Power Allocation for the Base Matrix of Spatially Coupled Sparse Regression Codes

Nian Guo, Shansuo Liang, Wei Han Nian Guo, Shansuo Liang, and Wei Han are with Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co. LTD., Hong Kong SAR, China. E-mail: {guonian4, liang.shansuo, harvey.huawei}@huawei.com.

Abstract

We investigate power allocation for the base matrix of a spatially coupled sparse regression code (SC-SPARC) for reliable communications over an additive white Gaussian noise channel. A conventional SC-SPARC allocates power uniformly to the non-zero entries of its base matrix. Yet, to achieve the channel capacity with uniform power allocation, the coupling width and the coupling length of the base matrix must satisfy regularity conditions and tend to infinity as the rate approaches the capacity. For a base matrix with a pair of finite and arbitrarily chosen coupling width and coupling length, we propose a novel power allocation policy, termed V-power allocation. V-power allocation puts more power to the outer columns of the base matrix to jumpstart the decoding process and less power to the inner columns, resembling the shape of the letter V. We show that V-power allocation outperforms uniform power allocation since it ensures successful decoding for a wider range of signal-to-noise ratios given a code rate in the limit of large blocklength. In the finite blocklength regime, we show by simulations that power allocations imitating the shape of the letter V improve the error performance of a SC-SPARC.

I Introduction

For reliable communications over an additive white Gaussian noise (AWGN) channel, Joseph and Barron [1] designed the sparse regression code (SPARC). It forms a codeword by multiplying a design matrix by a sparse message. The message is sparse as it is segmented into several sections and each section contains only one non-zero entry. The codeword is passed through an AWGN channel subject to an average power constraint. With uniform power allocation across the non-zero entries of a message and a maximum likelihood decoder, a SPARC asymptotically achieves the channel capacity of the AWGN channel [1]. To overcome the complexity barrier of the maximum likelihood decoder, the approximate message passing (AMP) decoder with polynomial complexity has been proposed [2]–[4]. Its decoding error is closely tracked by the state evolution (SE) and it outperforms other low-complexity decoders [5][6] in terms of the finite-blocklength error rates. By judiciously allocating power to the non-zero entries of a sparse message, SPARCs with AMP decoding continue to achieve the channel capacity [3]. For example, iterative power allocation [7] uses the asymptotic SE of the AMP decoder to decide the power allocation for a message section by section.

By introducing a spatial coupling structure to the design matrix, SC-SPARCs with AMP decoding not only achieve the channel capacity [8][9] but also display a better error performance compared to power-allocated SPARCs [4][10]. Similar to the graph-lifting of SC-LDPC codes [11][12], the design matrix of a SC-SPARC is constructed from a base matrix. Each entry of the base matrix is expanded as a Gaussian submatrix in the design matrix, and the variance of the Gaussian entry is determined by the corresponding entry in the base matrix. The coupling structure of the base matrix is determined by a coupling pair comprising a coupling width and a coupling length.

Existing works on SC-SPARCs commonly assumed that the power is uniformly allocated to the non-zero entries of the message as well as the base matrix, e.g., [7][9][10][13]. For such uniform power allocation (UPA), a decoding phenomenon termed sliding window is observed [9][13], namely, the decoding propagates from two sides to the middle of a message in a symmetric fashion. Once the outer parts of a message are successfully decoded, they act as perfect side information that facilitates the decoding of the inner parts of the message. This phenomenon is used as a decoding techinque termed seed to boost the decoding performance of SC-SPARCs [8].

While UPA is sufficient for a SC-SPARC with AMP decoding to achieve the channel capacity, the coupling pair of the base matrix must satisfy regularity conditions and tend to infinity as the rate approaches the channel capacity [8][9]. Yet, in practical implementations, the coupling pair is finite and arbitrary. Given a finite coupling pair, it has been observed that UPA might be inefficient and causes AMP decoding failure. Thus, it is of practical interest to design a power allocation policy for a base matrix with a finite coupling pair to ensure successful decoding for a wide range of power and code rates.

We propose a novel power allocation policy–V-power allocation (VPA)–for the base matrix of a SC-SPARC with AMP decoding. Its power allocation is non-increasing from the outer columns to the middle column of the base matrix, resembling the shape of the letter V. Similar to iterative power allocation [7], VPA leverages the asymptotic SE of the AMP decoder to tell whether a SC-SPARC ensures successful decoding in the limit of large blocklength. Dissimilar to conventional power allocation policies that vary the non-zero coefficients of a message, VPA only varies the non-zero entries of a base matrix. To measure the performance of a power allocation policy for the base matrix, we define a power-rate function (PRF). Given a finite coupling pair, a channel noise variance, and a rate, the PRF quantifies the minimum power so that a SC-SPARC with a power allocation policy ensures successful decoding for all power above it. We derive the PRFs for UPA and VPA, respectively, and we show that VPA outperforms UPA in terms of the PRF, meaning that VPA ensures successful decoding for a larger range of power. While VPA is designed in the infinite blocklength regime, we use simulations to show that a VPA-like power allocation improves the finite-blocklength block error rates of a SC-SPARC.

Notations: For a positive integer $n$ , we denote $[n]\triangleq\{1,2,\dots,n\}$ . For a matrix $\mathsf{W}$ , we denote by $\mathsf{W}_{rc}$ the entry at the $r$ -th row and the $c$ -th column. For a sequence $a_{1},a_{2},\dots$ , we denote $\{a_{i}\}_{i=p}^{q}\triangleq\{a_{p},a_{p+1},\dots,a_{q}\}$ .

II Spatially coupled sparse regression codes

II-A Encoder

The encoder of a SC-SPARC forms a codeword $\bm{x}\in\mathbb{R}^{n}$ by multiplying a message vector $\bm{\beta}\in\mathbb{R}^{ML}$ by a design matrix $\mathsf{A}\in\mathbb{R}^{n\times ML}$ ,

\displaystyle\bm{x}=\mathsf{A}\bm{\beta},

(1)

and the codeword is subject to an average power constraint

\displaystyle\frac{1}{n}\mathbb{E}[||\bm{x}||^{2}]=P.

(2)

The message $\bm{\beta}$ is a sparse vector of length $ML$ . It consists of $L$ length- $M$ sections. In each section $\ell=1,2,\dots,L$ , there is only one non-zero entry, whose value is set a priori. Since the information is carried only by the indices of the non-zero entries, the alphabet size of $\bm{\beta}$ is $M^{L}$ . As we will vary the variances of the entries of design matrix $\mathsf{A}$ by varying the power allocation for the base matrix, we set all the non-zero coefficients of $\bm{\beta}$ to $1$ without loss of generality.

The design matrix $\mathsf{A}$ , as shown in Fig. 1, is constructed from a base matrix $\mathsf{W}$ . The base matrix serves as a protograph for the design matrix. Each entry $\mathsf{W}_{rc}$ of base matrix $\mathsf{W}$ is expanded as an $M_{R}\times M_{C}$ submatrix of design matrix $\mathsf{A}$ , whose entries are i.i.d. Gaussian random variables $\mathcal{N}\left(0,\frac{1}{L}\mathsf{W}_{rc}\right)$ . A column block in $\mathsf{A}$ corresponds to a set of $M_{C}$ columns that are expanded from one column in $\mathsf{W}$ . A row block in $\mathsf{A}$ corresponds to a set of $M_{R}$ rows that are expanded from one row in $\mathsf{W}$ . The design matrix $\mathsf{A}$ contains $L_{C}$ columns blocks and $L_{R}$ row blocks. It holds that $M_{C}L_{C}=ML$ , $n=M_{R}L_{R}$ .

The rate of a SC-SPARC is defined as

\displaystyle R=\frac{L\log M}{n}~{}\text{(nats per channel use)}.

(3)

In this work, we focus on a class of band-diagonal base matrices defined below, which is introduced in [9]. We denote by $\omega$ and $\Lambda$ the coupling width and the coupling length of the base matrix, respectively.

Definition 1.

An $\left(\omega,\Lambda,P\right)$ base matrix $\mathsf{W}$ is specified by the following properties.

i)

The base matrix $\mathsf{W}$ is of size $L_{R}\times L_{C}$ , where $L_{C}\triangleq\Lambda$ , $L_{R}\triangleq\omega+\Lambda-1$ , $\Lambda\geq 2\omega-1$ ;
ii)

Given any column $c\in[L_{C}]$ , the non-zero entries are only at rows $c\leq r\leq c+\omega-1$ ;
iii)

The entries of $\mathsf{W}$ satisfy the average power constraint (2),

$\displaystyle\frac{1}{L_{R}L_{C}}\sum_{r=1}^{L_{R}}\sum_{c=1}^{L_{C}}\mathsf{W}_{rc}=P.$ (4)

Refer to caption — Figure 1: Base matrix and design matrix of a SC-SPARC. The base matrix has coupling width $\omega=3$ and coupling length $\Lambda=7$ .

II-B Decoder

The codeword $\bm{x}$ (1) is transmitted through an AWGN channel yielding $\bm{y}=\bm{x}+\bm{w}$ , where $\bm{w}$ is a vector of $n$ i.i.d. Gaussian random variables each with zero mean and variance $\sigma^{2}$ . The AMP decoder iteratively estimates the message $\bm{\beta}$ from the channel output $\bm{y}\in\mathbb{R}^{n}$ as follows [9, Section III]. At iteration $t=0$ , the AMP decoder initializes the estiamte of $\bm{\beta}$ as $\bm{\beta}^{0}=\bm{0}$ and initilizes two vectors $\bm{v}^{0}=\bm{0}$ , $\bm{z}^{-1}=\bm{0}$ . At iterations $t=1,2,\dots$ , the AMP decoder calculates the estimate $\bm{\beta}^{t}$ as

	$\displaystyle\bm{z}^{t}=\bm{y}-\mathsf{A}\bm{\beta}^{t}+\bm{v}^{t}\otimes\bm{z}^{t-1},$		(5)
	$\displaystyle\bm{\beta}^{t+1}=\eta_{t}\left(\bm{\beta}^{t}+(\mathsf{S}^{t}\otimes\mathsf{A})^{*}\bm{z}^{t}\right),$		(6)

where $\otimes$ denotes the entry-wise product; function $\eta_{t}$ is the minimum mean square error estimator for $\bm{\beta}$ ; vector $\bm{v}^{t}$ and matrix $\mathsf{S}^{t}$ are determined by the SE parameters. In the asymptotic regime $M\rightarrow\infty$ , the SE parameters [9, (23)–(24)] at iterations $t=0,1,\dots$ are given by


		$\displaystyle\phi_{r}^{t}=\sigma^{2}+\frac{1}{L_{C}}\sum_{c=1}^{L_{C}}\mathsf{W}_{rc}\psi_{c}^{t},~{}\forall r\in[L_{R}],$		(7a)
		$\displaystyle\psi_{c}^{t+1}=1-\mathbbm{1}\left\{\frac{1}{RL_{R}}\sum_{r=1}^{L_{R}}\frac{\mathsf{W}_{rc}}{\phi_{r}^{t}}>2\right\},~{}\forall c\in[L_{C}],$		(7b)

where $\psi_{c}^{0}=1,~{}\forall c\in[L_{c}]$ . The SE parameter $\psi_{c}^{t}$ (7b) closely tracks the normalized mean-square error between the part of message $\bm{\beta}_{c}$ and the part of the estimate $\bm{\beta}_{c}^{t}$ corresponding to column block $c$ at iteration $t$ , i.e., $\psi_{c}^{t}\approx\frac{L_{C}}{L}||\bm{\beta}_{c}-\bm{\beta}_{c}^{t}||^{2}_{2}$ for all $c\in[L_{C}]$ . This is evidenced both by the simulations [9, Fig. 3] and the concentration inequality [9, Theorem 2].

III Power allocation and performance metrics

We define power allocation policies for a base matrix as well as the performance metrics.

For an $(\omega,\Lambda,P)$ base matrix $\mathsf{W}$ in Definition 1, a power allocation policy is a mapping $\Pi\colon\mathbb{R}\rightarrow\mathbb{R}^{L_{R}\times L_{C}}$ that gives a set of non-negaitve values $\Pi(P)=\{\mathsf{W}_{rc}\}_{r\in[L_{R}],c\in[L_{C}]}$ corresponding to the entries of the base matrix. The power allocation policy $\Pi$ for the base matrix does not affect the non-zero coefficients of message $\bm{\beta}$ .

We say that a SC-SPARC successfully decodes column block $c$ of the message, i.e., $\bm{\beta}_{c}$ , if there exists a time $T\in\mathbb{Z}_{+}$ such that $\psi_{c}^{T}=0$ (7b); we say that a SC-SPARC successfully decodes the entire message if there exists a time $T\in\mathbb{Z}_{+}$ ,

\displaystyle\psi_{c}^{T}=0,~{}\forall c\in[L_{C}].

(8)

We use the asymptotic SE parameter $\psi_{c}^{t}$ (7b) to define the performance metrics. The asymptotic SE parameter $\psi_{c}^{t}$ is fully determined by the coupling pair $(\omega,\Lambda)$ , the noise variance $\sigma^{2}$ , the rate $R$ , the power $P$ , and the power allocation policy $\Pi$ . Fixing the first three parameters, it becomes $\psi_{c}^{t}=\psi_{c}^{t}(R,P,\Pi)$ .

We measure the performance of a power allocation policy using the rate-power function (RPF) and the power-rate function (PRF) defined next.

Definition 2.

Fix a finite coupling pair $(\omega,\Lambda)$ , a noise variance of the AWGN channel $\sigma^{2}$ , and a power $P$ . The RPF $R_{\Pi}(P)$ for power allocation policy $\Pi$ is the largest rate so that for any rate $R<R_{\Pi}(P)$ , a SC-SPARC generated by an $(\omega,\Lambda,P)$ base matrix with power allocation $\Pi$ ensures successful decoding,

	$\displaystyle R_{\Pi}(P)\triangleq\sup\{R^{*}\colon$	$\displaystyle\forall R<R^{*},\exists T\in\mathbb{Z}_{+},$
		$\displaystyle\psi_{c}^{T}(R,P,\Pi)=0,\forall c\in[L_{C}]\}.$		(9)

Fix a finite coupling pair $(\omega,\Lambda)$ , a noise variance of the AWGN channel $\sigma^{2}$ , and a rate $R$ . The PRF $P_{\Pi}(R)$ for power allocation policy $\Pi$ is the minimum power so that for any power $P>P_{\Pi}(R)$ , a SC-SPARC generated by an $(\omega,\Lambda,P)$ base matrix with power allocation $\Pi$ ensures successful decoding,

	$\displaystyle P_{\Pi}(R)\triangleq\inf\{P^{*}\colon$	$\displaystyle\forall P>P^{*},\exists T\in\mathbb{Z}_{+},$
		$\displaystyle\psi_{c}^{T}(R,P,\Pi)=0,\forall c\in[L_{C}]\}.$		(10)

We aim to find a power allocation policy $\Pi$ that leads to a large $R_{\Pi}(R)$ , or equivalently, a small $P_{\Pi}(R)$ .

IV Uniform power allocation

We say that an $(\omega,\Lambda,P)$ base matrix in Definition 1 has uniform power allocation (UPA) if

\displaystyle\mathsf{W}_{rc}=\begin{cases}P\frac{L_{R}}{\omega},&c\leq r\leq c+\omega-1,\\ 0,&\text{otherwise}.\end{cases}

(11)

We show the RPF (2) and the PRF (10) for UPA.

Theorem 1.

Fix a finite coupling pair $(\omega,\Lambda)$ and an AWGN channel with noise variance $\sigma^{2}$ . The RPF $R_{U}(P)$ for UPA is given by

\displaystyle R_{U}(P)=\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{1}{r+\frac{L_{C}}{L_{R}}\frac{\sigma^{2}}{P}\omega};

(12)

the PRF $P_{U}(R)$ for UPA is given by

\displaystyle P_{U}(R)=\begin{cases}R_{U}^{-1}(R),&R<\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{1}{r},\\ \infty,&\text{otherwise},\end{cases}

(13)

where $R_{U}^{-1}$ is the inverse function of $R_{U}$ .

Proof.

Appendix -A. ∎

We compare $R_{U}(P)$ (12) with the channel capacity $C(P)=\frac{1}{2}\log\left(1+\frac{P}{\sigma^{2}}\right)$ of the AWGN channel with noise variance $\sigma^{2}$ . Using Right-endpoint approximation, we upper bound (12) as

\displaystyle R_{U}(P)\leq\frac{L_{C}}{2L_{R}}\log\left(1+\frac{P}{\sigma^{2}}\frac{\omega}{\frac{L_{C}}{L_{R}}\omega+\frac{P}{\sigma^{2}}}\right).

(14)

The right side of (14) is smaller than $C(P)$ for a finite coupling pair, implying that a SC-SPARC with a finite coupling pair no longer achieves the channel capacity. The gap closes if and only if $\omega,\Lambda\rightarrow\infty$ and $\frac{\omega}{\Lambda}\rightarrow 0$ .

For rates $R_{U}(P)\leq R<C(P)$ , a SC-SPARC fails to ensure successful decoding, and the reason is shown in Proposition 1 stated below. We denote the index of the middle column of the base matrix by $\theta\triangleq\left\lceil\frac{\Lambda}{2}\right\rceil$ .

Proposition 1.

Consider a SC-SPARC generated by an $\left(\omega,\Lambda,P\right)$ base matrix with UPA (11). At iteration $t=1$ , if the AMP decoder successfully decodes $2g$ column blocks of the message,

\displaystyle\psi_{c}^{1}=\psi_{\Lambda-c+1}^{t}=0,~{}\forall c\leq g,

(15)

for some $0\leq g\leq\omega$ , then at iterations $t=2,3,\dots$ , the AMP decoder continues to decode $2g$ column blocks of the message,

\displaystyle\psi_{c}^{t}=\psi_{\Lambda-c+1}^{t}=0,~{}\forall c\leq\min\left\{gt,\theta\right\}.

(16)

Proof.

Appendix -B. ∎

Proposition 1 states that if $g=0$ , the decoder fails to decode even a single column block of the message; otherwise, the entire message is decoded within $\frac{\theta}{g}$ iterations. Here, it suffices to limit $g\leq\omega$ because $g\geq\omega$ means that the entire message is successfully decoded in the first iteration (Appendix -C). Proposition 1 indicates that a SC-SPARC with UPA fails to decode at $R_{U}(P)\leq R<C(P)$ because the power (11) allocated to columns $1$ and $\Lambda$ of the base matrix is smaller than the power needed to make the event in $\psi_{1}^{1}$ (7b) occur.

V V-power allocation

V-A VPA Algorithm

Fixing an AWGN channel with noise variance $\sigma^{2}$ and a rate $R$ , we present VPA for an $(\omega,\Lambda,P)$ base matrix.

In the extreme, a power allocation policy can allocate a different power to every non-zero entry of the base matrix $\mathsf{W}$ . The output $\{\mathsf{W}_{rc}\}_{r\in[L_{R}],c\in[L_{C}]}$ of VPA satisfy:

a)

The power does not change with rows, i.e., $\forall c\in[L_{C}]$ ,

$\displaystyle\mathsf{W}_{rc}\triangleq\mathsf{W}_{c},~{}\forall c\leq r\leq c+\omega-1;$ (17)
b)

The power is symmetric about the middle column index,

$\displaystyle\mathsf{W}_{c}=\mathsf{W}_{\Lambda-c+1},\forall c\in[L_{C}].$ (18)

We define function $\mathsf{f}_{t}\colon\mathbb{R}^{\theta-t+1}\rightarrow\mathbb{R}$ , $t=1,\dots,\theta$ as¹¹1Although the summation in the denominator of the right side of (19) may include $\mathsf{W}_{\theta+1},\dots,\mathsf{W}_{\Lambda-t+1}$ , $\mathsf{f}_{t}$ is still a function of variables $\mathsf{W}_{t},\dots,\mathsf{W}_{\theta}$ only, due to the symmetry assumption (18).

\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}\}_{i=t}^{\theta}\right)\triangleq\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}}.

(19)

Let $\{\delta_{t}\}_{t=1}^{\theta}$ be a sequence of positives chosen arbitrarily.

input :

\omega,\Lambda,R,P,\sigma,\{\delta_{t}\}_{t=1}^{\theta}

output :

\{\mathsf{W}_{c}\}_{c\in[L_{C}]}

1 for $t=\theta,\theta-1,\dots,1$ do

2 Solve

\mathsf{f}_{t}\left(\{\mathsf{W}_{i}\}_{i=t}^{\theta}\right)=2RL_{R}

for

\mathsf{W}_{t}

;

\mathsf{W}_{t}\leftarrow\mathsf{W}_{t}+\delta_{t}

;

\mathsf{W}_{\Lambda-t+1}\leftarrow\mathsf{W}_{t}

5 end for

6if $\frac{1}{L_{R}L_{c}}\sum_{c=1}^{L_{c}}\omega\mathsf{W}_{c}>P$ then

7 Declare failure

8 end if

9if $\frac{1}{L_{R}L_{c}}\sum_{c=1}^{L_{c}}\omega\mathsf{W}_{c}\leq P$ then

\text{residual}\leftarrow P-\frac{1}{L_{R}L_{c}}\sum_{c=1}^{L_{c}}\omega\mathsf{W}_{c}

;

\mathsf{W}_{1}\leftarrow\frac{\text{residual}L_{R}L_{c}}{2\omega}

;

\mathsf{W}_{\Lambda}\leftarrow\mathsf{W}_{1}

13 end if

Algorithm 1 VPA

Proposition 2, stated next, shows that VPA follows a shape of V, namely, $\mathsf{W}_{c}$ is non-increasing on $1\leq c\leq\theta$ and is non-decreasing on $\theta+1\leq c\leq\Lambda$ by symmetry (18).

Proposition 2.

Power allocation $\mathsf{W}_{1}^{(V)},\mathsf{W}_{2}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}$ that ensure $\mathsf{f}_{t}=2RL_{R}$ (line 2 of Algorithm 1) for all $t=1,2,\dots,\theta$ are unique and satisfy

\displaystyle\mathsf{W}_{1}^{(V)}\geq\mathsf{W}_{2}^{(V)}\geq\dots\geq\mathsf{W}_{\theta}^{(V)}.

(20)

Proof.

Appendix -D. ∎

Although the sequence $\{\mathsf{W}_{t}^{(V)}\}_{t=1}^{\theta}$ does not perfectly coincide with the sequence $\{\mathsf{W}_{t}\}_{t=1}^{\theta}$ formed at the end of line 5, it reflects the trend of $\{\mathsf{W}_{t}\}_{t=1}^{\theta}$ for abitrarily small $\{\delta_{t}\}_{t=1}^{\theta}$ .

V-B VPA performance

Before we show the PRF for VPA, we introduce Lemma 1 below. It states that if a column block of the message is decoded at some iteration, then it remains decoded in the subsequent iterations, and that the asymptotic SE $\psi_{c}^{t}$ (7b) can be expressed in terms of $\mathsf{f}_{t}$ (19) under some conditions.

Lemma 1.

Consider a SC-SPARC generated by an $(\omega,\Lambda,P)$ base matrix. Fix a noise variance $\sigma^{2}$ and a rate $R$ .

1.

If $\exists~{}c\in[L_{C}],t\geq 1$ , $\psi_{c}^{t}=0$ , then $\psi_{c}^{s}=0$ , $\forall s\geq t$ .
2.

For a power allocation policy satisfying a)–b), at $t=1$ ,

$\displaystyle\psi_{1}^{1}=1-\mathbbm{1}\left\{\mathsf{f}_{1}\left(\{\mathsf{W}_{i}\}_{i=1}^{\theta}\right)>2RL_{R}\right\};$ (21)

if $\psi_{c}^{c}=0,\forall c\leq t-1$ , then at iterations $2\leq t\leq\theta$ ,

$\displaystyle\psi_{t}^{t}\leq 1-\mathbbm{1}\left\{\mathsf{f}_{t}\left(\{\mathsf{W}_{i}\}_{i=t}^{\theta}\right)>2RL_{R}\right\}.$ (22)

Proof.

Appendix -E. ∎

We present the PRF for VPA.

Theorem 2.

Fix a finite coupling pair $(\omega,\Lambda)$ and an AWGN channel with noise variance $\sigma^{2}$ . The PRF $P_{V}(R)$ (10) for VPA (Algorithm 1) is given by

\displaystyle P_{V}(R)=\begin{cases}\frac{2\omega}{L_{R}L_{C}}\sum_{c=1}^{\theta}\mathsf{W}_{c}^{(V)},R<\frac{L_{C}(\omega+1)}{4L_{R}},~{}\Lambda~{}\text{is even}\\ \frac{2\omega}{L_{R}L_{C}}\sum_{c=1}^{\theta-1}\mathsf{W}_{c}^{(V)}+\mathsf{W}_{\theta}^{(V)},R<\frac{L_{C}(\omega+2)}{4L_{R}},\Lambda~{}\text{is odd}\\ \infty,~{}\text{otherwise}.\end{cases}

(23)

Proof sketch.

The proof is divided into two steps.

(i) We show that VPA outputs $\{\mathsf{W}_{c}\}_{c\in[L_{c}]}$ , or equivalently it does not declare failure, if and only if $P>P_{V}(R)$ and $R$ less than the upper bound in (23). Appendix -F.

(ii) We show that the output $\{\mathsf{W}_{c}\}_{c\in[L_{c}]}$ of VPA ensures successful decoding. Appendix -G. ∎

The working principle of VPA is to allocate sufficient power to the outer columns of the base matrix in order to jumpstart the wave-like decoding process that propagates from the sides to the middle of the message, and to allocate lower power (but not too low that prohibits the decoding process) to the inner columns of the base matrix.

V-C VPA outperforms UPA

Proposition 3.

Fix a finite coupling pair $(\omega,\Lambda)$ and an AWGN channel with noise variance $\sigma^{2}$ . The rate that ensures $P_{V}(R)<\infty$ also ensures $P_{U}(R)<\infty$ , i.e.,

\displaystyle\{R\colon P_{U}(R)<\infty\}\subseteq\{R\colon P_{V}(R)<\infty\}.

(24)

For a rate $R$ that belongs to both sets in (24), it holds that

\displaystyle P_{V}(R)\leq P_{U}(R).

(25)

Proof.

Appendix -H. ∎

In fact, UPA is a special case of VPA by carefully selecting $\{\delta_{t}\}_{t=1}^{\theta}$ (Appendix -I).

VI Simulations

We use an example to illustrate (25). Consider $\omega=2$ , $\Lambda=5$ , $P=3$ , $\sigma=1$ , and $R=0.45$ . For UPA, we have $\psi_{1}^{1}=1-\mathbbm{1}\{5.1708>5.4\}=1$ , and Proposition 1 implies that a SC-SPARC with UPA fails to decode the message. We now determine the power allocation using VPA. Choosing $\delta_{t}=0.01$ , $\forall t=1,2,3$ , and following lines 1–5 of VPA, we obtain $\mathsf{W}_{1}=9.87,\mathsf{W}_{2}=8.74,\mathsf{W}_{3}=5.88$ . We check that line 9 of VPA is satisfied, and we transfer the residual power to the boundary columns yielding $\mathsf{W}_{1}=10.82$ . Since the power $P>P_{V}(R)$ in Theorem 2, the output of VPA ensures successful decoding.

While VPA is designed in the limit of large section length $M\rightarrow\infty$ , we show by simulations that power allocation imitating the shape of the letter V (20) also improves the finite-blocklength error performance of a SC-SPARC. We consider a SC-SPARC of parameters $M=512,L=30,L_{C}=15,L_{R}=18,M_{R}=12$ and an AWGN channel of variance $\sigma^{2}=1$ . Fig. 2 compare the SC-SPARC with UPA (11) and that with a VPA-like power allocation chosen empirically in Table I. Fig. 2 shows that the BLER of the VPA-like power allocation is smaller than that of UPA, especially in the middle part of the waterfall region. Fig. 3 shows the convergence of the BLERs. To reduce the complexities, we use the Hadamard design matrix as in [4][9], instead of using the i.i.d. Gaussian design matrix. The simulations may not perfectly match our theoretical results since the asymptotic SE is accurate only for an i.i.d. Gaussian design matrix and $M\rightarrow\infty$ .

TABLE I: VPA-like power allocation (20)

SNR(dB)	Outer columns	Inner columns
9.5	$\mathsf{W}_{1}=\dots=\mathsf{W}_{3}=42.51$	$\mathsf{W}_{4}=\dots=\mathsf{W}_{8}=38.51$
10.0	$\mathsf{W}_{1}=\dots=\mathsf{W}_{5}=46.67$	$\mathsf{W}_{6}=\dots=\mathsf{W}_{8}=41.67$
10.5	$\mathsf{W}_{1}=\dots=\mathsf{W}_{4}=52.36$	$\mathsf{W}_{5}=\dots=\mathsf{W}_{8}=48.36$
11.0	$\mathsf{W}_{1}=\dots=\mathsf{W}_{5}=58.32$	$\mathsf{W}_{6}=\dots=\mathsf{W}_{8}=53.32$
11.5	$\mathsf{W}_{1}=\dots=\mathsf{W}_{6}=64.56$	$\mathsf{W}_{7}=\dots=\mathsf{W}_{8}=59.56$
12.0	$\mathsf{W}_{1}=\dots=\mathsf{W}_{6}=72.52$	$\mathsf{W}_{7}=\dots=\mathsf{W}_{8}=66.52$

VII Conclusion

In this paper, we propose V-power allocation for the base matrix of a SC-SPARC with a finite coupling pair. It yields power allocation that descends from the outer columns to the inner columns of the base matrix, resembling the shape of the letter V. By analyzing the PRFs, we show that given a code rate, V-power allocation ensures successful decoding for a wider range of power compared to uniform power allocation. Numerical simulations indicate that power allocation following the shape of the letter V reduces the finite-blocklength block error rates of a SC-SPARC.

References

[1] A. Joseph and A. R. Barron, “Least squares superposition codes of moderate dictionary size are reliable at rates up to capacity,” in IEEE Trans. Inf. Theory, vol. 58, no. 5, pp. 2541–2557, May 2012.
[2] J. Barbier and F. Krzakala. “Replica analysis and approximate message passing decoder for superposition codes,” in 2014 IEEE Int. Symp. Inf. Theory, Honolulu, HI, USA, July 2014, pp. 1494-1498.
[3] C. Rush, A. Greig and R. Venkataramanan, “Capacity-achieving sparse superposition codes via approximate message passing decoding,” in IEEE Trans. Inf. Theory, vol. 63, no. 3, pp. 1476–1500, March 2017.
[4] J. Barbier and F. Krzakala, “Approximate message-passing decoder and capacity achieving sparse superposition codes,” in IEEE Trans. Inf. Theory, vol. 63, no. 8, pp. 4894–4927, Aug. 2017.
[5] A. Joseph and A. R. Barron, “Fast sparse superposition codes have near exponential error probability for $R<C$ ,” in IEEE Trans. Inf. Theory, vol. 60, no. 2, pp. 919–942, Feb. 2014.
[6] S. Cho, and A. Barron. “Approximate iterative Bayes optimal estimates for high-rate sparse superposition codes,” in Sixth Workshop on Information-Theoretic Methods in Science and Engineering, 2013.
[7] R. Venkataramanan, S. Tatikonda, and A. Barron, “Sparse regression codes,” in Foundations and Trends in Communications and Information Theory, vol. 15, nos. 1–2, pp. 1–195, 2019.
[8] J. Barbier, M. Dia and N. Macris, “Proof of threshold saturation for spatially coupled sparse superposition codes,” in 2016 IEEE Int. Symp. Inf. Theory, Barcelona, Spain, 2016, pp. 1173–1177.
[9] C. Rush, K. Hsieh and R. Venkataramanan, “Capacity-achieving spatially coupled sparse superposition codes with AMP decoding,” in IEEE Trans. Inf. Theory, vol. 67, no. 7, pp. 4446–4484, July 2021.
[10] K. Hsieh, C. Rush and R. Venkataramanan, “Spatially coupled sparse regression codes: design and state evolution analysis,” in 2018 IEEE Int. Symp. Inf. Theory, Vail, CO, USA, 2018, pp. 1016–1020.
[11] S. Kudekar, T. J. Richardson and R. L. Urbanke, “Threshold saturation via spatial coupling: why convolutional LDPC ensembles perform so well over the BEC,” in IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803–834, Feb. 2011.
[12] D. G. M. Mitchell, M. Lentmaier and D. J. Costello, “Spatially Coupled LDPC Codes Constructed From Protographs,” in IEEE Trans. Inf. Theory, vol. 61, no. 9, pp. 4866–4889, Sept. 2015.
[13] C. Rush, K. Hsieh and R. Venkataramanan, “Spatially coupled sparse regression codes with sliding window AMP decoding,” in 2019 IEEE Inf. Theory Workshop, Visby, Sweden, 2019, pp. 1–5.
[14] N. Guo, S. Liang, W. Han, “Power allocation for the base matrix of spatially coupled sparse regression codes”, Arxiv Preprint, May 2023.

-A Proof of Theorem 1

-A1 Proof of $R_{U}(P)$

Before we prove $R_{U}(P)$ in (12), we first show that a SC-SPARC with UPA succesfully decodes the entire message if and only if $\psi_{1}^{1}=0$ . If $\psi_{1}^{1}=0$ , then $g\geq 1$ in (15), and Proposition 1 implies that the decoding is successful. To prove the reverse direction, we prove its equivalent, namely, if $\psi_{1}^{1}=1$ , then a SC-SPARC with UPA cannot decode successfully. Since $\psi_{c}^{1}$ (7b) is non-decreasing on $1\leq c\leq\theta$ , we conclude that $\psi_{1}^{1}=1$ implies $\psi_{c}^{1}=1$ for all $c\in[L_{C}]$ . Thus, $g=0$ in (15), and Proposition 1 implies the decoding failure.

We proceed to prove $R_{U}(P)$ (12). We write it as


$\displaystyle R_{U}(P)$	$\displaystyle=\sup\{R^{}\colon\forall R<R^{},\psi_{1}^{1}(R,P,\text{UPA})=0\}$	(26a)
	$\displaystyle=\sup\{R^{}\colon\psi_{1}^{1}(R^{},P,\text{UPA})=0\}$	(26b)
	$\displaystyle=\left\{R^{}\colon\sum_{r=1}^{\omega}\frac{P\frac{L_{R}}{\omega}}{\sigma^{2}+\frac{r}{L_{C}}P\frac{L_{R}}{\omega}}=2R^{}L_{R}\right\},$	(26c)

where (26a) holds as we have proved that a SC-SPARC with UPA decodes successfully if and only if $\psi_{1}^{1}=0$ ; (26b) holds since

\displaystyle\psi_{1}^{1}(R,P,\text{UPA})=1-\mathbbm{1}\left\{\sum_{r=1}^{\omega}\frac{P\frac{L_{R}}{\omega}}{\sigma^{2}+\frac{1}{L_{C}}rP\frac{L_{R}}{\omega}}>2RL_{R}\right\}

(27)

is non-decreasing as $R$ increases; (26c) holds since (26b) is equivalent to the supremum of $R$ that makes the event in (27) occur. Thus, (12) follows.

-A2 Proof of $P_{U}(R)$

Before we prove $P_{U}(R)$ (12), we calcuclate the derivative of the left side of the event in (27) with respect to $P$ as


$\displaystyle\frac{\partial\sum_{r=1}^{\omega}\frac{P\frac{L_{R}}{\omega}}{\sigma^{2}+\frac{1}{L_{C}}rP\frac{L_{R}}{\omega}}}{\partial P}$	$\displaystyle=\sum_{r=1}^{\omega}\frac{\frac{L_{R}}{\omega}\sigma^{2}}{\left(\sigma^{2}+\frac{1}{L_{C}}rP\frac{L_{R}}{\omega}\right)^{2}}$	(28a)
	$\displaystyle>0.$	(28b)

Since the left side of the event in (27) increases as $P$ increases, we conclude that $\psi_{1}^{1}(R,P,\text{UPA})$ is non-increasing as $P$ increases.

To express $P_{U}(R)$ in terms of the inverse function $R_{U}^{-1}$ of $R_{U}(R)$ , we show that the inverse function $R_{U}^{-1}$ exists. We calculate the derivative of $R_{U}(P)$ with respect to $P$ as


$\displaystyle\frac{dR_{U}(P)}{dP}$	$\displaystyle=\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{\frac{L_{C}}{L_{R}}\frac{\sigma^{2}}{P^{2}}\omega}{(r+\frac{L_{C}}{L_{R}}\frac{\sigma^{2}}{P}\omega)^{2}}$	(29a)
	$\displaystyle>0.$	(29b)

Since $R_{U}(P)$ is differentiable and its derivative is positive, we conclude that $R_{U}(P)$ is continuous and monotone, thus $R_{U}(P)$ is bijective and has an inverse function.

To demonstrate the domain and the range of the inverse function $R_{U}^{-1}$ , we show the range of $R_{U}(P)$ . Since $R_{U}(P)$ increases as $P$ increases by (29), it holds that for $\forall P<\infty$ ,


$\displaystyle R_{U}(P)$	$\displaystyle<R_{U}(\infty)$	(30a)
	$\displaystyle=\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{1}{r},$	(30b)

meaning that the inverse function satisfies

\displaystyle R_{U}^{-1}(R)<\infty,\text{if and only if}~{}R<\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{1}{r}.

(31)

We proceed to show $P_{U}(R)$ (12). We write it as


$\displaystyle P_{U}(R)$	$\displaystyle=\inf\{P^{}\colon\forall P>P^{},\psi_{1}^{1}(R,P,\text{UPA})=0\}$	(32a)
	$\displaystyle=\inf\{P^{}\colon\psi_{1}^{1}(R,P^{},\text{UPA})=0\}$	(32b)
	$\displaystyle=\left\{P^{}\colon\sum_{r=1}^{\omega}\frac{P^{}\frac{L_{R}}{\omega}}{\sigma^{2}+\frac{r}{L_{C}}P^{*}\frac{L_{R}}{\omega}}=2RL_{R}\right\}$	(32c)
	$\displaystyle=R^{-1}(R),$	(32d)

where (32a) holds as we have shown in Appendix -A1 that a SC-SPARC with UPA decodes successfully if and only if $\psi_{1}^{1}=0$ ; (32b) holds as we have shown that $\psi_{1}^{1}(R,P,\text{UPA})$ is non-increasing as $P$ increases below (28); (32c) holds since (32b) is equivalent to the infimum of $P$ that makes the event in (27) occur; (32d) holds by noticing that the objective functions in (32c) and (26c) are the same and by the fact that $R_{U}^{-1}$ exists.

Plugging (31) into (32d), we obtain (13).

-B Proof of Proposition 1

We show (16) by mathematical induction. We denote by $\bar{\mathsf{W}}$ the non-zero value of a base matrix with UPA (11). Plugging (7a) into (7b), we write the asymptotic SE parameter $\psi_{c}^{t}$ as

\displaystyle\psi_{c}^{t}=1-\mathbbm{1}\left\{\sum_{r=c}^{c+\omega-1}\frac{\mathsf{W}_{rc}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=\underline{c}_{r}}^{\bar{c}_{r}}\mathsf{W}_{rc^{\prime}}\psi_{c^{\prime}}^{t-1}}>2RL_{R}\right\}

(33)

where Definition 1 i)–ii) implies

		$\displaystyle\underline{c}_{r}\triangleq\max\{1,r-\omega+1\},$		(34)
		$\displaystyle\bar{c}_{r}\triangleq\min\{\Lambda,r\}.$		(35)

Initial step: At iteration $t=1$ , by the assumption of Proposition 1, $\psi_{c}^{1}=0$ , $\forall c\leq g$ . Plugging $t\leftarrow 1$ into (33) and using $g\leq\omega$ , we write $\psi_{c}^{1}$ , $c\leq g$ as


$\displaystyle\psi_{c}^{1}$	$\displaystyle=1-\mathbbm{1}\left\{\sum_{r=c}^{c+\omega-1}\frac{\bar{\mathsf{W}}}{\sigma^{2}+\frac{1}{L_{c}}\min\{\omega,r\}\bar{\mathsf{W}}}>2RL_{R}\right\}$	(36a)
	$\displaystyle=0.$	(36b)

Induction step: Assuming that (16) holds at iteration $t$ , we show that it continues to hold at iteration $t+1$ . If $t\geq\frac{\theta}{g}$ , then Lemma 1 item 1) implies that (16) holds at iteration $t+1$ . If $t<\frac{\theta}{g}$ , the asymptotic SE $\psi_{c}^{t+1}$ can be upper bounded as


		$\displaystyle\psi_{c}^{t+1}$
	$\displaystyle\leq~{}$	$\displaystyle 1-\mathbbm{1}\left\{\sum_{r=c}^{c+\omega-1}\frac{\bar{\mathsf{W}}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=r-\omega+1}^{r}\bar{\mathsf{W}}\psi_{c^{\prime}}^{t}}>2RL_{R}\right\}$	(37a)
	$\displaystyle=~{}$	$\displaystyle 1-\mathbbm{1}\left\{\sum_{r=c}^{c+\omega-1}\frac{\bar{\mathsf{W}}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=\max\{r-\omega+1,gt+1\}}^{r}\bar{\mathsf{W}}}>2RL_{R}\right\}$	(37b)
	$\displaystyle=~{}$	$\displaystyle 1-\mathbbm{1}\left\{\sum_{r=c}^{c+\omega-1}\frac{\bar{\mathsf{W}}}{\sigma^{2}+\frac{1}{L_{c}}\min\{\omega,r-gt\}\bar{\mathsf{W}}}>2RL_{R}\right\}$	(37c)
	$\displaystyle=~{}$	$\displaystyle 1-\mathbbm{1}\left\{\sum_{r=c-gt}^{c-gt+\omega-1}\frac{\bar{\mathsf{W}}}{\sigma^{2}+\frac{1}{Lc}\min\{\omega,r\}\bar{\mathsf{W}}}>2RL_{R}\right\},$	(37d)
where (37a) holds by plugging $t\leftarrow t+1$ , $\bar{c}_{r}\leq r$ , and $\underline{c}_{r}\geq r-\omega+1$ into (33); (37b) holds by the induction assumption and the fact $t<\frac{\theta}{g}$ ; (37c) holds by rewriting the summation in the denominator of (37b); (37d) holds by change of measure $r\leftarrow r+gt$ . Comparing (36) and (37d), we conclude that

	$\displaystyle\psi_{c}^{t+1}=0,~{}\forall gt\leq c\leq g(t+1).$		(37e)

Using (37e), the induction assumption, and Lemma 1 item 1), we conclude that (16) holds at iteration $t+1$ .

-C $g\leq\omega$ is sufficient

We show that in Proposition 1, it suffices to limit $g\leq\omega$ since $g\geq\omega$ implies that the entire message is successfully decoded in the first iteration. Indeed, for UPA (11), $\psi_{c}^{1}$ (36a) is non-decreasing on $1\leq c\leq\omega$ , remains constant on $\omega\leq c\leq\theta$ , and is symmetric about $c=\theta$ . As a result, $\psi_{\omega}^{1}=0$ implies $\psi_{c}^{1}=0$ for all $c\in[L_{C}]$ .

-D Proof of Proposition 2

In Appendix -D1, we first show that the sequence of power allocation $\left\{\mathsf{W}_{i}^{(V)}\right\}_{i=1}^{\theta}$ is unique, and we then show that it is non-increasing (20). In Appendices -D2–-D3, we prove the lemmas used in Appendix -D1.

-D1 Main proof

To show the uniqueness, we introduce Lemma 2 below.

Lemma 2.

Fixing $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ , function $\mathsf{f}_{t}$ is continuous in $\mathsf{W}_{t}$ and is monotonically increasing as $\mathsf{W}_{t}$ increases.

Proof.

Appendix -D2. ∎

Lemma 2 indicates that fixing $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ , $\mathsf{f}_{t}$ is a bijective function of $\mathsf{W}_{t}$ . Thus, there exist unique power allocation $\mathsf{W}_{\theta}^{(V)},\dots,\mathsf{W}_{1}^{(V)}$ that satisfy $\mathsf{f}_{\theta}=\dots=\mathsf{f}_{1}=2RL_{R}$ .

We proceed to show that the sequence $\{\mathsf{W}_{i}^{(V)}\}_{i=1}^{\theta}$ is non-increasing (20).

Lemma 3.

For any $t=1,2,\dots,\theta-1$ , given $\mathsf{W}_{t+1}\geq\mathsf{W}_{t+2}\geq\dots\geq\mathsf{W}_{\theta}$ , if $\mathsf{W}_{t}=\mathsf{W}_{t+1}$ , it holds that $\mathsf{f}_{t}(\{\mathsf{W}_{i}\}_{i=t}^{\theta})\leq\mathsf{f}_{t+1}(\{\mathsf{W}_{i}\}_{i=t+1}^{\theta})$ .

Proof.

Appendix -D3. ∎

The sequence $\{\mathsf{W}_{i}^{(V)}\}_{i=1}^{\theta}$ satisfies

\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t}^{\theta}\right)=\mathsf{f}_{t+1}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}\right)

(38)

for all $t=1,2,\dots,\theta-1$ . At $t=\theta-1$ , Lemmas 2–3 and (38) imply that $\mathsf{W}_{\theta-1}^{(V)}\geq\mathsf{W}_{\theta}^{(V)}$ . Similarly, iteratively applying Lemmas 2–3 to $t=\theta-2,\theta-3,\dots,1$ in the backward manner, we conclude (20).

-D2 Proof of Lemma 2

We compute the derivative of $\mathsf{f}_{t}$ with respect to $\mathsf{W}_{t}$ . If $t+\omega-1<\Lambda-t+1$ ,

\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}=\sum_{r=t}^{t+\omega-1}\frac{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t+1}^{r}\mathsf{W}_{c^{\prime}}}{(\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t}^{r}\mathsf{W}_{c^{\prime}})^{2}};

(39)

if $t+\omega-1\geq\Lambda-t+1$ ,

	$\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$	$\displaystyle=\sum_{r=t}^{\Lambda-t}\frac{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t+1}^{r}\mathsf{W}_{c^{\prime}}}{(\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t}^{r}\mathsf{W}_{c^{\prime}})^{2}}$
		$\displaystyle+(2t+\omega-\Lambda-1)\frac{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t+1}^{\Lambda-t}\mathsf{W}_{c^{\prime}}}{(\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t}^{\Lambda-t+1}\mathsf{W}_{c^{\prime}})^{2}}$		(40)

Since $\mathsf{f}_{t}$ is differentiable and its derivative is positive, we conclude that Lemma 2 holds.

-D3 Proof of Lemma 3

Given $\mathsf{W}_{t+1}\geq\dots\geq\mathsf{W}_{\theta}$ , function $\mathsf{f}_{t+1}$ can be written as


	$\displaystyle\mathsf{f}_{t+1}\left(\{\mathsf{W}_{i}\}_{t+1}^{\theta}\right)$
$\displaystyle=~{}$	$\displaystyle\sum_{r=t+1}^{t+\omega}\frac{\mathsf{W}_{t+1}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t+1}^{\min\{r,\Lambda-t\}}\mathsf{W}_{c^{\prime}}}$	(41a)
$\displaystyle=~{}$	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t+1}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t+1}^{\min\{r+1,\Lambda-t\}}\mathsf{W}_{c^{\prime}}}$	(41b)
$\displaystyle=~{}$	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t+1}}{\sigma^{2}+\frac{1}{L_{c}}\left(\mathsf{W}_{t+1}+\sum_{c^{\prime}=t+2}^{\min\{r+1,\Lambda-t\}}\mathsf{W}_{c^{\prime}}\right)}$	(41c)

where (41a) holds by definition (19); (41b) holds by change of measure $r\leftarrow r+1$ ; (41c) holds by expanding the summation in the denominator of (41b). Function $\mathsf{f}_{t}$ with $\mathsf{W}_{t}\leftarrow\mathsf{W}_{t+1}$ can be written as

		$\displaystyle\mathsf{f}_{t}\left(\mathsf{W}_{t+1},\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}\right)$
	$\displaystyle=$	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t+1}}{\sigma^{2}+\frac{1}{L_{c}}\left(\mathsf{W}_{t+1}+\sum_{c^{\prime}=t+1}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}\right)}.$		(42)

To compare (41c) and (42), it suffices to compare the summations in their denominators. We denote by $D_{t}$ and $D_{t+1}$ the summations in the denominators of (42) and (41c), respectively, i.e.,

	$\displaystyle D_{t}$	$\displaystyle\triangleq\sum_{c^{\prime}=t+1}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}$		(43)
	$\displaystyle D_{t+1}$	$\displaystyle\triangleq\sum_{c^{\prime}=t+2}^{\min\{r+1,\Lambda-t\}}\mathsf{W}_{c^{\prime}}.$		(44)

Fix $r=t,\dots,t+\omega-1$ .
Case 1: If $r\leq\Lambda-t+1$ and $r+1\leq\Lambda-t$ , it holds that

	$\displaystyle D_{t}-D_{t+1}$	$\displaystyle=\mathsf{W}_{t+1}-\mathsf{W}_{r+1}$		(45)
		$\displaystyle\geq 0$		(46)

where (46) holds by the fact $t+1\leq r+1\leq\Lambda-t$ and the fact $\mathsf{W}_{t+1}\geq\dots\geq\mathsf{W}_{\theta}$ .
Case 2: If $r\leq\Lambda-t+1$ and $r+1>\Lambda-t$ , it holds that

$\displaystyle D_{t}$	$\displaystyle=\sum_{c^{\prime}=t+1}^{r}\mathsf{W}_{c^{\prime}}$	(47)
	$\displaystyle\geq\sum_{c^{\prime}=t+1}^{\Lambda-t}\mathsf{W}_{c^{\prime}}$	(48)
	$\displaystyle\geq D_{t+1},$	(49)

where (48) holds since the assumptions on $r$ in Case 2 imply $r\in\{\Lambda-t,\Lambda-t+1\}$ .
Case 3: If $r>\Lambda-t+1$ and $r+1>\Lambda-t$ , it holds that

\displaystyle D_{t}-D_{t+1}=\mathsf{W}_{t+1}+\mathsf{W}_{\Lambda-t+1}\geq 0.

(50)

Since cases 1–3 indicate $D_{t}\geq D_{t+1}$ , we conclude that if $\mathsf{W}_{t}=\mathsf{W}_{t+1}$ , then $\mathsf{f}_{t}\leq\mathsf{f}_{t+1}$ .

-E Proof of Lemma 1

-E1 Proof of item 1)

We prove item 1) by mathematical induction. We denote the set of zero positions of $\psi_{c}^{t}$ by

\displaystyle\mathcal{N}^{t}\triangleq\{c\in[L_{c}]\colon\psi_{c}^{t}=0\}.

(51)

To show item 1), it suffices to show

\displaystyle\mathcal{N}^{0}\subseteq\mathcal{N}^{1}\subseteq N^{2}\subseteq\dots

(52)

Initial step: At $t=0$ , $\psi_{c}^{0}=1$ for all $c\in[L_{c}]$ , thus $\mathcal{N}^{0}$ is an empty set. It is trivial to conclude $\mathcal{N}^{0}\subseteq\mathcal{N}^{1}$ .

Induction step: Assuming that $\mathcal{N}^{t-1}\subseteq\mathcal{N}^{t}$ , we proceed to show $\mathcal{N}^{t}\subseteq\mathcal{N}^{t+1}$ . The asymptotic SE (7b) at iteration $t$ is given by (33). The induction assumption posits that if $\psi_{c}^{t-1}=0$ , we have $\psi_{c}^{t}=0$ . As a result, the denominator of the event in (33) at iteration $t$ is larger than or equal to that at iteration $t+1$ , and we obtain $\psi_{c}^{t}\geq\psi_{c}^{t+1}$ . Since $\psi_{c}^{t}\in\{0,1\}$ is binary for all $c\in[L_{C}]$ , $t=1,2,\dots$ , we conclude (52).

-E2 Proof of item 2)

The asymptotic SE $\psi_{1}^{1}$ can be written as (21) by comparing (21) and (33) with $c\leftarrow 1$ , $t\leftarrow 1$ . It remains to show that (22) holds for $t=2,\dots,\theta$ under the assumption that $\psi_{c}^{c}=0$ for $c\leq t-1$ in Lemma 1. The SE parameter $\psi_{t}^{t}$ is given by (33) with $c\leftarrow t$ and the left side of its event can be lower bounded as


	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=\underline{c}_{r}}^{\bar{c}_{r}}\mathsf{W}_{c^{\prime}}\psi_{c^{\prime}}^{t-1}}$	(53a)
$\displaystyle\geq~{}$	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=\max\{1,r-\omega+1,t\}}^{\min\{r,\Lambda,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}}$	(53b)
$\displaystyle=~{}$	$\displaystyle\sum_{r=t}^{t+\omega-1}\frac{\mathsf{W}_{t}}{\sigma^{2}+\frac{1}{L_{c}}\sum_{c^{\prime}=t}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}}$	(53c)
$\displaystyle=~{}$	$\displaystyle\mathsf{f}_{t}(\{\mathsf{W}_{i}\}_{i=t}^{\theta}),$	(53d)

where (53b) holds by (34)–(35), the assumption $\psi_{c}^{c}=0$ for $c\leq t-1$ , and the symmetry of $\psi_{c}^{c}$ (18); (53c) holds since $\Lambda-t+1\leq\Lambda$ , $t\geq 2$ , and $r-\omega+1\leq t$ ; (53d) holds by definition (19). The equality of (53b) is achieved if and only if $\psi_{c}^{t-1}=1$ for all $t\leq c\leq\theta$ . Replacing the left side of the event of $\psi_{t}^{t}$ by its lower bound in (53d), we obtain (22).

-F Proof of Theorem 2: step (i)

Given power $P<\infty$ and rate $R<\infty$ , we show that VPA outputs $\{\mathsf{W}_{c}\}_{c\in[L_{C}]}$ , i.e., it does not declare failure, if and only if $P>P_{V}(R)$ and $R$ is less than the upper bound in (23). To this end, we first introduce useful lemmas and notations in Appendix -F1; we prove the ‘if’ direction in Appendix -F2; we prove the ‘only if’ direction in Appendix -F3; the proof of the lemmas in Appendix -F1 are presented in Appendices -F4–-F7.

-F1 Lemmas and notations

We introduce Lemmas 4–7. We denote by $\bar{R}$ the upper bound on $R$ in (23), i.e.,

\displaystyle\bar{R}\triangleq\begin{cases}\frac{L_{C}(\omega+1)}{4L_{R}},&\Lambda~{}\text{is even},\\ \frac{L_{C}(\omega+2)}{4L_{R}},&\Lambda~{}\text{is odd}.\end{cases}

(54)

Lemma 4, stated next, shows the existence of $\{\mathsf{W}_{i}^{(V)}\}_{i=1}^{\theta}$ .

Lemma 4.

If and only if $R<\bar{R}$ , there exists a sequence $\mathsf{W}_{1}^{(V)},\mathsf{W}_{2}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}<\infty$ that satisfies $\mathsf{f}_{t}=2RL_{R}$ (line 2 of VPA) simultaneously for all $t=1,2,\dots,\theta$ .

Proof.

Appendix -F4. ∎

Lemma 5 shows how $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ changes with $\mathsf{W}_{t}$ .

Lemma 5.

Fixing $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ , the derivative $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ (39)–(40) monotonically decreases as $\mathsf{W}_{t}$ increases.

Proof.

Appendix -F5. ∎

Lemma 6 below shows how $\mathsf{f}_{t}$ changes with $\mathsf{W}_{s}$ , $s\geq t+1$ .

Lemma 6.

Fixing $\mathsf{W}_{i}$ for $t\leq i\leq\theta$ , $i\neq s$ , $s\geq t+1$ , function $\mathsf{f}_{t}$ is continuous in $\mathsf{W}_{s}$ and is monotonically decreasing as $\mathsf{W}_{s}$ increases.

Proof.

Appendix -F6. ∎

Lemma 7 below shows how $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ changes with $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ .

Lemma 7.

Consider $\mathsf{W}_{i}=\mathsf{W}_{\Lambda-i+1}\in[0,b_{i}]$ , $t+1\leq i\leq\theta$ . If the upper bounds of the intervals satisfy

\displaystyle\sigma^{2}+\frac{1}{L_{C}}\sum_{i=t+1}^{\Lambda-t}b_{i}\leq\sqrt{\frac{\mathsf{W}_{t}}{L_{C}}},

(55)

then the derivative $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ (39)–(40) is non-decreasing as the elements in any non-empty subset of $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ increase on their corresponding intervals, $t=1,2,\dots,\theta$ .

Proof.

Appendix -F7. ∎

We introduce notations that will be used in the following proof. Fixing $\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}$ , we denote the derivative of $\mathsf{f}_{t}$ with respect to $\mathsf{W}_{t}$ at $\mathsf{W}_{t}=\bar{\mathsf{W}}_{t}$ by

\displaystyle\mathsf{f}_{t}^{\prime}(\bar{\mathsf{W}}_{t})\triangleq\frac{\partial\mathsf{f}_{t}(\mathsf{W}_{t},\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta})}{\partial\mathsf{W}_{t}}\Big{|}_{\mathsf{W}_{t}=\bar{\mathsf{W}}_{t}}.

(56)

Given a sequence of positive numbers $\{\gamma_{i}\}_{i=t}^{\theta}$ , we denote by $K_{s}^{(t)}$ a positive number that ensures

	$\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t}^{s-1},\{\mathsf{W}_{i}^{(V)}+\gamma_{i}\}_{i=s}^{\theta}\right)$
$\displaystyle-~{}$	$\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t}^{s},\{\mathsf{W}_{i}^{(V)}+\gamma_{i}\}_{i=s+1}^{\theta}\right)$
$\displaystyle\geq~{}$	$\displaystyle-K_{s}^{(t)}\gamma_{s},$	(57)

for $s\geq t+1$ . Such $K_{s}^{(t)}$ always exists since $\mathsf{f}_{t}$ is continuously differentiable with respect to $\mathsf{W}_{s}$ , i.e., it is a Lipschitz function of $\mathsf{W}_{s}$ , and it decreases as $\mathsf{W}_{s}$ increases by Lemma 6. Given $\{\bar{\mathsf{W}}_{t}\}_{t=1}^{\theta}$ , we define sequence $\{g_{t}\}_{t=1}^{L_{C}}$ ,


		$\displaystyle g_{\theta}\triangleq 1,$		(58a)
		$\displaystyle g_{t}\triangleq\frac{\sum_{s=t+1}^{\theta}K_{s}^{(t)}g_{s}}{\mathsf{f}_{t}^{\prime}(\bar{\mathsf{W}}_{t})},~{}t\leq\theta,$		(58b)
		$\displaystyle g_{t}\triangleq g_{\Lambda-t+1},~{}t>\theta.$		(58c)

Given $\{\bar{\mathsf{W}}_{t}\}_{t=1}^{\theta-1}$ and an arbitrary positive number $\gamma_{\theta}^{(\theta)}$ , we define a non-increasing sequence $\gamma_{\theta}^{(t)}$ , $t=1,2,\dots,\theta-1$ , as

\displaystyle\gamma_{\theta}^{(t)}\triangleq\min\left\{\gamma_{\theta}^{(t+1)},\frac{1}{g_{t}}(\bar{\mathsf{W}}_{t}-\mathsf{W}_{t}^{(V)})\right\}.

(59)

We denote

\displaystyle\gamma_{\theta}^{\max}\triangleq\max\left\{\gamma_{\theta}>0\colon\frac{\omega}{L_{C}L_{R}}\sum_{t=1}^{L_{C}}g_{t}\gamma_{\theta}=P-P_{V}(R)\right\}.

(60)

We define the minimum of (59) and (60) as

\displaystyle\gamma_{\theta}^{*}\triangleq\min\{\gamma_{\theta}^{(1)},\gamma_{\theta}^{\max}\}.

(61)

We define a sequence of numbers $\{\gamma_{t}^{*}\}_{t=1}^{\theta-1}$ as

\displaystyle\gamma_{t}^{*}\triangleq g_{t}\gamma_{\theta}^{*}.

(62)

-F2 Proof of ‘If’ direction

We show that if $P>P_{V}(R)$ and $R<\bar{R}$ , then VPA does not declare a failure, equivalently, there exists a sequence $\{\mathsf{W}_{c}\}_{c\in[L_{C}]}$ that satisfies (lines 1–5 and line 9 of Algorithm 1):

		$\displaystyle\mathsf{f}_{t}(\{\mathsf{W}_{i}\}_{i=t}^{\theta})>2RL_{R},\forall t=1,2,\dots,\theta,$		(63)
		$\displaystyle\frac{\omega}{L_{R}L_{C}}\sum_{c=1}^{L_{C}}\mathsf{W}_{c}\leq P.$		(64)

For $P>P_{V}(R)$ and $R<\bar{R}$ , we set

\displaystyle\mathsf{W}_{t}=\mathsf{W}_{t}^{(V)}+\gamma_{t}^{*},~{}t=1,2,\dots,\theta,

(65)

where $\mathsf{W}_{t}^{(V)}$ exists due to Lemma 4; $\gamma_{t}^{*}$ is defined in (62); $\gamma_{\theta}^{(\theta)}$ defining $\gamma_{\theta}^{*}$ (61) via (59) is an arbitrary positive number; the sequence $\{\bar{\mathsf{W}}_{t}\}_{t=1}^{\theta-1}$ defining $\gamma_{\theta}^{*}$ (61) via (59) is chosen to be large enough so that

\displaystyle\bar{\mathsf{W}}_{t}>\mathsf{W}_{t}^{(V)},~{}\forall t=1,\dots,\theta-1,

(66)

and that $\bar{\mathsf{W}}_{t}$ satisfies (55) with $\mathsf{W}_{t}\leftarrow\bar{\mathsf{W}}_{t}$ and $b_{i}\leftarrow\mathsf{W}_{i}^{(V)}+g_{i}\gamma_{\theta}^{(t+1)}$ , $i=t+1,\dots,\theta$ .

We show that the power allocation in (65) satisfies (63)–(64), respectively. The power allocation (65) satisfies the power constraint (64) due to (60)–(61). To show that the power allocation satisfies (63), it suffices to show that the following statement:

\displaystyle\text{If}~{}0<\gamma_{\theta}^{*}\leq\gamma_{\theta}^{(t)},~{}\text{then \eqref{cons_1} holds at iteration}~{}t.

(67)

Since $0<\gamma_{\theta}^{*}\leq\gamma_{\theta}^{(t)}$ for all $t=1,2,\dots,\theta$ by (61), this statement allows us to conclude that the condition (63) is satisfied for all $t=1,2,\dots,\theta$ .

It remains to prove the statement (67). The statement trivially holds for $t=\theta$ , since $\mathsf{W}_{\theta}>\mathsf{W}_{\theta}^{(V)}$ ensures (63) according to Lemma 2. We proceed to prove the statement for $t\leq\theta-1$ . Taking the difference between two $\mathsf{f}_{t}$ with different $\mathsf{W}_{t}$ , we obtain


	$\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}+\gamma_{i}^{}\}_{i=t}^{\theta}\right)-\mathsf{f}_{t}\left(\mathsf{W}_{t}^{(V)},\{\mathsf{W}_{i}^{(V)}+\gamma_{i}^{}\}_{i=t+1}^{\theta}\right)$
$\displaystyle>~{}$	$\displaystyle\frac{\partial\mathsf{f}_{t}(\mathsf{W}_{t},\{\mathsf{W}_{i}^{(V)}+\gamma_{i}^{}\}_{i=t+1}^{\theta})}{\partial\mathsf{W}_{t}}\Big{\|}_{\mathsf{W}_{t}=\mathsf{W}_{t}^{(V)}+\gamma_{t}^{}}\gamma_{t}^{*}$	(68a)
$\displaystyle\geq~{}$	$\displaystyle\frac{\partial\mathsf{f}_{t}(\mathsf{W}_{t},\{\mathsf{W}_{i}^{(V)}+\gamma_{i}^{}\}_{i=t+1}^{\theta})}{\partial\mathsf{W}_{t}}\Big{\|}_{\mathsf{W}_{t}=\bar{\mathsf{W}}_{t}}\gamma_{t}^{}$	(68b)
$\displaystyle\geq~{}$	$\displaystyle\mathsf{f}^{\prime}_{t}(\bar{\mathsf{W}}_{t})\gamma_{t}^{*},$	(68c)

where (68a) holds by Mean Value Theorem and by Lemma 5; (68b) holds due to $\gamma_{t}^{*}=g_{t}\gamma_{\theta}^{*}\leq g_{t}\gamma_{\theta}^{(t)}\leq\bar{\mathsf{W}}_{t}-\mathsf{W}_{t}^{(V)}$ and Lemma 5; (68c) holds due to the fact that $\gamma_{i}^{*}=g_{i}\gamma_{\theta}^{*}\leq g_{i}\gamma_{\theta}^{(t)}\leq g_{i}\gamma_{\theta}^{(t+1)}$ for all $t+1\leq i\leq\theta$ , the choice of $\bar{\mathsf{W}}_{t}$ below (66), and Lemma 7. We then take the difference between two $\mathsf{f}_{t}$ with different $\mathsf{W}_{s}$ , $s\geq t+1$ , just like (57) with $\gamma_{i}\leftarrow\gamma_{i}^{*}$ , $\forall i=s,\dots,\theta$ . Summing (68) and (57) for all $s=t+1,t+2,\dots,\theta$ , we obtain


	$\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}+\gamma_{i}^{*}\}_{i=t}^{\theta}\right)-\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t}^{\theta}\right)$	(69a)
$\displaystyle>~{}$	$\displaystyle\mathsf{f}^{\prime}_{t}(\bar{\mathsf{W}}_{t})\gamma_{t}^{}-\sum_{s=t+1}^{\theta}K_{s}^{(t)}\gamma_{s}^{}$	(69b)
$\displaystyle=~{}$	$\displaystyle\mathsf{f}^{\prime}_{t}(\bar{\mathsf{W}}_{t})g_{t}\gamma_{\theta}^{}-\sum_{s=t+1}^{\theta}K_{s}^{(t)}g_{s}\gamma_{\theta}^{}$	(69c)
$\displaystyle=~{}$	$\displaystyle 0,$	(69d)

where (69c) holds by plugging (62) into (69b), and (69d) holds by plugging (58b) into (69c). Since the second term in (69a) is equal to $2RL_{R}$ , we conclude that (63) holds at iteration $t$ with the power allocation in (65).

-F3 Proof of ‘Only if’ direction

Given $P<\infty$ and $R<\infty$ , we show that if VPA does not declare failure, then $P>P_{V}(R)$ and $R<\bar{R}$ .

Not declaring failure implies that at the end of line 5 of Algorithm 1, VPA forms finite $\{\mathsf{W}_{t}\}_{t=1}^{\theta}$ that ensure $\mathsf{f}_{t}>2RL_{R}$ for all $t=1,2,\dots,\theta$ . We show by mathematical induction that there exist $\{\mathsf{W}_{t}^{(V)}\}_{t=1}^{\theta}$ that satisfy


		$\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}^{(V)}\}_{i=t}^{\theta}\right)=2RL_{R}$		(70a)
		$\displaystyle\mathsf{W}_{t}^{(V)}<\mathsf{W}_{t},t=1,2,\dots,\theta.$		(70b)

for all $1\leq t\leq\theta$ and for any $\{\mathsf{W}_{t}\}_{t=1}^{\theta}$ yielded by VPA.

Initial step: Since $\mathsf{f}_{\theta}(\mathsf{W}_{\theta})>2RL_{R}$ , $\mathsf{f}_{\theta}(0)=0$ , and $\mathsf{f}_{t}$ is continuously increasing by Lemma 2, we conclude that there exists $\mathsf{W}_{\theta}^{(V)}$ that satisfies (70) at $t=\theta$ .

Induction step: Assuming that there exist $\{\mathsf{W}_{t}^{(V)}\}_{t=s+1}^{\theta}$ that satisfy (70) for $t=s+1,\dots,\theta$ , we show that together with $\{\mathsf{W}_{t}^{(V)}\}_{t=s+1}^{\theta}$ , there exists $\mathsf{W}_{s}^{(V)}$ that satisfies (70) at $t=s$ . From Lemma 6 and the induction assumption, we conclude

\displaystyle\mathsf{f}_{s}\left(\mathsf{W}_{s},\{\mathsf{W}_{t}^{(V)}\}_{t=s+1}^{\theta}\right)>\mathsf{f}_{s}\left(\{\mathsf{W}_{i}\}_{i=s}^{\theta}\right).

(71)

Since $\mathsf{f}_{s}(\{\mathsf{W}_{i}\}_{i=s}^{\theta})>2RL_{R}$ , $\mathsf{f}_{s}(0,\{\mathsf{W}_{t}^{(V)}\}_{t=s+1}^{\theta})=0$ , and $\mathsf{f}_{s}$ is continuously increasing in $\mathsf{W}_{s}$ by Lemma 2, there exists $\mathsf{W}_{s}^{(V)}$ that satisfies (70).

The existence of $\{\mathsf{W}_{t}^{(V)}\}_{t=1}^{\theta}$ satisfying (70) implies $P>P_{V}(R)$ due to (70b) and implies $R<\bar{R}$ due to Lemma 4.

-F4 Proof of Lemma 4

Fixing $\mathsf{W}_{s}=0$ for all $s\geq t+1$ , we denote

\displaystyle R_{t}\triangleq\frac{1}{2L_{R}}\lim_{\mathsf{W}_{t}\rightarrow\infty}\mathsf{f}_{t}(\mathsf{W}_{t},0,0,\dots,0).

(72)

Before we prove Lemma 4, we show

\displaystyle\bar{R}=\min\{R_{t},t=1,2,\dots,\theta\}.

(73)

For $2t<\Lambda-\omega+2$ ,

\displaystyle R_{t}=\frac{\omega L_{C}}{2L_{R}};

(74)

for $\Lambda-\omega+2<2t<\Lambda+1$ ,

\displaystyle R_{t}=\frac{1}{2L_{R}}\left((\Lambda-2t+1)L_{C}+(2t+\omega-\Lambda-1)\frac{L_{C}}{2}\right);

(75)

for $2t=\Lambda+1$ ,

\displaystyle R_{t}=\frac{\omega L_{C}}{2L_{R}}.

(76)

If $\Lambda$ is even, $2\theta<\Lambda+1$ , the first two cases (74)–(75) describe $R_{t}$ for all $t=1,2,\dots,\theta$ , and the minimum in (73) is achieved at $2t=\Lambda$ , yielding

\displaystyle\bar{R}=R_{\frac{\Lambda}{2}}=\frac{L_{C}(\omega+1)}{4L_{R}}.

(77)

If $\Lambda$ is odd, $2\theta=\Lambda+1$ , the three cases (74)–(76) jointly describe $R_{t}$ for all $t=1,2,\dots,\theta$ , and the minimum in (73) is achieved at $2t=\Lambda-1$ , yielding

\displaystyle\bar{R}=R_{\frac{\Lambda-1}{2}}=\frac{L_{C}(\omega+2)}{4L_{R}}.

(78)

We begin to prove Lemma 4.

We show that if $R<\bar{R}$ , there exist $\mathsf{W}_{1}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}<\infty$ that satisfy $\mathsf{f}_{t}=2RL_{R}$ for all $t=1,2,\dots,\theta$ . We prove this by mathematical induction.

Initial step: Since $R<\bar{R}\leq R_{\theta}$ and $\mathsf{f}_{\theta}\in[0,2R_{\theta}L_{R})$ is continuously increasing in $\mathsf{W}_{\theta}$ , there exists $\mathsf{W}_{\theta}^{(V)}<\infty$ that satisfies $\mathsf{f}_{\theta}(\mathsf{W}_{\theta}^{(V)})=2RL_{R}$ .

Induction step: Assuming there exist $\mathsf{W}_{t+1}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}<\infty$ that satisfy $\mathsf{f}_{i}=2RL_{R}$ for all $i=t+1,\dots,\theta$ , we show that together with $\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}$ , there exists $\mathsf{W}_{t}^{(V)}<\infty$ that satisfies $\mathsf{f}_{t}=2RL_{R}$ . Since $\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}$ are finite by the induction assumption, it holds that

\displaystyle\lim_{\mathsf{W}_{t}\rightarrow\infty}\mathsf{f}_{t}\left(\mathsf{W}_{t},\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}\right)=2R_{t}L_{R}

(79)

Since $R<\bar{R}\leq R_{t}$ and $\mathsf{f}_{t}\in[0,2R_{t}L_{R})$ is continuously increasing in $\mathsf{W}_{t}\in[0,\infty)$ , there exists $\mathsf{W}_{t}<\infty$ that achieves $\mathsf{f}_{t}=2RL_{R}$ .

We show that if there exist $\mathsf{W}_{1}^{(V)},\mathsf{W}_{2}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}<\infty$ that satisfy $\mathsf{f}_{t}=2RL_{R}$ for all $t=1,2,\dots,\theta$ , then the rate satisfies $R<\bar{R}$ . It holds that


$\displaystyle 2RL_{R}$	$\displaystyle=\mathsf{f}_{t}\left(\mathsf{W}_{t}^{(V)},\mathsf{W}_{t+1}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}\right)$	(80a)
	$\displaystyle<\lim_{\mathsf{W}_{t}\rightarrow\infty}\mathsf{f}_{t}\left(\mathsf{W}_{t},\mathsf{W}_{t+1}^{(V)},\dots,\mathsf{W}_{\theta}^{(V)}\right)$	(80b)
	$\displaystyle=2R_{t}L_{R},$	(80c)

where (80b) is by Lemma 2; (80c) holds since $\{\mathsf{W}_{i}^{(V)}\}_{i=t+1}^{\theta}$ are finite. Since (80) holds for all $t=1,2,\dots,\theta$ , we conclude $R<\bar{R}$ , where $\bar{R}$ is defined in (73).

-F5 Proof of Lemma 5

Since $\mathsf{W}_{t}$ only appears in the denominator of $\frac{\partial\mathsf{f}_{t}}{\partial W_{t}}$ (39)–(40) as a summand, the increase of $\mathsf{W}_{t}$ leads to the decrease of $\frac{\partial\mathsf{f}_{t}}{\partial W_{t}}$ .

-F6 Proof of Lemma 6

We show that the derivative of $\mathsf{f}_{t}$ with respect to $\mathsf{W}_{s}$ for $s\geq t+1$ is negative. For $\Lambda-s+1>\min\{r,\Lambda-t+1\}$ ,

\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{s}}=\sum_{r=t}^{t+\omega-1}\frac{-\mathsf{W}_{t}\frac{1}{L_{C}}}{\left(\sigma^{2}+\frac{1}{L_{C}}\sum_{c^{\prime}=t}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}\right)^{2}}<0.

(81)

For $\Lambda-s+1\leq\min\{r,\Lambda-t+1\}$ ,

\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{s}}=\sum_{r=t}^{t+\omega-1}\frac{-\mathsf{W}_{t}\frac{2}{L_{C}}}{\left(\sigma^{2}+\frac{1}{L_{C}}\sum_{c^{\prime}=t}^{\min\{r,\Lambda-t+1\}}\mathsf{W}_{c^{\prime}}\right)^{2}}<0.

(82)

-F7 Proof of Lemma 7

We denote by $M_{r,t+1}\triangleq\sigma^{2}+\frac{1}{L_{C}}\sum_{i=t+1}^{r}\mathsf{W}_{i}$ , $t+1\leq r\leq\Lambda-t$ , and we rewrite $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ in (39)–(40) as follows. If $t+\omega-1<\Lambda-t+1$ ,


$\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$	$\displaystyle=\frac{\sigma^{2}}{(\sigma^{2}+\frac{1}{L_{C}}\mathsf{W}_{t})^{2}}$	(83a)
	$\displaystyle+\sum_{r=t+1}^{t+\omega-1}\frac{1}{(\sqrt{M_{r,t+1}}+\frac{1}{L_{C}}\frac{\mathsf{W}_{t}}{\sqrt{M_{r,t+1}}})^{2}};$	(83b)

if $t+\omega-1\geq\Lambda-t+1$ ,


		$\displaystyle\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}=\frac{\sigma^{2}}{(\sigma^{2}+\frac{1}{L_{C}}\mathsf{W}_{t})^{2}}$		(84a)
		$\displaystyle+\sum_{r=t+1}^{\Lambda-t}\frac{1}{(\sqrt{M_{r,t+1}}+\frac{1}{L_{C}}\frac{\mathsf{W}_{t}}{\sqrt{M_{r,t+1}}})^{2}}$		(84b)
		$\displaystyle+(2t+\omega-\Lambda-1)\frac{1}{(\sqrt{M_{\Lambda-t,t}}+\frac{2}{L_{C}}\frac{\mathsf{W}_{t}}{\sqrt{M_{\Lambda-t,t+1}}})^{2}}.$		(84c)

We observe that (i) each summand in (83b) and (84b) monotonically increases as $M_{r,t+1}$ increases on $M_{r,t+1}\in\left[0,\sqrt{\frac{\mathsf{W}_{t}}{L_{C}}}\right]$ ; (ii) (84c) increases as $M_{\Lambda-t,t+1}$ increases on $M_{\Lambda-t,t+1}\in\left[0,\sqrt{\frac{2\mathsf{W}_{t}}{L_{C}}}\right]$ ; (iii) $M_{r,t+1}$ increases as $r$ increases.

Since (55) means $M_{\Lambda-t,t+1}\leq\sqrt{\frac{\mathsf{W}_{t}}{L_{C}}}$ , observation (iii) implies that $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ satisfy $M_{r,t+1}\leq\sqrt{\frac{\mathsf{W}_{t}}{L_{C}}}$ for all $t+1\leq r\leq\Lambda-t$ . Thus, observations (i)–(ii) imply that $\frac{\partial\mathsf{f}_{t}}{\partial\mathsf{W}_{t}}$ (83)–(84) is non-decreasing as the elements in any non-empty subset of $\{\mathsf{W}_{i}\}_{i=t+1}^{\theta}$ increase on their corresponding intervals.

-G Proof of Theorem 2: step (ii)

We show that the output power allocation of VPA ensures successful decoding. The power determined at the end of line 5 of Algorithm 1 satisfies

\displaystyle\mathsf{f}_{t}\left(\{\mathsf{W}_{i}\}_{i=t}^{\theta}\right)>2RL_{R},\forall t=1,\dots,\theta,

(85)

since $\delta_{t}>0$ for all $t=1,2,\dots,\theta$ and Lemma 2, which states that $\mathsf{f}_{t}$ increases as $\mathsf{W}_{t}$ increases. Plugging (85) into Lemma 1 item 2), we conclude $\psi_{c}^{\theta}=0$ , $\forall c\in[L_{C}]$ , meaning that VPA ensures successful decoding within $\theta$ iterations.

Since the left sides of the inequalites in lines 6 and 9 are equal to the left side of (4), representing the resultant power, lines 6 and 9 check the satisfaction of the power constraint (4). After transferring the resiudal power to $\mathsf{W}_{1}$ and $\mathsf{W}_{\Lambda}$ in lines 9–13, the resultant power still satisfies (85) since $\mathsf{f}_{t},t\geq 2$ does not depend on $\mathsf{W}_{1}$ and $\mathsf{W}_{\Lambda}$ , and $\mathsf{f}_{1}$ by Lemma 2 monotonically increases as $\mathsf{W}_{1}$ increases.

-H Proof of Proposition 3

We first show (24). It suffices to show that the upper bound on $R$ in (13) is smaller than or equal to that in (23). For clarity, we denote the upper bounds on $R$ in (13) and (23) by $\bar{R}_{U}$ and $\bar{R}_{V}$ , respectively. We upper bound $\bar{R}_{U}$ as


$\displaystyle\bar{R}_{U}$	$\displaystyle=\frac{L_{C}}{2L_{R}}\sum_{r=1}^{\omega}\frac{1}{r}$	(86a)
	$\displaystyle\leq\frac{L_{C}}{2L_{R}}\left(1+\frac{\omega-1}{2}\right)$	(86b)
	$\displaystyle=\frac{L_{C}(\omega+1)}{4L_{R}}$	(86c)
	$\displaystyle\leq\bar{R}_{V}$	(86d)

where (86b) holds by lower bounding $r$ by $2$ for all $r\geq 2$ .

Given rate $R$ that ensures $P_{U}(R)<\infty$ and $P_{V}(R)<\infty$ , we proceed to show (25). We denote by $\bar{\mathsf{W}}\triangleq P_{U}(R)\frac{L_{R}}{\omega}$ the UPA at $P_{U}(R)$ . To show (25), it suffices to show

\displaystyle\mathsf{W}_{t}^{(V)}\leq\bar{\mathsf{W}}.

(87)

To this end, from (21) and (32), we conclude

\displaystyle\mathsf{f}_{1}(\bar{\mathsf{W}},\bar{\mathsf{W}},\dots,\bar{\mathsf{W}})=2RL_{R}.

(88)

Lemma 3 implies for all $t=2,3,\dots,\theta$ ,

\displaystyle\mathsf{f}_{t}(\bar{\mathsf{W}},\bar{\mathsf{W}},\dots,\bar{\mathsf{W}})\geq 2RL_{R}.

(89)

At $t=\theta$ , Lemma 2 and (89) imply (87). At $t=\theta-1$ , since Lemma 6 implies $\mathsf{f}_{\theta-1}(\bar{\mathsf{W}},\mathsf{W}_{\theta}^{(V)})\geq\mathsf{f}_{\theta-1}(\bar{\mathsf{W}},\bar{\mathsf{W}})$ , we conclude from Lemma 2 and (89) that (87) holds at $t=\theta-1$ . Simiarly, at $t=\theta-2,\theta-3,\dots,1$ , iteratively using Lemmas 2 and 6 and (88)–(89), we obtain (87).

-I UPA is a special case of VPA

We show that UPA is a special case of VPA. This is an alternative proof for (25). Consider any $P>P_{U}(R)$ and $R<\bar{R}_{U}$ with UPA $\bar{\mathsf{W}}^{\prime}\triangleq P\frac{L_{R}}{\omega}$ . Due to the fact that $\mathsf{f}_{1}$ in (88) increases as $\bar{\mathsf{W}}$ increases, $\bar{\mathsf{W}}^{\prime}>\bar{\mathsf{W}}$ , and Lemma 3, we conclude $\mathsf{f}_{t}(\bar{\mathsf{W}}^{\prime},\bar{\mathsf{W}}^{\prime},\dots,\bar{\mathsf{W}}^{\prime})>2RL_{R}$ for all $t=1,2,\dots,\theta$ . VPA recovers UPA by choosing $\delta_{t}=\bar{\mathsf{W}}^{\prime}-\mathsf{W}_{t}$ , where $\mathsf{W}_{t}$ is the output of line 2 of Algorithm 1. The difference $\delta_{t}$ is positive for all $t=1,2,\dots,\theta$ since Lemma 2 implies that the output of line 2 satisfies $\mathsf{W}_{t}<\bar{\mathsf{W}}^{\prime}$ .

Power Allocation for the Base Matrix of Spatially Coupled Sparse Regression Codes

Abstract

I Introduction

II Spatially coupled sparse regression codes

II-A Encoder

Definition 1.

II-B Decoder

III Power allocation and performance metrics

Definition 2.

IV Uniform power allocation

Theorem 1.

Proof.

Proposition 1.

Proof.

V V-power allocation

V-A VPA Algorithm

Proposition 2.

Proof.

V-B VPA performance

Lemma 1.

Proof.

Theorem 2.

Proof sketch.

V-C VPA outperforms UPA

Proposition 3.

Proof.

VI Simulations

VII Conclusion

References

-A Proof of Theorem 1

-A1 Proof of RU​(P)R_{U}(P)

-A2 Proof of PU​(R)P_{U}(R)

-B Proof of Proposition 1

-C g≤ωg\leq\omega is sufficient

-D Proof of Proposition 2

-D1 Main proof

Lemma 2.

Proof.

Lemma 3.

Proof.

-D2 Proof of Lemma 2

-D3 Proof of Lemma 3

-E Proof of Lemma 1

-E1 Proof of item 1)

-E2 Proof of item 2)

-F Proof of Theorem 2: step (i)

-F1 Lemmas and notations

Lemma 4.

Proof.

Lemma 5.

Proof.

Lemma 6.

Proof.

Lemma 7.

Proof.

-F2 Proof of ‘If’ direction

-F3 Proof of ‘Only if’ direction

-F4 Proof of Lemma 4

-F5 Proof of Lemma 5

-F6 Proof of Lemma 6

-F7 Proof of Lemma 7

-G Proof of Theorem 2: step (ii)

-H Proof of Proposition 3

-I UPA is a special case of VPA

-A1 Proof of $R_{U}(P)$

-A2 Proof of $P_{U}(R)$

-C $g\leq\omega$ is sufficient