在平行高斯通道下考慮固定碼長與碼率的編碼條件之吞吐量基準資源分配方式

國

立

交

通

大

學

電信工程研究所

碩

士

論

文

在帄行高斯通道下考慮固定碼長與碼率

的編碼條件之吞吐量基準資源分配方式

Throughput-Oriented Power Allocation Policies for Parallel Gaussian

Channels Under Finite-Length and Fixed-Rate Coding Constraints

研 究 生：張紊傑

指導教授：陳伯寧

在帄行高斯通道下考慮固定碼長與碼率

的編碼條件之吞吐量基準資源分配方式

Throughput-Oriented Power Allocation Policies for Parallel Gaussian

Channels Under Finite-Length and Fixed-Rate Coding Constraints

研究生：張紊傑 Student：Wen-Chieh Chang

指導教授：陳伯寧 Advisor：Po-Ning Chen

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

Submitted to Institute of Computer and Information Science College of Electrical Engineering and Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Computer and Information Science

June 2011

在平行高斯通道下考慮固定碼長與碼率的編碼條件之吞吐量基準資源分配

方式

學生：張紊傑

指導教授：陳伯寧

國立交通大學電信工程研究所碩士班

摘

要

傳統探討平行無記憶性高斯通道資源分配的問題都是以最大化整體通

道容量為目的，而所推得的最佳能量分配法為著名的充水(Water-Filling)

演算法。但是此一能量分配法則的問題為，各管道達到最佳能量分配的碼

率是總能量的函數，所得的最佳碼率常是一個實作上不易實現的實數，同

時隨著訊號雜訊比增大還需不斷更換最佳編碼方式，如此才能達到原本設

定的最大通道容量，另外最大通道容量是成立於碼長趨近於無限大的情

況，在實際有限的碼長下，此種能量分配方式是否可以達到最佳系統效能

(即系統有效吞吐量)值得探討，故而本論文直接探討在有限碼長、固定編

碼方式下以達到最大有效系統吞吐量為目的的傳送能量分配策略。結果顯

示，我們所提出的資源分配方式可以以類似充水演算法的概念以圖形詮

釋，並且幾乎達到最大系統有效吞吐量。

Throughput-Oriented Power Allocation Policies for

Parallel Gaussian Channels Under Finite-Length and

Fixed-Rate Coding Constraints

Student: Wen-Chieh Chang kkkAdvisor: Po-Ning Chen Institute of Communications Engineering

National Chiao Tung University

Abstract

The common criterion used in the power allocation problem for parallel memoryless Gaussian channels is to maximize overall mutual information (namely, to achieve the ca-pacity), resulting in the well-known water-filling policy. Such a capacity-achieving power allocation, although theoretically interesting and beneficial in conceptually elucidating the behavior of coding systems, does not match well with practical situations as capacity is an asymptotic rate requiring the codeword length to grow to infinity. In addition, the overall system capacity can only be achieved when the coding scheme of each channel is optimally and continuously adapted to the allotted power. However in a practical system, the adopted codes are by no means optimal in terms of achieving capacity and have only a finite number of rate choices. Furthermore, a common quantity of interest is the effective system through-put. In light of these observations, we study in this paper the problem of determining the power allocation strategy for a system of coded parallel Gaussian channels with the objective of maximizing effective throughput under finite-length and fixed-rate coding constraints. An approximating formula of the system’s effective throughput is proposed for the case of con-volutional codes and used to identify the optimal power allocation for each parallel channel. Our results show that the proposed power allocation policies can be graphically represented as a variation of the water-filling principle and achieves a near-optimal throughput.

Acknowledgements

I would like to express my heartily gratitude to my advisor, Professor Po-Ning Chen. This thesis would not be possible without his wisdom in guidance and his patience to me. I am deeply encouraged by his enthusiasm in research and the way he looks at life. To Professor Chung-Hsuan Wang and Professor Fady Alajaji, I deeply appreciate their invaluable instruc-tions in the research. To Dr. Shih-Wei Wang, I am truly thankful for all the solid suggesinstruc-tions from him.

I would also like to thank all the members in the NTL lab as well as all the people that have ever helped me in the past two years. Finally, I would like to dedicate this thesis to my family, who always provide me with love and spiritual support.

Contents

2.1 Capacity-Achieving Water-Filling Power Allocation Policy with Known Noise Variance in Each Channel . . . 4 2.2 Worst-Case-Capacity-Achieving Water-Filling Power Allocation Policy

Sub-ject to a Known Sum of Noise Variances . . . 6

4 Throughput-Oriented Water-Filling 13 4.1 Throughput-Oriented Water-Filling: Noise Variance in Each Channel is Known 13 4.2 Throughput-Oriented Water-Filling: Only Total Noise Variance is Available 17

5 Numerical and Simulation Results 24

5.1 Throughput-Oriented Water-Filling: Noise Variance in Each Channel is Known 24 5.2 Throughput-Oriented Water-Filling: Only Total Noise Variance is Available 34

6 Conclusion 40

List of Figures

2.1 An example of the water-filling power allocation for K = 3. . . 5

3.1 Pe and its approximations for a (4, 1, 6) convolutional code with generator

polynomial (in octal) being [177 127 155 171], dfree = 20 and Adfree = 2.

The adjusted parameters are d = 22.42 and A = 962.51. The frame size is N = 4(500 + 6). Eb/N0 is plotted in linear scale. . . 12

4.1 An example of the throughput-oriented water-filling with K = 3 and O = {1, 2}. . . 15 4.2 An example of the throughput-oriented water-filling with K = 3. . . 22 5.1 Case I: Effective throughputs for the seven choices of the active channel set O. 26 5.2 Case I: Optimal effective throughputs obtained from exhaustive search, the

throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy. . . 27 5.3 Case I: Optimal power ratio for channel 2. . . 28 5.4 Case I: Illustration of the optimal active set O changing from {1} to {1, 2}. 29 5.5 Case II: Effective throughputs for the seven choices of active channel set O. 30

5.6 Case II: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy. . . 31 5.7 Case II: Optimal power ratio for Channel 2. . . 32 5.8 Case III: Optimal effective throughputs obtained from exhaustive search, the

throughput-oriented water-filling based on the FER approximation, and the capacity-achiving water-filling policy. . . 33 5.9 Case I: The worst-case effective throughputs obtained from throughput-oriented

water-filling based on the FER approximation, and the worst-case-capacity-achieving equal power allocation. . . 35 5.10 Case II: The worst-case effective throughputs obtained from throughput-oriented

water-filling based on the FER approximation, and the the worst-case-capacity-achieving equal power allocation. . . 36 5.11 Case III: The worst-case effective throughput of using throughput-oriented

water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) codes are used in the first three channels, and (2, 1, 6) code is used again in the fourth channel. . . 38 5.12 Case III: The worst-case effective throughput of using the throughput-oriented

water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) code sare used in the first three channels, and (3, 2, 6) code is used in the fourth channel. . . 39

Chapter 1

Introduction

1.1

Overview

Finding the best strategy for allocating power over parallel independent additive white Gaus-sian noise (AWGN) channels is a classical problem in information theory (e.g., cf. [1, 6] and the references therein). For this problem, a typical optimization criterion for the distribution of power is to maximize the system’s mutual information (namely, to achieve the system’s capacity), which results in the well-known water-filling scheme. In this scheme, the capacity is achieved when the input of each parallel channel is Gaussian distributed and has a power allotment given by the “water level” of its respective “vessel” with base height equal to the channel’s noise variance [1]. In 2006, Lozano, Tulio and Verd´u re-visited this problem by judiciously constraining the input to be drawn from discrete modulation constellations used in practice such as phase-shift keying (PSK) [9] and quadrature amplitude modulation. They concluded the study with a refined optimal power allocation policy, referred to as mercury water-filling [6]. The result was obtained based on two key observations regarding parallel Gaussian channels: (i) both the mutual information and the minimum mean-square error are functions of the signal-to-noise ratios (SNRs), and (ii) the derivative of the former measure with respect to the SNR is equal to the latter one.

In literature, there is another challenging power allocation problem over parallel Gaussian channels. Instead of knowing the noise variance in each channel, only the total noise variance is known. In such case, the criterion becomes to maximize the so-called worst-case mutual information, defined as the smallest mutual information among all possible noise variance distributions with variance sum equal to a given constant. Signal power is then allotted to achieve the worst-case capacity, which is the maximum of the worst-case mutual information among all power allocations. The resulting power allocation policy is to allot equal signal power to each channel, regardless of the value of the total noise variance.

The capacity-oriented power allocation, although theoretically interesting and useful for the analysis of channel coded systems, is not realistic in several aspects. First, channel ca-pacity is a function of the total system power, and the optimal coding scheme that achieves capacity may be different for different capacity values. Hence, optimality can be achieved only when the coding scheme of each parallel channel can be optimally adapted to the power allotment, which is difficult to fulfill in practice. Secondly, the optimal rate obtained from a capacity-based power allocation is often a concretely unrealizable real number; this is in contrast with practical systems whose code rates are usually restricted to only a few rational numbers such as 1/2, 1/3, 2/3, 1/4, etc. Finally, capacity is an asymptotic quantity that re-quires the coding blocklength or frame size to grow without bound; yet, in practical systems, the blocklength is finite (typically preset as a function of the system’s delay requirements).

In view of the above points, we herein investigate power allocation policies that re-spectively achieve the maximum effective throughput (instead of capacity) and maximum worst-case effective throughput (instead of worst-case capacity) for convolutionally coded parallel memoryless Gaussian channels with finite-length and fixed-rate coding constraints, where effective throughput is defined as the number of successfully transmitted information bits per channel use. Since in general, there is no closed-form formula for the error rate (and

hence effective throughput) of a coded system, the optimal solution can only be obtained via case-by-case simulation. Our study however shows that it is possible to obtain a good approximating expression for the error rate of each coded channel and then use these approx-imations to derive the near-optimal power allocation policies as a function of the system’s total power and noise variances. The resulting near-optimal-throughput power allocation policies are reminiscent of a variation of the traditional water-filling principle, where the base width and height of each individual vessel (corresponding to each parallel channel) now become functions of the code characteristics. However, unlike the case of water-filling, we obtain that when a channel is in use (or active), a minimal power should be allocated to it. In the effective-throughput-optimizing problem, we show that the optimal power assigned to each channel may experience a sudden jump (or discontinuity) when the total system power increases. This is due to the practical constraint requiring the code rates to be fixed and pos-itive, and hence for a given channel, a power allotment that is smaller than a certain value can only result in an inferior overall throughput. In the worst-case-effective-throughput-optimizing problem, we provided a near-optimal power allocaton policy for system SNR greater than a certain threshold. We show that our proposed power allocation policy yields better gain than the traditional equal power allocation if the differences in the characteristics of the used codes between channels are larger.

The rest of the thesis is organized as follows. In Chapter 2, we prove the optimality of water-filling policy and equal power allocation policy for capacity-achieving and worst-case-capacity-achieving problems, respectively. In Chapter 3, we introduce the system model and define the throughput-optimizing and worst-case-throughput-optimizing power allocation problems. We then propose the near-optimal power allocation policies based on convolutional codes in Chapter 4 and present numerical and simulation results in Chapter 5. Finally, we conclude the thesis in Chapter 6.

Chapter 2

Preliminaries

As aforementioned, the traditional power allocation policy is to maximize the overall system mutual information. Under a common assumption that the noise variance of each individual channel is known, the resulting optimal power allocation that achieves the system capacity is the water-filling power allocation scheme. For an alternative case where only the total noise variance is known, the optimal power allocation that achieves the system worst-case capacity is to allocate equal power to each channel. For the two cases mentioned above, proofs of the optimality of the claimed power allocation policies will be respectively given in the two sections in this chapter.

2.1

Capacity-Achieving Water-Filling Power Allocation

Policy with Known Noise Variance in Each

Chan-nel

Consider a system with K parallel AWGN channels. Assume that the noises are independent of each other. Denote the noise variance for channel i by σ2

i, 1 ≤ i ≤ K. The capacity of K

parallel AWGN channels is then give by max PK i=1Pi≤Pt K X i=1 1 2log  1 + Pi σ2 i  . (2.1)

We denote for convenience P = {Pi} K

i=1 as its assemble format. Since (2.1) is concave over

P , the technique of Lagrange multiplier [7] can be applied as the following. Let f (P ) = K X i=1 1 2log  1 + Pi σ2 i  − λ K X i=1 Pi− Pt ! , (2.2)

where the constant λ is the so-called Lagrange multiplier and is always chosen such that the power-sum constraintPK

i=1Pi = Ptis satisfied. Taking the derivative of (2.2) with respective

to Pi, we have from the Kuhn-Tucker condition that

( 1 2(Pi+σ_i2) = 0, if Pi > 0; 1 2(Pi+σ2i) ≤ 0, if Pi = 0. Hence, Pi = ν − σi2
+ , (2.3)

where (x)+_{, max{0, x} and ν is chosen such that the power-sum constraint is satisfied (and}

is equal to − 1

2λ). (2.3) can be graphically interpreted via a water-filling scheme as shown in

Figure. 2.1. º º ¾2₁ ¾2₁ º º CH1 CH1 CH2CH2 CH3CH3 ¾₂2 ¾₂2 ¾₃2 ¾₃2 P1 P1 PP22 PP33 3 X i=1 Pi= Pt 3 X i=1 Pi= Pt

2.2

Worst-Case-Capacity-Achieving Water-Filling Power

Allocation Policy Subject to a Known Sum of Noise

Variances

Instead of knowing the noise variance, σ2

i, of each channel, we suppose that we obtain only

the information of total noise variance, σ2

t =

PK

i=1σi2. Lacking the knowledge of noise

variance in each channel, we need to consider the worst case scenario, where for any given power allocation, σ2

i is always chosen such that the system mutual information is minimized.

It is named the worst-case mutual information and is defined as min PK i=1σ2i=σ2t K X i=1 1 2log  1 + Pi σ2 i  . (2.4)

Based on (2.4), the optimal power allocation is chosen such that (2.4) is maximized subject to the power constraint PK

i=1Pi = Pt. The first step toward this problem is to

find the {σ2

i} that achieves the worst-case mutual information. It can be derived that the

second-order derivative of (2.4) with respect to each σ2

i is positive; hence, (2.4) is a convex

function of σ2

i. We can then apply the Lagrange multiplier technique to find the optimal

{σ2

i} that achieves the worst-case mutual information. Let

f1(P ) = K X i=1 1 2log  1 + Pi σ2 i  + λ1 K X i=1 σi2− σt2 ! , (2.5)

where λ1 is chosen such that

PK

i=1σ2i = σ2t. Taking derivative of (2.5) with respect to σ2i,

we have    Pi 2(σ_i2)2+Piσ2_i + λ1 = 0, if σ_i2 > 0; Pi 2(σ2 i) 2 +Piσ2_i + λ1 < 0, if σ_i2 = 0. (2.6)

(2.6) can be reorganized into a second-order polynomial function of σ2

i as the following: ( (σ2 i) 2 + Piσi2+ Pi 2λ1 = 0, if σ 2 i > 0; (σ2 i) 2 + Piσ2i + Pi 2λ1 < 0, if σ 2 i = 0. (2.7)

Hence, we obtain σi2 =   −Pi+ q (Pi)2− 2P_λ1i 2   + . (2.8)

We then replace the σ2

i in (2.4) by (2.8) and take away the minimization, and (2.4) becomes

n X i=1 1 2log   1 + Pi −Pi+ q (γPi)2−2Pi_λ1 2   . (2.9)

Since (2.9) is concave with respect to Pi, the Lagrange multiplier technique can be applied

again as the following. Let

f2(P ) = K X i=1 1 2log   1 + Pi −Pi+ q (Pi)2−2Pi_λ1 2   + λ2 K X i=1 Pi− 1 ! , (2.10)

where λ2 is chosen such that PKi=1Pi = Pt. The derivative of (2.10) with respect to Pi

becomes ∂f2(P ) ∂Pi =    1  −1 +q1 − _λ2 1Pi 2 + 2(−1 +q1 − 2_λ2 1Pi)      1 q (1 −_λ12_P iλ1P 2 i  + λ2. (2.11) (2.11) should satisfy ( _∂f 2(P ) ∂Pi = 0, if Pi > 0; ∂f2(P ) ∂Pi < 0, if Pi = 0. For any j 6= i, we accordingly have that if Pi > 0 and Pj > 0,

∂f2(P )

∂Pi

= ∂f2(P ) ∂Pj

. (2.12)

This concludes that choosing Pi = Pj for every i and j will be one of the optimal power

Chapter 3

System Model and Problem

Formulation

Consider a system with K parallel channels or links, each of which has a binary-antipodal-input (realized via binary PSK modulation) and suffers independent AWGN noise. Let Ri

be the rate of the code adopted by channel i, and denote by Pe,i its corresponding frame

error rate for frame size Ni, 1 ≤ i ≤ K. The system effective throughput is then defined as

T (P ),

K

X

i=1

Ri(1 − Pe,i) , (3.1)

which corresponds to the successfully transmitted information bits per channel use. Note that in the above formula, Pe,i is a function of Ni, σ2i and Pi. To simplify the notations, we

do not explicitly write Pe,i as a function of Ni, Pi and σ2i.

Corresponding to the capacity-achieving problem that σ2

i in each channel is known to the

system, we will find Pisuch that (3.1) is maximized under the power constraintPK_i=1Pi = Pt.

Similarly, corresponding to the wort-case-capacity-achieving problem that only the total noise variance is available, we will find Pi such that the worst-case effective throughput

defined in (3.2) is maximized. Tw(P ), min PK i=1σ2i=σt2 K X i=1 Ri(1 − Pe,i) (3.2)

In this thesis, we implicitly assume that an error-detection scheme is applied to each frame such that information is successfully transmitted only when no decoding error occurs within a frame.1 _{We also assume that the time needed to transmit one code bit is identical}

for all channels.

In general, Pe,i does not exhibit a closed-form formula. Hence, the power allocation

that maximizes T (P ) (respectively, Tw(P )) can be obtained only via case-by-case

simula-tion studies. It is thus hard to establish a general power allocasimula-tion principle from such a simulation-based power allocation result. One possible solution is to derive a good approxi-mation for Pe,i with a structure that can facilitate its analysis.

When transmitting a convolutional code over an AWGN channel with noise variance σ2_,

the frame error rate at high SNRs can be well approximated by the event error rate [5] as Pe≈ Adfreee

−1₂dfree_σ2P _{, (3.3)}

where dfree is the free distance of the convolutional code, Adfree is the number of codewords with Hamming weight equal to dfree, and P is the transmission power. However, the

approx-imated Pe in (3.3) is far from accurate for moderate SNRs and finite frame sizes (cf. the

approx. Pe curve in Figure. 3.1 for a convolutional code with rate R = 1/4 and memory

order 6, in which REb = P and σ2 = N0/2). Instead of adding more rectifying terms to (3.3)

that may later introduce analytical obstacles, we choose to fix this inaccuracy by replacing Adfree and dfree with the refined parameters A and d respectively such that the adjusted curve

1_{Alternatively, one may define the effective throughput based on the (information) bit error rate (P} b) to

avoid considering the frame size, e.g., PK

i=1

# of info. bits successfully recovered at receiver i # of total info. bits transmitted via channel i =

PK

i=1(1 − Pb,i).

This however may introduce an impractical situation where a high bit error rate (e.g., nearly one half) at the receiver can still provide a non-trivial throughput to the system. Such an impractical situation can be avoided under the definition in (3.1) since almost all frames fail the error detection check under a high bit error rate.

defined below, log(Pe) ≈ min  0, log(A) − P 2σ2d  , (3.4)

is close to the true Pe in the least squares sense over the range of operating SNRs (cf. the

adjusted approx. Pe curve in Figure. 3.1). For details of the procedure for retrieving A and

d, please see Example 3.1.

Example 3.1. From (3.4), we know that the approximated Pe in log scale is a linear

com-bination of di and log A in the operating SNR region. Thus linear least square estimator

[8] can be applied to retrieve d and log A. We let x = [x [0] x [1] ...x [M − 1]]T be a vector composed by M true Pe values which are in log scale and g = [g [0] g [1] ...g [M − 1]]T denote

its corresponding Eb

N0. We also let s = [s [0] s [1] ...s [M − 1]]

T

denote the approximated Pe

values in log scale. From (3.4), the s [j] can be modelled by

s [j] = −R g [j] d + log A ∀ 1 ≤ j ≤ M or in matrix form s= Hθ, where H =         −Rg[0] 1 −Rg[1] 1 . . . −Rg[M − 1] 1         , θ =  d log A  .

The least square estimator is found by minimizing

J (θ) = (x − Hθ)T (x − Hθ) . (3.5) The gradient of (3.5) is ∂J (θ) ∂θ = −2H T x+ 2HTHθ

Setting the gradient to be zero yields the least square estimator ˆ

θ= HTH
−1HTx

The refined parameters A and d for the code is then found.

We denote the refined parameters in channel i by Ai and di, respectively. Note that

the effect of Ni to the frame error rate is included in the choice of di and Ai. Even if

we use the same code, di and Ai of a code will be different if we use different frame size.

Once di and Ai for a given code is determined, it can be later used universally to find the

throughput-optimizing power allocation policy for every value of Pt.

An immediate consequence from the adjusted approximation formula in (3.4) is that the contribution to the system effective throughput from channel i will be zero if

Pi < Pth,i,

2σ2 i

di

log(Ai).

In Chapter 5, our simulations will confirm that for a given code assigned to channel i, allocating a power value smaller than Pth,i indeed provides very limited contribution to the

system throughput since almost all frames will fail the implicitly assumed error-detection check.

0 0.5 1 1.5 2 2.5 3 10−6 10−5 10−4 10−3 10−2 10−1 100 101 E b/N0 P e R = 1/4 P e approx. P e adjusted approx. P e

Figure 3.1: Pe and its approximations for a (4, 1, 6) convolutional code with generator

poly-nomial (in octal) being [177 127 155 171], dfree = 20 and Adfree = 2. The adjusted parameters

are d = 22.42 and A = 962.51. The frame size is N = 4(500 + 6). Eb/N0 is plotted in linear

Chapter 4

Throughput-Oriented Water-Filling

In this chapter, instead of maximizing the overall mutual information, we suggest a power allocation that aims at maximizing the effective throughput defined in (3.1) and (3.2). The analyses in Sections 4.1 and 4.2 are mainly based on the approximated Pe defined in (3.4).

Interestingly, both the proposed power allocation policies in the two sections can be inter-preted by some variations of water-filling. At the end of each section, we will remark on the near-optimal power allocation policies we proposed when total power goes without bound, and several conclusions will be given.

4.1

Throughput-Oriented Water-Filling: Noise

Vari-ance in Each Channel is Known

Based on the adjusted approximation formula in (3.4), (3.1) becomes T (P ) = K X i=1 Ri  1 − min  1, Aie −diPi 2σ2_i  = X i∈O Ri  1 − Aie −diPi 2σ2_i  , (4.1)

provided that the optimal set of active channels in use, denoted by O, can be priori deter-mined. Since (4.1) is a concave function over Pi, the power allocation can be obtained by

using the Lagrange multiplier technique and the Kuhn-Tucker condition as follows. Let E (P ) = T (P ) − λ X i∈O Pi− Pt ! ,

where the constant λ is the Lagrange’s multiplier and is chosen such that P

i∈OPi = Pt.

Taking derivative with respect to Pi, we have

∂E (P ) ∂Pi = di 2σ2 i Aie −diPi 2σ2_{i − λ. (4.2)}

By the Kuhn-Tucker condition, (4.2) should satisfy ( _{∂E(P )}

Pi = 0, if Pi > 0,

∂E(P )

Pi < 0, if Pi = 0.

(4.3) By (4.2) and (4.3), the optimal Pi can be shown to have the following form:

Pi∗ = 2σ2 i di  ν − log σ 2 i diAiRi + (4.4) where ν is chosen such that P

i∈OP ∗

i = Pt. Note that ν should also satisfy

ν ≥ νmin , max i∈O log

σ2 i

diRi

for the reason that all the channels in O should be activated (i.e. P∗

i ≥ Pth,i ∀ i ∈ O).

Interestingly, the above result can be interpreted graphically as a variation of the water-filling principle. For channels outside O, zero power will be allocated. For each channel in O, a vessel with base width 2σ2i

di and base height log

σ2 i

diAiRi will be used for water filling. The resulting water level ν must be no less than the base height log σ2i

diAiRi plus log(Ai) for every i ∈ O. The water inside each vessel is then the optimal power to be allotted. An example is illustrated in Figure. 4.1.

Example 4.1. A three channels (K = 3) system is considered. Each channel has its base height and base width as stated in the paragraph immediately above this example. We assume that O = {1, 2} has already been given. Thus, zero power will be allocated to channel 3. And

at least Pth_,1 and Pth_,2 should be allocated to channels 1 and 2, respectively. The lowest water

level νmin should then be chosen as

νmin = max  log σ 2 1 d1R1 , log σ 2 2 d2R2  .

log

log

log

_A

log

_A

º

º

º

º

P

P

P

P

CH1

CH1

CH2

CH2

CH3

CH3

Figure 4.1: An example of the throughput-oriented water-filling with K = 3 and O = {1, 2}. By (4.4), we obtain that for i ∈ O (hence ν ≥ log σ2i

diRi),

diγi∗ = 2ν + 2 log(diAiRi) − 2 log(σi2) (4.5)

where γ∗

i , Pi∗/σi2 denotes the SNR of channel i. Equation (4.5) then indicates that the

optimal power allocation should make the SNR γ∗

i inversely proportional to the logarithm

of noise power σ2

i. This is in stark contrast with the capacity-achieving water-filling policy

(with Gaussian inputs), which results in an SNR that is inversely proportional to the noise power itself. For example, when two active channels i and j adopt the same code with

di = dj = 5, and σj2/σ2i = 2, (4.5) implies that γi∗ = γ ∗ j + 2 di logσ 2 j σ2 i = γj∗+ 0.12,

while the capacity-achieving power allocation formula P∗

i = (ν − σi2)+ requires that

γ∗

i = 2γj∗+ 1.

From our simulations, we indeed observe that the latter power assignment actually yields a poor system throughput.

When the total power Pt is adequately large, all channels become active. We then obtain

from (4.5) that the SNRs of any two channels, say channels i and j, are characterized by diγi∗ = djγj∗+ 2 log σ2 j σ2 i + log diAiRi djAjRj Thus lim Pt→∞ diγi∗ djγj∗ = 1 + lim Pt→∞ 2 logσj2 σ2 i + log diAiRi djAjRj djγj∗ = 1.

Hence, when Pt is large, our result indicates that the allotted powers should make the diγi∗

products equal across all channels. As in most cases, the approximate di is close to the free

distance of the code used by channel i; this suggests that, when Pt grows without bound,

the optimal SNR γ∗

i should in general be chosen as the reciprocal of the code’s free distance.

As already mentioned, our result also indicates that there is a minimum power required for each channel to be activated. In other words, if the allocated power Pi is less than Pth,i

then re-assigning this power to other channels will generally result in a better throughput. A remaining question is how to determine the optimal O. A straightforward approach is to examine each of the choices of O, which is by no means complex. To examine one possible choice of O, for a given total power, Pi is then determined by (4.4) since the relationship

of Pi and total power is a one-to-one mapping. The corresponding effective throughput is

4.2

Throughput-Oriented Water-Filling: Only Total Noise

Variance is Available

Based on the adjusted approximation formula in (3.4), (3.2) becomes Tw(P ) = min PK i=1σ2i=σ2t K X i=1 Ri  1 − min  1, Aie −diPi 2σ2_i  . (4.6)

We focus on the situation that all channels are active, which means that

Pi ≥ Pth,i ∀ 1 ≤ i ≤ K. (4.7) (4.6) becomes Tw(P ) = min PK i=1σ2i=σ2t K X i=1 Ri  1 − Aie −diPi 2σ2_i  . (4.8)

A straightforward approach to eliminate the minimization in (4.8) is to use the Lagrange multiplier technique and Kuhn-Tucker condition to find σ2

i such that the worst-case effective

throughput is achieved. However, in general, the worst-case effective throughput is not a concave function of σ2 i since ∂2_T w(P ) ∂ (σ2 i) 2 = RiAidiPi (σ2 i) 3 e −diPi 2σ2_i  1 −diPi 4σ2 i  . (4.9) Thus, by letting Pi ≥ 4σ2 i di , (4.10)

we further constrain our problem to be concave over σ2

i such that (4.9) becomes always

negative. Under this constraint, we know that the σ2

i to achieve Tw should be chosen as

either 0 or σ2

t. Since the total noise variance is a given value, only one channel will be

allocated the whole noise power and the rest of the channels is allocated zero noise power. Next we consider one possible power allocation Pi†, which is defined as

P_i†= 2σ 2 t d  ν − log 1 AR  , (4.11)

where ν is chosen such that

K

X

i=1

Pi†= Pt.

Our aim is to prove that this power allocation performs better than any other power alloca-tion policies and hence is optimal.

From the power constraints in (4.7) and (4.10), P_i† should satisfy Pi†≥ max  2σ2 i di log (Ai), 4σ2 i di  ∀ 1 ≤ i ≤ K. (4.12) Since σ2

i ≤ σt2, we can further increase Pt (equivalently, ν) such that

P_i†≥ max 2σ 2 t di log (Ai), 4σ2 t di  ∀ 1 ≤ i ≤ K. (4.13) Define the minimum power required in channel i as

P_th,i† _{, max} 2σ 2 t di log (Ai), 4σ2 t di  ∀ 1 ≤ i ≤ K. Replacing Pi† by (4.11), we further deduce (4.13) as a condition on ν,

2σ2 t di  ν − log 1 AiRi  ≥ max 2σ 2 t di log Ai, 4σ2 t di  , thereby implying

ν ≥ max {− log Ri, 2 − log (AiRi)} ∀ 1 ≤ i ≤ K. (4.14)

We let νmin denote the minimum value of the choice of ν, which satisfies (4.14) for every i.

From the definition of Pi† and (4.14), we have

Pi†≥ 2σ2 t di  νmin− log  1 AiRi  ∀ 1 ≤ i ≤ K. (4.15) (4.15) equivalently implies a constraint in system SNR by taking summation over P_i† and dividing it by σ2 t, which is 1 σ2 t K X i=1 Pi†≥ K X i=1 2 di  νmin− log  1 AiRi  = γ_th†,

where γ_th† is the threshold system SNR. From above, we have claimed that by using P_i† as power allocation, the optimal choice of σ2

i is either 0 or σt2, when system SNR is greater than

γ_th†. Using this result, the worst-case effective throughput due to {Pi†}, which is denoted by

T†

w can be computed as follows.

T† w P†
, min P σ2 i=σ 2 t K X i=1 Ri 1 − Aie −diP † i 2σ2_i ! = min P σ2 i=σ 2 t K X i=1 Ri   1 − Aie −di 2σ2_t di (ν−logRiAi1 ) 2σ2_i    (4.16)

We define the noise variance that achieves T† w as

σi2 †

, σt2, if i = m

0, if i 6= m , (4.17)

where m can be chosen to be 1 ≤ m ≤ K. We will later show that any value of m yields the same T†

w. We then take away the minimization in (4.16) by using σi2 †

as the noise power. (4.16) becomes K X i=1 Ri   1 − Aie −di 2σ2_t di (ν−logRiAi1 ) 2(σ†_i)2    = X i6=m Ri+ Rm− e−(ν−log 1 RmAm)+log RmAm = K X i=1 Ri− e−ν. (4.18)

By (4.18), it is noted that the worst-case effective throughput is independent of m. Thus definition of σ2

i †

is justified. Besides, we know that Rm− e−ν is always non-negative for all

possible value of m from (4.14).

The optimality of using P_i†as power allocation is proved by the method of contradiction. We will show that the Tw obtained from any other power allocation is less than or equal to

T†

w. The proof is as follows. Consider any power allocation ˆPi = Pi†+ 4P †

i, where 4P † i 6= 0

for at least one channel and

K

X

i=1

4P_i†= 0. (4.19)

We also consider a specific noise power allocation ˆ σi2 = σ 2 t if i = k 0 if i 6= k , where k, arg min 1≤i≤K  diPi 2σ2 t − log AiRi  . (4.20)

An upper bound for the worst-case effective throughput of ˆP can be found as the following: min P σ2 i=σ2t K X i=1 Ri  1 − min  1, Aie −di ˆPi 2σ2_i  ≤ K X i=1 Ri  1 − min  1, Aie −di ˆPi 2 ˆσi2  = X i6=k Ri+ Rk  1 − min  1, Ake −dk ˆPk 2σ2_t  . (4.21)

The first inequality holds for the reason that the minimization over the effective throughput is always less than or equal to the effective throughput using the noise power ˆσi2 in our case.

For the second equality, it is obvious that the minimization over 1 and Aie −di ˆPi

2 ˆσi2 _{is always} greater than zero. Moreover, if we have

min  1, Ake −dk ˆPk 2σ2_t  = 1, (4.21) becomes X i6=k

Ri which is less than or equal to T_w†.

Else if min  1, Ake −dk ˆPk 2σ2_t  = Ake −dk ˆPk 2σ2_{t ,}

(4.21) becomes X i6=k Ri+ Rk− e −  dk ˆPk 2σ2_t −log (RkAk)  . (4.22) By comparing (4.22) with T†

w in (4.18), the only difference is in the exponential term. From

the definition of k in (4.20), we know that dkPˆk 2 ˆσ2 k − log (RkAk) = min 1≤i≤K diPˆi 2σ2 t − log AiRi ! . (4.23) Note that diPˆi 2σ2 t − log RiAi = di  Pi†+ 4Pi  2 ˆσt2 − log RiAi = ν +di4Pi 2σ2 t ∀ 1 ≤ i ≤ K. (4.24) There always exists at least a channel that has its di4Pi

2σ2

t being negative since ˆPi 6= P

† i and

4Pi should satisfy the constraint in (4.19). Taking minimization over (4.25), we have

min 1≤i≤K diPˆi 2σ2 t − log AiRi ! < ν (4.25) Thus X i6=k Ri+ Rk− e −  dk ˆPk 2σ2_t −log (RkAk)  ≤X i6=k Ri+ Rk− e−ν
= T_w† P†

From the discussion above, we have proved that the worst-case effective throughput of any power allocation ˆPi is less than that of Pi†. The optimality of P

†

i is then justified. Thus,

we can claim that when system SNR is greater than γ_th† , the optimal power allocation that maximizes Tw is by using Pi† as the power allocation. It is also worth knowing that the

corresponding choice of σ2

i that achieves Tw is to put total noise power to any one of the

channel.

The power allocation scheme can also be interpreted as a variation of water filling princi-ple. For each channel, a vessel with base width 2σ2t _{and base height log} 1 _{will be used for}

water filling. From our constraints in power in (4.13), each channel should be allocated at least P_th,i† . The resulting ν must be no less than νmin. The water filling inside each vessel is

then the optimal power to be allotted. An example with three channels(K = 3) is illustrated in Fig. 4.2.

Example 4.2. A three channels (K = 3) system is considered. Each channel has its base height and base width as defined in the paragraph immediately above this example. At least Pth†_,1, P

†

th_,2 and P †

th_,3 should be allocated to three channels, respectively. The lowest water level

νmin should then be chosen as

νmin = max 1≤i≤3



max {log Ai, 2} + log

1 AiRi  . log 1 A1R1 log 1 A1R1 maxflog (A1); 2g maxflog (A1); 2g º º ºmin ºmin Pthy;1 Pthy;1 Pthy;2 Pthy;2 CH1 CH1 CH3CH3 2¾2 t d1 2¾2 t d1 Pthy;3 Pthy;3 CH2 CH2

Figure 4.2: An example of the throughput-oriented water-filling with K = 3.

From the definition of P_i†, we can see that the proposed power allocation depends on σ2 t.

if we look at the power allocation ratio for each channel, which is derived as the following: p†i = Pi† Pt = 2 diγt  ν − log 1 AiRi  , (4.26)

where γt = Pt/σ2t is the system SNR. In practice, the information of system SNR can be

obtained by applying feedback technique to the system. Besides, the effect of system SNR in the power allocation can be eliminated if the code used in each channel can be chosen such that the product of Ai and Ri is the same for different channels. In this way, the ratio

of power between different channels becomes a constant, which is as the following: p†_i : p†_j = 1

di

: 1 dj

,

for i 6= j. Thus, the allocated power ratio for channel i becomes p†i = 1 di PK m=1 d1m ,

which is only related to di. Furthermore, we look at p†i when the system SNR goes without

bound. From (4.26), we have

lim γt→∞ p†i p†j = 1 di 1 dj

The optimal allotted power to channel i should be inversely proportional to its di, which

is closed to its free distance. It coincides with the fact that the free distance dominates frame error rate and thus dominates worst-case effective throughput when SNR tends to be infinity. Finally, our proposed power allocation scheme can be simplified to the traditional equal power allocation scheme when all channels use the same code.

Chapter 5

Numerical and Simulation Results

In this chapter, we compare the effective throughput retrieved by using different power allocation policies. Several convolutional codes [2, 3] are adopted. The parameters in the sense of the approximation given by (3.4) for these codes are listed in Table 5.1.

5.1

Throughput-Oriented Water-Filling: Noise

Vari-ance in Each Channel is Known

In this section, three situations of parallel Gaussian channels with K = 3 are examined. They are respectively referred to as Cases I, II and III.

In Case I, the noise variances for the three channels are σ2

1 = 1, σ22 = 3.5 and σ32 = 6,

respectively. Here, codes with higher code rates are naturally assigned to less noisy channels; hence we have R1 = 1/2, R2 = 1/3 and R3 = 1/4. The frame sizes for the three channels

are N1 = 2(1000 + 6), N2 = 3(1000 + 6) and N3 = 4(1000 + 6), respectively. In Figure. 5.1,

we depict the effective throughputs for the seven possible choices of the active channel set O. The figure indicates that all the power should be allocated to channel 1 if Pt < 5.14,

and both channels 1 and 2 should be active when 5.14 < Pt < 10.05. Beyond the point

Table 5.1: The information of the used codes in the simulation.

adjusted adjusted codeword length generator polynomial

code dfree Adfree _{d A N (octal)}

10.63 1478.07 2(500+6) (2, 1, 6) 10 11 11.02 4750.45 2(1000+6) [133 171] 15.79 593.83 3(500+6) (3, 1, 6) 14 1 16.12 1449.97 3(1000+6) [133 171 145] 22.42 962.51 4(500+6) (4, 1, 6) 20 2 22.13 1401.29 4(1000+6) [117 127 155 171] (2, 1, 2) 5 1 5.31 111.56 2(500+2) [5 7] (3, 1, 11) 24 13 29.04 41373.67 3(500+11) [5475 6471 7553] (4, 1, 10) 29 3 35.54 11266.62 4(500+10) [2565 2747 3311 3723]

In Figure. 5.2, we compare the optimal effective throughput obtained from exhaustive search with that obtained from our throughput-oriented water-filling based on the FER approximation and from the capacity-achieving water-filling policy. We remark that our throughput-oriented water-filling can achieve a near-optimal effective throughput as an-ticipated. We also observe that the capacity-achieving water-filling policy yields a good throughput only when all the power is allocated to a single channel (which is the optimal choice only for small values of Pt).

In Figure. 5.3, we plot the optimal power ratio P∗

2/Pt with respect to different power

allocation policies. We note that a sudden increase for this ratio occurs in the exhaustive search curve at Pt = 4.98 which is exactly the instance the active channel set O changes

from {1} to {1, 2} as shown in Figure. 5.4. This jump occurs when the total power is a little bit larger than the total power corresponding to ν = νmin = log σ

2 2 d2R2 in Figure. 4.1, i.e., Pt > 2σ2 1 d1  log σ 2 2 d2R2 − log σ 2 1 d1A1R1  + 2σ 2 2 d2  log σ 2 2 d2R2 − log σ 2 2 d2A2R2  = 4.93. This is because Channel 2 can provide a solid contribution to the system effective throughput only when P2 is adequately larger than Pth,2. Figure 5.3 also indicates that the predicted

i.e., Pt = 5.14, is very close to the true jump point, Pt = 4.98, while the capacity-achieving

water-filling policy always suggests a continuous increase in the power ratio.

0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 total power effective throughput case 1: Ch1 only case 2: Ch2 only case 3: Ch3 only case 4: Ch1 & Ch2 case 5: Ch1 & Ch3 case 6: Ch2 & Ch3 case 7: Ch1 & Ch2 & Ch3

Figure 5.1: Case I: Effective throughputs for the seven choices of the active channel set O. For Case II, we exchange the codes used in Channels 1 and 3 in Case I. Hence, R1 = 1/4

and R3 = 1/2. The results are summarized in Figures. 5.5, 5.6 and 5.7. These figures point

out that using a lower code rate for a less noisy channel will yield a better throughput only when the total power is very small. For moderate to high total power, exchanging the codes between channels 1 and 3 never results in a better effective throughput. This confirms the common intuition that when a channel is less noisy, a code with a higher rate should be used. A side observation is that when assigning a code with lower rate to a less noisy channel, the set of active channels changes more often with respect to Pt. In particular, Channel 2 will

0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 total power effective throughput exhaustive search throughput−oriented I capacity−optimizing

Figure 5.2: Case I: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy.

have two cut-off regions given by Pt< 3.76 and 11.42 < Pt < 14.65 as shown in Figure. 5.5.

In addition, Figure. 5.5 shows that adopting a wrong O will noticeably degrade the effective throughput. Hence, exchanging the codes between Channels 1 and 3 will make complicated the optimization of the throughput.

Finally for Case III, the codes used for three channels are the same as those used in Case I, but the frame sizes are changed to N1 = 2(500 + 6), N2 = 3(500 + 6) and N3 = 4(500 + 6).

Thus di and Ai are changed simultaneously. Besides, the noise variances are changed to

σ12 = 2, σ22 = 8, σ32 = 9. Similar behaviors can be observed from Figure 5.8 except that

0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 total power power ratio in ch2 exhaustive search throughput−oriented I capacity−optimizing

Figure 5.3: Case I: Optimal power ratio for channel 2.

one for high values of the total power. This can be somehow anticipated from the discussion following (4.5) as when the noise variances of the active channels have larger gaps (between Channel 1 and Channels 2 or 3), the capacity-achieving water-filling policy will yield a power allocation closer to the throughput-oriented water-filling.

º º Pth;1 Pth;1 CH1 CH1 CH2CH2 CH3CH3 Pth;1 Pth;1 Pth;2 Pth;2 CH1 CH1 CH2CH2 CH3CH3

0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 total power effective throughput case 1: Ch1 only case 2: Ch2 only case 3: Ch3 only case 4: Ch1 & Ch2 case 5: Ch1 & Ch3 case 6: Ch2 & Ch3 case 7: Ch1 & Ch2 & Ch3

0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 total power effective throughput exhaustive search throughput−oriented I capacity−optimizing

Figure 5.6: Case II: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy.

0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 total power power ratio in ch2 exhaustive search throughput−oriented I capacity−optimizing

0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 total power effective throughput exhaustive search throughput−oriented I capacity−optimizing

Figure 5.8: Case III: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achiving water-filling policy.

5.2

Throughput-Oriented Water-Filling: Only Total Noise

Variance is Available

In this section, three situations of parallel Gaussian channels are examined. They are re-spectively referred to as Cases I, II and III.

In Case I, we consider K = 3 and use the (2, 1, 6), (3, 1, 6) and (4, 1, 6) convolutional code in three channels, respectively. The frame size of the codes are N1 = 2(500 + 6),

N2 = 3(500 + 6) and N3 = 4(500 + 6). The total noise variance σ2t is set to 10. For

convenience, we will plot the ratios of the effective throughput against the maximum rate, which is the sum of the rates of the three channels in the following figures.

In Figure. 5.9, we compare the ratios of the effective throughputs against the maxi-mum rate, obtained from the throughput-oriented water-filling in (4.11) and the traditional worst-case capacity-achieving equal power allocation. The γ†_th, at which value our proposed power allocation becomes optimal, is 4.72 dB. We can see that almost all of the power al-location methods achieve the maximum rate when system SNR is above γ_th† . Although we cannot guarantee the optimality of using the throughput-oriented water-filling for system SNR smaller than γ†_th, we can still see that it has around 1.4 dB gain over the equal power allocation when the effective throughput of the system is required to achieve 85% of the maximum rate.

We also observe the distribution of worst-case noise variances for system SNR varying from 2 dB to 6 dB when using throughput-oriented water-filling as the power allocation method. The result shows that we should always give total noise power to Channel 2. This confirms our claim that the worst-case effective throughput is achieved by giving total noise power to only one channel for system SNR greater than γ_th† .

2 2.5 3 3.5 4 4.5 5 5.5 6 40 50 60 70 80 90 100 110 120 SNR(P_t/S_t in dB)

effective throughput/maximum rate(%)

throughtpu−oriented II

worst−case capacity optimizing(equal power) 85% maximum rate

Figure 5.9: Case I: The worst-case effective throughputs obtained from throughput-oriented water-filling based on the FER approximation, and the worst-case-capacity-achieving equal power allocation.

channels, respectively. Compared with the codes used in Case I, the codes used in Case II has larger gaps in di, where d1 = 5.31, d2 = 29.40 and d3 = 35.54. It is anticipated

that by using this set of codes, throughput-oriented water-filling should yield a greater gain than equal power allocation, when being compared with Case I. A simple way to prove this anticipation is by looking at the situation when system SNR is large. The proposed power allocation policy suggests that Pi should be allocated inversely proportional to di. For larger

difference in the amount of di’s, the proposed power allocation deviates greatly from the equal

power allocation, and thus yields better gain. Figure 5.10 confirms our deduction. We see that throughput-oriented water-filling yields around 2 dB gain when the effective throughput

achieves 85% of the maximum rate. Besides, when we look at the situation when system SNR is equal to γ_th† = 5.19 dB, throughput-oriented water-filling almost achieves the maximum rate while the equal power allocation achieves only 83% of the maximum rate.

3 3.5 4 4.5 5 5.5 6 40 50 60 70 80 90 100 110 120 SNR(P t/St in dB)

effective throughput/maximum rate(%)

throughtpu−oriented II

worst−case capacity optimizing(equal power) 85% maximum rate

Figure 5.10: Case II: The worst-case effective throughputs obtained from throughput-oriented water-filling based on the FER approximation, and the the worst-case-capacity-achieving equal power allocation.

For Case III, we increase the number of channels to be K = 4. We use the (2, 1, 6), (3, 1, 6) and (4, 1, 6) convolutional codes in the first three channels as in Case I. Two different codes are chosen to be used in Channel 4 for comparison.

Firstly, we use the (2, 1, 6) in Channel 4, which is the same code as that used in channel 1. We yield only 0.89 dB gain when the effective throughput is required to achieve 85% of the maximum rate (See Figure 5.11), which is less than the gain in Case I. Secondly, we use

the (3, 2, 6) punctured convolutional code in Channel 4. It is punctured from (2, 1, 6) code with puncture pattern

 1 1 1 0

 .

The adjusted parameters for the punctured (3, 2, 6) code is d4 = 7.89 and A4 = 6469.15,

where d4 is much less than the di of other used codes. From Figure 5.12, we could see that

the gain enlarges to 1.89 dB when the effective throughput achieves 85% of the maximum rate, which is greater than the gain obtained in Case I. The result in this case confirms the anticipation that the throughput-oriented water-filling yields a larger gain from traditional equal power allocation when the characteristics of the used codes deviate largely from each other .

3 3.5 4 4.5 5 5.5 6 6.5 7 60 65 70 75 80 85 90 95 100 105 110 115 SNR(P t/St in dB) effective throughput throughput−oriented II

worst−case capacity optimizing(equal power) 85% maximum rate

Figure 5.11: Case III: The worst-case effective throughput of using throughput-oriented water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) codes are used in the first three channels, and (2, 1, 6) code is used again in the fourth channel.

4 4.5 5 5.5 6 6.5 7 7.5 8 50 60 70 80 90 100 110 SNR(P_t/S_t in dB) effective throughput throughput−oriented II

worst−case capacity optimizing(equal power) 85% maximum rate

Figure 5.12: Case III: The worst-case effective throughput of using the throughput-oriented water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) code sare used in the first three channels, and (3, 2, 6) code is used in the fourth channel.

Chapter 6

Conclusion

In this paper, two power allocation policies are respectively proposed for the two situations: (i) noise variance is known to each channel and (ii) only total noise variance is known. We aim to maximize the effective throughput and the so-defined worst-case effective throughput of the K coded parallel AWGN channels, subject to practical finite-length and fixed-rate coding constraints. These policies preserve the notion of the water-filling principle by additionally taking into consideration the code characteristics. Simulation and numerical results show that the proposed policy for the situation that noise variance is known to each channel can achieve a near-optimal effective throughput for all values of the total power. When only the total noise variance is known, the proposed policy can also achieve a near-optimal effective throughput for system SNR greater than a certain threshold.

In practice, standards usually provide a list of optional codes for each channel. For the case where noise variance in each channel is known, a natural future work is thus to provide a quick determination of the optimal active channel set O (instead of examining all (2K_{− 1)}

possibilities) such that our policy can readily determine the suitable code to be used in each channel. For the case where only total noise variance is available, the future work is to find the optimal power allocation policy for system SNR below the threshold system SNR.

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