Adaptive Transmitter Precoding for Time Division Duplex CDMA in Fading Multipath Channels: Strategy and Analysis

The recently developed blind adaptive techniques for multiuser detection in code-division multiple-access (CDMA) systems oﬁer an attractive compromise of performance and complexity. However, the desire to further reduce complexity at the mobile unit has led to the investigation of techniques that move signal processing from the mobile unit to the base station. In this paper we investigate transmitter precoding for downlink time division duplex (TDD) code-division multiple-access (CDMA) communications. In particular, we develop a linear MMSE-based precoding strategy using blind channel estimation for fading multipath channels that allows for simple matched ﬂltering at the mobile unit and is easy to make adaptive. We also present a performance analysis using tools developed for the analysis of conventional (receiver-based) linear blind multiuser detection in unknown channels. We compare the analytical and simulation results to traditional receiver-based blind multiuser detection. It is seen that transmitter precoding oﬁers a reasonable alternative for TDD-mode CDMA when minimizing computational complexity at the mobile unit is a priority.


Introduction
The demand for capacity and performance in multiple-access wireless systems has spurred the development of sophisticated signal processing techniques for signal reception [1,2].
However, the goal of maintaining low cost and complexity, especially at the mobile unit, is as important as ever. As a result, researchers have recently begun investigating signal processing techniques that move computational complexity from the mobile unit to the base station, where it can be managed more efficiently. Generally speaking, these techniques involve some kind of transmitter-based multiuser interference cancellation at the base station (precoding) and simple linear processing, e.g., matched filtering, at the mobile unit.
They are particularly appealing for time division duplex CDMA (TDD-CDMA) [3] since the same carrier is used for both uplink and downlink in different time slots. Hence, the downlink channel can be estimated at the base station using the uplink signals [4].
In [5], the authors considered transmitter precoding for synchronous CDMA over AWGN channels. They also present an extension to multipath channels, but a RAKE receiver is required and the channel is assumed perfectly known. A similar technique, pre-RAKE diversity combining, is investigated in [6]. However, RAKE reception is inherently sensitive to channel mismatch and performance is generally inferior to MMSE or decorrelator based multiuser interference rejection. The authors in [7] present a comparison of the techniques mentioned above using a typical vehicular channel. In [8], the authors consider transmitter precoding for multipath fading channels but, in contrast to the present work, their prefilter is applied to the output of the spread spectrum encoder, rather than applying the filter first, followed by spreading. It was shown that this approach has inferior average performance unless the spreading codes themselves are allowed to be adaptive.
In this paper we develop a linear MMSE-based transmitter precoding strategy for the CDMA downlink in synchronous multipath fading channels. No RAKE receiver is required at the mobile unit, only matched filtering. Since we require channel information to construct optimal precoding filters, we implement blind channel estimation at the base-station, where complexity can be managed more efficiently. We also present a performance analysis using tools developed for the analysis of blind multiuser detection with blind channel estimation [9]. Finally, we develop an adaptive implementation that is able to adjust the precoding matrix as users enter and leave the system.
The remainder of this paper is organized as follows. Section 2 describes the system under consideration. Section 3 develops the transmitter precoding strategy for CDMA over synchronous multipath channels. Section 4 presents a performance analysis. Section 5 discusses an adaptive implementation. Section 6 reports simulation results, and Section 7 concludes.

Uplink Signal Model and Blind Channel Estimation
We consider a K-user discrete-time synchronous multipath CDMA system with no intersymbol interference (ISI). Such a system is realized either by neglecting the ISI when the multipath delay spread is small compared with the symbol interval, or by inserting guard intervals between symbols when the delay spread is large. The path delays are also assumed to be an integral number of chip periods and are known. We first consider the chip-match filtered uplink signal received at the base station which, during the i-th symbol interval, can be written as where L is the number of resolvable paths, b k [i] is the i-th symbol for the k-th user, s l,k and f l,k are, respectively, the delayed versions of the spreading waveform (with zero-padding when a guard interval is inserted) and the complex channel fading gain corresponding to the l-th path of the k-th user; n[i] ∼ N c (0, σ 2 I N ) is a complex white Gaussian noise vector.
Then (1) can be written as where A block diagram of the uplink system appears in Figure 1.
Let the autocorrelation matrix of the received signal r[i] be where (9) is the eigendecomposition of C r . Since the matrix H has full column rank K, the matrix HH H in (8) has rank K. Therefore, in (9) Λ s contains the K largest eigenvalues of C r ; U s contains the corresponding orthonormal eigenvectors; and U n contains the (N −K) orthonormal eigenvectors that correspond to the smallest eigenvalue, σ 2 .
Suppose User 1 is the user of interest. Then since U H n h 1 = U H n S 1 f 1 = 0, we can estimate f 1 at the base station in the following way [10,11,12,13] = arg min = minimum eigenvector of Q.
Note that (12) specifies f 1 up to a scale and phase ambiguity and that, in practice, (12) can be implemented blindly in a batch or sequential adaptive manner. In batch mode, we simply replace the noise subspace parameters in (11) with parameters obtained from the eigendecomposition of the sample autocorrelation matrix of the received signal. In sequential adaptive mode, where we update the channel estimates at each time slot, we may employ a suitable subspace tracking algorithm and use the sequential Kalman filtering technique described in Section 5. Note that a necessary condition for our channel estimate to be unique is that H have rank K, which necessitates that this matrix be tall, i.e., K ≤ N .

Downlink Signal Model, Precoding and Receiver
The (downlink) signal transmitted from the base station during the i-th symbol interval can be written where is the matrix of spreading waveforms and M ∈ C K×K is a complex precoding filter which we will optimize in the following section. Throughout this paper, we assume that the CDMA system is operating in the time division duplex mode, so that the downlink and uplink operate using the same carrier frequency in different time slots. We also assume that the time elapsing between uplink and downlink transmissions is sufficiently small compared to the coherence time of the channel that the channel impulse response is the same for the uplink and downlink. Then from (2) and (3), the received signal at User 1's mobile unit can be written as where S 1 , S 2 , . . . , S K contain shifted versions of their respective signature waveforms as in (2) except that the L shifts are the same for each user's waveform since all spreading codes have been transmitted over User 1's downlink channel. Detection of the downlink information bits is accomplished via matched filtering of the received signal r 1 [i] with User 1's signature waveform, s 1 . Figure 2 contains a block diagram of the signal processing that takes place at the base station when an adaptive implementation is employed. We will see more details in Section 5.

Mobile
Differential Encoder Differential Encoder Figure 1: The uplink of a K-user CDMA system.

Transmitter Precoding for Synchronous Multipath CDMA
We seek to choose the precoding matrix M so as to provide the best downlink performance possible when the mobile units are constrained to the use of a matched filter receiver. We choose the minimum mean-square error criterion, so M is chosen to minimize where we have dropped the time index for clarity. It is easy to see that Then  The following proposition gives the optimal precoding matrix.
Proof: Although cumbersome, a direct proof be constructed via a complex analog to the proof in Appendix A of [5]. We offer the following simple alternative proof by contradiction.
Suppose there exists a choice of M , say M = M 0 , that results in a smaller J than We may evaluate the left-hand side of (19) as Then (19) implies However, the left-hand side of (21) is equal to E{ b − HM 0 b 2 } which can never be less than zero. Hence, we have a contradiction.
There remain amplitude and phase ambiguities inM that are addressed in the following

sections.
Remarks: We should expect the performance of transmitter-based multiuser detection to be somewhat inferior to traditional receiver-based approaches since the transmitter precoding filter must satisfy more requirements than the detector in a receiver-based approach.
More specifically, notice that the objective function in (16) requires that the choice of M maximize performance for all users simultaneously. Choosing M to minimize an objective of the form could provide better downlink performance for User 1 than J of (16), but at the expense of the other users. Hence J is not an acceptable cost function. In this sense, the simultaneity requirement means that transmitter precoding has fewer degrees of freedom for combatting interference than does MMSE or decorrelating receiver-based multiuser detection.
Notice that choosing the optimal precoding matrix by minimizing J places no explicit constraint on average transmit power. In fact, it was found in a related work on MMSE precoding [5] that unconstrained optimization with simple power scaling provides superior performance at high SNR to constrained optimization. As a result, we shall focus on the former. We will also suppress the power scale factor for simplicity.

Performance Analysis
In [9,14,15,16,17], the authors developed analytical tools to investigate the performance of blind and group-blind linear MMSE multiuser detection. In this section, we adapt these tools to the analysis of transmitter precoding with blind channel estimation. In particular, we will derive signal-to-interference-plus-noise (SINR) and BER expressions that take residual multiple-access interference and channel estimation error into account.
In Section 6, we will compare these expressions to simulation results.
Notice that the estimateM given by (22) is not a consistent estimate of M because of the unknown phase and scaling factors. However, there is a diagonal matrix Φ so that where φ k , k = 1, . . . , K are phase factors that depend on how the estimation is implemented.
With this in mind, we state the following result, which is proved in the appendix.
Theorem 1 LetM be given by (22), and let b be i.i.d. QPSK symbols independent ofM .
where the diagonal elements of D are given by and while the off-diagonal elements can be ignored with good accuracy. Here Q † i denotes the Moore-Penrose generalized inverse [18] The SINR at the output of the matched filter for User 1 is given by [9,14] Now suppose that the phase and amplitude factors in Φ have been determined. Write the estimated matrix,M , asM Φ −1 = M + ∆M , where ∆M is the estimation error.
Dropping the time index for clarity, the received signal can then be written as where the notation [M ] :,2:K indicates the matrix composed of columns 2 through K of the matrix M . According to Theorem 1, for large M the third term in (33) is also Gaussian distributed (independent of the other terms) with variance Since M represents an MMSE detector, we can also make the approximate assumption that the multiple-access interference is Gaussian distributed [19]. We can therefore calculate the BER via a single Q-function as Notice that the first term in the denominator of the SINR expression is due to residual multiple-access interference. The second term is the ambient noise, and the third term is due to the channel estimation error.

Adaptive Implementation
In this section we present an adaptive implementation of the transmitter precoding strategy discussed in Section 3 that updates the precoding matrix M at each time slot. This sequential updating allows the implementation to adapt as the channel changes and as users enter and leave the system. A block diagram of the signal processing at the base station appears in Figure 2. Note that we have suppressed the signal processing necessary for detection of the uplink bits. The uplink signal received at the base station is used in a signal subspace tracker, along with the known spreading codes of all users, to construct channel estimates by solving (45). Recall that since we are assuming TDD mode, the uplink channel estimates also serve as downlink channel estimates that can be used to construct M . As previously mentioned, these channel estimates have amplitude and phase ambiguities. Since nearly all cellular CDMA systems employ power control, it is likely that the base station has some knowledge of each users' transmit power. This information, coupled with estimates of the received power, can be used to estimate the channel amplitude for each user. More specifically, let the diagonal matrices A = diag(α 1 , α 2 , . . . , α K ) and . . , √ p K ) contain the unknown channel amplitudes and the known uplink transmit powers, respectively. Also defineH such that H =HAP , so that the columns ofH have unit norm We propose an estimator based on the following fact.

Proposition 2
where U s , Λ s are signal subspace components derived from an eigendecomposition of the autocorrelation matrix, C r , of the received signal, as in (8) and (9).

Proof: SinceH
it is easy to see thatHA 2 P 2H H = U s (Λ s − σ 2 I N )U H s . It can also be verified using the definition of the Moore-Penrose generalized matrix inverse that H and, solving for A −2 , we have and the proposition follows. 2 We may obtain an estimate,Â, of A by replacingH, U s , Λ s and σ 2 of (37) with their respective estimates obtained from subspace tracking and the solution to (45). The subspace tracker we have chosen to use for this adaptive implementation is NAHJ-FST (noise averaged Hermitian-Jacobi fast subspace tracking), which has complexity O(N K) floating operations per user per bit and which performs close to the lower bound for all SVDbased subspace trackers [20]. The application of NAHJ-FST to the current tracking problem is a straightforward modification of [20] and will not be discussed in detail. The channel phase ambiguity can be circumvented by the use of differential encoding and decoding of the data. After channel and amplitude estimation, the amplitude corrected channel information is then used, along with the known spreading codes, to construct the precoding matrixM .
Finally, the downlink information bits are differentially encoded and filtered withM before spreading and transmission. At the mobile unit, matched filtering and differential detection are performed to obtain estimates of the downlink information bits.

Blind Sequential Adaptive Channel Estimation
Here we describe the adaptive channel estimator used in Figure 2. Consider User 1 the user of interest and denote by z[i] the projection of received uplink signal r[i] onto the noise subspace, i.e., Since z[i] lies in the noise subspace, it is orthogonal to any signal in the signal subspace and, in particular, it is orthogonal to S 1 f 1 . Hence f 1 is the solution to the following constrained optimization problem: In order to obtain a sequential algorithm to solve the above optimization problem, we write it in the following (trivial) state space form Then the standard Kalman filter can be applied to the above system as with the initial condition Σ[0] = I L .

Algorithm Summary
We may summarize the adaptive implementation at the base station as follows. 2. Track the channels {f k } K k=1 as follows: 3. Calculate the channel amplitudes via (37) using the channel estimates, the signal subspace parameters, the known spreading codes, and the known transmit powers. (17)  7. Perform matched filtering and differential detection at the mobile units.

Analytical Performance Versus Simulated Performance
Here we compare the BER expression in (36) to simulation results. We will also compare the analytical and simulated performance to conventional receiver-based subspace blind MMSE multiuser detection. The simulated system is a QPSK modulated synchronous multipath CDMA system as described in Section 2, i.e., channel is generated from a complex Gaussian distribution. The channel parameters are fixed for all simulations. Blind channel estimation is performed on the uplink signals using (12), where the exact noise subspace is replaced by an estimated noise subspace obtained from an eigendecomposition of the received signal, as in [9]. The frame length used for estimating the channel (and the precoding filter) is either M = 200 or M = 2000, as noted on the figures. As mentioned previously, there is an amplitude and phase ambiguity inherent in (12). For now they are assumed known. These ambiguities are resolved in the adaptive implementation discussed next. Figure 3 illustrates the best and worst BER performance among the 10 users using (35) for the analytical performance of transmitter precoding and equation (98) of [9] for the analytical performance of receiver-based subspace blind MMSE multiuser detection (denoted rx-mud:min and rx-mud:max in the figure). The SNR is defined as E b /(2σ 2 ). Notice that there is a very good match between the simulated and analytical performance results for transmitter precoding. The previously mentioned performance penalty of transmitter precoding relative to receiver-based multiuser detection is also evident. The number of signal samples used to estimate the channels and, hence, M is 2000. The simulation parameters for Figure 4 are identical to those for Figure 3 except that the number of signal samples used to estimate M has been reduced from 2000 to 200. Although the performance of subspace blind MMSE multiuser detection is still superior, the error floor that appears its performance does not appear in the performance of transmitter precoding. It is clear that an error floor must exist for both techniques since the detector or precoder is estimated from noisy received signals, but this result suggests that the performance of transmitter precoding may degrade more gracefully than that of subspace blind MMSE multiuser detection when the number of signal samples available diminishes. Figure 5 contains plots of the analytical and simulated SINR versus the number of signal samples used to construct the precoder matrix for the best performing user. The analytical values are obtained from the SINR expression in (36). The simulated values are taken from where the expectations are replaced with time averaging and where z k [i] is the decision statistics for user k at time slot i. This figure also contains an indication of the theoretical SINR when perfect channel information is available, i.e., for M = ∞. As was the case with the BER plots, there is good agreement between the simulated and theoretical results.

Performance of Adaptation Implementation
Here we examine the simulated performance of the adaptive implementation of transmitter precoding discussed in Section 5. We consider the same K-user synchronous multipath    CDMA system used in the previous simulation results, except that the modulation used is BPSK instead of QPSK. The channel is kept constant during all simulations and the SNR is fixed at 18dB. Figure 6 contains a plot of the average SINR for the first four users versus the time index. Since we are using differential detection, the decision statistic for detecting bit i of user k is and the SINR is defined here as where the d k [i] represents the i-th differentially coded bit for user k. For these simulations, the expectation is replaced with time averaging. During the first 1000 time slots, there are 7 users in the system. At time slot 1001, 3 users are added to the system and at iteration 2001, 6 users are removed from the system. We see that the adaptive system rapidly adjusts (within 500 time slots) when users leave and enter the system. Figure 7 contains plots of the bit-error-rate versus SNR. The system has been allowed 500 received signal samples to reach steady state before errors are accumulated. For comparison, this figure also contains plots generated with perfect knowledge of the channel. It is tempting to compare this figure to Figures 3 and 4, but it must be noted that the blind adaptive implementation uses BPSK instead of QPSK and differential encoding/decoding rather than coherent detection.

Conclusions
In this paper, we have developed a transmitter precoding strategy for the downlink of a CDMA system in fading multipath channels. The technique presented precodes the downlink bits before spreading and transmission and is optimal in the mean-squared-error sense. We have presented a performance analysis using tools developed for the analysis of (receiver-based) blind multiuser detection and we have developed an adaptive implementation of the precoding strategy that is able to adjust the precoding matrix as users enter and leave the system. Simulation results indicate a very close match between simulated and analytical performance. We also see through simulation that the adaptive implementation is able to quickly and successfully adapt as users enter and leave the system. Future work will address asynchronous multipath channels and cases where the uplink and downlink channels are not identical, but only statistically highly correlated.  channel estimate given by (12). Then there exists a phase factor e jφ , such that and where the notation N c (µ, C,C) indicates a complex Gaussian distribution with mean µ, Hermitian covariance matrix C, and symmetric covariance matrixC.
Proof of Theorem 1: Let ∆M denote the differential of the functionM [14,21]. Then ∆H k = S 1 ∆f k S 2 ∆f k · · · S K ∆f k .
Now each ∆f k is asymptotically circularly Gaussian distributed by Lemma 1. It follows that the same holds for ∆M , and since b is independent ofM , this is also true for ∆M b,