Amplify-and-Forward in Wireless Relay Networks

# Amplify-and-Forward in Wireless Relay Networks

Samar Agnihotri, Sidharth Jaggi, and Minghua Chen

Email: {samar, jaggi, minghua}@ie.cuhk.edu.hk
Department of Information Engineering, The Chinese University of Hong Kong
###### Abstract

A general class of wireless relay networks with a single source-destination pair is considered. Intermediate nodes in the network employ an amplify-and-forward scheme to relay their input signals. In this case the overall input-output channel from the source via the relays to the destination effectively behaves as an intersymbol interference channel with colored noise. Unlike previous work we formulate the problem of the maximum achievable rate in this setting as an optimization problem with no assumption on the network size, topology, and received signal-to-noise ratio. Previous work considered only scenarios wherein relays use all their power to amplify their received signals. We demonstrate that this may not always maximize the maximal achievable rate in amplify-and-forward relay networks. The proposed formulation allows us to not only recover known results on the performance of the amplify-and-forward schemes for some simple relay networks but also characterize the performance of more complex amplify-and-forward relay networks which cannot be addressed in a straightforward manner using existing approaches.

Using cut-set arguments, we derive simple upper bounds on the capacity of general wireless relay networks. Through various examples, we show that a large class of amplify-and-forward relay networks can achieve rates within a constant factor of these upper bounds asymptotically in network parameters.

## I Introduction

Since their introduction in [1] Amplify-and-Forward (AF) relay schemes have been studied in the context of cooperative communication [2, 3], estimating the capacity of relay networks [4, 5, 6], and analog network coding [7, 8, 9, 10, 11]. For cooperative communication, AF schemes provide spatial diversity to fight against fading; for capacity estimation of relay networks, such schemes provide achievable lower bounds that are known to be optimal in some communication scenarios; and for analog network coding, given the broadcast nature of the wireless medium that allows the mixing of the signals in the air, these schemes provide a communication strategy that achieves high throughput with low computational complexity at internal nodes. In this paper, we concern ourselves mostly with the capacity analysis of a general class of Gaussian AF relay networks. Extensions of our method and results to cooperative communication and analog network coding scenarios is part of our future work.

In previous work, while analyzing the performance of AF schemes in relay networks one or more of the following assumptions have been made: networks with a small number of nodes [8, 10]; networks with simple topologies [4, 3, 5, 8, 10]; or relay operation in the high-SNR regime, [10]. However, for two reasons, we believe that it is important to characterize the performance of the AF schemes without such assumptions. First, we feel that for a scheme such as amplify-and-forward that allows one to exploit the broadcast nature of the wireless medium such assumptions on the size and topology may result in lower achievable performance than otherwise. Second, even in the low-SNR regimes amplify-and-forward can be capacity-achieving relay strategy in some scenarios, [5]. Therefore, a framework to address the performance of AF schemes in general wireless relay networks is desired.

However, one major issue with constructing such a framework is the following. In general wireless relay networks with AF relaying, the resulting input-output channel between the source and the destination is an intersymbol interference (ISI) channel ([4, 10]) with colored noise. This is because both the source signal and the noise introduced at the relay nodes may reach the destination via multiple paths with differing delays. Without the assumptions above, this results in a formidable problem to analyze with the existing methods [4].

Our main contribution is that we provide a framework to compute the maximum achievable rate with AF schemes for a class of general wireless relay networks, namely Gaussian relay networks. This framework casts the problem of computing the maximum rate achievable with AF relay networks as an optimization problem. Our work shows that amplifying the received signal to the maximum possible value at intermediate nodes might result in sub-optimal end-to-end throughput. Also, we establish the generality of the proposed formulation by showing that it allows us to derive in a unified and simple manner not only the various existing results on the performance of simple AF relay networks but also new results for more complex networks that cannot be addressed in a straightforward manner with existing methods. We show through various examples that for a large class of relay networks the AF schemes can achieve rates within a constant factor of the cut-set upper-bounds on the capacity of general wireless relay networks.

The paper is organized as follows. In Section II we introduce the general class of Gaussian AF relay networks addressed in this paper. In Section III we formulate the problem of maximum rate achievable via AF schemes in these networks. In Section IV we compute the rates achievable via AF schemes for two instances of such relay networks under various communication scenarios, and then in Section V we discuss the asymptotic behavior of the gap between these rates and the corresponding upper bounds on the capacity of general wireless relay networks computed there. Section VI concludes the paper with a summary.

## Ii System Model

Let us consider a -node wireless relay network with source destination and relays as a directed graph with bidirectional links, as shown in Figure 1. Each node in the network is assumed to have a single antenna. Let us assume that the degree of the source node is , with it being connected to the destination node and a subset of the relay nodes, . Similarly, let us assume that the degree of the destination node is , with it being connected to the source node and a subset of the relay nodes, . In general, .

At instant , the channel output at node , is

 yi[n]=∑j∈N(i)hjixj[n]+zi[n],−∞

where is the channel input of the node in the neighbor set of node . In (1), is a real number representing the channel gain along the link from relay to relay . It is assumed to be fixed (for example, as in a single realization of a fading process) and known throughout the network. Further, is a sequence (in ) of independently and identically distributed (i.i.d.) Gaussian random variables with zero mean and variance . We also assume that are independent of the input signal and of each other. The source symbols , are i.i.d. Gaussian random variables with zero mean and variance that satisfies an average source power constraint, . We assume that the relay node’s transmit power is constrained as:

 E[x2i[n]]≤Pi,−∞

In a general wireless relay network there may exists cycles. If a relay merely amplifies and forwards its received signal in such scenarios then it may be use a significant fraction of its power budget on forwarding the previously forwarded information. Therefore, motivated by some work on analog network coding, such as [12], we propose the following relay operation to allow the relays to expend their transmit power in forwarding only the “new” information.

Relay operation: We assume that each relay node maintains a buffer of signals it forwarded previously. Therefore, each relay node , after receiving the channel output at time instant executes the following series of steps:

Step 1: Obtain the residual signal from its input by subtracting the contributions of previously forwarded signals (if any) from .

Step 2: Compute the power of the residual signal .

Step 3: At instant transmit the scaled version of the residual signal of its input at time instant :

 xi[n+1]=βiy′i[n],0≤β2i≤β2i,max=Pi/P′R,i, (3)

where the is the scaling factor111Note that, in general, may depend on relay’s past observations. . However, due to practical considerations, such as low-complexity operation, we do not consider such scenarios here.. Let the network-wide amplification vector for the relay nodes be denoted as .

Remark 1: One of the major advantages of this relay operation is that by subtracting the previously forwarded signal from its input, a relay node expends its power in forwarding only the “new” information. In a general wireless relay network there may exists cycles. In such scenarios if a relay merely forwards its received signal then it may result in the relay forwarding the scaled version of the linear combination of two or more of its previously forwarded signals. This may provide higher achievable rate as in [13].

Using (1) and (3), the input-output channel between the source and destination can be written as an intersymbol interference (ISI) channel that at instant is given by

 yt[n] = hstxs[n]+zt[n] + Ds∑d=1[∑(i1,…,id)∈Kdhsi1βi1hi1i2…hid−1idβidhidt]xs[n−d] + D1∑d=1[∑(i1,…,id)∈K1,dβ1h1i1…hid−1idβidhidt]z1[n−d] ⋮ + DM∑d=1[∑(i1,…,id)∈KM,dβMhMi1…hid−1idβidhidt]zM[n−d],

where , is the set of -tuples of node indices corresponding to all paths from the source to the destination with path delay and is the length of the longest such path. Note that along such paths . Similarly, , is the set of -tuples of node indices corresponding to all paths from the relay to the destination with path delay , is the length of the longest such path from relay to the destination. It should be noted that .

Let us introduce modified channel gains as follows. For all the paths between the source and the destination :

 h0=hst (5) hd=∑(i1,…,id)∈Kdhsi1βi1hi1i2…hid−1idβidhidt,1≤d≤Ds

For all the paths between the -relay, , and the destination :

 hm,0=0, (6) hm,d=∑(i1,…,id)∈Km,dβmhmi1…hid−1idβidhidt,1≤d≤Dm

Remark 2: Note that though there may be exponentially large number of paths between the source and the destination as well as between a relay and the destination, the modified channel gains for all such paths as in (5) and (6) can be efficiently computed using the line-graphs [14].

In terms of these modified channel parameters, the source-destination ISI channel in (II) can be written as:

 yt[n]=Ds∑j=0hjxs[n−j]+D1∑j=0h1,jz1[n−j]+…+DM∑j=0hM,jzM[n−j]+zt[n] (7)

Before proceeding further, let us introduce two special cases of the general class of relay networks introduced in the beginning of this section. In the rest of this paper, we illustrate various concepts and derive some results using these two special networks.

Type A Network: For one source-destination pair and relays, Type A network is defined as: , where and . In other words, the source node shares an edge with the destination node and each of relay nodes. Similarly, the destination node shares an edge with the source node and each of relay nodes. However, no pair of relay nodes share an edge between themselves. Figure 2 illustrates an instance of Type A networks.

Type B Network: For one source-destination pair and relays, Type B networks are defined as: , where and . In other words, as in Type-A networks, the source node shares an edge with the destination node and each of relay nodes. Similarly, the destination node shares an edge with the source node and each of relay nodes. However, relay shares an edge with node, for all . Figure 3 illustrates an instance of Type B networks.

## Iii Achievable rates for the source-destination ISI channel in general AF Relay Networks

We first derive the expression for the achievable rate for the source-destination channel in (7) for a given amplification-vector and then formulate the problem of maximizing the achievable rate over the domain of feasible .

###### Lemma 1

For given length- vector , the achievable rate for the channel in (7) with i.i.d. Gaussian input is:

 (8)

where

 H(λ)=Ds∑j=0hje−ijλ,Hm(λ)=Dm∑j=0hm,je−ijλ,i=√−1 (9)
###### Proof:

In [15] a Discrete Fourier Transform (DFT) based formalism is developed to compute the capacity of Gaussian channel with source intersymbol interference (ISI). We compute the maximum achievable rate for the channel in (7) for a given amplification-vector by generalizing this formalism to also include the ISI channel for the Gaussian noise at each relay node resulting in colored Gaussian noise at the destination222Note that the capacity of discrete-time Gaussian ISI channel was known prior to [15], for example in [25, 26, 27]. However, the analysis in these papers, based on the asymptotic properties of Toeplitz forms [28, 29], is not easily amenable to derive the capacity results of the ISI channel with colored Gaussian noise. Therefore, like [30, 31], we too use more accessible DFT based formalism developed in [15] to compute the maximum achievable rate for the ISI channel with colored Gaussian noise in (7).. The details of the proof are in the Appendix A. \qed

Remark 3: The derivation of an expression for with jointly Gaussian inputs is similar to the proof of Lemma 1. However as such an expression does not aid in the presentation of our ideas we do not discuss it in this paper.

For a given network-wide amplification vector , the achievable information rate is given by . Therefore the maximum information-rate achievable in an AF relay network with i.i.d. Gaussian input is defined as the maximum of over all feasible , subject to per relay-node amplification constraint (3). In other words:

 (P1): IAF(Ps)def=maxβ:0≤β2i≤β2i,maxI(Ps,β) (10)

Substituting for and from (9) in (8), we can rewrite problem (10) equivalently as:

 (P2): IAF(Ps) = maxβ:0≤β2i≤β2i,maxI(Ps,β) (11) I(Ps,β) = 12π∫π0log[1+Psσ2∑Dsi=0Aicos(iλ)∑Dni=0Bicos(iλ)]dλ

The coefficients , and , are defined in Appendix B.

The formulation of the problem P2 is illustrated in Appendix C for the Type A and Type B relay networks introduced in the previous section.

In general AF relay networks the simultaneous relay transmissions may interfere and if the relays always amplify the received signals to the maximum possible then it may result in sub-optimal end-to-end throughput. Therefore the scaling factor for each relay must be optimally chosen to maximize the achievable rate. This is emphasized by P1 and its significance is illustrated by the following example.

Example 1: Let us consider the relay network in Figure 4. Let . Let and noise variance at each node. Therefore, we have

 β21,max=P1h2s1Ps+σ2=0.99 β22,max=P2h2s2Ps+σ2=50.0

In this case from (10), we have the following rate maximization problem:

 IAF=maxβ1,β212log[1+100(β1+0.1β2)21+β21+β22]

subject to constraints and .

The optimal solution of this problem is . The objective function is plotted in the Figure 5 for . Therefore, it follows that in this case is not the optimal amplification factor.  ∎

With this observation and the definition of the relay operation given in the previous section, it is appropriate to call the forwarding scheme proposed in this paper as subtract-scale-and-forward.

Remark 4 (the computational complexity of the problem P2): Writing the objective function of the problem P2 as sum of ratios, as in (A), we can show from [16] that each of these ratios is neither quasiconcave333A function defined on a convex subset S of real vector space is quasiconcave if whenever and , then , [16]. (therefore, more than one local optimum may exist) nor quasiconvex (therefore, the globally optimal solution may not exist at an extreme point or on the boundary of the domain of optimization). The sum of such non-quasiconcave and non-quasiconvex ratios is also non-quasiconcave and non-quasiconvex. Therefore, the problem P2 is a global optimization problem, specifically it belongs to a subset of global optimization problems, called sum-of-ratios problems, which are supposed be hard, in general, [17]. Further, in general, , depends on . This dependence of on makes the constraint set in (11) non-convex for all but Type A relay networks. Based on these arguments, we conjecture the problem P2 to be computationally hard. However, at present we do not have a formal computational complexity-theoretic proof and constructing such a proof is the part of our future work.

Remark 5 (the approximation schemes to solve the problem P2): The lack of the exact complexity classification of the optimization problem in (11) notwithstanding, we envision solving this problem to be an important step towards characterizing the maximum rates achievable with various other relaying schemes in general wireless relay networks as such schemes also result in problem formulations similar to (11). In [32], we concern ourselves with developing a unified framework to efficiently approximate the optimization problems such as (11).

In the following we show that the problem formulation above not only allows us to reproduce existing results on the achievable rates for some special classes of amplify-and-forward wireless relay networks in a simple manner, but also allows us to compute the achievable rates for much broader class of amplify-and-forward wireless relay networks which could not be hitherto addressed with existing methods.

## Iv Approximating the Maximum Achievable Rate IAF(Ps) for Amplify-and-Forward Relay Networks

Let us consider the problem P2 when the relays operate instantaneously as in [8], that is, the relays amplify-and-forward their input signals without delay. Therefore, for relay node , we have . Note the possible system instability resulting from this assumption is avoided by the relay-operation (buffering and subtracting of the previously forwarded signals) as given in the Section II. With this assumption, P2 reduces to (after setting in (11) and then integrating):

 (P3): IAF(Ps)=maxβ:0≤β2i≤β2i,max12log[1+Psσ2∑Dsi=0Ai∑Dni=0Bi] (12)

The problem P3 can also be derived directly from the channel model in (7), however we do not discuss that derivation here for the sake of brevity.

Given the particular form of functional dependence of s and s on the components of the amplification vector (as follows from (II), (5), (B.2), and (B)) and dependence of s on each other, the problem P3 is a Geometric Program (GP) [18], where both, the objective function as well as the constraints are expressed in terms of posynomials. However, minimizing or upper bounding (equivalently, maximizing or lower bounding) a ratio of two posynomials is a non-convex problem that is intrinsically intractable [19, 20]444Note that in [19, 20] the objective function can be written as an inverted posynomial in high-SINR regime. Therefore the problem is efficiently solvable using GP methods. It is only in low to moderate SINR regimes where the objective function cannot be so written, the problem is NP-hard. However, the objective function in (12) involves the ratio of posynomials for all SNR values. Therefore, in general, the problem is hard irrespective of SNR value.. Therefore, for a general relay network, it is not possible to exactly solve or lower bound the problem P3 in a computationally efficient manner. However, unlike the problem P2 in (11), for the problem P3, efficient approximation schemes exist that solve such problems iteratively by solving a series of GPs, as discussed in [19, Section 3.3]. Also, for some specific relay networks, under some assumptions, we can efficiently compute the lower bounds, as we show next.

Let us consider the problem P3 when the relay nodes are constrained to use the same amplification-factor, that is, , for all . In the practical setting, this assumption considerably simplifies the system-design with set to one particular value for all relay nodes. Then, (P3) reduces to

 (P4): IAF(Ps)=max0≤β2≤β2maxIAF(Ps,β) (13) IAF(Ps,β)=12log[1+Psσ2∑Dsi=0Ai∑Dni=0Bi] (14)

Note that the solution of (P3) cannot be smaller than the solution of (P4) because the set of feasible , for (P4) is a subset of the set of feasible for (P3).

Note 1: In general, . Therefore . However, as we are considering the case of equal , , so we have or .

Now we discuss solving the problem P4 for Type-A and Type-B relay networks introduced in Section II, in different communication scenarios. The proofs of various propositions and lemmas are provided in the Appendix D.

### Iv-a Type-A Relay Network

Let us first consider Type-A relay network as in Figure 2. For such networks, we solve the problem P4 in the following two scenarios.

Scenario 1 (No attenuation network): Let us assume that there is no attenuation along any link in the network, that is, for all . The problem P4 in this case is:

###### Proposition 1
 IAF(Ps)=max0≤β2≤β2max12log[1+Psσ2(1+Mβ)21+Mβ2]
###### Lemma 2

attains its global maximum at .

Now let us consider two particular ways in which varies with network size.

Scenario 1, Case A (Increasing relay power): For the given network with relay nodes, let us assume that the sum power of the relay nodes is constrained as follows:

 M∑m=1E[X2m]≤M∑m=1Pi≤Mu+1Q,u>0,Q=constant

So . From Lemma 2, we have for .

 IAF(Ps)=⎧⎪⎨⎪⎩12log[1+Psσ2(1+M)], if βmax≥1,12log[1+Psσ2(M+2βmax)], % otherwise (15)

Scenario 1, Case B (Constant total relay power): Let us consider the case where the sum power of relay nodes is fixed irrespective of the number of relay nodes, that is . Therefore, we set . As , for sufficiently large , . Therefore, from Lemma 2, maximizes the achievable rate and we have for

 IAF(Ps)=12log[1+Psσ2Q1+QM] (16)

Scenario 2 (Bounded channel gains): In Scenario 1, we considered no attenuation relay networks. Now, let us consider the scenario where the channel gains are arbitrary, but strictly bounded, , . The problem P4 in this case is:

###### Proposition 2
 IAF(Ps)=max0≤β2≤β2max12log[1+Psσ2(hst+β∑Mi=1hsihit)21+β2∑Mi=1h2it]
###### Lemma 3

attains its global maximum at .

Increasing relay power: Let us consider the increasing total relay power scenario as in Scenario 1, Case A. Let . In this case, following Lemma 3, we obtain the following lower bound on the achievable rate as :

 IAF(Ps) > 12log[1+Psσ2(Mhmin+1)h2s,maxhminhmax], if βmax≥∑Mi=1hsihithst∑Mi=1h2it, IAF(Ps) > 12log[1+Psσ2(Mhmin+1βmax)h2s,maxhminhmax], otherwise,

where , , and .

### Iv-B Type-B Relay Network

Let us consider the Type-B relay network as in Figure 3. For such networks, we consider the no-attenuation scenario where all channel gains are set to unity, that is, , as well as , . The problem P4 in this case is:

###### Proposition 3
 IAF(Ps,β) = 12log[1+Psσ2(1+βMβM−1β−1−MβM+1β−1+β2βM−1(β−1)2)21+∑Mi=1β2(βM−(i−1)−1β−1)2] = IAF(Ps) = max0≤β≤βmaxIAF(Ps,β) (18)

It can be proved that the objective function is quasiconcave, therefore a unique global maximum exists. Let that solves (18) be denoted as . However, obtaining a closed-form expression for does not appear straightforward, though it can be numerically computed for any .

Constant total relay power: Let us consider the case where the sum power of relay nodes is fixed irrespective of the number of relay nodes as in Scenario 1, Case B. Let . As , for sufficiently large , . Therefore, maximizes the achievable rate and we have the following rate achievable asymptotically as

 IAF(Ps)=12log[1+Psσ2QM1+Q1+√Q/M1−√Q/M] (19)

## V Asymptotic Capacity

In the previous two sections, we formulated the problem of maximum achievable rate for AF relay networks and then we computed explicit lower bounds to the capacity of two specific AF relay networks in various communication scenarios. In this section, we first derive an upper bound to the capacity of the general relay networks we address in this paper as introduced in the Section II. We then discuss the asymptotic behavior of the gap between this upper bound and the lower bounds computed for two specific relay networks in the previous section.

### V-a Upper bounds to Capacity of Relay Networks

In [4], an upper bound to the capacity of the relay network of Type A is computed using a weaker corollary of the cut-set theorem [22, Theorem 15.10.1] and the capacity formula for Gaussian vector channels with fixed transfer function [23]. Using this corollary and the capacity formula, we can also compute the upper bound to the capacity of the general relay network introduced in the Section II, as stated in the following proposition.

###### Proposition 4

The capacity of a general wireless relay network is upper-bounded as , where and are the upper bounds on the capacity of the broadcast cut and multiple-access cut respectively, as in the Figure 6, and are given as follows

 CBC = log[1+Psσ2(h2st+∑i∈Ssh2si)] CMAC = log[1+Ps+∑i∈StPiσ2(h2st+∑i∈Sth2it)]
###### Proof:

The details of the proof are provided in the Appendix E. \qed

Remark 6: Proposition 3 in [4] can be obtained as a special case of the above proposition by setting .

### V-B Type A Relay Networks

No attenuation, increasing relay power: In this case, the broadcast bound is always asymptotically smaller than the multiple-access bound , as follows from

 2CMAC−CBC≈1+Mu+1QP, for large M.

Therefore it suffices to compute the asymptotic gap between and the lower bound in (15). In fact, in this case we have

 CBC−2IAF(Ps)=0, for all M≥1

The actual capacity of the relay network in this case is bounded by .

No attenuation, constant total relay power: In this case also the broadcast bound is asymptotically smaller than the multiple access-bound , as shown below

 limM→∞2CMAC−CBC=1+QP

Therefore we only address the asymptotic gap between and the lower bound in (16). We have

 limM→∞CBC−2IAF(Ps)=12log(1+1/Q)

The actual capacity of the relay network in this case is bounded by and the bound gets tighter with increasing .

Bounded channel gains: The gap between and lower bound of achievable rate in (IV-A) is bounded asymptotically as:

 limM→∞CBC−2IAF(Ps)≤12log[hmaxh2min]

The apparent looseness of the gaps computed above compared to the corresponding gaps in [4] arises from the series of simplifications made to reduce the problem (P2) to the problem (P4) and the particular definition of lower bound used in [4].

### V-C Type B Relay Networks

No attenuation, constant total relay power: As claimed above, the upper-bound in Proposition 4 holds for Type B networks too. Therefore we only address the asymptotic gap between and the lower bound in (19). We have

 limM→∞CBC−2IAF(Ps)=12log(1+1/Q)

The actual capacity of the relay network in this case too is bounded by and bound gets tighter with increasing .

## Vi Conclusion and Future Work

We provide a framework to analyze the performance of the AF relay schemes in a general class of wireless relay networks. We demonstrate the effectiveness of the proposed framework in two ways. We first show that some well-established results for Gaussian relay networks (for example, those in [4]) can be derived in a much simpler manner. Then, we show that the achievable rates for more complex Gaussian relay networks, which cannot be addressed with existing methods, can be derived straightforwardly. The proposed framework allows for the computation of tighter amplify-and-forward lower bounds to the capacity of Gaussian relay networks.

Future Work: We have provided the analysis of AF scheme for different scenarios with the assumption of instantaneous relay operation. However, the resulting performance is not optimal when delay in relay operation is included, as the constant power allocation is not optimal for the original problem (11). We also want to formally prove the hardness of optimization problem (11) and develop efficient approximation schemes for it. An extension of our work also facilitates the computation of achievable rates for analog network coding scenarios for non-layered networks and in low to moderate SNR regimes that cannot be addressed with existing approaches. We plan to address it in detail in our future work.

## Acknowldegment

This work has been supported by the CERG grant 412207.

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## Appendix A Proof of Lemma 1

Following Gallager [24], we define the information rate for the channel in (7) as

 I(Ps,β)=limN→∞IN(Ps,β), (A.1)

where

 IN(Ps,β)=N−1I(xs[0,N−1];yt[0,N−1]), (A.2)

with the assumption

 x[−Ds,−1]=(0,…,0)

It should be noted that as is finite, this assumption does not affect , but it simplifies the analysis as we show later.

Following [15], we introduce a new channel model whose maximum achievable rate can be easily computed. The capacity of this channel can be, subsequently, used to compute . The output of the new channel at time instant is given as

 ~yt[n] = N−1∑j=0~hjxs[(n−j)modN] +N−1∑j=0~h1,jz1[(n−j)modN] +…+N−1∑j=0~hM,jzM[(n−j)modN] +zt[n],0≤n

where . Extending the unit-sample response of the channel for signal with zeros, we define . We can similarly extend the corresponding unit-sample responses of the channel for noise samples and define , , , , respectively. With the definitions of these extended sample responses, we can rewrite (A) symbolically as

 ~yt[0,N−1] = +zt[0,N−1],

where denotes the circular convolution operator. The transmit power constraint (2) leads to the power constraint

 E[x2i[n]]≤Pi,0≤n

for the new channel model.

The information rate for the channel defined by (A) and (A.5) is defined as

 ~IN(Ps,β)=N−1I(xs[0,N−1];~yt[0,N−1]) (A.6)

Taking the Discrete Fourier Transform (DFT) on both sides of (A) we have

 ~Yi=~HiXi+~H1,iZ1,i+…+~HM,iZM,i+Zi,0≤i

where , , , , , , , , and are the components of , , , , , , , and , respectively.

Dividing both sides of this by and transforming the resulting first components with the transformation [15, eq. (24)], we get the following equivalent form of (A) in the transform domain:

 Y′i=X′i+Z′1,i+…+Z′M,i+Z′i,0≤i

where , , , , are obtained from transform [15, eq. (24)] with , , , , and , respectively.

Note that

 ~H1,i/~Hi = ~H∗1,N−i/~H∗N−i,0≤i

where denotes the complex conjugate. Therefore, it follows from [15, Lemma 1] that , in (A.8) are statistically independent zero mean Gaussian random variables with variance

 σ2m,i=Nσ2|~Hm,i|2|~Hi|2,0≤i

Similarly, are statistically independent zero mean Gaussian random variables with variance

 σ2i=Nσ21|~Hi|2,0≤i

Therefore, the equivalent transform domain channel model (A.8) is a set of parallel discrete memoryless additive Gaussian noise channels with total noise variance

 σ2i+M∑m=1σ2m,i=Nσ21+∑Mm=1|~Hm,i|2|~Hi|2,0≤i

Further, using [15, Lemma 2], it follows that the transformed inputs , are i.i.d. zero mean Gaussian random variables with variance . Therefore, the equivalent transform-domain channel model in (A.8) is a set of parallel, independent, discrete, memoryless additive Gaussian noise channels with zero mean Gaussian inputs of variance and total noise variance given in (A.11). This implies that the average mutual information of component channel is then the capacity of point-to-point AWGN channel, given by

 I(X′i,Y′i)=12log[1+Psσ2|~Hi|21+∑Mm=1|~Hm,i|2] (A.12)

As the component channels are mutually independent, therefore in (A.6) is given by

 ~IN(Ps,β)=12NN−1∑i=0log[1+Psσ2|~Hi|21+∑Mm=1|~Hm,i|2] (A.13)

Further, [15, Theorem 4] states that

 (1−Ds/N)IN−Ds(Ps,β) ≤ ~IN(Ps,β) ≤ (1+Ds/N)IN+Ds(Ps,β)

Therefore, we finally have in (A.1) as

 I(Ps,β) = limN→∞~IN(Ps,β) = limN→∞12NN−1∑i=0log[1+P