A Joint Time-Invariant Filtering Approach to the Linear Gaussian Relay Problem

A joint time-invariant filtering approach to the linear Gaussian relay problem

Abstract

In this paper, the linear Gaussian relay problem is considered. Under the linear time-invariant (LTI) model the problem is formulated in the frequency domain based on the Toeplitz distribution theorem. Under the further assumption of realizable input spectra, the LTI Gaussian relay problem is converted to a joint design problem of source and relay filters under two power constraints: one at the source and the other at the relay, and a practical solution to this problem is proposed based on the projected subgradient method. Numerical results show that the proposed method yields a noticeable gain over the instantaneous amplify-and-forward (AF) scheme in inter-symbol interference (ISI) channels. Also, the optimality of the AF scheme within the class of one-tap relay filters is established in flat-fading channels.

L

inear Gaussian relay, linear time-invariant model, Toeplitz distribution theorem, projected subgradient method, filter design

1 Introduction

Relay networks have drawn extensive interest from research communities because they play an important role in enlarging the network coverage in wireless communications. Although the capacity of relay networks is not exactly known yet, many ingenious coding strategies including decode-and-forward (DF), compress-and-forward (CF), etc. beyond simple AF schemes have been developed [1, 2]. Recently, Zahedi et al. proposed an advanced linear scheme for relay networks based on (strictly-)causal linear processing at the relay to compromise the complexity and performance between the complicated coding strategies and the simple AF1 scheme [3, 4]. While information theorists approached the problem from the perspective of capacity and capacity-achieving schemes [5, 6, 7, 8], researchers in the signal-processing community also tackled this problem based on measures like the received signal-to-noise ratio (SNR) or minimum mean square error (MMSE). Most of their results are based on the setup in which linear processing is at the relay and destination but not at the source, e.g., [9, 10]. Although these works provide meaningful approaches to the relay problem, it is not optimal not to have processing at the source from the fundamental perspective of data-rate maximization. To this end the processing at the source such as the input covariance function design should be incorporated together with the processing at the relay. (Once the processing at the source and relay is fixed, the optimal destination processing is automatically given for several well-known criteria.) However, the joint design of source and relay processing is a hard problem even in the linear Gaussian case, as shown in [3, 4]. In [3, 4], the authors considered general time-varying linear processing at the relay in Gaussian channels. Although they obtained the capacity for frequency-division strictly-causal linear relaying, the general linear relay case was not explored fully [3, 4]. In the general linear relay case, the problem is a sequence of non-convex optimization problems, and it is seemingly intractable. To circumvent such difficulty, in this paper we consider tractable and practical LTI filtering at the source and relay. We find that it is still a hard problem to obtain the capacity with a single-letter characterization even in this case because the search space still has countably infinite dimensions; optimal source and relay filters may have infinite impulse responses (IIRs). However, we provide a practical solution to design the source and relay filters jointly to maximize the transmission rate for general ISI Gaussian relay networks.

Under the LTI framework, the linear Gaussian relay problem can be formulated in the frequency domain using the Toeplitz distribution theorem [11, 12]. When the relay filter is given and there is no power constraint on the relay, the problem reduces to the classical ISI channel problem for which the optimal strategy is known as water-filling in the frequency domain [13, pp. 407 - 430]. However, the freedom to design the relay filter and the power constraint at the relay make the problem far more difficult than the classical ISI channel problem, especially when stability and causality constraints are imposed on the source and relay filters. Our approach to this problem is that we first convert the problem to a constrained optimization problem in a finite dimensional space by restricting the source and relay filters to the class of finite impulse response (FIR) filters as in most practical filtering applications, and then apply the projected subgradient method, initially proposed by Polyak [14] and fully developed by Yamada et al. [15, 16], to this problem. Numerical results show that our method performs well and yields a noticeable gain over the AF scheme in ISI relay channels.

Notations and Organizations

We will make use of standard notational conventions. Vectors and matrices are written in boldface with matrices in capitals. All vectors are column vectors. For a scalar , denotes its complex conjugate. For a matrix , , and indicate the transpose, Hermitian transpose and trace of , respectively, and denotes the -th row and -th column element of . denotes a diagonal matrix with elements . stands for the identity matrix of size (the subscript is omitted when unnecessary), and denotes a vector of all zero elements. For a vector , denotes its 2-norm. The notation means that is Gaussian-distributed with mean vector and covariance matrix . denotes the expectation. For two signal processes and , denotes the convolution of the two processes. , , and denote the sets of real numbers, integers, nonnegative integers and natural numbers, respectively. For two sets and , denotes the set minus operation. .

This paper is organized as follows. The system model and background are described in Section 2. In Section 3, the rate formula in the frequency domain is derived under the LTI model, and the performance of LTI relaying in flat-fading channels is investigated in Section 4. In Section 5, a joint source and relay filter design method is proposed based on the projected subgradient method, and its performance in ISI channels is examined in Section 6, followed by conclusions in Section 7.

2 System Model and Background

We consider the general discrete-time additive white Gaussian noise (AWGN) relay network composed of source, relay and destination nodes, as shown in Fig. 1, where the source and relay nodes have maximum available average power and , respectively. We assume that all propagation channels (i.e., the source-to-relay (S-R), relay-to-destination (R-D) and source-to-destination (S-D) channels) are linear, time-invariant and causal, and their impulse responses are absolutely summable, i.e., , and , where , and are the S-R, R-D and S-D channel impulse responses, respectively. Due to the absolute summability, the -transforms of the propagation channel impulse responses are well-defined and given by , and . Then, the received signals at the relay and destination at the -th symbol time are given by

(1)
(2)

respectively, where is the transmitted signal process at the source; and are the transmitted and received signal processes at the relay, respectively; is the received signal process at the destination; and the noise processes at the relay and at the destination are independent zero-mean white Gaussian processes with variance .

{psfrags} \psfragx[r] \psfragz1[c] \psfragy1[c] \psfragx1[c] \psfragz[c] \psfragy[l] \psfraghsr[c] \psfraghrd[c] \psfraghsd[c]

Figure 1: System model

We consider the linear and causal processing at the relay. The general causal linear processing at the relay is given by

(3)

for arbitrary linear combination coefficients , as considered in [3, 4]. However, such linear processing requires time-varying filtering at the relay, and is not readily realizable. Thus, in this paper, we restrict ourselves to the case of LTI causal filtering at the relay, as shown in Fig. 2.

{psfrags} \psfragsc[c]Source \psfragrel[c]Relay \psfragsz[c] \psfragrz[c] \psfragxt[c] \psfragx[c] \psfragz1[c] \psfragz[c] \psfragy[c] \psfraga[l] \psfrag1[c] \psfragb[c]

Figure 2: System model with linear time-invariant filters

In this case, the relay output is given by

(4)

where is the time-invariant impulse response of the relay filter and its -transform is given by . (In the case of strict causality, we have .) The received signal (4) at the relay can be written in matrix form as (5), and the filtering matrix in (5) has a Toeplitz structure.

(5)

We assume the stability (i.e., ) and realizability2 for the relay filter. Since all processing from the source and to the destination is linear and time-invariant, the received signal at the destination in the -domain is given by

(6)

where and are the -transforms of noise processes and , respectively.

2.1 Background

In this subsection, we briefly summarize some relevant results including the eigen-structure of Toeplitz matrices and the spectral factorization for the development in later sections. For a zero-mean3 stationary random process , the covariance sequence and its -spectrum are given by

(7)

respectively, where . The covariance matrix of a finite collection is given by

(8)
Theorem 1 (Asymptotic eigen-structure of Toeplitz covariance matrices [12], p. 135)

Let be an absolutely summable autocovariance sequence of a stationary process , let be its power spectral density (PSD), i.e., , and let be the matrix,

where . Then, for the covariance matrix , the components of converge to zero uniformly as (i.e. ), where is the discrete Fourier transform (DFT) matrix.

For even , we have a similar result with a slight modification. Theorem 1 simply states that the eigenvalues of the Toeplitz covariance matrix of a stationary process are the uniform samples of its spectrum. Using Theorem 1, the following can easily be shown.

Theorem 2 (Toeplitz distribution theorem [11], p. 65)

Let be the eigenvalues of the Toeplitz covariance matrix of a stationary process . Then,

(9)

for any continuous function .

In addition to the asymptotic eigen-structure of Toeplitz covariance matrices, we need some background in the spectral theory for stationary random processes, especially canonical spectral factorization.

Definition 1 (Canonical Spectral Factorization [18], p. 197)

Let be a rational z-spectrum of a finite power process and assume that is strictly positive. Then, the canonical spectral factorization of is given by

(10)

where is a unique stable, causal, monic and minimum-phase (SCAMP) filter (i.e., the zeros and poles of are strictly inside the unit circle and (or equivalently )), and . Here, denotes the para-Hermitian conjugate.

3 The Rate Formula in Frequency-Domain for LTI Relays

First, note that the overall channel model (6) with LTI relay filtering is still a linear additive stationary Gaussian noise channel. Thus, for a given relay filter, the overall channel with the LTI relay filter reduces back to the classical ISI channel with stationary Gaussian noise4. In this case, stationary Gaussian signal processes with well-defined spectra are sufficient to achieve the capacity [13, pp. 407 - 430]. Hence, we assume that the source (or input) process is a stationary Gaussian process. By concatenating symbols at the source up to time , we have

(11)

and vectors , and are constructed similarly for the relay and destination nodes. Then, the power constraints for the source and relay are respectively given by

(12)
(13)

where is the filtering matrix for the S-R channel constructed based on similar to in (5). Thus, the maximum rate with LTI relaying for block size is given by maximizing the mutual information between and over and under power constraints (12) and (13), and the capacity with LTI relaying is given by its limit

(14)

as [4], where

(15)

Here, and are the filtering matrices for the S-D and R-D channels, respectively. Note that (14) is still valid for general linear time-varying relay filtering with given by an arbitrary lower triangular matrix. As mentioned in [4], the computation of capacity and the design of capacity-achieving (or at least reasonable) and are difficult problems in the case of general linear causal relay filtering. In the time-varying case, if we increase by one, at least new variables appear (see (3)), and thus the complexity of the problem increases with the order of to make the problem difficult [3, 4]. In the LTI case with a stationary source process, however, we have only two new variables and for the increase of the problem size from to because of the Toeplitz structure of the covariance matrix in (8) and the filtering matrix in (5). Following the best input covariance matrix and relay filter for the problem size is equivalent to designing the best infinitely long autocovariance sequence and infinitely long relay filter first and then increasing the problem size. Thus, in the LTI case, we have

(16)

where the respective dependence of and on and is explicitly shown. Here, taking the limit of simplifies the problem significantly due to Theorem 2 since the eigenvalues are strictly positive due to the additive noise term and since is a continuous function of for . By Theorems 1 and 2 we have

(17)

where the input spectrum , since the eigenvalues of a covariance matrix are the samples of its spectrum and the determinant of a covariance matrix is the product of its eigenvalues. Here, we define the overall channel-to-noise power ratio (CNR) density as

(18)

where and are the numerator and denominator of the CNR density, respectively. Note that the CNR density captures the overall channel response from source to destination. When the CNR density is multiplied by the input signal PSD, the product becomes the overall SNR density at the destination. (This quantity will be used in later sections.) In addition to the rate formula (17) in the frequency domain, the power constraints can also be expressed in the frequency domain as . As , again by Theorems 1 and 2, the power constraints (12) and (13) are respectively given by

(19)
(20)

since the trace of a matrix is the sum of its eigenvalues. Thus, the LTI relay problem is summarized by (17), (19) and (20). Note that for a given relay filter the problem without the power constraint (20) reduces to the well-known ISI channel problem and the solution of is given by water-filling in the frequency domain [19]. However, the freedom to design and the relay power constraint (20) make the problem far more difficult than the simple ISI channel problem. To construct a practical method to solve this problem, we further assume that the input spectrum is also realizable. That is, its canonical spectral factorization is given by

(21)

where the SCAMP filter has a rational transfer function and, thus, is a rational spectrum. In this case, the source process can be modelled as the output of the stable and causal ARMA filter driven by a white Gaussian process with unit variance, as seen in Fig. 2. Thus, the rate maximization problem under LTI relaying with realizable input spectra now reduces to a joint design problem of LTI source and relay filters. Obtaining the capacity in a closed form still seems to be a difficult problem even in the LTI relay case. However, we propose a very effective and practical solution to this joint filter design problem in Section 5. Before we tackle this problem, we investigate the problem in the case that all S-D, S-R and R-D channels have flat frequency responses in the next section.

4 Examination of LTI Relaying in Flat-Fading Channels

In the case of flat fading, we have the system model (6) in which each of S-R, R-D and S-D channels has only one tap, i.e., , and , as considered in [3, 4]. Then, the received signal model in the -domain is given by

(22)

4.1 The One-Tap Relay Filter Case

First, consider the well-known AF relaying. In this case, we have

where and to satisfy the power constraints, and the received signal model is given by

(23)

Due to the simple data model (23), the achievable rate in this case is known and given by

(24)

and the optimal value of is explicitly given by

(25)

Now consider the one-tap LTI relay filter with an arbitrary delay:

(26)

for some integer 5, where the relay gain can be optimized under the power constraints. Note in the system model (22) that the relay filter affects both the channel gain and noise spectrum. However, in the one-tap relay filter case, the problem is simplified because the overall noise spectrum is white. In this case, the overall channel gain is given by and the overall noise spectrum is given by

since . Note that the overall noise process in this case is white and equivalent to that in the AF data model (23); both have the same variance . Thus, the spectrum of is given by

(27)

and the channel frequency response is explicitly given by a raised-cosine function:

(28)

Since for , from (19) and (20) the power constraints are given by

(29)
(30)

which are the same as those of the AF scheme with .

The problem with the given relay filter reduces to the simple ISI channel problem, and the optimal input spectrum is obtained by water-filling under the two simple power constraints (29) and (30). In the following theorem, we establish the optimality of the AF scheme within the class of all one-tap relay filters.

Theorem 3

Among all one-tap linear relay filters, i.e., with , the AF scheme with maximizes the achievable rate.

Proof: For a given , let

(31)

Then,

(32)
(33)
(34)
(35)

Here, (32) is obtained because . (33) is obtained because the feasible set satisfying the power constraint for is the same as that for when . (See (29) and (30).) (34) is obtained by Jensen’s inequality. Finally, (35) is obtained by the definition of in (24).

Theorem 3 states that the AF scheme with performs best within the class of one-tap relay filters with arbitrary delays. This is because the AF scheme achieves coherent signal combining between the two signal paths S-D and S-R-D. Instead of using the optimal water-filling source filter, we can also consider a simple channel-equalizing source filter. However, the performance in this case is bad, as shown in the following theorem.

Theorem 4

The achievable rate by an equalizing source filter for the one-tap relay filter is given by

(36)

regardless of the value of . Further, the supremum is given by achieved when .

Proof: We have , where is the -transform of the white Gaussian process with unit variance, and the equalizing source filter is given by

When , the overall channel response is SCAMP and, thus, the channel-equalizing source filter is also SCAMP. By the power constraint at the source, we have

(37)

because . Since for every integer , regardless of the value of . With the channel-equalizing source filter , the data model is given by , where and , and the corresponding achievable rate is given by (36). Now consider in (36). Its derivative with respect to (w.r.t.) is given by for all . Thus, the rate is maximized when .

    (a) (b)

    (c) (d)

Figure 3: Achievable rates of several schemes in flat-fading channels (, ) - (a) and (b): , , and (c) and (d): ,

Theorem 4 states that it is optimal to turn off the (one-tap) relay filter when the channel-equalizing filter is to be used at the source. Thus, using the channel-equalizing source filter is not a proper choice for relay networks.

Fig. 3 shows the achievable rates of several relaying schemes. For Fig. 3 (a) and (b), which show the same curves with two different x-axis ranges, we set and , as in [4]. It is seen that simple linear strictly causal schemes (one based on the filtering matrix in [4] and the other based on one-tap filtering ) can outperform the CF scheme in the low SNR region, as already known from [4]. In this case of and , the AF scheme achieves the cut-set upperbound for [6, Proposition 9]. It is interesting to observe that the simple linear scheme in [4] with performs better than filtering in some low SNR values, although the latter outperforms the former eventually at high SNR. Fig. 3 (c) and (d), again showing the same curves in two different x-axis ranges, show the achievable rates when . In this case, it is seen that there is a gap between the cut-set bound and the AF scheme. In all the cases, it is seen that the two strictly causal linear schemes (one based on two-symbol concatenation in [4] and the other based on one-tap LTI filtering ) do not outperform the AF scheme, as expected by Theorem 3.

4.2 The Multiple-Tap Relay Filter Case: Insights from Ideal Low-Pass Filtering Relays

In Section 4.1, it is shown that one-tap relay filters do not outperform the AF scheme in flat-fading channels. This is because any one-tap relay filter with a causal or non-causal non-zero delay cannot change the noise spectrum, but destroys the coherent signal combining that is available in the AF scheme. However, this is not the case when the relay filter has multiple taps. In this case, the overall noise spectrum as well as the channel gain spectrum in (22) can be shaped by the relay filter, and the LTI relaying scheme with multiple taps can outperform the AF scheme in flat-fading channels. However, the performance analysis in this case is far more difficult than that in the one-tap relay case, especially when the causality constraint is imposed on the relay filter. To circumvent this difficulty, in this subsection we relax the causality constraint on and consider the tractable ideal low-pass6 relay filtering to gain insights into the interaction between the source and relay filters and to assess the rate gain that can be obtained by multiple-tap relay filtering. The frequency response of the ideal low-pass relay filter is given by

(38)

where is the passband gain and is the cutoff frequency. For a given , the optimization problem (17,19,20) with the ideal low-pass relay filter is expressed as

(39)

subject to

(40)
(41)
(42)

where the even symmetry of spectra is used. Note that the problem is not jointly convex w.r.t. and for a given . However, we can still apply the Karush-Kuhn-Tucker (KKT) conditions to this problem to obtain the necessary conditions for optimality [20]. The Lagrangian of this problem is given by

(45)