Mobile user demand for wireless data services has been increasing dramatically in recent years. As the conventional sub-6 GHz communications spectrum is packed with existing wireless services, the millimeter-wave (mmWave) frequency band has become a key asset for next-generation cellular networks. Along with increasing antenna array sizes at both sides of the transceiver, compressed sensing (CS) based algorithms have received great attention in estimating mmWave channels. Owing to the mobility of users and scattering obstacles (moving cars and so on) in the communication environment, parameters underlying mmWave channels vary dynamically over time. These variations can either be estimated from scratch, likely at the expense of significant training overhead, or tracked by making use of dynamic channel characteristics. The focus of our work is to perform channel estimation using approximate message passing (AMP) aided by side information (SI). Our AMP-SI approach to channel estimation utilizes the dynamic channel structure, leading to improved estimation quality and reducing training overhead.
, where one wishes to recover an unknown sparse vectorfrom noisy linear measurements modeled as
where is a measurement matrix with more columns than rows, and is independent and identically distributed (i.i.d.) noise. AMP iteratively estimates using a possibly non-linear denoiser function tailored to prior knowledge about . One key property of AMP is that under some technical conditions on the measurement matrix and signal , observations at each iteration of the algorithm, referred to as pseudo-data, are asymptotically (in the large system limit) equal in distribution to plus i.i.d. Gaussian noise.
AMP with Side Information (AMP-SI). Recently [7, 8], AMP-SI was introduced as an algorithmic framework that incorporates SI into AMP for CS tasks (1). AMP-SI has been empirically demonstrated to have good reconstruction quality, and is easy to use. For example, we have proposed to use AMP-SI for a toy model for channel estimation in emerging mmWave communication systems , where the time dynamics of the channel structure allow previous channel estimates to be used as SI when estimating the current channel structure . In Liu et al. , the nice empirical performance of AMP-SI was strengthened through a rigorous performance analysis. For these reasons, it is not surprising that our novel approach to channel estimation outperforms a benchmark based on orthogonal matching pursuit (OMP)  as evidenced in Sec. V, and it is unlikely that other non-AMP based approaches would yield further improvements.
Contributions. Our main insight in this paper is that the channel matrix can be represented sparsely over the domain of angles of arrival and departure. This insight leads us to develop a denoiser within AMP-SI that monitors and estimates paths with continuous angles of arrival and departure. We use 2D spectral estimation within AMP-SI for a simplified problem with a single drifting path, and will address increasingly complicated (and thus realistic) models.
Notation. Let and denote the complex conjugate and Hermitian operations, respectively. We use and to represent a zero vector of size
and identity matrix of size, respectively. Next, stands for the -th element of the vector x and for the -th element of the matrix M
. A complex Gaussian distribution with meanm and covariance C is denoted by , and
stands for the uniform distribution taking values betweenand . Finally, the set of integers is denoted by , and the Dirac delta function, , takes the value if , and otherwise.
Ii System Model
Consider point-to-point downlink communication in mmWave frequency spectrum (Fig. 1), where a base station (BS) communicates with a mobile user equipment (UE). The number of transmit and receive antennas at the BS and UE are and , respectively, both of which form a one-dimensional (1D) uniform linear array (ULA).
We study a blockwise transmission strategy that relies on block fading. The downlink channel in the -th transmission block, denoted by , does not change during the -th transmission block; the channel matrix takes a new value in the next transmission block. We will assume that across blocks , there is some dependence or structure, so that an estimate of the block can be used as side information (SI) in estimating . The relationship between and is detailed in Sec. II-B, and the channel model in Sec. II-A. Our goal will be to estimate for each block from received signals, with the signal model specified in Sec. III.
Ii-a Transmission and Channel Models
In our transmission strategy, the BS transmits pilot symbols (known to the UE) during the first time slots of each transmission block, which consists of time slots in total. The UE estimates the channel using these pilot symbols together with the SI, which is obtained from the channel during previous transmission blocks. The BS uses the remaining time slots to transmit payload data (unknown to the UE), which is decoded by the UE using the channel estimate of the current transmission block.
The respective downlink channel, , within the -th transmission block is given as follows,
where is the number of multipath components, is the signal-to-noise ratio (SNR), is the complex gain of the -th multipath component assumed to be distributed as with uncorrelated gains for different paths, i.e. where . In addition, and are the angle-of-arrival (AoA) and angle-of-departure (AoD), respectively, of the -th multipath component with and . Note that the distributions for the complex path gain and AoA/AoD folow from mmWave channel measurement studies . Furthermore, and represent the array steering vectors at the receive (UE) and transmit (BS) sides, respectively, where the -th element of a generic array steering vector is
for , with an arbitrary phase representing AoA/AoD, and elements representing the number of transmit/receive antenna elements, where is the antenna element spacing of the ULA, and is the carrier frequency wavelength.
Recall that our goal is to estimate for each block from received signals and SI. To this end, we will assume throughout that in our definition for (2) the scalar values , , , and are known, and that and are also available to the ULA antenna array. Note that the number of paths and SNR are relatively stationary for point-to-point communications, and can be separately obtained over many transmission blocks. As a result, estimating the channel block boils down to estimating the complex Gaussian random variables (RVs) and the uniform RVs and , from which the arrays and and thus can be recovered.
Ii-B Time Variation for Channel Parameters
We now provide a model for the dynamic relationship between and . Our model was discussed by numerous authors (c.f., [11, 12, 13] and references therein). The complex path gain varies from one transmission block to another following a first order auto regressive (AR) process,
where is the correlation coefficient, and is innovation with . Note that (4) represents variation in the complex path gains (i.e., small-scale fading) with correlation to the previous transmission blocks through . Moreover, both AoA and AoD follow Gaussian innovation processes in the next transmission block,
where and are the aggregate AoA and AoD vectors, respectively, and and are corresponding innovation vectors.
Iii mmWave Channel Estimation via AMP-SI
In this section, we consider mmWave channel estimation for the scenario described in Sec. II, and explain how SI from previous transmission blocks enhances the estimation quality. To this end, we define an complex-valued unit-energy vector , which represents the pilot symbol transmitted during the -th time slot within transmission block . We also assume that is selected from an uncorrelated dictionary, i.e., where .
From signal estimation to matrix estimation. The received signal vector at the UE is given by
where is measurement noise that follows . Considering pilot transmissions over time slots, the aggregate received signal during the -th transmission block, , where denotes a matrix obtained by concatenating the column vectors , is given by
Note that and are and , respectively. Our goal is to estimate using (7) given observations and pilot symbols and SI from previous transmission blocks.
which is more aligned with the AMP framework. In particular, we could modify (8) by vectorizing and and composing a measurement matrix having repeated on the diagonal. However, this modification is not necessary, because AMP provides favorable results even when applied to multi-dimensional signals [14, 15]. Therefore, we run AMP directly on the multi-dimensional problem.
AMP-SI with 2D denoisers. Consider a fixed block . We run AMP-SI on the matrix directly, employing a well-chosen 2D (matrix) denoiser. Each AMP iteration will have access to pseudo-data that is asymptotically (in the large system limit) equal in distribution to
plus a matrix of i.i.d. Gaussian noise, where the existing AMP theory allows us to calculate a good approximation for the noise variance. Importantly, the 2D denoiser we propose incorporates SI from the previous estimate at block, where this SI is our estimate for the multipath parameters in block , and our knowledge of the dynamics of these parameters per (4)-(5).
Denoising a 2D matrix within AMP, as opposed to a vector, is non-standard. That said, some related art (including by the authors) is encouraging. For example, previous applications of AMP with multi-dimensional denoisers have provided encouraging empirical results in image reconstruction [14, 15]. Beyond empirical results, a rigorous analysis by Ma et al.  provides performance guarantees for a family of multi-dimensional sliding window denoisers used within AMP.
Spectral estimation in 2D. Our proposed denoiser resembles work by Hamzehei and Duarte [17, 18], who performed analog denoising of 1D vectors within AMP in the context of spectral estimation. Our denoisers resemble theirs, except that we perform 2D instead of 1D spectral estimation, with improved performance owing to SI from block . Our main insight is that is sparse over the continuous (AoA, AoD) domain. This insight leads us to develop a denoiser that monitors and estimates paths with continuous and . Our work seems most related to Bellili et al. , where the authors sparsify a linear inverse problem using Fourier arguments. Our approach expands over theirs by performing 2D continuous spectral estimation within AMP.
To make the details of our denoiser tractable (Sec. IV), we begin with a simplified setting comprised of a single drifting path. That is, we assume in channel model (2). While this paper introduces AMP-SI and our 2D spectral estimation denoiser for this simplified version of the problem, we aim to leverage these results and develop a series of denoisers addressing increasingly complicated (and thus realistic) models. Our current work will be extended to multiple paths with birth-death-drift dynamics between blocks .
Iv One drifting path
As mentioned previously, we set in (2), and model the channel in the -th transmission block, , as
In this section, we introduce an AMP-SI algorithm for completing this parameter estimation task, and discuss some implementation details.
AMP-SI details. The AMP-SI algorithm for estimating in (9) takes the following form. Initialize the matrix estimate with
, a zeros matrix, and at iteration, compute
where we interpret as a residual, is our current estimate of , and is the pseudo-data, which is equal in distribution to plus i.i.d. Gaussian noise with variance . Our denoiser, denoted , takes as inputs the pseudo-data and SI from the previous block, denoted ; the form of is specified below. Finally, the residual uses the normalized divergence of the denoiser,
We highlight that the conjugate operator is applied elementwise to when computing as part of complex AMP .
Candidate Denoisers. Given the distributional properties of the pseudo-data, namely where has i.i.d. complex Gaussian entries, there are at least two plausible denoising styles for AMP-SI (10).
The first denoising style we consider is conditional expectation, where is calculated using
where we have explicitly stated that the SI at time takes the form of an estimate of the channel at the previous block, . Within each AMP iteration, conditional expectation provides a minimum mean squared error (MMSE) estimator of given the pseudo-data, , and SI, . Under some technical conditions, for large scale linear inverse problems , upon convergence, AMP with conditional expectation denoisers yields the overall MMSE signal estimator. Unfortunately, with our current understanding of the model, the conditional expectation denoiser appears computationally intractable when , so we did not consider it further.
The second denoising style we consider is maximum a posteriori (MAP), where we compute the triple that maximizes the posterior,
where denotes a generic density, and then use the estimated triple to produce an estimate of . In contrast to the conditional expectation denoiser, MAP signal estimation is sub-optimal in terms of MSE in individual AMP iterations, because conditional expectation is the MMSE estimator, and thus minimizes the noise variance for the next iteration, whereas MAP differs from conditional expectation. Additionally, MAP denoisers may not achieve the overall MMSE. Despite MAP having these estimation-theoretic drawbacks, we will see that it offers computational advantages in our analog denoising problem.
For the MAP denoiser, we need to further study the posterior distribution.. First, by Bayes’ rule,
where we have used the independence of the RVs and the fact that the pseudo-data at iteration , given by , is independent of given . In the numerator (12), the densities are Gaussian,
The denominator is a normalization constant that does not affect MAP optimization.
MAP Denoiser. Focusing on the MAP denoiser, the form of the conditional distribution given in (12) suggests,
We simplify the above using (13),
where is similar to , and
where for a matrix we have . Plugging into (14), we find
and we have written as with .
using the shorthand , , and and indicate real and imaginary parts, along with the fact that for any and ,
For any , the minimizing takes the form
We can similarly show
In implementing the denoiser in (17) within AMP, for any pair we solve for the optimal using the estimates of and given just above. Optimal
are computed using a grid search over values within four standard deviations of the SI. The remaining consideration in implementing (10) is to compute the divergence of the denoiser (11). While it is difficult to compute the divergence analytically, our implementation in Sec. V approximates it numerically.
We pause here to note that extending the MAP denoising technique just outlined to two drifting paths, i.e., in our channel model (2), seems feasible but computationally expensive, and it is unclear if these approaches will work when going beyond, i.e., . Extending our results here to these regimes will be pursued in future work.
V Numerical Results
We demonstrate the performance of the proposed AMP-SI algorithm by presenting numerical results based on Monte Carlo simulations. As a benchmark, we used Dirichlet orthogonal matching pursuit with local optimization (DOMP-LO) , which uses Dirichlet kernels while estimating the unknown mmWave channel. Our communications setting used , , , , and dB, which represent a mmWave channel with reasonable time variation. Fig. 2 shows numerical results using 300 AMP-SI iterations; DOMP-LO used a dictionary size of , which yields the best performance. It can be seen that the empirical MSE declines as more pilots, , are used. Moreover, a 20 dB SNR outperformed 10 dB, resulting in lower curves. Overall, the empirical MSE obtained by our AMP-SI approach was roughly an order of magnitude less than that of DOMP-LO. We also compared our setting to one without SI by increasing the variance of the drift. Larger variance reduced the estimation quality; we omit the details for brevity. Finally, our future work will evaluate the spectral efficiency of the proposed algorithm along with hybrid/digital beamforming schemes in comparison to training length, and compare the performance gains over existing methods (e.g., [22, 23]).
The authors thank Arian Maleki for graciously discussing his work on complex AMP  with us, and Chethan Anjinappa for help producing numerical results.
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