence of any user is independent of all other signature sequences and all the data symbol sequences. , can The transmitted signal for the th user, for be expressed as (3) is the transmitted power for the th signal, is the where is the delay that models the asyncarrier frequency, and chronous system. We consider a decentralized interference suppression approach in detecting the transmitted data symbol. Without loss of generality, we consider the signal from the zeroth user as the communicating signal and the signals from all other users as interfering signals throughout the paper. We assume time-selective fading and additive white Gaussian . noise (AWGN) with two-sided power spectral density The signal received at the intended receiver in complex baseband notation is given by (4) and represent the desired signal and the interferwhere ence components, respectively. The desired signal contribution is given by (5) is the sum of the AWGN component The interference and the MAI contribution , which is given by
(6) models the phase angle of the th user’s signal. In (5) where models the time-varying fading coefficient of and (6), the th user’s signal. Suppose the th transmitter is moving with in the direction of arrival of the th signal to relative speed the zeroth receiver. By properly scaling the receiver power , as a complex, zero-mean Gaussian we can model [1] random process with autocorrelation function (7) is the zeroth-order Bessel function of the In the above, is the maximum Doppler frequency of the first kind and th signal given by (8) 1The restriction to a time-limited chip waveform is just for simplicity of notation. All the results in this paper hold as long as the chip waveform satisfies the Nyquist criterion [15], which guarantees the absence of interchip interference.
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where is the speed of light. Hence, a time-varying flat fading , the th model for the channel is adopted. Moreover, symbol waveform of the th interferer, is given by (9)
Fig. 1. Receiving structure.
III. RECEIVING STRUCTURE We assume that we have achieved synchronization with the desired user signal, for example, through the use of pilots. and . In this and all the Hence, we may assume following sections, we assume that we have exact knowledge . Unless otherwise stated, all of the desired user sequence, . probabilistic arguments are ones that condition on To obtain the observations (see Section I for rationale) corresponding to the th data symbol, we despread the received , signal by passing it through the linear filter which is matched to the th signature waveform, as shown in at Fig. 1. Then, we sample the despread received signal for , times . We stack the samples together to obtain a where and denote this vector by . This vector of length sample vector is then fed into the MAP demodulator in Fig. 1. As shown in Section I, for the targeted practical systems, the fading coefficients are basically unchanged for about a symbol duration. Moreover, we also assume that the processing gain is large so that the approximations invoked in [15] are also valid here, i.e., we can neglect the effect due to the boundary chips.2 Following arguments similar to those used in [15], we can express as (10) where is the component due to the desired signal and is the component due to MAI and AWGN. The desired signal compois given by nent (11) where
(13) (14) and (15) is the sum of two vectors: The interference component due to AWGN and due to MAI, which is given by (16)
for . In (11) and (16), , for , can be expressed in forms We note that . Moreover, we have included the data symsimilar to that of of the interferers into the expressions of since we bols will not attempt to demodulate these symbols. It is easy to show , for , are zero-mean random vectors that or such that, for (17) From the conditional Gaussian approximation results in [17] and [18], when is large, it is reasonable to assume that are jointly Gaussian random vectors given all the interferers’ fading coefficients. Hence, from now on, we make the approximation that ’s are independent jointly Gaussian random vectors given all the interferers’ fading coefficients. IV. MAP DEMODULATOR
.. .
First, we rewrite the observations form. For
, and
in matrix
.. .
.. . (12)
.. 2Throughout this paper, all the equalities involving this approximation should be interpreted as limiting arguments as tends to infinity.
N
.
.. .
.. .
(18)
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where
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complexity, we approximate the sampled autocorrelation function by the autocorrelation function of a dis, i.e., crete-time moving average (MA) process of order for . Appendix A shows a way to obtain from . To start the development of the recursive algorithm, we note that
. Conditioning on
.. .
.. .
and , is a zero-mean Gaussian . random vector of length Our objective is to develop a -lag, decision-feedback MAP . More predemodulator based on the observation vector th data symbol of the zeroth cisely, we demodulate the user by maximizing its a posteriori probability (19) The interferers’ fading coefficients required in (19) can be obtained via explicit estimation using the methods described in is provided through Section V. The data symbol vector decision feedback. In the following development, we assume is given, i.e., the decision feedback gives perfect estimates. We define the system state at the th symbol by
(23) recurHence, it suffices to calculate are jointly Gaussian given sively. We know that and and . Hence, is conditionally Gaussian [19] given 3 , , and and
(24) (20) where Following the standard development in [4] and [5], we can write: (25) and
(26)
(21)
Hence,
We proceed to make several definitions that are essential to our development. Then, we state a recursive algorithm to calcuand give physical interpretations to late the algorithm. , define Definition 1: For for (27)
(22) Therefore, we have to calculate the values of for all the possible states in order to obtain the estimate . We note that the dimensions of the matrices involved in the increase by when a new obevaluation of servation vector comes in. Hence, the complexity of solving for increases with time. This is obviously undesirable. It turns were time-limited, out that if the autocorrelation function we could derive a recursive algorithm of fixed complexity to . Unfortunately, is not time-limcalculate ited as shown in (7). In order to obtain an algorithm with fixed
.. .
..
..
.
.
.. .
(28)
(29)
3The conditional density function for the special case of BPSK modulation with BPSK spreading cannot be expressed in the form of (24). For simplicity, we exclude this special case here.
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and ..
(30)
.
(31) and . where With these definitions, it is easy to show that (32) (33) Definition 2: For
(40) , since We note that for . From (24), we have . With all the definitions stated above, we show in Appendix can be B that the probability density function calculated by applying the following recursive algorithm. Algorithm 1: : Initialization
, define for
(34) (35) (36) (37) and for For define
, all of the above are set to zero. Moreover, for
(38) , (39)
Recursion : See (A) and (B), shown at the bottom of the page. We note that part (A) of the recursion in Algorithm and, hence, is the same for all system 1 is independent of states. On the other hand, (B) of the recursion depends on and, hence, has to be calculated for each system state. There are many ways to interpret the mechanics of Algorithm 1. First, from (24)–(26), we can see that there are two main operations in Algorithm 1: with in (25) 1) linear prediction of based on as the prediction filter; 2) noise whitening by premultiplying the linear prediction by . error
(A)
for for
(B)
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The linear prediction step works on predicting the desired user fading coefficient, while the noise-whitening step suppresses MAI. A more illustrative way to interpret Algorithm 1 is to look at the recursion steps themselves. Interference suppression, based on the linear MMSE suppression technique similar to that proposed in [15], is performed at the step to is the interference-suppressed received calculate . Hence, signal for the th symbol. The other main operation of Algorecursively. We notice that rithm 1 is to update is the desired signal component in . It is not hard to see is the linear prediction of from the from (37) that . Thus, is the linear prediction of observation from the observation , and
(41) instead of is the estimation error in by using estimate at the th symbol. Moreover, it is clear from (38) that 4 for (42) Thus
(43) by using the estimate is the expected error in calculating instead of at the th symbol. From (41) and (43), we see that the term
for , represents the effect on due to the estimation at the th symbol. error by using the estimate for all , only of these terms Since we assume is used for normalization. Thereare nonzero. We note that in Algorithm 1 can be interfore, the update equation of preted as follows: , as ; 1) start by using 0, the mean of by adding the effect on 2) adjust the estimator due to all the errors of previous estimations. Based on Algorithm 1 and the discussions before, it is easy to develop an algorithm to find the -lag decision-feedback by using instead of MAP demodulator estimate in (19). The following algorithm presents an effective , which is usually way to do the job for the case when ) should be kept true in practice since the number of states ( reasonably small. Some steps in Algorithm 1 are repeated here for easy reference.
4This
is an approximation since we use c~(j ) to approximate c(j ).
Algorithm 2: Initialization
5
:
for for for
for 6 : See (a)–(g), shown on the next page. Recursion It is easy to see that the amount of memory storage units required . To demodulate for Algorithm 2 is of order a data symbol, the amount of computations performed by Al, without counting the gorithm 2 is of order computations needed for the calculations of , , and (see Section V for a discussion on this). We point out that soft-outputs [9], i.e., the a posteriori probabilities, are also obtained by Algorithm 2. However, the soft-outputs given by Algorithm 2 are not the optimal ones discussed in [9] since we employ decision feedback, which is a kind of hard decision, to reduce the number of system states. Moreover, the can be taken from the decoder output to inestimate crease its reliability with the expense of increasing . In practice, it may be more desirable to insert pilot symbols in the data stream to assist the decision-feedback process. The pilot symbols can be used in the MAP demodulation process by setting the corresponding a priori probabilities to 0 and 1 in (22).
V. CHANNEL PARAMETER ESTIMATION In addition to delay and phase synchronization, which is assumed to be achieved through the use of pilots, we need to know , , and in order to apply Algorithm 2. From (8) and can Appendix A, we see that the autocorrelation function be calculated based on knowledge of the speed of the desired transmitter, which can be obtained, for example, via a control channel. According to (12), we only need to know the received of the desired user signal in order to calculate . power and are the channel parameters which need to be Hence, estimated from the received signal. , we sample the despread received First, for at time , for signal , to obtain the auxiliary observation vector 5We use capitalized alphabets in the typewriter font, such as Y , to denote memory storages. x ! Y means to fill the memory storage Y with the value x. Y ! Y means to fill the memory storage Y with the value inside the memory storage Y . When multiple assignments are shown on the same line, the assignment order is from the right to the left. For example, Y ! Y ! Y means to fill the memory storage Y with the value inside the memory storage Y first, and then fill the memory storage Y with the value inside the memory storage Y . 6The notation p( ~ j ~ ; ^ ), for example, represents a function of ^ . It does not represent a conditional density given ^ . This is a slight abuse of notation for convenience. Moreover, we emphasize functions that depend on the data symbols by explicitly writing out the arguments. For example, (d ) emphasizes that is a function of d . A and B in (b) are normalization constants to make the p(1)’s density functions.
D
RG D
D
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a) Calculate
b) Calculate for for for for
for c) Decide
d) for e)
for for
,
.. . for
,
for
.
f) for
for
,
.. .
for for
, .
g)
for
.
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. We note that these samples are separated from those corresponding to by integer multiples of a chip duration. It is for easy to show that each of the vectors can be written in the form (44) , for , are independent7 and We note that identically distributed random vectors having the same second if is small enough so that the interferer order statistics as fading coefficients are almost unchanged within the whole sampling duration, i.e., for (45) Moreover, by using properly designed signature sequences, the are much smaller than those magnitudes of the elements of of as it is a well-known fact that sampling off by a chip time at the output of the matched filter will give a small desired signal times smaller when avercomponent [21]. In fact, they are [16]. Therefore, aged over all possible signature sequence we have, for (46) Hence, an obvious estimator for
is (47)
As described in Section VI, we need to calculate , , and in Algorithm 2 when a new symbol comes in. Applying Lemma 1 with a computational to the estimate in (47), we can obtain . Hence, from (34) to (36) and (47), complexity of order we need the same order of computations to calculate , , and for each recursion. To estimate , we note that (48) , which does not depends on where , we have cause of (12). Averaging over all possible
be-
(49)
denotes the trace of a matrix. Using time averages where to approximate the ensemble averages in (49), we obtain the following estimator of :
(50)
j
7Clearly, n ^ ( ) are uncorrelated. By the Gaussian approximation we employed, they are independent.
Fig. 2.
Single user case with
M = 1.
VI. PERFORMANCE In this section, we evaluate the BER performance of the MAP demodulator developed in Section IV via Monte Carlo simulations. We consider the illustrative time-selective fading channel described in Section I, i.e., the Doppler spread rad/s and the symbol duration ms. We assume that the desired user signal and the interferers’ signals undergo independent time-selective fading with the same Doppler spread. The . The data symbols take the BPSK format, i.e., direct-sequence CDMA system employs QPSK spreading and a rectangular chip waveform. The processing gain of the system is 127. For the estimation of , we take . Simulation results show that the BER performance of the system is relatively insensitive to the estimation error of . For simplicity, below. Morewe assume that we have exact knowledge of data symbols to assist over, we insert a pilot symbol every the decision-feedback process. We note that the BER performance of a single user coherent system [20], which means that we know exactly the phase of the received signal for each symbol and perform coherent detection, provides a lower bound on the performance of the MAP demodulator. We use this single-user bound to gauge the peralso formance of the proposed MAP receiver. The case corresponds to using the sequence-matched filter followed by the MAP demodulator. Because a matched filter has no MAI suppression capability, we can use this special case to measure the MAI suppression capability of the proposed receiver with . We also define [20] the signal-to-thermal-noise ratio , where is (SNR) for systems to be the ratio the energy per bit. First, let us focus on the single user case. Figs. 2 and 3 show and , the BER performances against SNR with respectively. In each of the figures, we show the performances of , , systems with different configurations: . Since there is no MAI in this case, sampling and ) per symbol has no effect on the thermal more than once ( are more susnoise. However, since the systems with ceptible to making larger errors in estimating the instantaneous
WONG et al.: DF MAP RECEIVER FOR TIME-SELECTIVE FADING CDMA CHANNELS
Fig. 3. Single-user case with M = 7.
noise covariance matrices, the BER performances for the sysshould be a little bit worse than those for the tems with when the other parameters are the same. systems with This fact is readily observed by comparing Figs. 2 and 3. We also observe the following from the results. 1) The effect of increasing the decision delay from 5–7 is minimal. is a good enough approximation for this Doppler 2) spread and symbol rate. is an important parameter affecting the BER perfor3) for the best performance in mance, and we need the low SNR region. However, the information rate will be reduced. In summary, the MAP demodulator can achieve a BER performance within 0.5 dB of the optimal one with a reasonable complexity. Next, we consider the case with two users, namely, a desired user and an interferer. Figs. 4 and 5 show the BER curves of and . For all the systems in these systems with and . We vary the ratio (SIR) two figures, we set of the received power of the desired user signal to that of the to 10 dB. The interferer signal has a interferer signal from with respect to the desired user signal. For delay of , since the matched filter is used, the system does not have MAI cancellation capability. Hence, we observe from Fig. 4 that the BER curves level off due to the presence of MAI. However, for , the system can cancel MAI. This fact can be observed from Fig. 5, where the BER curves for the low SIR cases stay close to those for the high SIR cases. As mentioned in [15], the MAI cancellation capability of this and the kind of receiving structure depends on the value of relative signal delay between the interferer and the desired user. We investigate the effect of signal delay by looking at a sample dB, configuration, namely, a two-user system with dB, , , and . Fig. 6 shows the BER performances against the signal delay for different values of . We note that the system cannot reject MAI when the interferer and the desired user is chip-synchronous. To show the
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Fig. 4. Two-user case with M = 1,D = 5, L = 50, and an interferer delay of 0:5T .
Fig. 5. Two-user case with M = 7, D = 5, L = 50, and an interferer delay of 0:5T .
effect of , we look at another sample configuration, namely, dB, , , and a 4-user system with . The signal delays of the three interferers are , , and relative to the desired signal, respectively. for different Fig. 7 shows the BER performances against values of SIR.8 It is obvious from the figure that we need larger for stronger interferers to achieve a certain level of BER performance. dB, , Finally, we consider a system with , and to see the effect of increasing the number increases. of interferers, . Fig. 8 shows the BER’s when The delays of the interferers are spread evenly across the in. Comparing the curves for (no MAI canterval , the MAI rejection capability decreases as cellation) and the number of interferers increases (see the discussions in [15] 8When there are multiple interferers, SIR stands for the received power ratio of the desired user to an individual interferer.
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Fig. 6.
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Effect of interferer delay with two users. The parameters are SIR = D = 5, N = 5, and L = 50.
010 dB, SNR = 40 dB,
Fig. 8. Effect of increasing number K of interferers. The parameters are SNR = 40 dB, D = 5, N = 5, and L = 50. The interferer delays are spread evenly across the interval [0; T ].
using a sequence of observations. This receiver also rejects MAI and estimates channel fading coefficients implicitly to give good demodulation decisions. We have introduced some simplifications to the standard MAP demodulator to reduce its complexity to within practical ranges. Moreover, we have performed computer simulations which show that the BER performance of the receiver is good under various channel conditions. Finally, it is worthwhile to mention that the receiver can be easily modified to employ antenna arrays, which can provide spatial diversity to reject MAI. APPENDIX A
Fig. 7. Effect of increasing M with four users. The parameters are SNR = 40 dB, D = 5, N = 5, and L = 50. The interferer delays are 0:25T , 0:5T , and 0:75T .
In this appendix, we outline a simple method to approximate by the autocorrelation function of a finite-order MA of process. First, we note that the power spectral density is given by [1] the discrete-time process if
for more details). The case of nine interferers is a good approximation to show the trend in the limiting case, in which the peris still better than that of formance of the receiver with . With , the BER for the case of the receiver with -dB SIR is almost the same as that for the case of 0-dB SIR when the number of interferers is less than 4. Hence, we see that this receiver works best when there are a few strong interferers [15]. Nevertheless, we can still gain an order of improvement in for the cases shown. BER performance when VII. CONCLUSIONS In this paper, we have proposed a decision-feedback MAP receiver for time-selective fading CDMA channels. The receiver consists of a sequence-matched filter and a MAP demodulator. Output samples (more than one per symbol) from the matched filter are fed into the MAP demodulator. The MAP demodulator exploits the channel memory by delaying the decision and
(51)
if , and is periodic with period . Then, for the following procedure can be employed, which is based on the windowing method to design linear-phase FIR filters [22], . to obtain 1) Find the inverse discrete-time Fourier transform of and denote it as . (assuming is even), 2) Truncate to by is a chosen windowing function of length where , i.e., for or . 3) Obtain
for otherwise.
,
WONG et al.: DF MAP RECEIVER FOR TIME-SELECTIVE FADING CDMA CHANNELS
We note that are the coefficients of the MA process. In Section VI, we employ the rectangular windowing function for simplicity. Other windowing functions [22] can be used to obtain a smoother approximation.
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By Lemma 1, we have (60)
APPENDIX B can be calcuIn this appendix, we show that lated by applying Algorithm 1. We apply the following matrix inversion lemmas [19] repetitively. Lemma 1: Assuming all the required invertibilities
(61) Applying these results and Lemma 2 to (58), we get (62), shown at the bottom of the page. from (37), we have By partitioning
(52) Lemma 2: Assuming all the required invertibilities, see (53), shown at the bottom of the page. From (27) to (40), we have (54) (55) (63) (56)
(57)
and so on, we Continuing to partition shown in Algorithm 1. obtain the desired recursion for , by partitioning from Similarly, for (38), we have
and Hence, it suffices to derive recursive expressions for for . To do this, we start by finding using the inversion a recursive expression for the matrix from (31), we have lemmas. By partitioning
(64)
(58) where (59)
Continuing to partition so on, we obtain the desired recursion for Algorithm 1.
and shown in
(53)
(62)
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REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [2] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [3] Mobile Station–Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System, TIA/EIA/IS-95 Interim Standard, July 1993. [4] K. Abend and B. D. Fritchman, “Statistical detection for communication channels with intersymbol interference,” Proc. IEEE, vol. 58, pp. 779–785, May 1970. [5] G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363–378, May 1972. [6] G. D. Forney, “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268–278, Mar. 1973. [7] R. A. Iltis, “A Bayesian maximum-likelihood sequence estimation algorithm for a priori unknown channels and symbol timing,” IEEE J. Select. Areas Commun., vol. 10, pp. 579–588, Apr. 1992. [8] G. Lee, S. B. Gelfand, and M. P. Fitz, “Bayesian decision feedback techniques for deconvolution,” IEEE J. Select. Areas Commun., vol. 13, pp. 155–166, Jan. 1995. [9] Y. Li, B. Vucetic, and Y. Sato, “Optimum soft-output detection for channels with intersymbol interference,” IEEE Trans. Inform. Theory, vol. 41, pp. 704–713, May 1995. [10] D. W. Matolak and S. G. Wilson, “Detection for a statistically known, time-varying dispersive channel,” IEEE Trans. Commun., vol. 44, pp. 1673–1682, Dec. 1996. [11] J. P. Seymour and M. P. Fitz, “Near-optimal symbol-by-symbol detection schemes for flat Rayleigh fading,” IEEE Trans. Commun., vol. 43, pp. 1525–1533, Feb./Mar./Apr. 1995. [12] J. S. Lehnert and M. B. Pursley, “Error probabilities for binary directsequence spread-spectrum communications with random signature sequences,” IEEE Trans. Commun., vol. COM-35, pp. 87–98, Jan. 1987. [13] S. Vembu and A. J. Viterbi, “Two different philosophies in CDMA—A comparison,” in Proc. IEEE VTC, Atlanta, GA, Apr. 28–May 1, 1996, pp. 869–873. [14] S. Verdu, Multiuser Detection: Cambridge University Press, 1998. [15] T. F. Wong, T. M. Lok, and J. S. Lehnert, “Asynchronous multiple-access interference suppression and chip waveform selection with aperiodic random sequences,” IEEE Trans. Commun., vol. 47, pp. 103–114, Jan. 1999. [16] T. F. Wong, T. M. Lok, J. S. Lehnert, and M. D. Zoltowski, “A linear receiver for direct-sequence spread-spectrum multiple-access systems with antenna arrays and blind adaptation,” IEEE Trans. Inform. Theory, vol. 44, pp. 659–676, Mar. 1998. [17] T. M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA communication systems with random polyphase signature sequences,” IEEE Trans. Inform. Theory, vol. 42, pp. 129–136, Jan. 1996. [18] T. M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA communication systems with general linear modulation and error control coding,” IEEE Trans. Inform. Theory, vol. 44, pp. 870–881, Mar. 1998. [19] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis. Reading, MA: Addison-Wesley, 1991. [20] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [21] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [22] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
Tan F. Wong (S’96–M’97) received the B.Sc. degree (first class honors) in electronic engineering from the Chinese University of Hong Kong in 1991, and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1992 and 1997, respectively. He was a Research Engineer working on the highspeed wireless networks project in the Department of Electronics at Macquarie University, Sydney, Australia. He also served as a Postdoctoral Research Associate in the School of Electrical and Computer Engineering at Purdue University. He is currently an Assistant Professor of Electrical and Computer Engineering at the University of Florida. His research interests include spread-spectrum communication systems, multiuser communications, and wireless cellular networks.
Qian Zhang was born on July 16, 1964, in Beijing, China. He received the B.S. degree from the Tsinghua University, Beijing, China, in 1986, and the M.S. degree from Peking University, Peking, China, in 1989. He received the M.S.E.E. degree from the University of Central Florida, Orlando, in 1991, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1998. From 1992 to 1994, he was a Research Engineer and a Project Engineer working at PTR Technology, Inc., Palm Bay, FL, and at Ampro Corporation, Melbourne, FL, respectively. His work included hardware design and software programming for a radio transmitter geolocation system and for a big screen projection television system. From 1994 to May 1998, he was a Ph.D. student and a Research Assistant in the School of Electrical and Computer Engineering at Purdue University. In May 1998, he joined Globespan Semiconductor, Inc., Red Bank, NJ, to develop xDSL modem chipsets for high-speed digital communications over telephone lines. His current research interests and work responsibilities include CDMA systems, adaptive coding, diversity techniques, wireless cellular networks, discrete multi-tone modulation, equalization, xDSL modems, and the development of cable modems.
James S. Lehnert (S’83–M’84–SM’95–F’00) received the B.S. (highest honors), M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign in 1978, 1981, and 1984, respectively. From 1978 to 1984, he was a Research Assistant at the Coordinated Science Laboratory, University of Illinois, Urbana, IL. Dr. Lehnert was a University of Illinois Fellow from 1978 to 1979 and an IBM Predoctoral Fellow from 1982 to 1984. He has held summer positions at Motorola Communications, Schaumburg, IL, in the Data Systems Research Laboratory, and Harris Corporation, Melbourne, FL, in the Advanced Technology Department. He is currently a Professor of Electrical and Computer Engineering at Purdue University, West Lafayette, IN. His current research work is in communication and information theory with emphasis on spread-spectrum communications. Dr. Lehnert has served as the Editor for Spread Spectrum for the IEEE TRANSACTIONS ON COMMUNICATIONS and as Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS.
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Decision-Feedback MAP Receiver for Time-Selective Fading CDMA Channels Tan F. Wong, Member, IEEE, Qian Zhang, and James S. Lehnert, Fellow, IEEE
Abstract—A decision-feedback maximum a posteriori (MAP) receiver is proposed for code-division multiple-access channels with time-selective fading. The receiver consists of a sequence-matched filter and a MAP demodulator. Output samples (more than one per symbol) from the matched filter are fed into the MAP demodulator. The MAP demodulator exploits the channel memory by delaying the decision and using a sequence of observations. This receiver also rejects multiple-access interference and estimates channel fading coefficients implicitly to give good demodulation decisions. Moreover, computer simulations are performed to evaluate the bit-error rate performance of the receiver under various channel conditions. Index Terms—Code-division multiaccess, fading channels, feedback communication, linear prediction, MAP estimation.
I. INTRODUCTION
T
IME-SELECTIVE fading due to Doppler spread is a major difficulty in wireless mobile communication systems [1]. Because code-division multiple-access (CDMA) systems tend to have larger symbol durations than time-division multiple-access systems, the effect of time-selective fading can be more severe in CDMA systems. For example, consider an IS-95 CDMA system [2], [3] operating in the personal communication systems (PCS) band at a carrier frequency around 2 GHz. For a mobile user moving at 60 mi/h, the Doppler spread is about 200 Hz. According to Jakes model [1] (see Section II), this corresponds to a coherence time of 2 ms. For a symbol rate at 9600 symbols/s, fading coefficients separated by 20 symbols are almost uncorrelated. For this example, it is no longer reasonable to approximate the channel as time invariant. Hence, the detection and interference cancellation schemes based on the time-invariant channel assumption may not be applicable. Demodulation schemes based on the methods of maximumlikelihood sequence estimation (MLSE) and maximum a posPaper approved by U. Mitra, the Editor for Spread Spectrum/Equalization of the IEEE Communications Society. Manuscript received April 8, 1998; revised June 15, 1999 and September 22, 1999. This work was supported by the Defense Advanced Research Projects Agency under Contract DAAB07-97-CD016. This paper was presented in part at the IEEE Military Communications Conference (MILCOM’98), Boston, MA, October 1998. T. F. Wong was with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA. He is now with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130 USA (e-mail: [email protected]). Q. Zhang was with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA. He is now with Globespan Semiconductor, Inc., Red Bank, NJ 07701 USA (e-mail: [email protected]). J. S. Lehnert is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)04019-8.
teriori (MAP) estimation [4]–[10] have been proposed for narrowband channels with intersymbol interference. It is shown in [9] that MAP algorithms outperform MLSE algorithms because MAP algorithms can give soft-outputs [9], which are useful for outer decoders. The applicability of the MAP technique for narrowband time-selective fading channels has also been demonstrated in [11]. By performing the demodulation process based on a sequence of observations, these techniques try to make use of the channel memory in order to lower the error probability. Although extending the MAP technique to CDMA systems seems natural, the presence of multiple-access interference (MAI) in CDMA systems presents two major difficulties, which hamper the application of the MAP demodulation technique. First, the MAI has a complex distribution function [12], which makes the development of the MAP demodulator difficult. Second, in near–far situations, the received power of the MAI can be stronger than that of the desired signal. In this case, the MAP method will not give good performance unless some kind of MAI cancellation is performed. In this paper, we develop an efficient way to apply the MAP technique to direct-sequence CDMA systems for time-selective flat fading channels. We assume that users in the system communicate asynchronously. For synchronous CDMA systems, the situation reduces to the narrowband case by employing orthogonal sequences. Hence, our focus is on asynchronous CDMA systems. Details of the system configuration and the fading model are given in Section II. We consider the use of long spreading sequences [13], [14] in this paper. As a result, most MAI rejection techniques developed for short spreading sequences are not applicable [15], [16]. Assuming no other system resources, such as receiving antenna arrays, are available, we employ the simple decentralized receiver considered in [15] to achieve MAI cancellation as a practical alternative. The receiver contains a sequence-matched filter whose impulse response is matched to the signature waveform of the desired user. Output samples from the matched filter are fed into a delayed-decision forward-recursive MAP demodulator similar to the ones described in [4], [9], and [11]. Although the output of the matched filter does not provide a sufficient statistic for the detection of the transmitted symbols, processing that is limited to observing the matched filter output greatly simplifies the complexity of the receiver and provides a certain degree of immunity to MAI when long sequences are used [15]. Based on the results in [17] and [18], we note that the MAI components in these output samples are asymptotically Gaussian distributed when the number of chips approaches infinity. Since is designed to be per symbol large in most practical CDMA systems, it is reasonable for us to assume that the matched filter output samples are Gaussian
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random variables. This Gaussian approximation simplifies the development of the MAP receiver. The receiving structure and the MAP demodulator are developed in Section III and IV, respectively. The complexity of the conventional MAP demodulator in [4] grows exponentially with the number of observations with our channel model. To reduce complexity, we employ the decision feedback technique, and approximate the autocorrelation function of the fading process by that of a finite-order moving average process. Based on these two complexity reduction techniques, a recursive algorithm whose complexity depends exponentially on the decision delay is developed in Section IV. The resulting algorithm performs two essential operations implicitly, namely, MAI rejection by noise-whitening [15] and linear prediction of the fading coefficients of the desired user signal. The recursive algorithm requires an estimation of the instantaneous covariance matrices of the interference components in the matched filter output samples. In Section V, we describe a method based on the delayed sampling technique in [16] to accomplish this estimation. In Section VI, we conduct Monte Carlo simulations to evaluate the bit-error rate (BER) performance of this decision-feedback MAP receiver with the time-selective fading channel model described in Section II for both single-user and multiple-user systems. Finally, we draw conclusions from the development of this receiver in Section VII. II. SYSTEM MODEL In this section, we describe the DS-CDMA system and the channel model. simultaneous users in the We assume that there are , generates a sequence of system. The th user, for . We assume that the data symbols are indata symbols dependent random variables taking values from a finite alphabet such that . We note that both binary phase-shift ) and quadrature phase-shift keying (BPSK) ( ) satisfy this assumpkeying (QPSK) ( tion. Generalizations to other signaling schemes, such as -ary phase-shift keying (MPSK) and quadrature amplitude modulation, are straightforward. Moreover, the data symbols are not assumed to be identically distributed to allow the possible use of pilot-assisted modulation schemes. The th user is provided a randomly generated signature segiven by quence (1) are independent random variables taking The elements or with values from either is used to generate the equal probabilities. The sequence spectrally spread signal given by (2) is the chip duration and is the common chip where is time-limwaveform for all signals. The chip waveform
ited to ,1 and is normalized so that . , where is the The symbol duration is given by number of chips per symbol. Moreover, we assume that the signature sequence of any user is independent of all other signature sequences and all the data symbol sequences. , can The transmitted signal for the th user, for be expressed as (3) is the transmitted power for the th signal, is the where is the delay that models the asyncarrier frequency, and chronous system. We consider a decentralized interference suppression approach in detecting the transmitted data symbol. Without loss of generality, we consider the signal from the zeroth user as the communicating signal and the signals from all other users as interfering signals throughout the paper. We assume time-selective fading and additive white Gaussian . noise (AWGN) with two-sided power spectral density The signal received at the intended receiver in complex baseband notation is given by (4) and represent the desired signal and the interferwhere ence components, respectively. The desired signal contribution is given by (5) is the sum of the AWGN component The interference and the MAI contribution , which is given by
(6) models the phase angle of the th user’s signal. In (5) where models the time-varying fading coefficient of and (6), the th user’s signal. Suppose the th transmitter is moving with in the direction of arrival of the th signal to relative speed the zeroth receiver. By properly scaling the receiver power , as a complex, zero-mean Gaussian we can model [1] random process with autocorrelation function (7) is the zeroth-order Bessel function of the In the above, is the maximum Doppler frequency of the first kind and th signal given by (8) 1The restriction to a time-limited chip waveform is just for simplicity of notation. All the results in this paper hold as long as the chip waveform satisfies the Nyquist criterion [15], which guarantees the absence of interchip interference.
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where is the speed of light. Hence, a time-varying flat fading , the th model for the channel is adopted. Moreover, symbol waveform of the th interferer, is given by (9)
Fig. 1. Receiving structure.
III. RECEIVING STRUCTURE We assume that we have achieved synchronization with the desired user signal, for example, through the use of pilots. and . In this and all the Hence, we may assume following sections, we assume that we have exact knowledge . Unless otherwise stated, all of the desired user sequence, . probabilistic arguments are ones that condition on To obtain the observations (see Section I for rationale) corresponding to the th data symbol, we despread the received , signal by passing it through the linear filter which is matched to the th signature waveform, as shown in at Fig. 1. Then, we sample the despread received signal for , times . We stack the samples together to obtain a where and denote this vector by . This vector of length sample vector is then fed into the MAP demodulator in Fig. 1. As shown in Section I, for the targeted practical systems, the fading coefficients are basically unchanged for about a symbol duration. Moreover, we also assume that the processing gain is large so that the approximations invoked in [15] are also valid here, i.e., we can neglect the effect due to the boundary chips.2 Following arguments similar to those used in [15], we can express as (10) where is the component due to the desired signal and is the component due to MAI and AWGN. The desired signal compois given by nent (11) where
(13) (14) and (15) is the sum of two vectors: The interference component due to AWGN and due to MAI, which is given by (16)
for . In (11) and (16), , for , can be expressed in forms We note that . Moreover, we have included the data symsimilar to that of of the interferers into the expressions of since we bols will not attempt to demodulate these symbols. It is easy to show , for , are zero-mean random vectors that or such that, for (17) From the conditional Gaussian approximation results in [17] and [18], when is large, it is reasonable to assume that are jointly Gaussian random vectors given all the interferers’ fading coefficients. Hence, from now on, we make the approximation that ’s are independent jointly Gaussian random vectors given all the interferers’ fading coefficients. IV. MAP DEMODULATOR
.. .
First, we rewrite the observations form. For
, and
in matrix
.. .
.. . (12)
.. 2Throughout this paper, all the equalities involving this approximation should be interpreted as limiting arguments as tends to infinity.
N
.
.. .
.. .
(18)
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complexity, we approximate the sampled autocorrelation function by the autocorrelation function of a dis, i.e., crete-time moving average (MA) process of order for . Appendix A shows a way to obtain from . To start the development of the recursive algorithm, we note that
. Conditioning on
.. .
.. .
and , is a zero-mean Gaussian . random vector of length Our objective is to develop a -lag, decision-feedback MAP . More predemodulator based on the observation vector th data symbol of the zeroth cisely, we demodulate the user by maximizing its a posteriori probability (19) The interferers’ fading coefficients required in (19) can be obtained via explicit estimation using the methods described in is provided through Section V. The data symbol vector decision feedback. In the following development, we assume is given, i.e., the decision feedback gives perfect estimates. We define the system state at the th symbol by
(23) recurHence, it suffices to calculate are jointly Gaussian given sively. We know that and and . Hence, is conditionally Gaussian [19] given 3 , , and and
(24) (20) where Following the standard development in [4] and [5], we can write: (25) and
(26)
(21)
Hence,
We proceed to make several definitions that are essential to our development. Then, we state a recursive algorithm to calcuand give physical interpretations to late the algorithm. , define Definition 1: For for (27)
(22) Therefore, we have to calculate the values of for all the possible states in order to obtain the estimate . We note that the dimensions of the matrices involved in the increase by when a new obevaluation of servation vector comes in. Hence, the complexity of solving for increases with time. This is obviously undesirable. It turns were time-limited, out that if the autocorrelation function we could derive a recursive algorithm of fixed complexity to . Unfortunately, is not time-limcalculate ited as shown in (7). In order to obtain an algorithm with fixed
.. .
..
..
.
.
.. .
(28)
(29)
3The conditional density function for the special case of BPSK modulation with BPSK spreading cannot be expressed in the form of (24). For simplicity, we exclude this special case here.
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and ..
(30)
.
(31) and . where With these definitions, it is easy to show that (32) (33) Definition 2: For
(40) , since We note that for . From (24), we have . With all the definitions stated above, we show in Appendix can be B that the probability density function calculated by applying the following recursive algorithm. Algorithm 1: : Initialization
, define for
(34) (35) (36) (37) and for For define
, all of the above are set to zero. Moreover, for
(38) , (39)
Recursion : See (A) and (B), shown at the bottom of the page. We note that part (A) of the recursion in Algorithm and, hence, is the same for all system 1 is independent of states. On the other hand, (B) of the recursion depends on and, hence, has to be calculated for each system state. There are many ways to interpret the mechanics of Algorithm 1. First, from (24)–(26), we can see that there are two main operations in Algorithm 1: with in (25) 1) linear prediction of based on as the prediction filter; 2) noise whitening by premultiplying the linear prediction by . error
(A)
for for
(B)
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The linear prediction step works on predicting the desired user fading coefficient, while the noise-whitening step suppresses MAI. A more illustrative way to interpret Algorithm 1 is to look at the recursion steps themselves. Interference suppression, based on the linear MMSE suppression technique similar to that proposed in [15], is performed at the step to is the interference-suppressed received calculate . Hence, signal for the th symbol. The other main operation of Algorecursively. We notice that rithm 1 is to update is the desired signal component in . It is not hard to see is the linear prediction of from the from (37) that . Thus, is the linear prediction of observation from the observation , and
(41) instead of is the estimation error in by using estimate at the th symbol. Moreover, it is clear from (38) that 4 for (42) Thus
(43) by using the estimate is the expected error in calculating instead of at the th symbol. From (41) and (43), we see that the term
for , represents the effect on due to the estimation at the th symbol. error by using the estimate for all , only of these terms Since we assume is used for normalization. Thereare nonzero. We note that in Algorithm 1 can be interfore, the update equation of preted as follows: , as ; 1) start by using 0, the mean of by adding the effect on 2) adjust the estimator due to all the errors of previous estimations. Based on Algorithm 1 and the discussions before, it is easy to develop an algorithm to find the -lag decision-feedback by using instead of MAP demodulator estimate in (19). The following algorithm presents an effective , which is usually way to do the job for the case when ) should be kept true in practice since the number of states ( reasonably small. Some steps in Algorithm 1 are repeated here for easy reference.
4This
is an approximation since we use c~(j ) to approximate c(j ).
Algorithm 2: Initialization
5
:
for for for
for 6 : See (a)–(g), shown on the next page. Recursion It is easy to see that the amount of memory storage units required . To demodulate for Algorithm 2 is of order a data symbol, the amount of computations performed by Al, without counting the gorithm 2 is of order computations needed for the calculations of , , and (see Section V for a discussion on this). We point out that soft-outputs [9], i.e., the a posteriori probabilities, are also obtained by Algorithm 2. However, the soft-outputs given by Algorithm 2 are not the optimal ones discussed in [9] since we employ decision feedback, which is a kind of hard decision, to reduce the number of system states. Moreover, the can be taken from the decoder output to inestimate crease its reliability with the expense of increasing . In practice, it may be more desirable to insert pilot symbols in the data stream to assist the decision-feedback process. The pilot symbols can be used in the MAP demodulation process by setting the corresponding a priori probabilities to 0 and 1 in (22).
V. CHANNEL PARAMETER ESTIMATION In addition to delay and phase synchronization, which is assumed to be achieved through the use of pilots, we need to know , , and in order to apply Algorithm 2. From (8) and can Appendix A, we see that the autocorrelation function be calculated based on knowledge of the speed of the desired transmitter, which can be obtained, for example, via a control channel. According to (12), we only need to know the received of the desired user signal in order to calculate . power and are the channel parameters which need to be Hence, estimated from the received signal. , we sample the despread received First, for at time , for signal , to obtain the auxiliary observation vector 5We use capitalized alphabets in the typewriter font, such as Y , to denote memory storages. x ! Y means to fill the memory storage Y with the value x. Y ! Y means to fill the memory storage Y with the value inside the memory storage Y . When multiple assignments are shown on the same line, the assignment order is from the right to the left. For example, Y ! Y ! Y means to fill the memory storage Y with the value inside the memory storage Y first, and then fill the memory storage Y with the value inside the memory storage Y . 6The notation p( ~ j ~ ; ^ ), for example, represents a function of ^ . It does not represent a conditional density given ^ . This is a slight abuse of notation for convenience. Moreover, we emphasize functions that depend on the data symbols by explicitly writing out the arguments. For example, (d ) emphasizes that is a function of d . A and B in (b) are normalization constants to make the p(1)’s density functions.
D
RG D
D
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a) Calculate
b) Calculate for for for for
for c) Decide
d) for e)
for for
,
.. . for
,
for
.
f) for
for
,
.. .
for for
, .
g)
for
.
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. We note that these samples are separated from those corresponding to by integer multiples of a chip duration. It is for easy to show that each of the vectors can be written in the form (44) , for , are independent7 and We note that identically distributed random vectors having the same second if is small enough so that the interferer order statistics as fading coefficients are almost unchanged within the whole sampling duration, i.e., for (45) Moreover, by using properly designed signature sequences, the are much smaller than those magnitudes of the elements of of as it is a well-known fact that sampling off by a chip time at the output of the matched filter will give a small desired signal times smaller when avercomponent [21]. In fact, they are [16]. Therefore, aged over all possible signature sequence we have, for (46) Hence, an obvious estimator for
is (47)
As described in Section VI, we need to calculate , , and in Algorithm 2 when a new symbol comes in. Applying Lemma 1 with a computational to the estimate in (47), we can obtain . Hence, from (34) to (36) and (47), complexity of order we need the same order of computations to calculate , , and for each recursion. To estimate , we note that (48) , which does not depends on where , we have cause of (12). Averaging over all possible
be-
(49)
denotes the trace of a matrix. Using time averages where to approximate the ensemble averages in (49), we obtain the following estimator of :
(50)
j
7Clearly, n ^ ( ) are uncorrelated. By the Gaussian approximation we employed, they are independent.
Fig. 2.
Single user case with
M = 1.
VI. PERFORMANCE In this section, we evaluate the BER performance of the MAP demodulator developed in Section IV via Monte Carlo simulations. We consider the illustrative time-selective fading channel described in Section I, i.e., the Doppler spread rad/s and the symbol duration ms. We assume that the desired user signal and the interferers’ signals undergo independent time-selective fading with the same Doppler spread. The . The data symbols take the BPSK format, i.e., direct-sequence CDMA system employs QPSK spreading and a rectangular chip waveform. The processing gain of the system is 127. For the estimation of , we take . Simulation results show that the BER performance of the system is relatively insensitive to the estimation error of . For simplicity, below. Morewe assume that we have exact knowledge of data symbols to assist over, we insert a pilot symbol every the decision-feedback process. We note that the BER performance of a single user coherent system [20], which means that we know exactly the phase of the received signal for each symbol and perform coherent detection, provides a lower bound on the performance of the MAP demodulator. We use this single-user bound to gauge the peralso formance of the proposed MAP receiver. The case corresponds to using the sequence-matched filter followed by the MAP demodulator. Because a matched filter has no MAI suppression capability, we can use this special case to measure the MAI suppression capability of the proposed receiver with . We also define [20] the signal-to-thermal-noise ratio , where is (SNR) for systems to be the ratio the energy per bit. First, let us focus on the single user case. Figs. 2 and 3 show and , the BER performances against SNR with respectively. In each of the figures, we show the performances of , , systems with different configurations: . Since there is no MAI in this case, sampling and ) per symbol has no effect on the thermal more than once ( are more susnoise. However, since the systems with ceptible to making larger errors in estimating the instantaneous
WONG et al.: DF MAP RECEIVER FOR TIME-SELECTIVE FADING CDMA CHANNELS
Fig. 3. Single-user case with M = 7.
noise covariance matrices, the BER performances for the sysshould be a little bit worse than those for the tems with when the other parameters are the same. systems with This fact is readily observed by comparing Figs. 2 and 3. We also observe the following from the results. 1) The effect of increasing the decision delay from 5–7 is minimal. is a good enough approximation for this Doppler 2) spread and symbol rate. is an important parameter affecting the BER perfor3) for the best performance in mance, and we need the low SNR region. However, the information rate will be reduced. In summary, the MAP demodulator can achieve a BER performance within 0.5 dB of the optimal one with a reasonable complexity. Next, we consider the case with two users, namely, a desired user and an interferer. Figs. 4 and 5 show the BER curves of and . For all the systems in these systems with and . We vary the ratio (SIR) two figures, we set of the received power of the desired user signal to that of the to 10 dB. The interferer signal has a interferer signal from with respect to the desired user signal. For delay of , since the matched filter is used, the system does not have MAI cancellation capability. Hence, we observe from Fig. 4 that the BER curves level off due to the presence of MAI. However, for , the system can cancel MAI. This fact can be observed from Fig. 5, where the BER curves for the low SIR cases stay close to those for the high SIR cases. As mentioned in [15], the MAI cancellation capability of this and the kind of receiving structure depends on the value of relative signal delay between the interferer and the desired user. We investigate the effect of signal delay by looking at a sample dB, configuration, namely, a two-user system with dB, , , and . Fig. 6 shows the BER performances against the signal delay for different values of . We note that the system cannot reject MAI when the interferer and the desired user is chip-synchronous. To show the
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Fig. 4. Two-user case with M = 1,D = 5, L = 50, and an interferer delay of 0:5T .
Fig. 5. Two-user case with M = 7, D = 5, L = 50, and an interferer delay of 0:5T .
effect of , we look at another sample configuration, namely, dB, , , and a 4-user system with . The signal delays of the three interferers are , , and relative to the desired signal, respectively. for different Fig. 7 shows the BER performances against values of SIR.8 It is obvious from the figure that we need larger for stronger interferers to achieve a certain level of BER performance. dB, , Finally, we consider a system with , and to see the effect of increasing the number increases. of interferers, . Fig. 8 shows the BER’s when The delays of the interferers are spread evenly across the in. Comparing the curves for (no MAI canterval , the MAI rejection capability decreases as cellation) and the number of interferers increases (see the discussions in [15] 8When there are multiple interferers, SIR stands for the received power ratio of the desired user to an individual interferer.
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Fig. 6.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 5, MAY 2000
Effect of interferer delay with two users. The parameters are SIR = D = 5, N = 5, and L = 50.
010 dB, SNR = 40 dB,
Fig. 8. Effect of increasing number K of interferers. The parameters are SNR = 40 dB, D = 5, N = 5, and L = 50. The interferer delays are spread evenly across the interval [0; T ].
using a sequence of observations. This receiver also rejects MAI and estimates channel fading coefficients implicitly to give good demodulation decisions. We have introduced some simplifications to the standard MAP demodulator to reduce its complexity to within practical ranges. Moreover, we have performed computer simulations which show that the BER performance of the receiver is good under various channel conditions. Finally, it is worthwhile to mention that the receiver can be easily modified to employ antenna arrays, which can provide spatial diversity to reject MAI. APPENDIX A
Fig. 7. Effect of increasing M with four users. The parameters are SNR = 40 dB, D = 5, N = 5, and L = 50. The interferer delays are 0:25T , 0:5T , and 0:75T .
In this appendix, we outline a simple method to approximate by the autocorrelation function of a finite-order MA of process. First, we note that the power spectral density is given by [1] the discrete-time process if
for more details). The case of nine interferers is a good approximation to show the trend in the limiting case, in which the peris still better than that of formance of the receiver with . With , the BER for the case of the receiver with -dB SIR is almost the same as that for the case of 0-dB SIR when the number of interferers is less than 4. Hence, we see that this receiver works best when there are a few strong interferers [15]. Nevertheless, we can still gain an order of improvement in for the cases shown. BER performance when VII. CONCLUSIONS In this paper, we have proposed a decision-feedback MAP receiver for time-selective fading CDMA channels. The receiver consists of a sequence-matched filter and a MAP demodulator. Output samples (more than one per symbol) from the matched filter are fed into the MAP demodulator. The MAP demodulator exploits the channel memory by delaying the decision and
(51)
if , and is periodic with period . Then, for the following procedure can be employed, which is based on the windowing method to design linear-phase FIR filters [22], . to obtain 1) Find the inverse discrete-time Fourier transform of and denote it as . (assuming is even), 2) Truncate to by is a chosen windowing function of length where , i.e., for or . 3) Obtain
for otherwise.
,
WONG et al.: DF MAP RECEIVER FOR TIME-SELECTIVE FADING CDMA CHANNELS
We note that are the coefficients of the MA process. In Section VI, we employ the rectangular windowing function for simplicity. Other windowing functions [22] can be used to obtain a smoother approximation.
839
By Lemma 1, we have (60)
APPENDIX B can be calcuIn this appendix, we show that lated by applying Algorithm 1. We apply the following matrix inversion lemmas [19] repetitively. Lemma 1: Assuming all the required invertibilities
(61) Applying these results and Lemma 2 to (58), we get (62), shown at the bottom of the page. from (37), we have By partitioning
(52) Lemma 2: Assuming all the required invertibilities, see (53), shown at the bottom of the page. From (27) to (40), we have (54) (55) (63) (56)
(57)
and so on, we Continuing to partition shown in Algorithm 1. obtain the desired recursion for , by partitioning from Similarly, for (38), we have
and Hence, it suffices to derive recursive expressions for for . To do this, we start by finding using the inversion a recursive expression for the matrix from (31), we have lemmas. By partitioning
(64)
(58) where (59)
Continuing to partition so on, we obtain the desired recursion for Algorithm 1.
and shown in
(53)
(62)
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REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [2] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [3] Mobile Station–Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System, TIA/EIA/IS-95 Interim Standard, July 1993. [4] K. Abend and B. D. Fritchman, “Statistical detection for communication channels with intersymbol interference,” Proc. IEEE, vol. 58, pp. 779–785, May 1970. [5] G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363–378, May 1972. [6] G. D. Forney, “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268–278, Mar. 1973. [7] R. A. Iltis, “A Bayesian maximum-likelihood sequence estimation algorithm for a priori unknown channels and symbol timing,” IEEE J. Select. Areas Commun., vol. 10, pp. 579–588, Apr. 1992. [8] G. Lee, S. B. Gelfand, and M. P. Fitz, “Bayesian decision feedback techniques for deconvolution,” IEEE J. Select. Areas Commun., vol. 13, pp. 155–166, Jan. 1995. [9] Y. Li, B. Vucetic, and Y. Sato, “Optimum soft-output detection for channels with intersymbol interference,” IEEE Trans. Inform. Theory, vol. 41, pp. 704–713, May 1995. [10] D. W. Matolak and S. G. Wilson, “Detection for a statistically known, time-varying dispersive channel,” IEEE Trans. Commun., vol. 44, pp. 1673–1682, Dec. 1996. [11] J. P. Seymour and M. P. Fitz, “Near-optimal symbol-by-symbol detection schemes for flat Rayleigh fading,” IEEE Trans. Commun., vol. 43, pp. 1525–1533, Feb./Mar./Apr. 1995. [12] J. S. Lehnert and M. B. Pursley, “Error probabilities for binary directsequence spread-spectrum communications with random signature sequences,” IEEE Trans. Commun., vol. COM-35, pp. 87–98, Jan. 1987. [13] S. Vembu and A. J. Viterbi, “Two different philosophies in CDMA—A comparison,” in Proc. IEEE VTC, Atlanta, GA, Apr. 28–May 1, 1996, pp. 869–873. [14] S. Verdu, Multiuser Detection: Cambridge University Press, 1998. [15] T. F. Wong, T. M. Lok, and J. S. Lehnert, “Asynchronous multiple-access interference suppression and chip waveform selection with aperiodic random sequences,” IEEE Trans. Commun., vol. 47, pp. 103–114, Jan. 1999. [16] T. F. Wong, T. M. Lok, J. S. Lehnert, and M. D. Zoltowski, “A linear receiver for direct-sequence spread-spectrum multiple-access systems with antenna arrays and blind adaptation,” IEEE Trans. Inform. Theory, vol. 44, pp. 659–676, Mar. 1998. [17] T. M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA communication systems with random polyphase signature sequences,” IEEE Trans. Inform. Theory, vol. 42, pp. 129–136, Jan. 1996. [18] T. M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA communication systems with general linear modulation and error control coding,” IEEE Trans. Inform. Theory, vol. 44, pp. 870–881, Mar. 1998. [19] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis. Reading, MA: Addison-Wesley, 1991. [20] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [21] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [22] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
Tan F. Wong (S’96–M’97) received the B.Sc. degree (first class honors) in electronic engineering from the Chinese University of Hong Kong in 1991, and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1992 and 1997, respectively. He was a Research Engineer working on the highspeed wireless networks project in the Department of Electronics at Macquarie University, Sydney, Australia. He also served as a Postdoctoral Research Associate in the School of Electrical and Computer Engineering at Purdue University. He is currently an Assistant Professor of Electrical and Computer Engineering at the University of Florida. His research interests include spread-spectrum communication systems, multiuser communications, and wireless cellular networks.
Qian Zhang was born on July 16, 1964, in Beijing, China. He received the B.S. degree from the Tsinghua University, Beijing, China, in 1986, and the M.S. degree from Peking University, Peking, China, in 1989. He received the M.S.E.E. degree from the University of Central Florida, Orlando, in 1991, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1998. From 1992 to 1994, he was a Research Engineer and a Project Engineer working at PTR Technology, Inc., Palm Bay, FL, and at Ampro Corporation, Melbourne, FL, respectively. His work included hardware design and software programming for a radio transmitter geolocation system and for a big screen projection television system. From 1994 to May 1998, he was a Ph.D. student and a Research Assistant in the School of Electrical and Computer Engineering at Purdue University. In May 1998, he joined Globespan Semiconductor, Inc., Red Bank, NJ, to develop xDSL modem chipsets for high-speed digital communications over telephone lines. His current research interests and work responsibilities include CDMA systems, adaptive coding, diversity techniques, wireless cellular networks, discrete multi-tone modulation, equalization, xDSL modems, and the development of cable modems.
James S. Lehnert (S’83–M’84–SM’95–F’00) received the B.S. (highest honors), M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign in 1978, 1981, and 1984, respectively. From 1978 to 1984, he was a Research Assistant at the Coordinated Science Laboratory, University of Illinois, Urbana, IL. Dr. Lehnert was a University of Illinois Fellow from 1978 to 1979 and an IBM Predoctoral Fellow from 1982 to 1984. He has held summer positions at Motorola Communications, Schaumburg, IL, in the Data Systems Research Laboratory, and Harris Corporation, Melbourne, FL, in the Advanced Technology Department. He is currently a Professor of Electrical and Computer Engineering at Purdue University, West Lafayette, IN. His current research work is in communication and information theory with emphasis on spread-spectrum communications. Dr. Lehnert has served as the Editor for Spread Spectrum for the IEEE TRANSACTIONS ON COMMUNICATIONS and as Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS.
