Politecnico di Milano
School of Industrial and Information Engineering
Master of Science in Telecommunications Engineering
Capacity Bounds for
Amplitude-Constrained MIMO Channels
Supervisor: Prof. Luca BARLETTA
Co-Supervisor: Prof. Maurizio MAGARINI Co-Supervisor: Andrei-Stefan NEDELCU
Master Thesis of: Saswati MITRA Student ID 872280 Academic Year 2017-2018
Contents
1 Introduction 1
1.1 Overview . . . 1
1.2 Objectives and Motivation . . . 2
1.3 Thesis Structure and Outline . . . 3
2 Preliminaries 6 2.1 Fundamentals of Information Theory . . . 6
2.1.1 Entropy . . . 6
2.1.2 Joint and Conditional Entropy . . . 7
2.1.3 Mutual Information . . . 8
2.2 AWGN Channel Capacity . . . 9
2.2.1 Channel Capacity with Average Power Constraint 11 2.3 Multiple Input Multiple Output Channel . . . 12
2.3.1 Advantages and Disadvantages of MIMO Systems 14 2.3.2 MIMO Model . . . 15
2.3.3 SVD for MIMO Channel . . . 17
2.3.4 Linear Precoding and Postcoding . . . 20
3 Prior Works 22
CONTENTS 3.2 Capacity of MIMO Channel with Amplitude Constraints 26
3.2.1 Moment Upper Bound . . . 29
3.2.2 Duality Upper Bounds . . . 29
3.2.3 EPI Lower Bounds . . . 31
3.2.4 Ozarow-Wyner Type Lower Bound . . . 32
3.2.5 Jensen’s Inequality Lower Bounds . . . 33
4 Capacity Bounds for Arbitrary Channel Matrices 35 4.1 Duality Upper Bounds . . . 35
4.1.1 Duality Upper Bound with Postcoding . . . 39
4.2 Jensen’s Lower Bound . . . 40
4.2.1 Optimal ‘b’ for the Lower Bound . . . 41
5 Experimental Results 45 5.1 Introduction . . . 45
5.2 Simulation Results . . . 46
5.2.1 Duality Upper Bounds . . . 46
5.2.2 Comparison of the Upper and Lower Capacity Bounds 47 6 Gap between Upper and Lower Capacity Bounds 51 6.1 Gap to Capacity for 2 × 2 MIMO Case . . . 52
6.1.1 Analytical Approach: . . . 54
6.2 Experimental Approach . . . 57
7 Conclusion and Future Avenues 60 7.1 Conclusion . . . 60
7.2 Future Avenues . . . 61
CONTENTS
List of Figures
2.1 AWGN channel . . . 9
2.2 Diversity scheme . . . 13
2.3 Principle of beamforming . . . 14
2.4 MIMO channel model . . . 16
2.5 SVD of the channel matrix . . . 18
2.6 MIMO channel after application of SVD . . . 20
2.7 Transmitter and receiver scheme with linear transforma-tions . . . 21
3.1 Comparison of the upper and lower bounds for a 2 × 2 MIMO system with per-antenna amplitude constraints A1 = A2 = A and fixed channel matrix H. [1] . . . 33
4.1 Constraint bounds for the case without postcoding: A · β0( √ nt· A), B · Λβ0( √ nt· A), C · β0(σmaxβ0( √ nt· A)), D · Box(a0) . . . 43
4.2 Comparison of constraint bounds . . . 44
LIST OF FIGURES 5.1 Comparison of Duality Upper Bounds for a 2 ∗ 2 MIMO
system with per-antenna amplitude constraints A1 = A2 =
A = 1 for different ratios of singular values of arbitrary channel matrix H, with and without postcoding. . . 47 5.2 Upper and Lower Bounds A = 20 in linear scale, N = 2 48 5.3 Comparison of Jensen Lower Bound and Cdual2 Upper
Bound with postcoding. . . 49 5.4 Comparison of Jensen Lower Bound and Cdual2 Upper
Bound with postcoding for 4 different ratios of singular values of H and SNR in dB . . . 50 6.1 3D plot of 4C = f (x) − g(x) for 0.1 ≤ r ≤ 0.9, and
Sommario
I sistemi di comunicazione wireless MIMO (Multiple-Input-Multiple-Output) hanno riscosso un enorme interesse nel campo della ricerca e nel settore grazie alle sue capacit`a di migliorare l’efficienza spettrale e l’affidabilit`a operativa. `E stato dimostrato dall’analisi dei modelli di canali classici sotto l’ipotesi di vincoli che la tecnologia MIMO `e in grado di aumentare linearmente la velocit`a di trasferimento delle informazioni, facendo uso di pi`u antenne di trasmissione e ricezione che operano nella stessa larghezza di banda di frequenza.
L’obiettivo di questa tesi `e quello di studiare la capacit`a dei canali MIMO con limiti di ampiezza. In particolare, vengono proposti dei bound su-periore e inferiore alla capacit`a di canale quando la realizzazione della matrice di fading `e nota sia al trasmettitore che al ricevitore. Un ul-teriore miglioramento delle prestazioni `e stato dimostrato attraverso la post-codifica sul ricevitore. Infine, un limite finito e analitico al gap tra i bound di capacit`a viene fornito per il caso 2 × 2 MIMO.
Abstract
Multiple-Input-Multiple-Output (MIMO) wireless communication sys-tems has garnered immense interest in the field of research as well as in the industry due to its capabilities to improve spectral efficiency and operational reliability. It has been shown by the analysis of classic chan-nel models under the assumptions of constraints that MIMO technology is able to linearly increase the rate of information transfer by making use of multiple transmit and receive antennas operating in the same fre-quency bandwidth.
The objective of this thesis is to study the capacity of MIMO channels with amplitude constraints. In particular, upper and lower bounds on the capacity have been evaluated for arbitrary, fixed channel matrices and further improvement in performance has been shown through post-coding at the receiver. Finally, a finite, analytical upper bound to the gap between the capacity bounds is provided for 2 × 2 MIMO case.
Chapter 1
Introduction
1.1
Overview
In today’s world the demand for high data rates and better quality of ser-vice (QOS) under tight constraints of power, bandwidth and complexity is constantly increasing in wireless communication systems. Moreover, factors like multipath fading and distortion create problems in trans-mission. To enable effective ways of transmission and reception of sig-nal, needs the introduction of techniques like Multiple Input Multiple Output technology which provides the capabilities to improve spectral efficiency and operational reliability. MIMO technology makes use of multiple transmit and receive antennas operating in the same frequency bandwidth to achieve high spectral efficiency under the assumption of independent and rich scattering [2], [3].
For a long time spatial diversity combining has been used with multiple antennas to reduce multipath fading effect. But modern MIMO systems
1.2. OBJECTIVES AND MOTIVATION take advantage of multipath fading so that with the increase of the num-ber of antennas the spectral efficiency is increased linearly. Due to this increase in spectral efficiency, a lot of research activity has been focused on the capacity analysis, modeling of propagation environments, mod-ulation techniques for multiantenna systems, etc. In spite of a lot of ongoing research in this area, there are many open problems.
1.2
Objectives and Motivation
The capacity of a MIMO channel is one of the major characteristics to measure its performance. Capacity is the maximum rate of information transfer for which the probability of error goes asymptotically to zero. Capacity gives an important information theoretic bound which provides a benchmark for practical wireless systems.
There have been extensive studies on the capacity of various MIMO chan-nels. The capacity of MIMO channels with an average power constraint is also available in literature [3]. But the capacity of MIMO channels with amplitude constraints has been studied only for special cases and offers several open problems. In [1], several upper and lower bounds have been proposed for MIMO channels with amplitude constraints and their performance has been evaluated for invertible channel matrices.
The objective of this thesis is to tighten the bounds for arbitrary con-straints on the support of the input distribution. We also try to generalize the tightest bounds for arbitrary channel matrices and evaluate their per-formance.
1.3. THESIS STRUCTURE AND OUTLINE The motivation behind this objective comes from the importance of un-derstanding the capacity limits to gain insights on the performance of wireless communication systems. The capacity bounds help to give a comparison of the different systems in terms of improvement in spectral efficiency evaluated for different channel models, input constraints, etc. These insights can then be used for design of coding techniques, channel estimation and feedback techniques. It can also help network operators to make important choices on investments for infrastructure by considering financial expenditure for the performance gain expected in return.
1.3
Thesis Structure and Outline
The thesis is organized as follows. • Chapter 2: Preliminaries
In this chapter an introduction to the main concepts of informa-tion theory is given. This will be important for the realizainforma-tion and understanding of our work. The main information theoretic met-rics like entropy and mutual information are defined along with the basic understanding of channel capacity. Then we give an overview of the MIMO systems, its categories, list its advantages and disad-vantages and give a detailed description of the channel model. • Chapter 3: Literature Review of Prior Works
In this chapter we focus on giving a brief overview of the prior re-search works done on the MIMO channel capacity. We describe the MIMO channel capacity with average power constraints from the
1.3. THESIS STRUCTURE AND OUTLINE results of Telatar’s seminal paper [3]. Then we focus on the capac-ity bounds for MIMO channel with amplitude constraints which is a relatively new area of research and has been evaluated for special cases only.
• Chapter 4: Capacity Bounds for Arbitrary Channel Ma-trices
In this chapter we start presenting the contribution of this thesis. We focus on the Duality upper bound and the Jensen’s inequal-ity lower bound as explained in the prior works and try to extend the bounds for arbitrary channel matrices. We also improve the previous bounds with postcoding technique.
• Chapter 5: Experimental Results
In this chapter we present the numerical results of our evaluation. We compare the performance of our theoretical upper and lower bounds for 2 × 2 MIMO channel by simulating them in Matlab. We anticipate that our results with tighter bounds are promising. • Chapter 6: Gap between Upper and Lower Capacity Bounds
In this chapter we prove analytically that the capacity gap for the 2 × 2 MIMO channel is upper bounded by a finite 2.23 bits per channel use and this is independent of the input amplitude or the channel coefficients. We also provide the simulation based valida-tion of the upper bound to the capacity gap to end this chapter. • Chapter 7: Conclusion and Future Avenues
Finally, here we conclude the thesis and give an idea about future developments.
1.3. THESIS STRUCTURE AND OUTLINE • Appendix
Chapter 2
Preliminaries
2.1
Fundamentals of Information Theory
Information theory is the basis of studying the performance limits in communication. Shannon established the concepts of information theory in his seminal paper [4] in 1948, in which he defines the notions of channel capacity, i.e. the highest achievable information rate with arbitrarily low probability of error, and gives a mathematical model to compute it. In this section, to convey the concept of information, we give the definition and properties of the quantities entropy and mutual information [5]. For the notations used in this chapter, we refer the reader to the Appendix.
2.1.1
Entropy
Entropy gives the measure of uncertainty of a random variable x. Definition 1. The differential entropy H(x) of a continuous random
2.1. FUNDAMENTALS OF INFORMATION THEORY variable x with probability density function p(x), is defined as [6],
H(x) = − Z +∞
−∞
p(x) log p(x) dx (2.1)
2.1.2
Joint and Conditional Entropy
Definition 2. The joint differential entropy H(x, y) of continuous ran-dom variables x and y with joint probability density function p(x, y), is defined as [6], H(x, y) = − Z +∞ −∞ Z +∞ −∞ p(x, y) log p(x, y) dxdy (2.2) Definition 3. The conditional differential entropy H(y|x) of continuous random variables x and y is defined as [6],
H(y|x) = − Z +∞
−∞
Z +∞
−∞
p(x, y) log p(y|x) dxdy (2.3) Theorem 1. The relation between conditional and joint entropy is given by the chain rule of entropies [6],
H(x, y) = H(x) + H(y|x) = H(y) + H(x|y) (2.4)
Theorem 2. Entropy of a complex multivariate normal distribution: Let x1, x2, ...., xn have a multivariate normal distribution Nn(µ, Σ) with mean
µ and n × n covariance matrix Σ. Its entropy is given by,
2.1. FUNDAMENTALS OF INFORMATION THEORY Theorem 3. Entropy of a real multivariate normal distribution: Let x1, x2, ...., xn have a multivariate normal distribution with mean µ and
n × n covariance matrix Σ. Its entropy is given by, H(Nn(µ, Σ)) =
1
2log((2πe)
ndet(Σ)) (2.6)
Corollary 1. The entropy of a random variable x with normal distribu-tion having zero mean and variance σ2 is given by,
H(x) = 1
2log(2πeσ
2
) (2.7)
2.1.3
Mutual Information
The mutual information I(x; y) is the measure of dependency between two random variables x and y and is the reduction of uncertainty of one variable by the presence of the other.
Definition 4. The mutual information I(x; y) between two random vari-ables x and y with joint probability density function p(x,y) is defined as,
I(x; y) = Z +∞ −∞ Z +∞ −∞ p(x, y) log p(x, y) p(x)p(y)dxdy (2.8) From the definition we can also see that,
2.2. AWGN CHANNEL CAPACITY I(x; y) = Z +∞ −∞ Z +∞ −∞ p(x, y) logp(x|y) p(x) dxdy (2.9) = Z +∞ −∞ Z +∞ −∞ p(x, y) log p(x|y)dxdy − Z +∞ −∞ Z +∞ −∞ p(x, y) log p(x)dxdy = H(x) − H(x|y) (2.10)
2.2
AWGN Channel Capacity
The capacity of a channel is the basic measure of performance. The capacity of a channel can be described as the maximum rate of infor-mation transfer over a communication channel for which we can achieve arbitrarily low probability of error [6]. But it is impossible to drive the probability of error to zero if our rate of communication is greater than the capacity [4].
Figure 2.1: AWGN channel
2.2. AWGN CHANNEL CAPACITY of AWGN (additive white Gaussian noise) channel. The AWGN chan-nel model is important to study as it produces simple and manageable mathematical models which are useful for understanding the behavior of communication systems, although the AWGN model does not account for fading, frequency selectivity, interference or dispersion.
The AWGN channel is a discrete time channel, with input Xi, added
noise Zi and output Yi at time i. The channel model is depicted in
Figure 2.4. The noise Zi is independent and identically distributed and
drawn from a normal distribution with zero mean and variance N . Yi = Xi+ Zi, Zi ∼ N (0, N ). (2.11)
The noise is further assumed to be independent of the signal Xi.
The capacity of this channel is infinite if the noise variance is zero, in which case we can perfectly detect the transmitted symbol from the out-put. Even if the noise variance is non zero, the channel capacity can be infinite if there is no constraint on the input, as we can choose the inputs arbitrarily far apart which can be easily detected at the output with arbitrarily low probability of error.
However, in practical systems there exist some constraints. The most common of them is the limitation on the power or energy of the input. We assume an average power constraint.
2.2. AWGN CHANNEL CAPACITY
2.2.1
Channel Capacity with Average Power
Con-straint
Let P be the maximum channel power per channel use. Then for a codeword (x1, x2, . . . , xk) transmitted through the channel, we have:
1 k k X i=1 x2i ≤ P. (2.12)
Shannon proved that the channel capacity for most channels equals the mutual information of the channel maximized over all possible input distributions:
C = max
f (x) I(X; Y ) (2.13)
Definition 5. The information capacity of the Gaussian channel with average power constraint P is [5]
C = max
f (x):EX2≤PI(X; Y ) (2.14)
From (2.10), we get
I(X; Y ) = H(Y ) − H(Y |X) (2.15) = H(Y ) − H(X + Z|X) (2.16) = H(Y ) − H(Z|X) (2.17)
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL As Z is independent from X, hence,
I(X; Y ) = H(Y ) − H(Z) (2.18)
Since E[Y2] = P + N , so from (2.7) and (2.18) we can derive,
I(X; Y ) ≤ 1 2log(2πe)(P + N ) − 1 2log(2πe)N (2.19) = 1 2log(1 + P N) (2.20)
This upper bounds the mutual information which gives the maximum rate, i.e., capacity, which can be achieved when X ∼ N (0, P ).
2.3
Multiple Input Multiple Output
Chan-nel
In this section we give a overview of multiple antenna systems, or MIMO (Multiple Input Multiple Output). In wireless system we use multiple antennas to increase the performance of the system.
We consider a MIMO system with nt transmit antennas and nr antennas
at the receiver as shown in Figure 2.2.
MIMO can be studied under different aspects:
• Diversity: Diversity is related to the process of obtaining more versions of the transmitted signal at the receiver (with independent realizations of the channel fading). Diversity techniques are used when there is no Channel State Information (CSI) at the
transmit-2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL
Figure 2.2: Diversity scheme
ter. With diversity we can increase the SNR at the receiver and consequently decrease the probability of error. Diversity can be obtained spatially by receiving the transmitted signal by multiple antennas (separated by appropriate distances) at the receiver, i.e., (nt = 1 and nr > 1). Spatial diversity is most important against
small scale fading. In general, a MIMO system shows a maximum diversity order equal to nt× nr. The diversity scheme is depicted
in Figure 2.2.
• Spatial Multiplexing: Spatial multiplexing is a very important technique for increasing channel capacity at high SNR. We will see subsequently that a MIMO link can be decomposed into r parallel SISO ( Single Input Single Output) links if there is perfect CSI at the receiver. Here r is limited by r ≤ min(nt, nr). In this case we
are able to increase the capacity of the MIMO link by a factor of r with respect to an equivalent SISO link (spatial multiplexing gain), but to obtain this gain it is necessary to have both ntand nr to be
greater than 1. Spatial multiplexing can also be used transmitting simultaneously to multiple users ( multi-user MIMO).
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL • Beamforming: By beamforming we transmit the signal from each of the transmit antennas with specific phase and gain weight such that the signals add constructively at the receiver and SNR is max-imized. This technique functions as a spatial filter by rejecting interference coming from other directions. Apart from increasing SNR at receiver, beamforming can also be used to separate chan-nels for multiple users in the same frequency band. The principle of beamforming is depicted in Figure 2.3.
Figure 2.3: Principle of beamforming
2.3.1
Advantages and Disadvantages of MIMO
Sys-tems
The advantages of MIMO can be summarized as follows. Firstly, there is increase in spectral efficiency (bits/s/Hz) due to multiplexing gain. Secondly, the error rate decreases as a result of the diversity gain by
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL combating multipath fading. Thirdly, interference is reduced by the spa-tial filtering. Last but not the least, it is energy efficient due to the array gain from the multiple antennas.
The main disadvantages of MIMO systems are consumption of more power, increased hardware costs from the multiple RF ends, increased computational complexity, etc.
2.3.2
MIMO Model
Considering the case where the channel is not frequency or time selective, the complex discrete baseband linear model of the channel can be derived directly from the SISO model:
y = h · x + z (2.21) where x and y are the channel input and output respectively, and z is the noise added at the receiver.
Let us consider a MIMO system with nt∈ N transmit and nr ∈ N receive
antennas.
Assumptions: We assume a memoryless MIMO channel model, hence the output depends only on the current input and is independent from the previous inputs or outputs.
We assume a flat fading channel with fixed channel matrix H which is known at the receiver. The elements of H are independent identically distributed (i.i.d) and complex circularly symmetric channel gains. Z is complex Gaussian noise with zero mean and independent real and imaginary part. We also assume that the noise samples at each receiver
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL
Figure 2.4: MIMO channel model antenna are independent.
For this MIMO channel with flat fading, the input-output relation (per channel use) is given by the complex baseband model,
Y = HX + Z (2.22)
where X is the (nt×1) transmit vector, Y is the (nr×1) (array response)
receive vector, H is the (nr × nt) channel matrix and Z is the (nr× 1)
additive white Gaussian noise (AWGN) vector.
The complex scalar hmn is the channel gain between transmit antenna n
and receive antenna m and the channel matrix H is given by:
H = h11 h12 . . h1nt h21 . . . . . . . . . hnr1 . . . hnrnt (2.23)
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL The covariance matrix of Z is,
E[ZZH] = N0Inr (2.24)
where Inr is a nr× nr identity matrix.
Since the trace of the covariance matrix (i.e., the sum of the diagonal element) represents the total power, the total transmitted power is given by
tr(E[XXH]) = PT (2.25)
and the total noise power is given by,
tr(E[ZZH]) = PN = N0 (2.26)
2.3.3
SVD for MIMO Channel
Singular value decomposition (SVD) is an important mathematical tech-nique for computing the capacity of MIMO systems [3].
By applying SVD to the fixed channel matrix H, we can decompose it into the product of three matrices as shown in Figure 2.5,
H = UΛVH (2.27)
where U and V are unitary matrices (UUH = Inr and VV
H
= Int)
with dimensions (nr × nr) and(nt × nt) respectively. Λ is a diagonal
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL
Figure 2.5: SVD of the channel matrix
elements. The singular values are denoted as √λi for i = 1, 2, ..., r. The
rank of H is given by r, which is equal to the number of singular values that are not zero.
The rank is limited by,
r ≤ min(nt, nr) (2.28)
λi’s are the eigenvalues of the matrix HHH(or HHH). The columns of
the unitary matrix U are the nr eigenvectors of HHH. The columns of
the unitary matrix V are the nt eigenvectors of HHH.
We compute the Shannon capacity of the MIMO channel by converting into a parallel of r SISO channels.
From equations (2.22) and (2.27) we have,
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL Multiplying both sides of equation (2.29) by UH we get,
UHY = ΛVHX + UHZ (2.30) We consider the following vectors, obtained by a linear transformation of the original ones: ˜Y = UHY, ˜X = VHX and ˜Z = UHZ. Now from
equation (2.30) we get, ˜
Y = Λ ˜X + ˜Z (2.31)
As Λ is a diagonal matrix, therefore equation (2.31) is equivalent to r SISO channels:
˜ yi =
p
λix˜i+ ˜zi (2.32)
where i is the index for each SISO channel.
The scheme for MIMO channel after SVD decomposition is shown in Figure 2.6.
If we check the power relations of the equivalent parallel channels, we observe that for the noise,
E[Z ˜˜ZH] = UH E[ZZH] U (2.33) = PNUHInrU (2.34)
2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL
Figure 2.6: MIMO channel after application of SVD For the transmitted signal,
E[X ˜˜XH] = E[XXH] (2.36)
We can see that even the total transmitted power remains the same as the trace of the covariance matrices remains unchanged by the transfor-mation with the unitary matrices due to their properties.
With this SVD technique we can achieve Shannon’s capacity for MIMO channels as will be seen in the next chapter.
2.3.4
Linear Precoding and Postcoding
In the scenario where the channel is known at both the transmitter and receiver, it is possible to derive the unitary matrices U and V from the channel matrix H. Now the linear transformations in (2.30) can be
ap-2.3. MULTIPLE INPUT MULTIPLE OUTPUT CHANNEL
Figure 2.7: Transmitter and receiver scheme with linear transformations plied in a real scheme to make ˜X and ˜Y accessible at the transmitter and receiver respectively. The transformation done at the transmitter is called linear precoding, due to which it becomes possible to directly modulate the parallel channels in ˜X. The receiver then performs a post-coding transformation to recover the original signal. The main goal of pre and postcoding is to diagonalize the channel matrix, so that the MIMO channel is separated into independent single input single output (SISO) channels without co-channel interference. In Figure 2.7, we depict a sim-ple transmitter and receiver scheme with the application of the linear transformations with the unitary matrices of the SVD.
Chapter 3
Prior Works
In this chapter, we review the previous works. For the notations used in this chapter, we refer the reader to the Appendix.
3.1
Capacity of MIMO Channel with
Av-erage Power Constraints
The capacity of a channel can be described as the maximum rate of information transfer over a communication channel for which we can achieve arbitrarily low probability of error [6]. For a communication channel with input X and output Y , we can write the capacity as
C = max FX I(X; Y ) (3.1) C(X, H) = max FX I(X; HX + Z) (3.2)
3.1. CAPACITY OF MIMO CHANNEL WITH AVERAGE POWER CONSTRAINTS where FX is the cumulative distribution of X.
The results of Telatar’s paper [3] define the channel capacity with aver-age power constraints.
We show the derivations of the flat-fading, MIMO channel capacity con-sidering the assumptions from section 2.3.
The mutual information is given as,
I(X; Y ) = H(Y ) − H(Y |X) (3.3) = H(Y ) − H(HX + Z|X) (3.4) = H(Y ) − H(Z|X) (3.5) = H(Y ) − H(Z) (3.6)
Equation (3.5) follows from the assumption that the channel matrix is fully known at the receiver, hence there is no uncertainty of HX condi-tioned on X, and the equation (3.6) follows from the assumption that Z is independent from X.
Since Z ∼ CN (0, N0Inr), we get the entropy of the Gaussian noise from
theorem 2 as,
H(Z) = log((πe)nrN
0) (3.7)
To find the capacity, we need to maximize the mutual information and hence the entropy H(Y ). Since the normal distribution maximizes the entropy over all distributions with the same covariance, the mutual infor-mation is maximized when Y represents a multivariate Gaussian random
3.1. CAPACITY OF MIMO CHANNEL WITH AVERAGE POWER CONSTRAINTS variable. Then the covariance matrix of Y is,
E[Y YH] = E[(HX + Z)(HX + Z)H] (3.8) = E[(HX + Z)(XHHH + ZH)] (3.9) = E[HXXHHH] + E[HXZH] + E[ZXHHH] + E[ZZH] = E[HXXHHH] + E[HX]E[ZH] + E[Z]E[XHHH] + E[ZZH] = HΣHH + N0Inr (3.10)
where Σ is the covariance matrix of X. The output entropy is given by,
H(Y ) = log((πe)nrdet(HΣHH + N
0Inr)) (3.11)
From equations 3.6, 3.7 and 3.11, we get the MIMO channel capacity as,
C = log((πe)nrdet(HΣHH + N 0Inr)) − log((πe)nrdet(N 0Inr) (3.12) = log(det(HΣHH + N0Inr)) − log(det(N0Inr)) (3.13) = log(det(HΣHH + N0Inr)det((N0Inr) −1 )) (3.14) = log(det(Inr+ HΣHH N0 )) (3.15)
We have assumed that the CSI is not known at the transmitter, so it is optimal to distribute the total transmit power PT equally between the
3.1. CAPACITY OF MIMO CHANNEL WITH AVERAGE POWER CONSTRAINTS given by, Σ = PT nt Inr (3.16)
So the MIMO channel capacity can be written as,
C = log(det(Inr+
PT
ntN0
HHH)) (3.17) The SVD of the channel matrix H = UΛVH (as explained in subsection 2.3.3) is used to write the capacity as,
C = log(det(Inr+ PT ntN0 UΛVHVΛHUH)) (3.18) = log(det(Inr+ PT ntN0 Λ2)) (3.19) = r X i=1 (1 + PT ntN0 λi) (3.20)
where r is the rank of H and√λi’s are the singular values of the diagonal
matrix Λ. Remarks:
The channel capacity with total power constraints across all transmit antennas is well established in the literature. But MIMO capacity un-der per-antenna power constraints is more important in practice due to the power constraints on the individual antenna’s RF end. Also in the case of distributed MIMO system, where the transmit antennas are not physically located in the same node, and hence cannot share power, the per-antenna power constraint becomes more relevant.
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS But MIMO channel capacity with per-antenna power constraint is still an open problem and has been solved for some special cases only. In [7], the capacity for MISO channel with per-antenna power constraint has been established in closed form. In [8], a closed-form expression for the capacity of the point-to-point static Gaussian MIMO channel under per-antenna power constraints and special constraints on the channel matrix and input signal, is derived.
3.2
Capacity of MIMO Channel with
Am-plitude Constraints
Although there has been extensive research on the capacity of MIMO channels with average power constraints, the literature on the capacity with amplitude constraints on the input is quite limited. We want to focus more on this relatively less known, but more important case of the capacity where the channel inputs are amplitude constrained.
In his seminal paper [9], Smith shows that the capacity of an amplitude-constrained scalar Gaussian channel is achieved by a discrete input ran-dom variable with a finite number of values. In [10], Smith’s results were generalized for peak power limited quadrature Gaussian channels. From the generalization of the approach of [10] for a deterministic MIMO channel with identity channel matrix and peak and average power con-straints, it is shown in [11], that the capacity achieving input distribution is discrete. In [12], McKellips develops an analytic approach to obtain an upper bound on the capacity of an amplitude-constrained scalar channel by bounding the channel output entropy, which is especially tight for
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS high SNR. The McKellips’ bound is improved by [13] which finds upper bounds for scalar and vector AWGN channels by using a dual capacity expression.
In [1], they have derived several upper and lower bounds on this capac-ity, such as the Moment Upper Bounds, Duality Upper Bound (which is based on results from [13]), EPI Lower Bounds, Ozarow-Wyner Type Lower Bound, Jensen’s Inequality Lower Bound.They have evaluated the performance of these capacity bounds by considering diagonal channel matrix H with fixed coefficients, and we see in Fig. 1 of [1] that the Duality Upper Bound and the Jensen’s Lower Bound are considerably tighter than the other bounds, especially in the high amplitude regime. In this section we give a brief overview of the capacity bounds of MIMO channels with amplitude constraints as in the paper [1]. In the next chapter we will improve their performance leading to tighter bounds.
Notations:
Here essential notations for the following section are provided which are also given in the appendices.
n-dimensional ball centered at X ∈ Rnand of radius r ∈ R+is defined
as the set
Bx(r) := {y : ||y − x|| ≤ r}
The Volume of the ball is given by V ol(Bx(r)) =
πn2
Γ(n2 + 1)r
n
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS Rn and which contains the origin, is defined as
rmax(D) := min{r ∈ R+: D ⊂ B0(r)}
rmin(D) := max{r ∈ R+ : B0(r) ⊆ D}
We define for a given vector a = (a1, ..., an) ∈ Rn+,
Box(a) := {a ∈ Rn : |xi| ≤ ai, i = 1, ..., n}
The smallest box that contains the set D is defined as, Box(D) := inf{Box(a) : D ⊆ Box(a)}
Assumptions:
We consider a convex and compact input space X ⊂ Rnt which contains
the origin.
We consider the two special amplitude constraints:
• Per antenna amplitude constraint, X = Box(a), where a ∈ Rnt
+ is
given as a = (A1, A2, ..., Ant),
• nt-dimensional amplitude constraint, X = β0(A), where A ∈ R+ is
given.
We assume real valued channel model with fixed channel matrix H ∈ Rnr×nt which is known at the receiver, and Z is complex Gaussian noise with zero mean and independent real and imaginary part. We also as-sume that the noise at each receiver antenna are independent, hence the covariance matrix of Z is E[ZZH] = N
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS matrix.
3.2.1
Moment Upper Bound
Theorem 4. For any input space X and any fixed channel matrix H, the Moment Upper Bound is given as in [1],
C(X , H) ≤ CM(X , H) := inf p>0nrlog( knr,p (2πe)12 n 1 p r||˜x + Z||p) (3.21)
where ˜x ∈ HX is chosen such that ||˜x|| = rmax(HX ), let n ∈ N and
p ∈ (0, ∞) be arbitrary and, kn,p:= √ πe1p(p n) 1 pΓ(n p + 1) 1 n Γ(n2 + 1)n1 (3.22)
3.2.2
Duality Upper Bounds
Theorem 5. For any input space X and any fixed channel matrix H, the Duality Upper Bounds are given as in [1],
C(X , H) ≤ CDual,1(X , H) := log(cnr(d) + V ol(β0(d)) (2πe)nr/2 ) (3.23) where d := rmax(HX ), cnr(d) := Pnr−1 i=0 nr−1 i Γ(nr −i2 ) 2i/2Γ(nr 2 ) di, and C(X , H) ≤ CDual,2(X , H) := nr X i=1 log(1 + √2Ai 2πe) (3.24) where a = (A1, ...., Anr) such that Box(a) = Box(HX ).
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS Proof: The proof is identical to that in [1] and we also include it here as we will refer to it to improve the results in the paper.
For any n-dimensional ball with radius r ∈ R+, it has been shown in [13],
max FX:X∈β0(r) I(X; X + Z) ≤ log(cn(r) + V ol(β0(r)) (2πe)n/2 ) (3.25) where, cn(r) = n−1 X i=0 n − 1 i Γ(n−i 2 ) 2i/2Γ(n 2) ri (3.26)
From the expression of capacity in terms of entropy we know,
C(X , H) = max FX:X∈X H(HX + Z) − H(HX + Z|HX) (3.27) = max FX:X∈X I(HX; HX + Z) (3.28) = max FX˜: ˜X∈HX I( ˜X; ˜X + Z) (3.29)
To find the upper bound to the capacity, the optimization domain is increased from HX to a nr-dimensional ball β0(d) with d := rmax(HX ).
Hence,
C(X , H) ≤ max
FX˜: ˜X∈β0(d),d:=rmax(HX )
I( ˜X; ˜X + Z) (3.30)
From equations 3.25 and 3.30 we get, C(X , H) ≤ log(cnr(d) +
V ol(β0(d))
(2πe)nr/2 ) (3.31)
This proves the upper bound in equation 3.23.
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS of 3.29 by: C(X , H) = max FX˜: ˜X∈HX I( ˜X; ˜X + Z) (3.32) ≤ max FX˜: ˜X∈Box(HX ) I( ˜X; ˜X + Z) (3.33) ≤ max FX˜: ˜X∈Box(HX ) nr X i=1 I( ˜Xi; ˜Xi+ Zi) (3.34) = nr X i=1 max FXi˜ :| ˜Xi|≤Ai I( ˜Xi; ˜Xi+ Zi) (3.35) ≤ nr X i=1 log(1 + √2Ai 2πe) (3.36) where inequality (3.33) follows from expanding the optimization domain from HX to Box(HX ); (3.34) by separating the mutual information for each parallel channel; equation (3.35) by choosing per-antenna con-straints such that Box(a) = Box(HX ); and (3.36) by putting n = 1 in (3.25).
3.2.3
EPI Lower Bounds
Theorem 6. For any fixed channel matrix H and any channel input space X with continuous X, we have from [1],
C(X , H) ≥ CEP I(X , H) := max FX:X∈X nr 2 log(1 + 2nr2 H(HX) 2πe ) (3.37) For the special case of when nt = nr = n, invertible channel matrix H
and X uniformly distributed over X , then
C(X , H) ≥ CEP I(X , H) := n 2 log(1 + |det(H)|2nVol(X ) 2 n 2πe ) (3.38)
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS
3.2.4
Ozarow-Wyner Type Lower Bound
Theorem 7. Let XD ∈ supp(Xd) ⊂ Rnt be a discrete random vector
of finite entropy, g : Rnr → Rnt a measurable function, and p > 0.
Furthermore, let Kp be a set of continuous random vectors, independent
of XD, such that for every U ∈ Kp we have H(U ) < ∞, ||U ||p < ∞, and
supp(U + xi) ∩ supp(U + xj) = ∅ for all xi, xj ∈ supp(Xd), i 6= j. Then
we have from [14], C(X , H) ≥ COW(X , H) := [H(Xd) − G]+, (3.39) where, G := inf U ∈Kp,p>0 ((G1,p(U, XD, g) + G2,p(U ))) (3.40) with G1,p(U, XD, g) := ntlog( ||U + XD − g(Y )||p ||U ||p ) (3.41) G2,p(U ) := ntlog( knt,pn 1 p||U ||p t 2nt1H(U ) ) (3.42) kn,p := √ πep1(p n) 1 pΓ(n p + 1) 1 n Γ(n2 + 1)n1 (3.43)
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS
3.2.5
Jensen’s Inequality Lower Bounds
Theorem 8. For any fixed channel matrix H and any channel input space X with continuous X, we have from [1]
C(X , H) ≥ CJ ensen(X , H) := max FX:X∈X log+((2 e) nr 2 E[exp(−||H(X − X 0)||2 4 )] −1 )(3.44)
where X0 is an independent copy of X.
Figure 3.1: Comparison of the upper and lower bounds for a 2 × 2 MIMO system with per-antenna amplitude constraints A1 = A2 = A and fixed
channel matrix H. [1]
In [1] they have evaluated some of the above bounds for the special case of diagonal channel matrices and simulated their performance for a 2 × 2 MIMO system with per-antenna amplitude constraints A1 = A2 = A and
3.2. CAPACITY OF MIMO CHANNEL WITH AMPLITUDE CONSTRAINTS fixed channel matrix H =
0.3 0 0 0.1 as seen in Figure 3.1.
In the Figure 3.1, we can observe that the Duality upper bound and the Jensen’s inequality lower bound are considerably tighter than the Moment upper bound and the EPI lower bound, especially in the high amplitude region. So in the next chapter we consider the Duality and Jensen’s bounds and try to evaluate them for arbitrary channel matrices and also try to tighten them.
Chapter 4
Capacity Bounds for
Arbitrary Channel Matrices
In this chapter we start presenting the contribution of our work. We try to apply the Duality Upper Bound for arbitrary channel matrices. We further try to tighten the bounds by considering postcoding at the receiver. We also evaluate the performance of the Jensen’s Lower bound for arbitrary channel matrices.
4.1
Duality Upper Bounds
As seen in Theorem 5, the Duality Upper Bounds are given as,
CDual,1(X , H) = log(cnr(d) +
V ol(β0(d))
4.1. DUALITY UPPER BOUNDS where d = rmax(HX ), cnr(d) = Pnr−1 i=0 nr−1 i Γ(nr −i2 ) 2i/2Γ(nr 2 ) di and CDual,2(X , H) = nr X i=1 log(1 +√2Ai 2πe) (4.2) where a = (A1, ...., Anr) such that Box(a) = Box(HX ).
To evaluate the Duality Upper Bounds of capacity for MIMO channel with arbitrary channel matrix, we have to consider the singular value decomposition of the channel as shown in chapter 2. So we consider H = UΛVH. From the previous theorem, we derive the following theo-rem:
Theorem 9. Let H ∈ Rnr×nt be fixed arbitrary channel matrix, n
min :=
min(nr, nt), and the input space X = Box(a), a = (A, . . . , A) for
suffi-ciently large A ∈ R+. Moreover, let σi be the i-th singular value of H,
where i = 1, ...., nmin and σmax be the maximum singular value among
them. Then the Duality upper bound is given by,
CDual,1(X , H) = log(cnr(d 0 ) + V ol(β0(d 0 )) (2πe)nr/2 ) (4.3) where d0 = σmax √ nt· A and cnr(d 0 ) as defined in (4.1) CDual,2(X , H) = nrlog(1 + 2√ntAσmax √ 2πenr ) (4.4) Proof:
4.1. DUALITY UPPER BOUNDS In the proof of the CDual,1 (4.1), as shown in section 3.2, to derive the
upper bound, the optimization domain has been expanded from HX to a nr-dimensional ball β0(d) with d := rmax(HX ), and for CDual,2 (4.2),
the optimization domain has been expanded from HX to Box(HX ). We have considered the input space X = Box(a), a = (A, . . . , A) for sufficiently large A ∈ R+. Hence,
Box(HX ) = Box(HBox(a)) (4.5)
Now the smallest ball that contains the space Box(HBox(a)) is given by,
Box(HBox(a)) ⊆ β0(rmax(HBox(a))) (4.6)
Again the smallest ball that contains Box(a) is given by β0(
√
nt · A).
Hence equation (4.6) can be bounded as, Box(HBox(a)) ⊆ β0(rmax(Hβ0(
√
nt· A))) (4.7)
The enlarged constraint set of the bounds can be expressed as follows:
β0(rmax(HBox(a))) ⊆ β0(rmax(Hβ0(
√ nt· A))) (4.8) = β0(rmax(UΛVHβ0( √ nt· A))) (4.9) = β0(rmax(UΛβ0( √ nt· A)) (4.10) = β0(rmax(Λβ0( √ nt· A))) (4.11) = β0(rmax(σmaxβ0( √ nt· A))) (4.12) = β0(σmax √ nt· A) (4.13)
4.1. DUALITY UPPER BOUNDS where (4.9) follows after applying SVD to H; (4.10) and (4.11) as the ball β0(
√
nt· A) remains unchanged by the rotation by unitary matrices VH
and U; and (4.12) as the maximum singular value σmax scales the ball
β0(
√
nt· A) to the maximum radius. From (4.1) and (4.13), the CDual,1
bound for arbitrary channel matrix can be given as,
CDual,1(X , H) = log(cnr(σmax
√ nt· A) + V ol(β0(σmax √ nt· A)) (2πe)nr/2 ) (4.14)
This concludes the proof of (4.3).
Now we proceed with the proof for CDual,2. Using (4.7) and (4.13) we
can further show,
Box(HX ) = Box(HBox(a)) (4.15) ⊆ β0(r) (4.16) ⊆ Box(a0) (4.17) where r = σmax √ nt· A and a0 = (√nr min, r √ nmin, . . . ) ∈ R nmin + .
(4.17) follows from the observation that Box(a0) contains the ball β0(r).
We assume nt >> nr. Therefore, nmin = nr.
Using these constraint sets in (4.2) we have,
CDual,2(X , H) ≤ nrlog(1 +
2√ntAσmax
√ 2πenr
) (4.18)
This concludes the proof of (4.4).
4.1. DUALITY UPPER BOUNDS CDual upper bounds.
4.1.1
Duality Upper Bound with Postcoding
We have seen how linear transformations can be achieved at the trans-mitter and receiver if the channel state is known both at the transtrans-mitter and receiver. Here though we have assumed that the CSI is known, still precoding is not possible at the transmitter due to the per-antenna ampli-tude constraints applied at the input. However we can apply postcoding at the receiver by multiplying the received signal by UH.
Theorem 10. Let H ∈ Rnr×nt be a fixed arbitrary channel matrix,
nmin := min(nr, nt) = nr, and the input space X = Box(a), a =
(A, . . . , A) for sufficiently large A ∈ R+. Moreover, let σi be the i’th
singular value of H, where i = 1, . . . , nmin. Then the Duality upper
bound with postcoding is given by, C0(Box(a), UH · H) ≤ nr X i=1 log(1 + 2 √ ntAσi √ 2πenr ) (4.19)
Proof: Similar to the previous proofs, we enlarge the constraint set by, Box(UHHX ) = Box(UHUΛVHBox(a)) (4.20)
= Box(IΛVHBox(a)) (4.21) ⊆ Box(IΛVHβ0( √ ntA)) (4.22) = Box(Λβ0( √ ntA)) (4.23) = Box(a00) (4.24)
4.2. JENSEN’S LOWER BOUND where a00 = (√r1 nr, r2 √ nr, . . . , ri √ nr), i = 1, 2, . . . , nmin = nr. and ri = √ ntAσi.
Using these constraint sets in (4.2) we have, C0(Box(a), UH · H) ≤ nr X i=1 log(1 + 2 √ ntAσi √ 2πenr ) (4.25)
This concludes the proof of Theorem 10. In Figure 4.2 we compare the constraint bounds for the cases without and with postcoding respectively. We will also see in the experimental results in the next chapter that as the singular values of H become equal to each other, CDual,2 bound becomes
same with or without postcoding. We are able to tighten the bounds considerably with postcoding for cases where the singular values of our channel matrix are far apart from each other.
4.2
Jensen’s Lower Bound
Theorem 11. Let H ∈ Rnr×nt be a fixed, arbitrary channel matrix,
nmin := min(nr, nt), and X = Box(a) for a = (A1, ..., Ant) ∈ R
nt
+.
More-over, let σi is the i’th singular value of H. Then the Jensen’s Inequality
lower bound for this arbitrary channel matrix is given by [1],
CJ ensen(X , H) = max(log(( 2 e) nmin 2 1 ψ(H, b∗)), 0) (4.26) where, ψ(H, b∗) = min b∈Box(a) nmin Y i=1 ϕ(σiBi) (4.27)
4.2. JENSEN’S LOWER BOUND ϕ(x) = 1 x2(e −x2 − 1 +√πx(1 − 2Q(√2x))) (4.28) for b := (B1, ..., Bnt) and ϕ : R+→ R+.
4.2.1
Optimal ‘b’ for the Lower Bound
In [1], Page 13, Remark 3 suggests further studies regarding the choice of the optimal b and speculates that it may be similar to the process of waterfilling for power allocation in the case of average power constraints. However, here we will propose a simpler approach to choose b∗ for the lower bound. For that we first need to prove that ϕ(x) is a monotonically decreasing function for non-negative x.
Lemma 1. For x > 0 and ϕ : R+ → R+,
ϕ(x) = 1 x2(e
−x2
− 1 +√πx(1 − 2Q(√2x) (4.29)
is a monotonically decreasing function. Proof:
If we are able to show that the derivative of the function dϕdx = ϕ0(x) ≤ 0 for all values of x ≥ 0, then it will be proved that ϕ(x) is monotonically decreasing for our useful range.
From (6.8) we have, ϕ(x) = e −x2 x2 − 1 x2 + √ π x − 2√πQ(√2x) x (4.30)
4.2. JENSEN’S LOWER BOUND Taking the derivative of ϕ(x) with respect to x we get,
ϕ0(x) = p0(x) + q0(x) (4.31) where p0(x) = −2e −x2 x3 − 2e−x2 x + 2 x3 − √ π x2 (4.32) q0(x) = √ π x2 − √ π x2 Erf(x) + 2e−x2 x (4.33)
Plotting ϕ0(x) we observe that ϕ0(x) ≤ 0 for our useful range of x. Hence the lemma is proved.
To find the lower bound, we need to minimize ψ(H, b∗) which will be achieved by minimizing each of the terms ϕ(σiBi) individually in the
product of (4.27). Since ϕ(σiBi) is monotonically decreasing, so we can
minimize its value by maximizing σiBi. Since the singular value σi is
fixed for fixed H, so we need to maximize Bi.
We have considered our input space X = Box(a) for a = (A1, ..., Ant)
for sufficiently large A ∈ R+. So we choose Bi = Ai, i.e., the highest
4.2. JENSEN’S LOWER BOUND
Figure 4.1: Constraint bounds for the case without postcoding: A · β0( √ nt· A), B · Λβ0( √ nt· A), C · β0(σmaxβ0( √ nt· A)), D · Box(a0)
4.2. JENSEN’S LOWER BOUND
Figure 4.2: Comparison of constraint bounds
0 10 20 30 40 50 60 70 80 90 100 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 phi
Chapter 5
Experimental Results
5.1
Introduction
In this chapter we will evaluate the performance of the theoretical bounds and compare them by implementing them in Matlab. As a performance indicator we will use spectral efficiency in bits/s/Hz. We have simulated the bounds by taking into consideration the following special constraints: • We perform the simulations for a simple case of 2 × 2 MIMO
chan-nel, i.e., nt= nr = N = 2.
• We consider fixed, non-diagonal, arbitrary channel matrix with real coefficients. Moreover, while choosing the coefficients of H, we have set the channel power equal to nr× nt, i.e., tr(H.HH) = N2. Now
if we increase the dimension of the channel matrix, power in the channel also increases proportionately. If we fix the channel power for a particular dimension of the channel matrix, then we are able
5.2. SIMULATION RESULTS to properly compare how the bounds change and their dependency on the ratio between the singular values of the channel matrix and the input amplitude.
• We consider the amplitude constraint on the input space X = Box(a), a = (A, . . . , A) for sufficiently large A ∈ R+.
• We have considered complex Gaussian noise with power 2, hence the SNR as A2/2.
5.2
Simulation Results
5.2.1
Duality Upper Bounds
In this subsection we compare the Duality Upper Bounds from (4.3) and (4.4) and Duality Upper Bound with postcoding (4.19) by plotting them as a function of the ratio of the singular values of H as shown in Figure 5.1. Here the ratio is given by r = σ1
σ2.
Observations:
From Figure 5.1 we see that CDual2,postcoded gives a much tighter upper
bound than the Duality Bounds without postcoding. So we have been successful in tightening the bounds considerably.
We also observe that CDual2,postcoded gives much tighter bounds when σ1
and σ2 are very different from each other. As they become equal, that
is, r = 1, the CDual,2 bounds with and without postcoding also become
5.2. SIMULATION RESULTS
10-2 10-1 100
Singular Value Ratio r 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Rate [bit/s/Hz] Cdual Cdual2 Cdual2postcoded
Figure 5.1: Comparison of Duality Upper Bounds for a 2 ∗ 2 MIMO system with per-antenna amplitude constraints A1 = A2 = A = 1 for
different ratios of singular values of arbitrary channel matrix H, with and without postcoding.
5.2.2
Comparison of the Upper and Lower Capacity
Bounds
In this subsection we compare the Duality Upper Bounds with the Jensen’s Inequality Lower Bound and lower bound for fixed arbitrary channel ma-trix as a function of r, as shown in Figure 5.2.
Observations:
From Figure 5.2 it is evident that the Jensen’s Inequality Bound gives a much tighter lower bound compared to the EPI Lower Bound (as also seen for the diagonal channel matrices shown in [1]).
5.2. SIMULATION RESULTS
10-3 10-2 10-1 100
Singular Value Ratio r 0 1 2 3 4 5 6 7 8 9 10 Rate [bit/s/Hz] Dual 7a Dual 7b NoCoding Dual7bPostCoding Jensen EPI
Figure 5.2: Upper and Lower Bounds A = 20 in linear scale, N = 2 We also observe that the Jensen’s Lower Bound closely follows the Dual-ity Upper Bound with postcoding with a gap not more than 1 bit/s/Hz. This gap seems to be quite tight.
In Figure 5.4 we compare the Jensen’s Lower Bound with the Cd-ual2 Upper Bound with postcoding for a fixed ratio between the singular values but different values of the amplitude A. Next in Figure 5.4, we compare the bounds as a function of SNR(dB), but for four different singular value ratios.
5.2. SIMULATION RESULTS 100 101 102 A in linear scale 0 2 4 6 8 10 12 Rate [bit/s/Hz] ratio=0.34 Cdual2postcoded Jensen
Figure 5.3: Comparison of Jensen Lower Bound and Cdual2 Upper Bound with postcoding.
5.2. SIMULATION RESULTS -20 0 20 40 0 2 4 6 8 ratio=0.01 -20 0 20 40 0 5 10 15 ratio=0.34 -20 0 20 40 0 5 10 15 ratio=0.67 -20 0 20 40 SNR in dB 0 5 10 15 Rate [bit/s/Hz] ratio=1 Cdual2postcoded Jensen
Figure 5.4: Comparison of Jensen Lower Bound and Cdual2 Upper Bound with postcoding for 4 different ratios of singular values of H and SNR in dB
Chapter 6
Gap between Upper and
Lower Capacity Bounds
The capacity of MIMO channels with average power constraints is well known. But the research on the capacity of the MIMO channels with amplitude constraints is still an open problem. Throughout this work we have studied and proposed several upper and lower bounds for this channel capacity. Our objective is to tighten the bounds so that we can get an estimate as close as possible to the true channel capacity. The gap between the upper and lower capacity bounds can help us gain insights about the performance of systems evaluated for different channel models, input constraints, etc, which can then be used for design of coding techniques, channel estimation and feedback techniques.
6.1. GAP TO CAPACITY FOR 2 × 2 MIMO CASE
6.1
Gap to Capacity for 2 × 2 MIMO Case
We are considering in the following section the special case of 2×2 MIMO channel with nt = nr = N = 2. We also consider the channel power to
be constrained by, nr X i=1 σ2i = N2 (6.1) In our case, σ12+ σ22 = 22 (6.2) We consider the ratio of the two singular values as,
σ1/σ2 = r (6.3)
From (6.2) and (6.3), we get σ1 = 2r √ r2+ 1; σ2 = 2 √ r2+ 1 (6.4)
We have proposed that Duality Bound with postcoding for arbitrary channel matrix is given by,
C0(Box(a), UH · H) ≤ nr X i=1 log(1 +2 √ ntAσi √ 2πenr ) (6.5)
6.1. GAP TO CAPACITY FOR 2 × 2 MIMO CASE For our specific case it becomes,
C0(r, A) ≤ log(1 + √ 4Ar
2πe√r2+ 1) + log(1 +
4A √
2πe√r2+ 1) (6.6)
For arbitrary, invertible fixed channel matrix H, the lower bound is given by [1], CJ ensen(X , H) = max log (2 e) nmin 2 1 ψ(H, b∗), 0 (6.7)
where X = Box(a), a = (A, ..., A) ∈ Rnt
+, and ψ(H, b∗) = min b∈Box(a) nmin Y i=1 ϕ(σiBi) (6.8) and ϕ(x) = 1 x2(e −x2 − 1 +√πx(1 − 2Q(√2x))) (6.9)
Since ϕ(x) is monotonically decreasing in our useful range (which we have proved before as the maxima of ϕ0(x) is always less than zero in our useful range), hence each element in the product of (6.8) is minimized by choosing Bi = A.
So for our special case, (6.8) becomes, ψ(r, A) = (r 2+ 1)2 16r2A4 (e −4r2A2 r2+1 − 1 + 2√πA√ r r2+ 1(1 − 2Q( 2√2Ar √ r2+ 1))) ×(e−r2+14A2 − 1 + 2 √ πA√ 1 r2+ 1(1 − 2Q( 2√2A √ r2+ 1))). (6.10)
6.1. GAP TO CAPACITY FOR 2 × 2 MIMO CASE
6.1.1
Analytical Approach:
Here we try to propose a finite analytical upper bound for the capacity gap for the 2 × 2 MIMO channel which is independent of the input am-plitude or the channel coefficients.
Upper bound : C0(r, A) = log(1 + √ 4Ar 2πe√r2+ 1) + log(1 + 4A √ 2πe√r2+ 1). (6.11) Lower bound: CJ ensen(r, A) = log( 2 e) − log(ψ(r, A)) (6.12) where ψ(r, A) = (r 2+ 1)2 16r2A4 (e −4r2A2 r2+1 − 1 + 2 √ πA√ r r2+ 1(1 − 2Q( 2√2Ar √ r2+ 1))) ×(e−r2+14A2 − 1 + 2 √ πA√ 1 r2+ 1(1 − 2Q( 2√2A √ r2+ 1))). (6.13)
Let us introduce the variable x = √2A
6.1. GAP TO CAPACITY FOR 2 × 2 MIMO CASE the capacity as follows
∆(x, r) ≤ C0− CJensen (6.14) = log(1 +√2xr 2πe) + log(1 + 2x √ 2πe) − log( 2 e) + log( 1 x4r2) + log(e−r2x2 − 1 +√πrxErf(rx)) + log(e−x2 − 1 +√πxErf(x)) (6.15) = ∆(x) + ee ∆(rx) − log( 2 e) (6.16) where e ∆(x) = log 1 + √2x 2πe + log 1 x2 + loge−x2 − 1 +√πxErf(x). (6.17) Let ¯x be an arbitrary positive real number, then for x ≥ ¯x we can upper-bound e∆(x) as follows: e ∆(x) = log 1 x + 2 √ 2πe + log e −x2 − 1 x + √ πErf(x) ! (6.18) ≤ log 1 x+ 2 √ 2πe + log √π (6.19) ≤ log 1 ¯ x+ 2 √ 2πe + log √π , x ≥ ¯x, (6.20)
6.1. GAP TO CAPACITY FOR 2 × 2 MIMO CASE While for x ≤ ¯x the upper bound is
e ∆(x) = log 1 + √2x 2πe + log 1 x2 + loge−x2 − 1 +√πxErf(x) (6.21) ≤ log 1 + √2¯x 2πe , x ≤ ¯x (6.22)
where the inequality in (6.22) is because
log(ϕ(x)) = log e −x2 − 1 +√πxErf(x) x2 ! (6.23)
is a decreasing function of x, as we previously proved that ϕ is monoton-ically decreasing, and therefore log(ϕ(x)) ≤ log(ϕ(0)) = 0. The value of ¯
x that minimizes the overall gap is chosen by equating (6.20) and (6.22) to get:
¯
x =√π (6.24)
for which we have
e
6.2. EXPERIMENTAL APPROACH Using (6.25) into (6.16) yields:
∆(x, r) ≤ 2 e∆(¯x) − log 2 e
(6.26) = 2 log√2 +√e− 2 log(√e) − log 2
e
(6.27) = 2 log√2 +√e− log(2) (6.28)
≤ 2.23 bpcu (6.29)
where bpcu stands for bits per channel use. Note that this upper bound to the capacity gap is independent of both A and r.
6.2
Experimental Approach
In the previous section we have found a finite bound to the capacity gap which is independent of the input amplitude and the singular values of H. In this section we do not consider any approximations and simulate the gap between the upper and lower capacity bounds to get an under-standing of how the gap behaves with the variation of input amplitude and with the ratio between the singular values.
From (6.5) we have the upper bound for 2 × 2 MIMO as, C0(r, A) ≤ log(1 + √ 4Ar 2πe√r2+ 1) + log(1 + 4A √ 2πe√r2+ 1)(6.30) = f (x) (6.31)
6.2. EXPERIMENTAL APPROACH for 2 × 2 MIMO, from (6.32),
CJ ensen(r, A) ≥ log( 2 e) − log(ψ(r, A)) = g(x) (6.32) where ψ(r, A) = (r 2+ 1)2 16r2A4 (e −4r2A2 r2+1 − 1 + 2 √ πA√ r r2+ 1(1 − 2Q( 2√2Ar √ r2+ 1))) ×(e−r2+14A2 − 1 + 2 √ πA√ 1 r2+ 1(1 − 2Q( 2√2A √ r2+ 1))) (6.33) 0 200 0.2 0.4 150 1 0.6 0.8 0.8 100 1 0.6 1.2 0.4 50 0.2 0 0
Figure 6.1: 3D plot of 4C = f (x) − g(x) for 0.1 ≤ r ≤ 0.9, and 1 ≤ A ≤ 200
6.2. EXPERIMENTAL APPROACH is given by,
4C = max(f (x) − g(x)) (6.34)
We plot the function 4C = f (x) − g(x) as Matlab 3D plot, as function of r and A, for 0.1 ≤ r ≤ 0.9, and 1 ≤ A ≤ 200. We observe the plot in Figure 6.1. We observe that the function 4C attains maximum value of 1.1693 bits for r = 0.9 and A = 1 (i.e. for low SNR and maximum ratio of singular values).
Chapter 7
Conclusion and Future
Avenues
7.1
Conclusion
In this thesis, inspired by the upper and lower capacity bounds for MIMO channels with amplitude constraints, we have evaluated the performance of the bounds for arbitrary channel matrices. We have also shown that with the help of postcoding we can tighten the upper bound to the ca-pacity considerably.
For the Jensen’s Inequality Lower Bound we are able to find the optimal choice of input so that the lower bound is maximized. By comparing the bounds for different ratios of the singular values, by simulation, we can observe that the Jensen’s Lower Bound closely follows the Duality Upper Bound which is an interesting result.
7.2. FUTURE AVENUES to the gap between the capacity bounds of 2.24 bits per channel use.
7.2
Future Avenues
From Figure 3.1, for the diagonal channel matrix case, we can observe that the Jensen’s Lower Bound is tight at high SNR region, but quite loose in the low SNR region as it goes to zero very fast at low SNR. Even in the case of arbitrary channels, further studies should be done to find tighter lower bounds. One possible direction is to pursue the performance of Ozarow-Wyner Type Lower Bound [15] for the given constraints. We have given several upper and lower capacity bounds, as well as a finite bound on the gap to the capacity for the 2 × 2 MIMO case. The next step would be to try and extend to the general n × n MIMO.
Appendix A
Notation
In this work we have used the following notation:
Scalars are written as lowercase italic characters, e.g. x Vectors are lowercase bold-faced characters, e.g. x Random Vectors are uppercase characters, e.g. X Matrices are bold-faced uppercase characters, e.g. X Hermitian transpose is denoted by (.)H
Trace of a matrix is denoted by tr(.) Determinant of a matrix is det(.) Expectation is denoted by E[.]
Logarithm to the base 2 is denoted by log(.) Support is denoted by supp(.)
Euclidean norm of vector x is given by ||x||.
Normal distribution with mean µ and variance σ2is denoted by N (µ, σ)
Multivariate normal distribution with mean µ and n × n covariance matrix Σ is denoted by Nn(µ, Σ)
Error function is denoted by Erf(x) for real x. The set of all real numbers is denoted by R. The set of all integers is denoted by Z.
The set of all natural numbers is denoted by N. The set of non-negative real numbers is defined by
R+ := {x ∈ R : x ≥ 0}
n-dimensional ball centered at X ∈ Rnand of radius r ∈ R
+is defined
as the set
Bx(r) := {y : ||y − x|| ≤ r}
The Volume of the ball is given by V ol(Bx(r)) =
πn2
Γ(n2 + 1)r
n
The maximum and minimum radius of a set D which is a subset of Rn and which contains the origin, is defined as
rmax(D) := min{r ∈ R+: D ⊂ B0(r)}
rmin(D) := max{r ∈ R+ : B0(r) ⊆ D}
We define for a given vector a = (a1, ..., an) ∈ Rn+,
Box(a) := {a ∈ Rn : |xi| ≤ ai, i = 1, ..., n}
The smallest box that contains the set D is defined as, Box(D) := inf{Box(a) : D ⊆ Box(a)}
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