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Iterative Demodulation and Channel Estimation for Asynchronous Joint Multiple Access Reception Christian Schlegel, Fellow, IEEE, and Marcel Jar
Abstract—Iterative demodulation with integrated channel estimation is investigated for multiple access systems as alternative to separate estimation/demodulation, which is the current stateof-the-art method. While basic theoretical system performance is well understood, practical aspects such as those arising from estimating the time-varying channels due to parameter drifts or inherent channel dynamics are not so well explored. The integration of adaptive estimators for these time-varying channels into the iterative receiver is studied, and it is shown that simple correlation-based estimators are sufficient to allow adequate tracking even for time-varying channels in a multiple access environment, and that near-ideal performance of the receiver is achievable. The requirements of the estimators and their performance in the demodulation loops are investigated via system convergence functions. Quantitative analytical results are verified with selected simulation examples. Spread-spectrum access is used to illustrate the principles of iterative demodulation and other potential areas for application are identified and discussed. Index Terms—Interference cancelation, iterative channel estimation, iterative processing, joint detection, multiuser reception.
I. INTRODUCTION
M
ULTIPLE access and joint detection are promising methods to alleviate the pressure on spectral resources and substantially improve the utilization of wireless systems. Joint detection systems have been primarily proposed for direct-sequence spread-spectrum (DSSS) communications, due to the simplicity of separating different users via their individual signature sequences [5], [10], [12], [13], [23]. In this paper we use direct-sequence spread spectrum (DSSS) communications operated in code-division multiple access (CDMA) mode as our main example for the use of iterative demodulation and channel estimation. In DSSS-CDMA, the spreading waveforms have two functions: 1) they suppress interference by the system’s processing gain, and 2) the spreading sequences are used to separate and identify individual users or terminals. One of the apparent disadvantages of DSSS-CDMA
Manuscript received October 27, 2014; revised April 10, 2015 and July 20, 2015; accepted August 03, 2015. Date of publication August 14, 2015; date of current version November 02, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jian-Kang Zhang. This work was presented in part at Milcom 2014, Baltimore, MD, USA, October 2014. C. Schlegel is with the Ultra Maritime Digital Research Center, Dalhousie University, Halifax, NS B3S 1K4, Canada (e-mail:
[email protected]). M. Jar was with HCDC Group LLC, Salt Lake City, UT 84101 USA. He is now with Ultra Maritime Digital Research Center, Dalhousie University, Halifax, NS B3S 1K4, Canada (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2015.2468681
is that the signal bandwidth is used jointly and possibly concurrently by a multitude of transmitters, thus creating a dense interference situation where multiple transmissions interfere with each other. Despite the processing gain of DSSS, one can show that without extra measures the achievable spectral efficiencies of DSSS-CDMA is rather modest, and not competitive with other modulation methods, for example orthogonal frequency multiplexing (OFDM), used in next generation commercial cellular telephony systems. Nonetheless, there may be important reasons to advocate for DSSS-CDMA, which can be operated in a virtually completely uncoordinated manner, such as a mobile (intermittently operated) packet data or sensor network, or in networks with long round-trip delays, such as satellite packet, or underwater acoustic networks. In these situations, the fact that the network does not need to be synchronized on a network-wide basis can be a huge operational advantage. There is a growing body of work that theoretically illustrates that joint detection for DSSS-CDMA can achieve high levels of spectral efficiency, and combined with power layering or spatial coupling can even achieve the capacity of the multiple access channel [17], [19]. However, not much work has been reported in the open literature regarding how to operate such systems in real-world environments where channels are affected by independent phase and frequency shifts, as well as channel distortions. In this paper we examine iterative demodulation using partitioned signaling (PS) [15], which has been shown to be capacityachieving in theory [17], [18], [21]. Specifically, we incorporate channel estimators into the iterative demodulation loop. We show that relatively simple channel estimators, if operated as part of the iteration process, are adequate to provide accurate channel estimates and that such detectors can approach the theoretical limits for a wide operating regime. This is shown to be true even in situations which would not allow for an effective separation of the processes of demodulation and estimation. While a variety of iterative multiple access systems have been studied in the literature [5], [10], [12], [25], collectively known as turbo demodulation, we choose to use PS for the following reasons. PS can be viewed as repetition coding in the iterative loop and therefore is a special case of the more general turbo demodulation method. By avoiding a complex FEC code in the iterative loop, PS has a much reduced complexity and is attractive for implementation. Even quite large PS multiple access demodulators fit onto small-sized FPGA platforms as shown in [4]. Furthermore, in combination with external FEC coding, PS can provably achieve the capacity of the random multiple access channel [18] and therefore represents an optimal signaling and demodulation methodology.
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Fig. 1. Illustration of partitioned DSSS-CDMA for
partitions. Only one partitioned symbol is shown for each user for clarity.
Iterative estimation of transmission channels has also been explored before in the context of iterative decoding in a variety of papers. In [14], for example, a simple iterative correlation estimator is used in combination with differential space-time coding. In [6], [7], [11] a similar concept is used to track a noisy phase in a serially concatenated trellis-coded modulated system. However, most papers in the literature either assume perfectly known channel state information, or use simple channel estimation techniques in conjunction with training sequences or pilot symbols. Some exceptions are [9], [22], which use iterative estimators to find the fading gains in flat channels, similar to [14]. In this paper, while referring to earlier concepts, we consider systems which differ in the sense that we have multiple channels associated with different users. As a consequence the interference these users present to each other has to be modeled by time-varying interchip-interference channels due to the asynchronicity among the users. This important fact leads to a complex, yet accurate, signal processing model which needs to be incorporated in any multi-user detector study aimed at real-world channels. Furthermore, we present a convergence analysis which will allow us to gauge the requirements that the estimators have to fulfill, rather than relying on simple simulations of the system. The paper is organized as follows: Section II briefly reviews partitioned signaling (PS), Section III presents the iterative demodulation methodology and develops the basic convergence equations for the combined receiver. Section IV discusses the estimator design and functionality as part of the loop, Section V presents numerical results of the iterative estimators, and Section VI presents a concluding discussion. II. PARTITIONED SIGNALING We consider a method called partitioned signaling (PS) in conjunction with iterative demodulation as our multiple-access transmission method [15]. PS has evolved from work with integrating error control coding with multiple access interference suppression. The basic concept of PS is that of introducing redundant signals, which can then be used to create a network of dependencies among symbols which are utilized by a message-passing receiver to achieve a performance far superior that of known multiuser detectors. It can be shown that the performance of a PS detector is always better than that of an equivalent MMSE detector [17], and, in conjunction with spatial coupling,
PS can achieve universally the capacity of the multiple access channel [20]. Let the signal space be populated with times the number of original signals. For conceptual convenience, we may think of each signal being decomposed into orthogonal components, components individually instead of the and we consider the original signals. A system model is now given by (1) is the where is the vector or received signal samples, symbol that modulates the -th component of the -th bit of the -th user of a total of users with —due to the PS principle all partitions of a symbol represent the same data bit— is a vector of unit-variance zero-mean Gaussian is the channel noise variance. The length noise samples, and of is samples, where is the number of samples (or chips) per symbol, and is the length of a data frame. The have non-zero samples in the apsignal vectors propriate positions. These vectors are the result of multiplying the modulated symbols by a partitioning sequence and sending the result through an interleaver, as seen in Fig. 1. In this paper is a discrete chip or sample sequence used to modulate symbol partition (see Fig. 1). We note that the are , since its different not in successive order for a given symbol partitions are interleaved over the entire frame prior to transmission. In all comparisons we will normalize performance to energy per information bit and the spectral efficiency to bits per unit bandwidth, or per signal dimension, since the two measures are equivalent [26]. We assume in this paper, however this is neither necessary, nor is it a significant restriction. In [16] the authors show that binary signals can be easily superimposed to form higher modulation alphabets. The transmitted signal of each user is therefore a sequence of rearranged chips (samples), and we will need the following chip-based formulation of the transmit sequence for user
(2) where is the chip index running through the entire transmission frame, and are the individual chips of .
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Fig. 2. The interlocking iteration loops for demodulation and estimation discussed in this paper. The APP detectors are simple LLR aggregators in the case of partitioned signaling (PS), which amounts to repetition coding, for which APP detection results in the addition of extrinsic LLR values.
While PS is most easily discussed for DSSS-CDMA, its application is more general, and the method can be tailored to many other relevant transmission systems, such as multiple-input multiple-output (MIMO) channels [16], and others.
from all users is used to generate the canceled signal at each subsequent iteration, given as the sample vector
III. ITERATIVE DEMODULATION AND CHANNEL TRACKING
A. Ideal Iterative Demodulation Iterative demodulation with built-in estimation consists of two interacting iteration loops as shown in Fig. 2. Basic iterative demodulation (gray boxes) is well understood [17], [19]. For large systems, the residual variance in the loop is well described by the iteration equation
(3) where the effective noise at iteration is a function of that at the previous iteration and the system loading , typically expressed in bits/signaling dimension. The function captures the variance of the estimation error of the transmitted binary symbols and can be computed in semi-closed form1. We in the sequel to simplify (3). will use Asynchronous sampling and other channel effects cause interchip interference, such that a transmitted partition appears at the receiver as , which is the convolution of the partition with a discrete equivalent channel impulse (see Section IV-B). The estimator in the -th loop now needs to form an estimate of this discrete channel response , which is used for coherent demodulation as shown in Fig. 2. This estimator has two inputs, from the cancelation device, and the sequence of soft symbols from the APP detector. The remodulation of signals 1The function is the error variance of a binary soft bit estimate of a bipolar signal 1, 1 embedded in Gaussian noise with variance [19], and is a zero-mean unit variance Gaussian random is analyzed variable, over which the expectation is computed. The function in [2].
(4) where is the estimated channel response vector to , i.e., , the operator denotes convolution, and is the effective noise variance, which depends on the iteration , that is, on the stage of the cancelation process. The effect that non-ideal estimation has on the performance of the iterative demodulator is primarily that the residual iteration variance is increased w.r.t (3). This effect can be decomposed into two parts: a first part which is driven by the incomplete cancelation—called “reconstruction error”—and a second part which is due to the estimation error, given by (5) where the first equality results from . The reconstruction of the received signal is used for cancelation and shows up in the cancelation (4) as (6) We can assume that the estimation error variance . This allows us to develop
is white with (7)
which has variance2 (8) 2We used the identity if , and is a log-likelihood ratio of a binary symbol in Gaussian noise with variance , see [2]. In short, this goes as follows: , is the statistic from which is computed. Now where , which is what we need.
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The first term in (7) results from incomplete estimation of the transmitted symbols, and the second term from inaccuracies in the estimation of the effective discrete transmission channel. The term is the estimation error of the channel, i.e., the variance of . While not explicitly noted in (8), too, will depend on residual noise and interference. With these definitions, we can express the performance cancelation loop as the system of functions (9) (10) where (9) expresses the noise plus interference variance at the -th iteration, which is an extension of (3) that includes the estimation errors in the channel estimators that operate in parallel, and (10) is a function that describes the mean squared error of the estimator if driven by and . Since the mean squared error in the estimate of the transmitted symbols also depends soley on , we will show below that the overall convergence equation only depends on the free parameter . This is address in the next section. The system of equations (9) and (10) describes the statistical convergence behavior of an iterative demodulator with an integrated iterative estimator for the transmission channels, under the assumption that the estimator performance depends only on the residual noise and interference variance in the cancelled datastream . IV. INTEGRATED ESTIMATION A. The General Picture One of the main issues in order to successfully operate multiuser signaling, which is often glossed over in the academic literature, is the fact that there are a multitude of transmission channels which need to be estimated concurrently. Additionally, each channel experiences interference from the signals of other users, compounding the problem where accurate estimation in low signal-to-noise ratios is required. While multiuser detectors have been proposed based on a multitude of processing techniques [23], the effects of inaccurate channel estimation affect all the techniques similarly. In fact, it is easy to show that, from the chain rule of information theory [3], signal cancelation is an optimal process for the decoding of joint signals. In this sense, our cancelation-based receiver is not compromising optimality. Furthermore, all the linear multiuser receiver structures can be cast into the form of cancelation receivers by applying iterative matrix inversion methods [1]. Any estimation errors that occur or remain after estimation will directly translate into enhanced linear interference at the input of our decoders. Making the mild assumption that this interference is adequately modeled by Gaussian noise, the receiver simply sees a signal embedded in noise which is the sum of the original channel noise and the error in the channel estimators, since, as developed in the previous section, these errors directly add to the unresolvable interference. B. Channel Model In the target application discussed in this paper, namely the uncoordinated multiple access system with transmitters,
there is an aggregate of the signals that are seen at the receiver. The resulting composite signal is formally given as (11) where is a random delay of the -th signal, and is the transmitter--receiver carrier mismatchfor signal . That is, (11) models the minimal distortion that arises from asynchronicity. However, other linear channel distortions can be captured with the same model. Each user’s signal is the superposition of data and pilot waveforms3
(12) where is the data waveform for symbol time and partition , and is the pilot component, which repeats in every partition interval of seconds. and are the data and pilot energies per symbol interval, respectively, and are normalized to to maintain consistency with (3). Each partition waveform and the pilot waveform is composed of a sequence of sample pulses, typically those of a spreading sequence. That is, the signal , for example, is generated as the interpolation (13) is some shaping pulse, for example a root-Nyquist where pulse with bandwidth . Technically, any sufficient-statistics sampling can be used, such as Nyquist sampling. However, given that we envision sample-based processing and not necessarily Nyquist-rate processing, we assume that a filter matched to is used at the receiver, even though this is technically not critical. The sequence is a predefined discrete sequence, in our case a random binary phase-shift keying (BPSK) or quaternary phase-shift keying (QPSK) DSSS sequence. The data waveform is defined completely analogously, but we allow it to vary with the symbol time and partition . The -th received signal after receiver matched filtering is then given by
(14) where the second line is simply the condensed sample-based model of the -th received signal, and 3We are using the concept of an embedded pilot [27], but any other pilot and now method can be used without conceptual changes. Also, contain data and pilot, an extension of (2).
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time variations for radio signaling applications. In other applications, such as underwater acoustic communications, the drift of and variations in are of the same order of magnitude [8]. In radio applications the high carrier frequencies will likely require a course frequency synchronization and/or doppler offset compensation to bring the values of into the range where they can be effectively tracked by the filter estimators used in this paper.
Fig. 3. Tap values for a model with brick wall transmission channel, i.e.,
, and an ideal .
is the overall system impulse response, and the sample index . It is important to realize that this model is still relevant even if channel distortions are added, and the overall pulse is given by instead. This is so because we are not exploiting the Nyquist property of in our model. We now assume that our receiver front-end will asynchronously sample the received signal. Information optimality is preserved if the sampling rate obeys the Nyquist criterion, but in many practical cases it is sufficient to sample at the chip rate , and little loss is incurred, primarily from useful signal loss, if the receiver sampling clock and transmitter clocks are offset around . Note that making an assumption of a synchronous sampling in a multiuser scenario is inherently incorrect, for even if we were to synchronize the receiver clock to a given transmitter clock, all the other user signals would still arrive asynchronously, and in order to cancel these signals, we effectively have to deal with asynchronous sampling. Sampling at , we therefore obtain a sampled version of the received signal as
C. The Estimator Since each of the signals is distorted by its own equivalent discrete filter and there is no direct correlation between the filter taps of different signals, the received signals will contribute very little to the estimation of the taps . The iterative demodulator, on the other hand, will remove the interference and thus provide the main contribution to the estimation of . We therefore develop the estimator for using the signal model (18) where contains the noise and interference, now treated can be pilot symbols, data purely as noise. The symbols symbols, or a combination of pilot and data in this context. Recalling the sample vector of the -th signal stream for a “frame” of size symbols, and letting be the impulse response of the discrete channel filter, then
.. .. .
.. .
.. .
.. .
.
.. .
(15) which can be put into a finite-impulse response (FIR) form
..
.
.. .
(16) where the taps of the filter that models this asynchronous sampling process are now given as
(19) If is large enough, we can ignore the edge effects, and the -th section of is simply given by the circulant sub matrix
(17) The sum in (16) is over the span of the filter, which can contain a large number of taps. Fig. 3 illustrates the tap values that result in the filter model from a delay of . The addition of multi-path components would simply generate a more spread out collection of filter tap values. However, the approach used in this paper would still be valid for such a scenario. The time variations of these taps are not explicitly expressed in the notation, but are caused primarily by and any drift of . Since results from carrier frequency mismatches and/or doppler shifts, it is usually the dominant source of the
.. .
.. .
.. .
(20)
such that ; these are the shiftWe can design orthogonal sequences used in Section V. The minimum mean-square error (MMSE) estimator for the stationary model (19) is standard, and is given by
(21)
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which is used in the estimator in Fig. 4, is formed using (2) with the actual data symbols replaced by their estimates , which are then used as input to the correlator. Recalling that the sample sequence is composed of pilot and data as in (24) we examine the estimator (23) as follows:
(25)
Fig. 4. Correlation-based channel tap estimator, which uses as input the combination of the pilot and the soft pilot signal recovered during iterations.
The difficulty is to obtain accurate tap correlation functions , which are dependent on the unknown , as well as channel distortions. To fully implement the estimator in (21) we would need to build correlation matrices from the received signal at a considerable cost in complexity. A more promising strategy in many situations is to exploit sparseness in the channel impulse response, a topic that is beyond the scope of this paper. If we disregard the intertap dependence, becomes diagonal with the entries . These, however, are will typically die off also unknown a priori, but the taps rather rapidly away from a central tap (Fig. 3). If the pilot sequences are designed well, we have , and the estimator collapses to
(22) where is sparsity selector which will select only components that are known a priori not to be negligible, see for example Fig. 3. We have also used the approximation , primarily due to the difficulties of obtaining more precise estimates. Note that this implies the filter impulse response normalization . Sparsity concepts can thus be introduced to improve performance and exploit extended channel information, but for now we assume no such knowledge and therefore let . Equation (22) is basically a correlator bank, and its signal flow implementation is illustrated in Fig. 4. The development of the estimator in (22) was carried out for a known sequence , but in reality contains pilots and partially known data, which signifies that the estimator has to be modified into (23) where the local samples used for correlation are generated as
(24) by including the pilot. What this means in operational practice is simply that the regenerated sample sequence at the receiver,
where we have assumed that pilot and signal are orthogonal, i.e., , and are implicitly defined above with (24). This orthogonality between pilot and data can be achieved in a number of ways, for example, by using discrete pilot frequencies which are not used in the signal, or simply by letting and ensuring that the are uncorrelated with the . requires that the periodic Furthermore, pilot sequence is shift-orthogonal. This can be ensured, for example, by using the inverse discrete Fourier transform of a random, unit-amplitude sequence in the frequency domain as the pilot. Maximum-length sequences can be used since they have small, although not zero autocorrelation, and regular conventional spreading sequences also work as long as is sufficiently large. Lastly, for large
(26) where the last step was discussed in Footnote 2. Putting these steps together we obtain for the filter taps in (25)
(27) , and we also have assumed that , i.e., the sparsity selector does not eliminate any actual filter tap with non-zero values. We notice that the right hand side of (27) is biased, and in order to have an unbiased estimator, we therefore set where
(28) D. Performance The estimator in Fig. 4 is a rake-type matched filter, that is, each tap of the equivalent discrete channel model is correlated against the reference sequence in (28), so the -th tap is estimated as (29) where was defined in (25). Assessing the performance of the estimator condenses now into computing the variance of the noise term. Noting that the noise power of is , the variance of the noise term in (29) is given as . If taps have to be estimated, and using ,
SCHLEGEL AND JAR: ITERATIVE DEMODULATION AND CHANNEL ESTIMATION FOR ASYNCHRONOUS JOINT MULTIPLE ACCESS RECEPTION
we obtain for the channel estimation error in (5) the following expression:
To interpret this equation, we use the latter iterations, where simplification can be used
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, and consider . Then the following (34)
(30) quantifies the filter where the noise reduction factor dynamics. It depends primarily on the number of parameters that are to be estimated and how much time is available for doing so. While equation (30) was derived for a specific estimator, namely performing a simple average, the general form of the equation is true for any estimator whose performance depends linearly on the variance of the noise in the signal. This is the case for all the linear estimators, such as MMSE, Wiener or Kalman filters. In all these cases, a convergence equation involving only second moments can be established, as shown in the next section. Clearly, expression (30) is strictly correct only for a time-invariant channel response and a finite frame size of , and therefore not typically of practical interest. If the channel taps are time-dependent due to operation over a continuously varying channel, the sums are replaced by smoothing filters. Such time variations are due to many causes, but we are concerned here with the time dependency of and , which introduce shifts . typand rotations into the discrete filter tap values ically contains the higher frequencies and should therefore be used to determine the filter bandwidth to be used. Avoiding details outside the scope of this paper, we can nonetheless estimate the achievable signal-to-noise ratio enhancement by this filter as follows. The filter needs to let frequencies pass unattenuated, and block all higher frequencies. We normalize numerical values to the symbol frequency , for which the noise samples have variance . The power of the noise is therefore reduced proportionally due to the filsamples tering of . With ideal filters we obtain the reduced variance . This value shall serve as a lower bound on the noise power. How closely this bound can be approached depends on the complexity of the filter design. In practice, we are likely going to see a filter implementation loss of several dB with respect to the optimal, see Section V. Since we can bound the noise reduction factor by (31) the channel estimation error
is now bounded as (32)
Clearly, we need for the estimator to work, which represents a first limitation on the number of filter taps that can be tracked or estimated. With (30) used in (9) and (10), we now obtain (33)
which means that the variance transform function of the APP decoder is altered by the addition of a linear term, whose magnitude is determined by the filter requirements needed to track the channel changes. That is, when the data detectors have converged, the channel estimators operate at their maximum signalto-noise ratio where the entire data sequence functions as pilot signal. With this basic estimator model developed in this section, we can now study the interaction of the demodulation/decoding loop and the estimator. E. Convergence Functions In this section we examine how the system can achieve the “ideal” operating point of (34). The convergence behavior of iterative systems is often characterized by single-parameter iterative equations, such as (3). If we replace by the place holder variable , we note that successive iterations continue to reduce the residual single user equivalent noise, as long as . This means that (35) or that the
, where (36)
is the convergence function of the iterative system. The final operating signal-to-noise ratio achieved by the iterative detector corresponds to the largest solution of (36), that is, iterations start with and proceed until the first zero of (36) is reached. For an iterative estimator of the form (33), where the estimator only uses the pilot signal in its repeated estimation the system convergence equation is given by (37) and, finally, an iterated estimator which uses the emerging data information for estimation follows the convergence equation (38) These convergence functions therefore describe the dynamics of large iterative systems by describing the evolution of the residual noise and interference variance . The assumption made in the derivation of these equations is that the systems are large enough, such that signal correlations cannot build up over the course of the iterations considered. For details, see, for example [18]. In the multiple access case discussed here, we made the additional assumption that the interfering signals were sufficiently random, such that their contribution acted essentially as white noise, a case that is customarily made for spread spectrum multiple access systems based on direct-sequence spreading [24]. Assuming these premises hold, the system performance now converges to the rightmost of the at most three fixed points of the equation (zero-crossings of ). If the equation has three
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Fig. 6. Convergence functions for ideal, iterative, single-pass, and iterated . channel estimation using only the pilot signal, all with Fig. 5. Convergence functions for ideal iterative, single-pass, and iterated . channel estimation using only the pilot signal, all with
zeros (fixed points), this typically means that the final noise variance is too large for operation, and the system load has to be decreased until only one fixed point remains—see subsequent example scenarios. Fig. 5 shows the convergence functions (37), (37), and (38) at a signal-tofor a system with a moderate load of noise rato of . As discussed above the iterative receiver system converges to the largest solution of from above , and this solution determines the final signal-to-interference ratio of the system. A load of represents the limit for the single-pass estimator, and the final signal-to-interference ratios (SIR) achieved is 8.79 dB. For iterated demodulation a final SIR of 9.70 dB at a somewhat larger . For the lower load typical in current systems the difference becomes negligible, and single-pass estimation is sufficient for a filter noise reduction factor of or better. A value of might be typical for a simple radio system with the following parameter ranges. Using a symbol rate of , a carrier frequency of and 0.1 ppm oscillators we would see a frequency drift of . In this . With filter taps, for example, case simple single-pass estimation would be sufficient, even in a dense multiuser multiple access system. In many highly mobile scenarios, a filter suppression coefficient of 30 dB may not be achievable due to high parameter dynamics. Our next set of illustrations shows examples for instead. Now, while repeated estimation can achieve a load of at a final SIR of , ideal iterative demodulation would achieve a load of , while single-pass estimation fails at these at values and would achieve a load of only 0.5 at . Convergence curves for are shown in Fig. 6. While pilot-only iterative demodulation was sufficient to achieve near optimal performance in the scenario of Fig. 5, for the parameters in Fig. 6 the improvements of using the soft information in the iterative demodulation loop is quite evident. Fig. 7 shows the achievable spectral efficiencies for different noise reduction factors. The curves compare achievable values, assuming that the iterative demodulator is followed by an ideal
Fig. 7. Achievable spectral efficiencies of iterative receivers with full iterative channel estimation compared to ideal performance and channel capacity.
Shannon-capacity achieving code of rate , and full iterative estimation is used with partially demodulated data acting as additional pilot signal. Fig. 8 presents a different perspective and shows the achievable spectral efficiencies as a function of the relative frequency error , where (32) was used. The figure again shows ideal, full iterative demodulation, iterative demodulation using the pilot only, and single-pass demodulation, where the channel is estimated once at the beginning only. While full iterative demodulation can maintain not only high spectral efficiencies, but also high post-cancelation SIRs, the comparison methods degrade substantially with increasing frequency error. The post-cancelation SIR values of the signal are marked at the various performance points, where the initial SNR is 10 dB, and ideal cancelation achieves and SIR of 9.7 dB. We note that single-pass and pilot-only iterative demodulation, while achieving fairly high spectral efficiencies, show low to very low post-cancelation SNR, indicating that cancelation is not really effective anymore, and that the spectral efficiencies are achieved by low-rate coding with many interfering users. In fact, it is known that simply tolerating the interference and apply powerful low-rate error control coding can achieve a limit spec[13, Section 3.4], but this tral efficiency of
SCHLEGEL AND JAR: ITERATIVE DEMODULATION AND CHANNEL ESTIMATION FOR ASYNCHRONOUS JOINT MULTIPLE ACCESS RECEPTION
Fig. 8. Spectral efficiencies of various cancelation approaches as a function of . The operating point is at a signal-to-noise ratio of 10 dB.
requires , and is not a scenario of immediate practical importance. The regimes where the post-cancelation SNR has dropped by more then 6 dB is indicated by dotting the curves, signifying that cancelation is no longer effectively in separating the users. V. ESTIMATION IN PRACTICE It can be appreciated that the iterative receiver structure presented here has the potential for application in a large variety of systems where either the channel parameters change rapidly, or where the parameter space is large, such as in the asynchronous multiple access communications system discussed in this paper. In practice, however, the estimator in (21) cannot be built directly, since it requires information that may not be readily available, such as the cross-correlation of the tap values. In many actual system implementations, therefore, the task of channel estimation is therefore handled by adaptive filters. Given receiver filter taps , the error between the received and its estimate, i.e., signal (39) is used to update the filter coefficients, where from (28). The gradient of (39) is given by (40) which is used in a standard gradient-based stochastic update algorithm. That is, the filter taps are updated at each sample time with the LMS step size factor as
(41) Note that other update schedules are also feasible, such as bidirectional, block, or windowed updates, but the principle remains the same. This standard form of the least mean-squares (LMS) algorithm will also adapt to amplitude changes in and unknown
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Fig. 9. LMS estimator using a soft-pilot to operate the update equations (41).
as long as gains in , and in general track the dynamics of is properly adjusted. The algorithm in (41) is illustrated in Fig. 9. The advantage of the structure of this cancelation receiver is that it can be shown that the canceled signal is exactly Gaussian for large , see [17]. We therefore only need to understand the behavior of the estimator in Gaussian noise. For this case the mean-squared error minimizing filters all have an expected squared error which depends linearly on the (residual) noise variance, i.e., is of the form (31), which enables the convergence equations presented in Section IV-E. We now wish to verify that the performance of the LMS estimator in Fig. 9 is sufficient to operate the iterative demodulation loop of the receiver. As before we use a pilot sequence of length that is shift orthogonal4, as discussed above, and statistically independent of the data signal as well as all other users’ signals and pilots. Fig. 10 shows a tracking example of the LMS estimator , with the following test parameters: Symbol size , i.e., —which is a fairly large clock mismatch—inducing a rapidly varying FIR channel , , and model. The doppler offset is is a combination of channel noise and incomplete data cancelation. The behavior of the iterative estimator is now examined with the structure of Fig. 9, where the iterative demodulation loop is emulated by the test channel at the top of the Figure. This test channel accepts the transmitted samples and adds Gaussian noise of variance . The LLR statistics generated by the non-linearity at the output of the test channel are now exactly identical to those generated by a large iterative demodulator, and can thus be used to replace the demodulator in order to study the estimator. Fig. 11 verifies the basic linear model (30) of the estimator for our LMS-based channel estimator. There is ideal agreement in the region of most interest, but two regimes of deviation need some explanations. For a very small value of , the error not only depends on the pilot SNR, but also on the parameter , or 4The exact process we used to generate the pilot was to choose a random and applied the inverse discrete fourier transform QPSK symbol set of size . to that data set to obtain the pilot symbols
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example is intended as a generic illustration of a highly-mobile channel model, not necessarily an actual system implementation. VI. CONCLUDING DISCUSSION
Fig. 10. Evolution of the filter taps , and LMS-tracked estimated taps over a time window of 2 ms; compare Fig. 3.
Fig. 11. Simulated relationship (30) for the LMS tracker as function of with The noise reduction factor of this example estimator is . Other parameters are discussed in the text.
.
more precisely, the tracking error of this simple estimator represents a lower bound on . (Such a tracking error is also visible in Fig. 10). Second, for very small values of , the distribution of , which acts as the additional noise due to incomplete knowledge of , assumes an asymmetric distribution which increasingly deviates from the Gaussian model used in the analysis. However, this appears to have no observable impact as evidenced in Fig. 11. The straight line shows the error performance with Gaussian noise with variance . The simulated points show , with , where the was varied to achieve a given effective test noise variance pilot SNR on the -axis according to the simulation setup in Fig. 9. The inset in the figure shows the distribution of the soft symbol estimate for the specific operating point indicated in the figure. With a measured noise reduction value of , we would expect to lose no more then about 40% capacity w.r.t. ideal channel knowledge, see Fig. 7, using only a simple LMS estimator to compensate for a clock offset of 1000 ppm. Clearly, even an approximate compensation of such large frequency offsets should easily allow us to recuperate that capacity loss. The
We studied iterative demodulation with channel estimators in the loop. We have presented a methodology based on convergence functions which unites the behavior of the estimator and demodulator when both are operated inside the decoding loop in a fully iterative receiver. We have further argued that the performance of the estimator is important over an entire range of the signal-to-noise ratio, and that its primary performance criterion is whether it does enable the iterative receiver to converge. Furthermore, the mean-square error measure of linear estimators is proportional to the signal-to-noise ratio the estimator sees, and we have introduced a noise reduction factor of the estimator, which was explicitly developed for correlation-based estimators, and numerically verified for a basic least-mean square (LMS) estimator model. This noise reduction factor depends critically on the channel over which the communications system is operated. Finally we have shown that, for highly dynamic channels, an iterative receiver including estimation in the iterative loop can provide convergence where traditional separated processing fails, and that very basic estimators are often sufficient if operated inside the loop. In a sense, including the estimator in the loop strongly enhances its performance, and the effect of re-estimating the channel is very powerful due to two effects: (i) as estimates of the data symbols become available, more of the transmitted signal can act as “training pilot”, and (ii) as the interference in each data stream decreases with iterations, estimators provide increasingly better performance. A key issue of how much pilot is necessary to successfully start the combined demodulation and estimation process has also been discussed. The overall system performance is dominated by the dynamics of the combined iteration process, and not so much by the performance of the estimator as an individual component. Is is noted that not including the estimator in the iteration process can be very costly in terms of performance, and is a useful option only for systems that are not resource stressed. Finally, we wish to acknowledge that there is a potentially large choice of channel estimators, and other then the meansquare error based versions, these estimators’ performance may not necessarily follow the simple model (30). In such cases, a more adequate estimation error versus estimator input SNR transfer characteristics needs to be worked out. In any case, our methodology establishes two regimes for which the channel estimators need to be designed: (i) In the “pilot mode”, where much of the data is reliably available, the estimator operates with essentially full knowledge of the transmitted signal, and its performance determines the final symbol and bit error rates that the transmission method can achieve. (ii) In the “convergence mode” the function of the estimator is to provide a level of channel estimation that is sufficiently accurate for the soft symbol estimator in the demodulation loop to operate without significant loss in performance. At this stage, the accuracy of the channel estimator required is not stringent. For the linear estimators discussed here, we have shown that a simple linear model is adequate, and that performance of the estimator during the iterations can be included in the system’s
SCHLEGEL AND JAR: ITERATIVE DEMODULATION AND CHANNEL ESTIMATION FOR ASYNCHRONOUS JOINT MULTIPLE ACCESS RECEPTION
convergence function via the introduction of a noise reduction factor, which captures the behavior of the estimator.
REFERENCES [1] O. Axelsson, “Milestones in the development of iterative solution methods,” J. Electr. Comput. Eng., vol. 2010, 10.1155/2010/972794. [2] M. V. Burnashev et al., “Analysis of the dynamics of iterative cancellation decoding,” J. Prob. Inf. Transmission, no. 4, pp. 3–25, 2004. [3] T. Cover and J. A. Thomas, Elements of Information Theory. Hoboken, NJ, USA: Wiley, 1991. [4] R. Dodd, C. Schlegel, and V. Gaudet, “DS-CDMA implementation with iterative multiple access interference cancelation,” IEEE Trans. Circuits Syst. I, vol. 60, no. 1, 2013. [5] P. D. Alexander, A. Grant, and M. C. Reed, “Iterative detection on code-division multiple-access with error control coding,” Eur. T. Telecommun. (ETT), vol. 9, no. 5, pp. 419–426, 1998. [6] P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, no. 6, pp. 837–843, Jun. 1999. [7] S. Howard and C. Schlegel, “Differential turbo-coded modulation with APP channel estimation,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1397–1406, Aug. 2006. [8] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics (Modern Acoustics and Signal Processing). New York, NY, USA: Springer, 1997. [9] C. Komninakis and R. Wesel, “Joint iterative channel estimation and decoding in flat correlated Rayleigh fading,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1706–1717, Sep. 2001. [10] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, no. 7, pp. 870–880, 1998. [11] M. Peleg, S. Shamai, and S. Galán, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. Inst. Electr. Eng.—Commun., vol. 147, pp. 87–95, Apr. 2000. [12] M. C. Reed et al., “Iterative multiuser detection for CDMA with FEC: Near-single-user performance,” IEEE Trans. Commun., vol. 46, no. 12, pp. 1693–1699, 1998. [13] C. Schlegel and A. Grant, Coordinated Multiuser Communications. New York, NY, USA: Springer, 2006. [14] C. Schlegel and A. Grant, “Differential space-time turbo codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2298–2306, Sep. 2003. [15] C. Schlegel, “CDMA with partitioned spreading,” IEEE Commun. Lett., vol. 12, Dec. 2007. [16] C. Schlegel, D. Truhachev, and Z. Bagley, “Transmitter layering for multi-user MIMO systems,” EURASIP J. Wireless Commun. Netw., vol. 2008, Jan. 2008, Article ID 372078. [17] C. Schlegel, Z. Shi, and M. Burnashev, “Asymptotically optimal power allocation and code selection for iterative joint detection of coded random CDMA,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4286–4295, Sep. 2006. [18] C. Schlegel and D. Truhachev, “Multiple Access demodulation in the lifted signal graph with spatial coupling,” IEEE Trans. Inf. Theory, vol. 59, no. 4, pp. 2459–2470, 2013. [19] D. Truhachev, C. Schlegel, and L. Krzymien, “A two-stage capacity-achieving demodulation/decoding method for random matrix channels,” IEEE Trans. Inf. Theory, vol. 55, no. 1, Jan. 2009. [20] D. Truhachev, “Universal multiple access via spatially coupling data transmission,” presented at the ISIT, Istanbul, Turkey, Jun. 2013. [21] D. Truhachev and C. Schlegel, “Coupling data transmission for capacity-achieving multiple-access communications,” Dec. 2012, arXiv:1209.5785v2 [cs.IT] [Online]. Available: http://arxiv.org/pdf/ 1209.5785.pdf
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[22] M. Valenti and B. Woerner, “Iterative channel estimation and decoding of pilot-symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1691–1706, Sep. 2001. [23] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [24] A. Viterbi, CDMA: Principles of Spread Spectrum Communication. Reading, MA, USA: Addison-Wesley, 1995. [25] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, 1999. [26] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. Long Grove, IL, USA: Waveland Press, 1990. [27] H. Zhu and B. Farhang-Bourougheny, “Pilot embedding for joint channel estimation and data detection in MIMO communication systems,” IEEE Commun. Lett., vol. 7, no. 1, pp. 30–32, Jan. 2003. Christian Schlegel (S’86–M’88–SM’97–F’10) held the CORE Chair for Digital Communications at the University of Alberta, Canada, from 2002 to 2012. Prior to that, he held academic appointments at the University of Hawaii at Manoa, USA, (visiting), the University of South Australia in Adelaide, Australia, the University of Texas at San Antonio, USA, and from 1996 to 2002 at the University of Utah, Salt Lake City, USA. From 2004 to 2008, he was also serving as CTO (part-time) of Aquantia Corporation, Milpitas, CA, USA, a start-up company building 10 Gbit/s Ethernet transceivers. He is the author of Trellis Coding (1997, IEEE Press), Trellis and Turbo Coding (2004, 2015, Wiley), as well as Coordinated Multiple User Communications (2006, Springer). Dr. Schlegel was associate editor for coding theory and techniques for the IEEE TRANSACTIONS ON COMMUNICATIONS from 1999 to 2007, guest editor for the PROCEEDINGS OF THE IEEE, and currently serves on the editorial board of Editorial Board of Hindawi Publishing. He served as technical program chair of the IEEE Information Theory Workshop 2001, the IEEE International Symposium on Information Theory (ISIT’05) 2005, as general chair of the 2005 IEEE Communication Theory Workshop (CTW’05), the 2013 IEEE Conference on Wireless On-Demand Network Systems and Services (WONS’13), and of the 2016 Symposium on Information Theory and its Applications (ISITA’16), to be held in California, USA. He received a U.S. National Science Foundation Career Award in 1997, and a Canada Research Chair in 2001. He currently holds an NSERC Industrial Research Chair at Dalhousie University, , Halifax, NS, Canada, and focuses on reliable digital communications for complex transmission environments. He was named IEEE Distinguished Lecturer in 2007 and 2011, and is an IEEE Fellow.
Marcel Jar was born on May 31, 1982, in Recife, Brazil. He received the B.Sc. and M.Sc degrees from the Federal University of Pernambuco, Brazil, in 2005 and 2006, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Alberta, Canada, in 2011. He was a postdoctoral fellow at the Technische Universität Dresden, German, from 2011 to 2012. Currently, he holds a full-time professorship at Seneca College, Toronto, ON, Canada, as well as an Adjunct Professor position at Dalhousie University, Halifax, NS, Canada. He has received a full graduate CNPq scholarship (Brazil) in 2005, during his M.Sc. studies, and a J. Gordin Kaplan Graduate Student Award in 2010, during his Ph.D. studies. He served as chair of the 2008 Canadian Summer School on Communications and Information Theory. His research interests include information theory, digital communications, iterative processing, and hardware implementation of communication systems.