Design of Estimation-Assisted Detector and Space-Time ...kostas/slam/files/mimo-report.pdf · eigenvalues, diag(·) is the diagonal matrix, vec(·) is the vectorization operation

Design of Estimation-Assisted Detector and

Space-Time Block Codes for MIMO System

S. Lam, K. N. Plataniotis, and S. Pasupathy

[email protected], [email protected], [email protected]

Department of Electrical and Computer Engineering

University of Toronto, Canada

Techical Report

September 14, 2005

Contents

1 Introduction 5

2 System Model 9

3 Proposed Estimation-Assisted Receiver 14

3.1 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Data Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Probability of Detection Error 21

5 Space-Time Block Codes 24

5.1 Coherent Space-Time Block Codes . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Differential Space-Time Block Codes . . . . . . . . . . . . . . . . . . . . . 27

5.3 Performance of Square Unitary Space-Time Block Codes . . . . . . . . . . 27

6 Performance Degradation in a CE/DD System 30

6.1 Symmetry of Space-Time Block Codes . . . . . . . . . . . . . . . . . . . . 30

6.2 Unitary Transform Property of CE/DD . . . . . . . . . . . . . . . . . . . . 31

1

CONTENTS 2

6.3 Isometry in MIMO System . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Space-Time Block Code Design 34

7.1 Asymmetric Space-Time Block Codes . . . . . . . . . . . . . . . . . . . . . 34

7.2 Estimation Based Space-Time Block Codes . . . . . . . . . . . . . . . . . . 37

7.2.1 Code Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.2.2 Self-Matching Space-Time Block Codes . . . . . . . . . . . . . . . . 40

7.2.3 Detection Performance of Self-Matching Space-Time Block Codes . 46

8 Results 48

9 Conclusions 59

A Matrix KF Recursion Algorithm for MIMO Matrix State-Space Model 62

B Derivation of the ML Space-Time Block Code Detector 66

C Derivation of the Detection Pairwise Error Probability 68

D Derivation of Detection Performance for Square Unitary Codes 70

E Proofs for Lemmas, Propositions, and Corollaries 74

F Derivation of Code Design Criterion 76

G Derivation of Distance Properties 80

H Derivation of Detection Performance for Self-Matching Codes 83

List of Tables

2.1 Definitions of the matrices in the observation equation for both continuous

fading model and quasi-static fading model. . . . . . . . . . . . . . . . . . 13

3.1 Specifics of state equation matrices for T = 2 and T = 4. . . . . . . . . . . 16

8.1 Various linear space-time block codes and their corresponding properties.

∆αβ = Cα −Cβ where Cα,Cβ ∈ ΩC : Cα 6= Cβ. N/A indicates that these

properties are not defined for the STBC. . . . . . . . . . . . . . . . . . . . 49

8.2 Summary of simulation parameters. . . . . . . . . . . . . . . . . . . . . . . 50

8.3 Comparison between theoretical and experimental detection performance of

various space-time block codes. (An experimental BER of 0 indicates that

no error resulted in the 5000 independent iterations.) . . . . . . . . . . . . 57

3

List of Figures

2.1 System diagram of the MIMO joint channel estimation and data detection

in fading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7.1 The construction of asymmetric space-time block code from a group code. . 37

8.1 Detection performance comparison among the different receivers and mod-

ulation schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2 Bit error rate (BER) of the joint estimation and detection scheme for various

space time block codes and isometry breaking solutions. . . . . . . . . . . . 53

8.3 Estimation mean square error (MSE) of the joint estimation and detection

scheme for various space time block codes and isometry breaking solutions. 54

4

Chapter 1

Introduction

Multi-input multi-output (MIMO) systems are gaining popularity due to their higher

capacity [1]. In MIMO systems, space-time block code (STBC) utilizes the diversity of

multiple antenna to mitigate deep fading [2]. Various coherent and differential STBC have

been developed to optimize performance in terms of error probability or capacity [3–13].

However, perfect channel state information (CSI) is generally not available at the receiver,

and differential demodulation substantially reduces the capacity and degrades detection

performance [14]. Therefore, a joint channel estimation and data detection (CE/DD)

scheme is often needed [3, 15–19].

Due to the importance of estimation-assisted MIMO detection, a novel CE/DD pro-

cedure is proposed and analyzed in this report. Other MIMO CE/DD schemes have been

discussed in [3, 17–19]. The schemes in [3, 18] estimate the channel using pilot training.

They assume that the fading coefficients remain constant from one pilot matrix to the

next. This assumption is even more restrictive than the quasi-static fading model where

5

CHAPTER 1. INTRODUCTION 6

the fading coefficients are assumed to remain constant for one block duration only. Thus,

the channel estimates are already out-of-date when detection occurs and the detection per-

formance may suffer. Motivated by the principle of minimizing the probability of error,

a decision-directed CE/DD scheme, where the channel estimates are recursively updated,

is desirable. Furthermore, a CE/DD scheme where the channel is modeled using the con-

tinuous fading model is also beneficial. Even though the schemes in [17, 19] are decision-

directed, they employ employ matrix quantities in the detector, but vector quantities in

the estimator. This means, to employ these schemes, the channel state vector provided

by the estimator must be converted to the channel state matrix for the detector, and vice

versa. Moreover, these schemes and their corresponding system models do not depicit the

temporal and spatial dimensions of MIMO systems. Motivated by the principle of creating

a consistent and natural representation of the MIMO system, a matrix state-space model,

where the temporal and spatial dimensions are naturally and intuitively represented in

matrix quantities, and its corresponding matrix CE/DD scheme, where matrix quantities

are consistently used in the estimator and the detector, are derived. Using a matrix state-

space model based on the continuous fading model, the CE/DD proposed in this report

recursively estimates the channel using the matrix Kalman filter (KF) and detects the data

using the normalized innovations-based maximum likelihood (ML) detector. Ordinary KF

algorithm only works with vector state-space model. Hence, this report introduces the

technique developed in [20, 21] and the definition of the covariance of a matrix developed

in [22] to derive the matrix KF for the matrix state-space model. The detector is shown to

be a Mahalanobis distance-square (i.e. weighted Euclidean distance-square) detector where


the weighting factor represents the effective signal-to-noise ratio (SNR) and is calculated

recursively by the matrix KF.

The probability of error of the CE/DD is derived and is shown to be intricately linked

to not only the properties of the model, the estimator and the detector, but also the

properties of the STBC. Linear unitary STBCs, a very common class of STBCs, have

been shown to be optimal STBCs when CSI is known [1, 6, 7, 15, 16]. However, they are

‘symmetric’. In a CE/DD scheme in multiplicative fading, symmetry causes isometric data

sequences and leads to a detection error floor [15,16,23,24]. To minimize the probability of

error, asymmetric STBCs are introduced in this report to be used with symmetric STBCs

to break isometry and to improve detection performance. To further improve the detection

performance, a self-matching STBC (SM-STBC) is proposed to minimize the probability

of error by adapting its code properties according to the estimation performance, reducing

the estimation error and hence the detection error with embedded training, and breaking

isometry with asymmetry. The SM-STBC in this report generalizes the one in [25] by

replacing the fixed unitary matrix, which provides training but reduces transmission rate,

with the information-bearing asymmetric STBC, which generalizes training.

The unifying framework of transceiver and STBC designs based on continuous fading

channel model in order to minimize detection error and to maximize capacity is the major

contribution in this report. This report is organized as follows. The characteristics of the

fading channel and the derivation of the discrete system model are discussed in Section

2. In Sections 3 and 4, an estimation-assisted detector is proposed and the corresponding

probability of detection error is derived. Section 5 discusses the use of common space-time


block codes (STBCs). The degradation of the detection performance due to symmetry

of STBCs is discussed in Section 6. Section 7 discusses the use of asymmetric and self-

matching STBCs to mitigate this degradation. Simulation results are discussed in Section

8 and conclusions are drawn in Section 9.

Chapter 2

System Model

In the sequel, scalars are in italic, MATRICES are in bold, I is the identity matrix, 0

is the zero matrix, E· is the expectation, ℜ· is the real part, ℑ· is the imaginary

part, ·∗ is the conjugate, ·T is the transpose, ·H is the Hermitian transpose, rank(·) is

the rank, tr(·) is the trace, det(·) is the determinant, det′(·) is the product of non-zero

eigenvalues, diag(·) is the diagonal matrix, vec(·) is the vectorization operation [20,21], ⊗

is the Kronecker product [26], CN (M,Σ) denotes a complex Gaussian random variable

with mean M and covariance Σ, the covariance of a zero mean matrix M is defined as

cov (M) = Evec(M)vec(M)H

[22], a r× c matrix M is said to be unitary if MHM = I,

a r × c space-time block code (STBC) X is said to be unitary if CHC = rI, ML1 is

the sequence of matrices [M(1), . . . ,M(L)], ML1 N is [M(1)N, . . . ,M(L)N], and NML

1 is

[NM(1), . . . ,NM(L)].

In Rayleigh fading, the theoretical capacity of a MIMO system with M transmit an-

tennas and N receive antennas grows linearly with min(M,N) [1, 27, 28]. Hence, MIMO

9

CHAPTER 2. SYSTEM MODEL 10

Pulse Shaping

Pulse Shaping

Thermal Noise

Thermal Noise

Space Time Block

Coding

Data Bits

Fading Channel

Matched Filtering

Matched Filtering

Data Detection

Channel Estimation

Detected Bits

Symbol-rate Sampling

Symbol-rate Sampling

Figure 2.1: System diagram of the MIMO joint channel estimation and data detection in

fading channels.

systems are gaining popularity, over SISO systems, among high data rate wireless appli-

cations. The MIMO system considered in this report is shown in Fig. 2.1.

For slow flat Rayleigh fading, matched filtering and symbol-rate sampling are per-

formed to obtain sufficient statistics, and the observation of the n-th receive antenna at

the k-th symbol duration, yn(k), can be modeled as follows [15, 29]:

yn(k) =

M∑

m=1

xm(k)hm,n(k) + vn(k), ∀n = 1, . . . , N (2.1)

where M is the number of transmit antennas, N is the number of receive antennas, xm(k) is

the transmitted symbol from them-th antenna, hm,n(k) ∼ CN (0, 1) is the fading coefficient

from the m-th transmit antenna to the n-th receive antenna, and vn(k) ∼ CN (0, r) is the

additive white Gaussian noise (AWGN). Instead of assuming that the fading is temporally


flat, this report considers a more realistic scenario: the MN fading coefficients, hm,n(k),

are temporally correlated but spatially independent and identically distributed (IID), i.e.

Eha,b(k)h∗c,d(k + τ) = δ(a− c)δ(b− d)J0(2πτfDTs), ∀1 ≤ a, c ≤ M and ∀1 ≤ b, d ≤ N ,

where δ(k) =

1, k = 0

0, k 6= 0

and fDTs is the normalized fading rate [19]. The assumption

of spatial independence holds when the antennas are sufficiently spaced [30]. Furthermore,

the AWGN samples are also IID, i.e. E vb(k)v∗d(k + τ) = δ(b−d)δ(τ)r. For convenience,

it is assumed that E|xm(k)|2 = 1; hence, the signal-to-noise ratio (SNR) is ρ = M/r.

For each receive antenna, the observations over T symbol durations in (2.1) can be

collected as follows:

Yn(l) =

M∑

m=1

Xm(l)Hm,n(l)+Vn(l) = [X1(l) . . .XM(l)] [H1,n(l) . . .HM,n(l)]T+Vn(l) (2.2)

where Yn(l) = [yn((l + 1)T − 1) . . . yn(lT )]T, Xm(l) = diag ([xm((l + 1)T − 1) . . . xm(lT )]),

Hm,n(l) = [hm,n((l + 1)T − 1) . . . hm,n(lT )]T, and Vn(l) = [vn((l + 1)T − 1) . . . vn(lT )]T.

Then, the observations over all N receive antennas in (2.2) can be collected as follows:

[Y1(l) . . .YN(l)]︸︷︷︸

Y(l)

= [X1(l) . . .XM(l)]︸︷︷︸

X(l)

H1,1(l) . . . H1,N(l)

.... . .

...

HM,1(l) . . . HM,N(l)

︸︷︷︸

H(l)

+ [V1(l) . . .VN(l)]︸︷︷︸

V(l)

Y(l) = X(l)H(l) + V(l) (2.3)

Remark—Fading model: The fading model in the matrix observation equation (2.3)

corresponds to the continuous fading model, not the quasi-static or block fading model [31].

Quasi-static or block fading model assumes that each fading coefficient is constant over


the duration of each block of T symbols and the adjacent blocks experience uncorrelated

fading [8, 11]. The block fading model can be used when T < Tc where Tc is the channel

coherence time. However, in a realistic mobile environment, the fading process changes

gradually without piece-wise jumps and the continuous fading model is more accurate

[14, 19]. The definitions of the matrices in (2.3) for both quasi-static and continuous

fading models are compared in Table 2.1. As seen in Table 2.1, the quasi-static fading

model is simply a special case of the continuous fading model when hm,n((l + 1)T − 1) =

hm,n((l + 1)T − 2) = . . . = hm,n(lT + 1) = hm,n(lT ) ≡ hm,n, ∀m,n. Thus, the continuous

fading model generalizes the quasi-static fading model by allowing gradual changes and

temporal correlation of the fading coefficients. Because the continuous fading model is

more realistic, accurate and general, it is considered in this report.

Given the MIMO system model in (2.3), three types of receivers can be employed

based on the availability of the channel state information (CSI) at the receiver: coherent

(perfect CSI), differential (no CSI), and estimation-assisted (estimated CSI). Perfect CSI

is unavailable at the receiver in realistic applications. In the absence of CSI, differential

demodulation drastically reduces the capacity and the detection performance in terms

of probability of error [1, 14, 18, 27]. The capacity and the detection performance of an

estimation-assisted detector can approach those of the coherent receiver, especially when

the estimation error is small [17–19,27]. Therefore, the estimation-assisted receiver in Fig.

2.1 is considered in this report.


Table 2.1: Definitions of the matrices in the observation equation for both continuous

fading model and quasi-static fading model.

Quasi-static Fading Model Continuous Fading Model

Y(l)

y1((l + 1)T − 1) . . . yN ((l + 1)T − 1)

.... . .

...

y1(lT ) . . . yN (lT )

y1((l + 1)T − 1) . . . yN ((l + 1)T − 1)

.... . .

...

y1(lT ) . . . yN (lT )

X(l)

x1((l + 1)T − 1) . . . xM ((l + 1)T − 1)

.... . .

...

x1(lT ) . . . xM (lT )

diag

x1((l + 1)T − 1)

...

x1(lT )

· · ·diag

xM ((l + 1)T − 1)

...

xM (lT )

H(l)

h1,1 . . . h1,N

..

.. . .

..

.

hM,1 . . . hM,N

h1,1((l + 1)T − 1) . . . h1,N ((l + 1)T − 1)

.

... . .

.

..

h1,1(lT ) . . . h1,N (lT )

..

....

hM,1((l + 1)T − 1) . . . hM,N ((l + 1)T − 1)

.... . .

...

hM,1(lT ) . . . hM,N (lT )

V(l)

v1((l + 1)T − 1) . . . vN ((l + 1)T − 1)

.... . .

...

v1(lT ) . . . vN (lT )

v1((l + 1)T − 1) . . . vN ((l + 1)T − 1)

.... . .

...

v1(lT ) . . . vN (lT )

Chapter 3

Proposed Estimation-Assisted

Receiver

3.1 Channel Estimation

From (2.3), because the fading coefficients are circularly symmetric Gaussian distributed

and the thermal noise is AWGN, a second-order linear filter is needed. Among the various

second-order linear filters, a recursive adaptive version is considered because (a) a recursive

filter can be used in a decision-directed mode; (b) an adaptive filter can effectively track

the temporal variation of the fading coefficients; (c) a more complex batch filter, such as

the Wiener filter, is not really necessary because the discrete-time system model in (2.1)

is free of intersymbol interference (ISI) [17, 19, 32]. The Kalman filter (KF), a recursive

adaptive second-order linear estimator, is used in this report because it is optimal in

the linear minimum mean square error (MMSE) sense [17, 19, 32, 33]. To use the KF, a

14

CHAPTER 3. PROPOSED ESTIMATION-ASSISTED RECEIVER 15

state equation, where the temporal variation of the fading is approximated by a rational

hypermodel, is needed [17, 19]. Among the various hypermodels, the following second-

order autoregressive (AR-2) model is used in this report because it offers excellent channel

tracking performance for a reasonable complexity [17, 32–35]:

hm,n(k) = −a1hm,n(k − 1) − a2hm,n(k − 2) + wm,n(k) (3.1)

where wm,n(k) ∼ CN (0, β), Ewab(k)w∗cd(k+τ) = δ(a−c)δ(b−d)δ(τ)β, a1 = −2rd cos(

√2πfDTs),

a2 = r2d, rd = 1 − 0.2πfDTs, and β =

[(1+a2)2−a21](1−a2)

1+a2.

The fading coefficients in (3.1) over T symbol durations from them-th transmit antenna

to the n-th receive antenna can be collected as follows:

Hm,n(l) = AHm,n(l − 1) + Wm,n(l) (3.2)

For examples, the specifics of the matrices A and Wm,n(l) for T = 2 and T = 4 are given

in Table 3.1.

Then, the fading coefficients over all M transmit antennas and N receive antennas in

(3.2) can be collected as follows:

H1,1(l) . . . H1,N(l)

.... . .

...

HM,1(l) . . . HM,N(l)

︸︷︷︸

H(l)

=(

I ⊗ A)

︸︷︷︸

A

H(l − 1) +

W1,1(l) . . . W1,N(l)

.... . .

...

WM,1(l) . . . WM,N(l)

︸︷︷︸

W(l)

H(l) = AH(l − 1) + W(l) (3.3)

where A is the state transition matrix, and W(l) ∼ CN (0,Q) is the driving noise.

Remark—Time varying mobile environment: It is important to note that (3.3) is con-

structed using the statistical knowledge of the channels, i.e. fDTs, which can vary in a


Table 3.1: Specifics of state equation matrices for T = 2 and T = 4.

T = 2 T = 4

A

a21 − a2 a1a2

−a1 −a2

a41 − 3a2

1a2 + a22 a3

1a2 − 2a1a22 0 0

−a31 + 2a1a2 −a2

1a2 + a22 0 0

a21 − a2 a1a2 0 0

−a1 −a2 0 0

Wm,n(l)

wm,n(k + 1) − a1wm,n(k)

wm,n(k)

wm,n(k + 3) − a1wm,n(k + 2) + (a21 − a2) × . . .

wm,n(k + 1) + (−a31 + 2a1a2)wm,n(k)

wm,n(k + 2) − a1wm,n(k + 1) + (a21 − a2)wm,n(k)

wm,n(k + 1) − a1wm,n(k)

wm,n(k)


realistic mobile environment. Using the technique in [36], fDTs can be estimated accu-

rately and periodically. Thus, the matrices A and Q in (3.3) can be easily updated when

needed, and the resulting state-space model can adapt to a time-varying mobile environ-

ment.

Remark—Matrix nature of the state-space model: Due to the MIMO nature, the system

consists of temporal and spatial dimensions. Thus, the matrix state-space model in (2.3)

and (3.3) most naturally and intutitively characterizes the system. Other models have been

used in the past. Liu et. al. and Komninakis et. al. employ matrices in the observation

equation and the detector, but vectors in the state equation and the estimator [17, 19].

Furthermore, Liu et. al. allows the observation to be characterized in a matrix equation

simpler than (2.3) by modelling the state equation with a first-order autoregressive (AR-

1) model [19]. Naguib et. al., Zhu et. al., and Baccarelli and Biagi employ a simpler

matrix model by assuming the fading to be quasi-static [18, 30, 37]. Guey et. al. and

Giese and Skoglund avoid the spatial dimension in the observation by employing only

one receive antenna [3, 38]. In the case of more than one antenna, Giese and Skoglund

employ a vectorized model by stacking the columns of the observation matrix and the state

matrix [38]. Zhu and Murch avoid the temporal dimension by using vector modulation

instead of matrix modulation, such as space-time block code [39].

The matrix state-space model in (2.3) and (3.3) is used in this report because (a) it

naturally and intuitively represents the temporal and spatial dimensions of the system and

is a more insightful formulation than the equivalent vectorized model [20,21]; (b) it offers

a consistent state-space model, and hence a consistent CE/DD scheme, where matrices


are used in both the observation and the state equations; (c) it poses no restriction on the

number of receive antennas; (d) although the state-space model in this report is based on

the AR-2 model, the formulation can be done easily using any hypermodels; (e) it employs

the more realistic and general continuous fading model, not the quasi-static fading model;

(f) it employs matrix modulation, which generalizes vector modulation, and offers better

detection performance when STBC is used [2].

However, the regular KF only works with scalar or vector state-space model. Therefore,

a matrix version of the KF is needed. This is done by first vectorizing the matrix state-

space model. Then, the ordinary KF recursion is applied to the vectorized state-space

model. Finally, the applied KF recursion is converted back to the matrix form, called the

matrix KF recursion [15, 20, 21] (Appendix A):

[

H(l + 1|l),P(l + 1|l)]

= MKF(

H(l|l − 1),P(l|l− 1),X(l),Y(l))

(3.4)

The initial condition and its corresponding covariance are H(0|0) and P(0|0). In the

absense of a priori information, they are assumed to match the statistical properties of

the spatially IID Rayleigh fading channel, i.e. H(0|0) = 0 and P(0|0) = I.

Remark—Matrix KF recursion: The use of the matrix state-space model dictates that

a matrix estimator is needed. The matrix KF recursion, along with the vectorization

technique developed in [20, 21] and the notation of covariance of a matrix defined in [22],

is applied for the first time in the area of communications, to estimate fading channels

[20–22]. It is important to note that the vectorization and conversion processes do not

lose any information or impose any restriction on the matrix state-space model [20, 21].


3.2 Data Detection

Given the channel state prediction H(l|l − 1) and its covariance P(l|l − 1) calculated by

the matrix KF, the maximum likelihood (ML) detector is then given as follows (Appendix

B):

X(l) = arg minΞ∈ΩX

ZP

(

Y(l),Ξ, H(l|l − 1),P(l|l− 1))

(3.5)

where ZP

(

Y,X, H,P)

=

vec(

Y −XH)H (

(I ⊗X)P (I ⊗X)H + R)−1

vec(

Y − XH)

is the normalized innovation calculated by the KF recursion (A.14)-(A.18), and ΩX is the

set of possible matrices for X(l). Thus, the ML detector is hereafter called the matrix

normalized innovations-based detector.

Remark—Matrix normalized-innovations: The metric used in the ML detector in (3.5)

is the Mahalanobis distance-square, which is the Euclidean distanace-square weighted

adaptively by the innovations covariance provided by the matrix KF recursively [40].

Hence, the matrix normalized innovations-based detector generalizes the usual Euclidean

distance-square-based detector. The innovation covariance matrix represents the effec-

tive SNR at the receiver, which consists of the covariances of the thermal noise and the

estimation error.

Remark—Operation of the CE/DD: For estimation-assisted detection, the matrix KF

operates in a decision-directed mode, so X(l) in (3.4) is replaced by X(l) provided by the

detector. The matrix KF provides the ML detector with H(l|l − 1) for the detection of

X(l), and the detector provides the matrix KF with X(l) for the estimation of H(l+ 1|l).

The CE/DD recursively estimates the channel and detects the transmitted data for each


l, and the channel estimates are always up-to-date, unlike the CE/DD schemes in [3, 18]

where the channel estimates are obtained during the training phase prior to detection.

Chapter 4

Probability of Detection Error

For the matrix state-space model in (2.3) and (3.3) and the matrix CE/DD scheme in

(3.4) and (3.5), given H(l|l− 1) and P(l|l− 1), the bit error rate (BER) is approximated

by the following union bound (UB) [3, 39] (Appendix C):

Pe,UB[Estimation] ≤∑

α6=β

Pα→β[Estimation]

(# error bits in the error event

# bits per code matrix

)

(4.1)

where Pα→β[Estimation] is the pairwise error probability (PEP) assuming that Xα is sent

but Xβ is erroneously detected

Pα→β[Estimation] =1

(1 − λαβ+

λαβ−

)2NT−1

NT−1∑

l=0

(

2NT − 1

l

)(

−λαβ+

λαβ−

)l

, (4.2)

λαβ+ and λαβ− are respectively the positive and negative eigenvalues of

cov(Υ)Θα,β =

Xαcov(H(l))XHα + R Xαcov

(

H(l|l − 1))

cov(

H(l|l − 1))H

XHα cov

(

H(l|l − 1))

·

P−1

Z,β− P−1

Z,αP−1

Z,αXα − P−1

Z,βXβ

XHαP−1

Z,α− XH

β P−1

Z,βXH

β P−1

Z,βXβ − XH

αP−1

Z,αXα

(4.3)

21

CHAPTER 4. PROBABILITY OF DETECTION ERROR 22

PZ,α = XαP(l|l−1)XH

α +R, PZ,β = XβP(l|l−1)XH

β +R, Xα = I⊗Xα and Xβ = I⊗Xβ .

The asymptotic bound (AB) of (4.2) is

Pα→β,AB[Estimation] = limρ→∞

Pα→β[Estimation] =

(

2NT − 1

NT − 1

)(

−λαβ+

λαβ−

)−NT

(4.4)

Remark—Complete and consistent package of matrix state-space model, CE/DD and

probability of error equation: It should be noted that the matrix state-space model, matrix

KF, matrix normalized innovations-based detector, and the probability of error equation

all fit naturally together. They consistently employ the same matrix quantities, which

intuitively and naturally characterize the time and space dimensions of the MIMO system.

Furthermore, there is no need to translate any quantity between the matrix form and the

vector form.

Remark—Parameters affecting the detection performance: Equations (4.1)–(4.4) are

too complex for a general quantitative analysis. Thus, qualitative discussion is offered

here and quantitative analysis for specific STBC examples are given in Sections 5 amd

7. From (4.3), the detection performance is linked, either explicitly or implicitly, to the

following parameters [15]:

1. Properties of the space-time block code: The matrix in (4.3) contains vari-

ous forms of weighted distances between the transmitted data matrix Xα and the

detected data matrix Xβ. Hence, code properties affect the detection performance.

More discussion follows in Sections 5–7.

2. The fading channel model: The innovations covariance PZ(l|l − 1) is calculated

recursively by the matrix KF according to the matrix state-space model. Hence, we

CHAPTER 4. PROBABILITY OF DETECTION ERROR 23

expect the detection performance to be affected by the model parameters, such as

R, A, Q, and the assumed initial conditions, H(0|0) and P(0|0). Thus, unknown or

mismatched fading rate, SNR, initial conditions, etc. degrade detection performance.

3. The channel estimation performance: The channel prediction covariance, which

is part of the innovations covariance, indicates the performance of the matrix KF.

Thus, poor estimation performance degrades the detection performance.

4. The detection performance of past codewords: The detection performance

equation is derived assuming that H(l|l− 1) is accurate. If the previous codeword is

detected incorrectly, then H(l|l − 1) becomes inaccurate and an error is more likely

to occur with the detection of the current codeword. This causes error propagation.

Therefore, past detection performance will affect the current detection performance.

Chapter 5

Space-Time Block Codes

As discussed previously, properties of the transmitted data matrix and the detected data

matrix are expected to affect the detection performance. In this section, these properties

and their effects are examined. In an uncoded system where independent symbols are

sent via different transmit antennas, deep fading might render the transmitted symbols

undetectable. The use of space-time codes where the symbols are coded and sent across

different transmit antennas, exploits transmit diversity and allows the system to mitigate

the effects of deep fading [2]. Two types of space-time codes are available: block code and

trellis code, and the former is the focus of this report.

A space-time block code (STBC) can be expressed as a set of T ×M complex matrices

ΩC =C ∈ CT × CM

, where each matrix contains the coded symbols xm(k)’s to be sent

over M transmit antennas and T symbol periods. To accomodate continuous fading in the

discrete-time system model in (2.3), the codeword C as a compact T ×M matrix needs

to be mapped uniquely into the transmit matrix X(l) as a sparse T ×MT matrix. Since

24

CHAPTER 5. SPACE-TIME BLOCK CODES 25

the performance of the CE/DD scheme should lie between that of the coherent receiver

and that of the differential demodulator, the coherent and differential STBCs serve as the

basis for the discussion of code design.

5.1 Coherent Space-Time Block Codes

Assuming that the fading is quasi-static and the CSI is known at the receiver, the Chernoff

bound (CB) for the pairwise error probability (PEP) is

Pα→β,CB[coherent] ≤ det(I + ρ∆H

αβ∆αβ

)−N

≈

det′(ρ∆H

αβ∆αβ

)−N, high ρ

1 + ρ tr(∆H

αβ∆αβ

)− 1

2ρ2[

tr((

∆Hαβ∆αβ

)2)

−(tr(∆H

αβ∆αβ

))2]−N

, low ρ

(5.1)

where ∆αβ = Cα − Cβ; Cα,Cβ ∈ ΩC; and Cα 6= Cβ [1]. In the presence of CSI, STBCs

should be designed to minimize the probability of error, and the rank, determinant and

trace criteria have been derived from (5.1) [1, 3–5, 41]. (Table 8.1).

A linear STBC, the most common type of STBCs, can be expressed as a linear function

of Q complex symbols, cq = z2q−1 + jz2qQq=1, drawn from some constellation:

C(

cqQq=1

)

=

Q∑

q=1

ℜcqB2q−1 + ℑcqB2q =

Q∑

q=1

z2q−1B2q−1 + z2qB2q (5.2)

where Bq2Qq=1 is a set of 2Q constant T ×M complex matrices, called a basis. An impor-

tant characteristic of a linear STBC is that the design criteria, expressed in code differences,

are related directly to the form of the code: ∆αβ = Cα−Cβ = C(

αq − βqQq=1

)

[1]. Four

additional criteria for designing linear STBCs can be derived from (5.1) and (5.2) [1, 6]

(Table 8.1).


If the STBC is linear and unitary (complex orthogonal), i.e. CHC =∑Q

q=1 |cq|2 I, then

the basis must be complex Radon-Hurwitz (RH) orthogonal: BHp Bq +BH

q Bp = 2δ(p− q)I,

where the RH equation can be solved using Clifford algebra [1, 7, 42]. It has been shown

that, with known CSI, linear unitary STBCs is optimal in terms of minimizing (5.1) [1,6,7].

Alamouti’s code is a well-known example of a linear unitary STBC for M = T = 2 [8]:

Cunitary

(

cq2q=1

)

=

c1 c2

−c∗2 c∗1

(5.3)

Remark—Square versus rectangular codes: The square codes in the literature, including

the example in (5.3), were developed using the quasi-static model. However, the system

model in (2.3) assumes the continuous fading model. When the fading rate is slow, the

fading model can be approximated by the quasi-static model. Assuming that the fading

is quasi-static, to minimize (5.1), the rank criterion states that the STBC must achieve

maximal diversity, i.e. rank (∆αβ), ∀Cα 6= Cβ , must be maximized [1]. Since rank (∆αβ) ≤

min(T,M) and the parameter M is usually fixed prior to code design, only codes with

T ≥ M are considered. A STBC is square if T = M , and a STBC is rectangular if

T > M . Almost all coherent STBCs, including the example in (5.3), are square because

no transmit diversity can be gained by choosing T > M . diversity for continuous fading

channels.] Hence, only square STBCs are discussed in this section. And square STBCs,

including the one in (5.3), used as a benchmark in this report.


5.2 Differential Space-Time Block Codes

When the CSI is ignored, differential demodulation and differential STBCs are used. To

use differential demodulation, the channels need to be approximately constant for 2T

symbol intervals, i.e. 2T < Tc. A differential STBC is a unitary group STBC, and it is

obtained if a set of unitary matrices forms a group, either cyclic or dicyclic [10, 12, 13].

Due to the nature of differential encoding, group codes are square and non-linear. Unitary

group STBCs can also be used for coherent detection also [11]. The absence of CSI,

however, increases the detection error probability by a factor ofP

α→β,CB[Differential]

Pα→β,CB[Coherent]

=

1

2(4

ρT )MN

1

2(2

ρT )MN = 2MN [14]. Since differential demodulation assumes that the fading remains

constant for a period of 2T symbol intervals, which is only valid in very slow fading, it is

likely thatP

α→β,CB[Differential]

Pα→β,CB[Coherent]> 2MN in general.

5.3 Performance of Square Unitary Space-Time Block

Codes

Since (4.1) is rather complicated, simplification is needed to gain better insights on how the

square unitary codes discussed thus far affect the detection performance. Approximating

the fading with the quasi-static fading model, the ratio −λαβ+

λαβ−

in (4.2) and (4.4) is simplied

to the following for square unitary codes (Appendix D):

−λαβ+

λαβ−

= 1 +T1

2MN+

√(

T1

2MN

)2

+T1

MN(5.4)

where T1 = tr(cov(Υ)Θα,β) = (1 − σ2) Nr+σ2T

d2αβ and d2

αβ = tr[

(Xα − Xβ) (Xα − Xβ)H]


Remark—Parameters affecting the performance of square unitary codes: From (4.1)–

(4.2), the detection performance improves when (5.4) increases. From (5.4), −λαβ+

λαβ−

in-

creases when increases. Therefore, d2αβ, or alternatively d2

min = min(d2αβ), should be maxi-

mized to minimize (4.1). The distance property of the square unitary STBC, d2min, affects

the detection performance directly and describes how closely the different codewords of

a STBC are packed on the multi-dimensional sphere. This is similar to the d2min in SISO

systems where it describes how closely the different signal points of a signal constellation

are packed on the complex plane [15, 43]. The farther apart the different codewords are

packed, the larger d2min and d2

αβ are, and the better the detection performance is.

The effect of M and N on the detection performance through (5.4) is now examined.

Since T1 contains N in its numerator, changing N does not change (5.4). However, the

effect ofM is not as obvious. When M increases, the size of the space-time sphere increases

and it becomes easier to pack the codewords to obtain a larger d2min. Therefore, when M

increases, T1 generally increases. However, it is difficult to calculate the rate in which T1

increases with M because it depends on the signal constellation the symbols are drawn

from, the number of codewords in the STBC, and the exact formulation of the STBC.

The effect of the last factor is especially difficult to quantify because the formulation of

the STBC can vary quite significantly for different M ’s. However, as M increases, the

code rate of a square linear unitary code decreases and there is proportionally more room

for the codewords to spread out in the space-time sphere [1]. Hence, (5.4) is expected to

increase as M increases.

Experimentally, when −λαβ+

λαβ−

≥ 4, as NT increases,(

−λαβ+

λαβ−

)−NT

decreases faster than


(

2NT − 1

NT − 1

)

increases and (4.4) decreases exponentially and monotonically. Hence, assum-

ing that −λαβ+

λαβ−

≥ 4, increasing the number of antennas, M = T or N , improves detection

performance. However, the marginal improvement with each additional antenna decreases

monotonically to zero. That is, after a certain number of antennas, each additional an-

tenna contributes very little improvement in detection performance.

Finally, as shown in (5.4), the factors, (1 − σ2) and 1r+σ2T

, affect T1 and hence the

detection performance. From (5.4), it is deducted that square unitary codes are most

suitable for coherernt detection when σ2 = 0 or when estimation is near perfect σ2 ≈ 0.

When σ2 is not negligible, the factors, (1−σ2) and 1r+σ2T

, reduces T1, especially when T is

large. Hence, even a small estimation error can degrade the detection performance when

square unitary codes are used.

Remark—Euclidean distance-square property of the CE/DD system and the detection

performance of the square unitary codes: From (5.4), the detection performance for a

square unitary code is related to the Euclidean distance square property of the code. This

agrees with the CE/DD scheme because the matrix KF minimizes the mean Euclidean

distance-square between the actual channel states and the estimated channel state and

the matrix normalized innovations-based detector minimizes the Mahalanobis distance-

square, which is the weighted Euclidean distance-square, of the innovations sequence.

Chapter 6

Performance Degradation in a

CE/DD System

This section shows that the symmetry of a STBC causes detection and estimation error

floors.

6.1 Symmetry of Space-Time Block Codes

A STBC ΩC is said to be symmetric if a unitary rotation leaves the STBC invariant [23]:

∃U 6= I, UHU = I∀C ∈ ΩC : CUH ∈ ΩC

(6.1)

Given any linear unitary STBC, if the symbols are drawn from a rotationally invariance

constellation, then there always exists a unitary tranform that leaves the STBC invariant

[15,43]. Therefore, all linear unitary STBCs with rotationally invariant constellations are

symmetric. Furthermore, group codes are, by definition, symmetric. So, the STBCs in

30

CHAPTER 6. PERFORMANCE DEGRADATION IN A CE/DD SYSTEM 31

Section 5 are all symmetric. Even though only square codes have been discussed so far, a

symmetric code can also be rectangular.

6.2 Unitary Transform Property of CE/DD

Similar to the behavior observed in the SISO case examined in [43], given that a sym-

metric STBC is used (i.e. X(l),X(l)UH ∈ ΩX), the matrix KF in (3.4) preserves unitary

transforms and introduces a counter-unitary transform in the channel prediction H(l+1|l)

in response to a unitary transform in the transmit matrix X(l):1

Lemma 1: Given some square unitary transform U where UHU = I

[

UH(l + 1|l), (I ⊗ U)P(l + 1|l) (I ⊗ U)H]

= . . .

MKF(

UH(l|l − 1), (I ⊗ U)P(l|l − 1) (I ⊗ U)H ,X(l)UH,Y(l))

(6.2)

Similar to the behavior observed in the SISO case examined in [43], given that a sym-

metric STBC is used, the matrix normalized innovations-based detector in (3.5) preserves

unitary transforms and introduces a counter-unitary transform in the detected matrix X(l)

in response to a unitary transform in the channel prediction H(l|l − 1):

Lemma 2: Given Y(l), the pair of detected matrix and channel prediction(

X(l), H(l|l − 1))

and its unitary transformed counterpart(

X(l)UH,UH(l|l − 1))

both minimize (3.5).

When any symmetric STBC is used (square or rectangular, linear or non-linear), Propo-

sition 3 shows that the CE/DD system in (3.4) and (3.5) propagates unitary rotation and

1The proofs for all lemmas, propositions and corollaries are found in Appendix E.


code symmetry recursively, and generates ambiguity in detected data and channel esti-

mate.

Proposition 3: Let XL1 =

[

X(1), . . . , X(L)]

and HL1 =

[

H(1|0), . . . , H(L|L− 1)]

. For

the CE/DD scheme in (3.4) and (3.5), given the observation sequence YL1 = [Y(1), . . . ,Y(L)],

the set of detected code sequence, channel estimate sequence, and initial condition(

XL1 , H

L1 , H(0|0)

)

and its unitary transformed counterpart(

XL1 U

H,UHL1 ,UH(0|0)

)

both minimize (3.5).

Corollary 4: In most realistic applications, the actual initial condition and its co-

variance are not available, and they are assumed to match the statistical properties of

the channels. For the CE/DD system in (3.4) and (3.5), given YL1 , if H(0|0) = 0 and

P(0|0) = I, then both XL1 and XL

1 UH are equally likely and cannot be differentiated.

6.3 Isometry in MIMO System

Since the discrete-time system model in (2.3) is assumed to be ISI free, the normalized-

innovations in (3.5) is a white process, and the path metric is simply the sum of the branch

metrics:

p(XL

1

)=

L∑

l=1

p(

X(l))

=L∑

l=1

ZP

(

Y(l), X(l), H(l|l − 1),P(l|l− 1))

Two sequences of block codes are isometric if their path metrics are identical.

From Corollary 4, XL1 and XL

1 UH are isometric because

p(

XL1

)

=

L∑

l=1

ZP

(

Y(l), X(l), H(l|l − 1),P(l|l− 1))

=L∑

l=1

ZP

(

Y(l), X(l)UH,UH(l|l − 1),UP(l|l − 1)UH)

= p(

XL1 U

H)

(6.3)


Therefore, symmetric STBCs (6.1) cause isometry in CE/DD. In Section 6.1, it has been

shown that a large number of STBCs used for various receivers are symmetric; hence,

isometry affects a wide variety of applications. From Corollary 4, isometry causes estima-

tion and detection ambiguity. Selection of the wrong isometric sequence results in detec-

tion error floor (X(l) 6= X(l)UH) and increases the estimation error (H(l) − UH(l|l) ≫

H(l) − H(l|l)) [15, 43, 44].

Chapter 7

Space-Time Block Code Design

7.1 Asymmetric Space-Time Block Codes

Since isometry is induced by the symmetry of the STBCs, it is natural to consider a

solution based on asymmetric STBC design. To ensure that XL1 and XL

1 UH are not

isometric, it is sufficient that the sub-sequences [X(a),X(b)] and[X(a)UH,X(b)UH

], for

some 1 ≤ a < b ≤ L, are not isometric [15, 45]:

∀(X(a) ∈ ΩX(a), X(b) ∈ ΩX(b)

), ∄(UH 6= I

) X(a)UH ∈ ΩX(a),X(b)UH ∈ ΩX(b)

(7.1)

This can be accomplished by designing ΩX(b) to be asymmetric according to (6.1), which

can be achieved by using a code structure that is variant to unitary rotation or by using

an asymmetric signal constellation (ASC) that is rotationally variant [15, 43]. Training,

which breaks isometry, is equivalent to having only one element in the set ΩX(b). Hence,

training can be seen as a special case of asymmetric STBC.

Two easy ways to design a pair of P -ary symmetric STBC ΩC(a) and P -ary asymmetric

34

CHAPTER 7. SPACE-TIME BLOCK CODE DESIGN 35

STBC ΩC(b), so that the corresponding transmit matrices X(a) and X(b) together break

isometry, as indicated in (7.1), are described in Algorithms 1 and 2.

Algorithm 1: Retain the linear unitary code structure but employ ASC.

1. Select a linear unitary code structure.

2. For ΩC(a), employ regular Q√P -PSK for each of the Q uncoded symbols.

3. For ΩC(b), employ asymmetric Q√P -PSK for each of the Q uncoded symbols [15,43].

For example, using the Alamouti’s code structure in (5.3), BPSK 1,−1, and asymmetric

BPSK 1, j [15, 43], the following pair of STBCs ΩC(a) and ΩC(b) break isometry as

described in (7.1):

ΩC(a) =

1 −1

1 1

,

1 1

−1 1

,

−1 −1

1 −1

,

−1 1

−1 −1

ΩC(b) =

1 −1

1 1

,

1 j

j 1

,

j −1

1 −j

,

j j

j −j

Remark—Trade-off with Algorithm 1: Both the symmetric and asymmetric STBCs

contain the same number of codewords. However, due to the employment of ASC, the code-

words of the asymmetric STBC are not uniformly spread out in the space-time sphere.

Thus, the asymmetric STBC has a smaller d2min than its symmetric counterpart. The

smaller d2min contributes unfavorably to the detection performance of the asymmetric code-

word. However, when isometry is broken, the sequence of codewords and channel estimates

are uniquely identified. This means that the performance improvement in channel esti-

mation and the detection of past codewords contribute favorably to the overall detection


performance of the entire sequence. Since asymmetric STBC is only used once in each

transmit sequence, as long as d2min is not too small to break isometry, the use of asymmetric

STBCs improves overall detection performance.

Algorithm 2: Unitary group code can be seen as generalized PSK in the space-time

domain [46]. Hence, Algorithm 1 in [43] can be modified to construct a pair of P -ary

symmetric STBC and P -ary asymmetric STBC from a P 2-ary unitary group code.

1. Select a P 2-ary unitary group code, either cyclic or dicyclic.

2. Arrange the codewords clockwise in a circular sequence where each codeword is the

UH rotated version of the previous codeword.

3. Label the codewords clockwise sequentially with 1, . . . , P .

4. Select a set of codewords with the same label to obtain the P -ary symmetric STBC.

5. Select one codeword from each set of codewords with the same label to obtain the

P -ary asymmetric STBC.

For example, Fig. 7.1 illustrates how a pair of symmetric 2-ary STBC and asymmetric

2-ary STBC that break isometry can be constructed from a 4-ary cyclic group code.

Remark—Trade-off with Algorithm 2: Since the P -ary symmetric and asymmetric

STBCs are sub-sets of the P 2-ary group code, the code rate is reduced. This can be

compensated by using a group code with a higher code rate, but smaller d2min. Again, as

in Algorithm 1, the smaller d2min is usually not a problem.

Two algorithms to construct asymmetric STBCs, one linear and one non-linear, have

been proposed. Asymmetry breaks isometry and provides the ML detector with the correct


HU

(1)

(2)

(1) (1)

HU ( )3HU

− 11

11

−−−

11

11

−

=01

10HU

HU

− 11

11

−11

11

−−

−11

11

HU

−11

11

( )2HU

( )2HU −

11

11

−−

−11

11

Quadrature group code Asymmetric binary code Symmetric binary code

(1) (1)

(2) (2)

Figure 7.1: The construction of asymmetric space-time block code from a group code.

channel estimates H(l|l − 1) instead of the unitarily rotated version UH(l|l − 1). This

leads to the mitigation of detection error floor in (4.1).

Because the coherent and differential STBCs in Section 5 are all square, the examples

of asymmetric STBCs shown in this section are all square as well. However, asymmetric

STBCs can also be rectangular, and the analysis in this section apply to rectangular

STBCs as well.

7.2 Estimation Based Space-Time Block Codes

Thus far, Algorithms 1– 2 construct asymmetric STBCs from linear unitary STBCs and

non-linear group STBCs. They improve detection performance by breaking isometry to

mitigate the detection and estimation error floors. From (4.2), it is clear that estimation

error affects the detection performance, and a good STBC should perform well in various

levels of estimation error. Hence, further improvement of detection performance can be


obtained by designing a STBC that not only breaks isometry, but also adapts to the

different levels of estimation error.

7.2.1 Code Design Criteria

In general (i.e. for any STBCs, whether they are square or rectangular, linear or non-

linear, symmetric or asymmetric, unitary or not, etc.), to minimize (4.1), (4.2) must be

minimized, and thusλαβ+

−λαβ−

must be maximized due to the negative exponent in (4.4).

Since λαβ− is the negative eigenvalue, it is necessary that λαβ+ ≫ −λαβ−, which implies

λαβ+ + λαβ− ≫ 0. Therefore, it is necessary to maximize

MN (λαβ+ + λαβ−) = tr (cov(Υ)Θα,β) = T1

= tr[(

Xαcov(H(l))XHα + R

)(

P−1

Z,β− P−1

Z,α

)

+ Xαcov(

H(l|l − 1))(

XHαP−1

Z,α− XH

β P−1

Z,β

)]

+tr

[

cov(

H(l|l − 1))H

XHα

(

P−1

Z,αXα − P−1

Z,βXβ

)

+ cov(

H(l|l − 1))(

XHβ P−1

Z,βXβ − XH

αP−1

Z,αXα

)]

= tr[(

Xαcov(H(l))XHα + R

)(

P−1

Z,β− P−1

Z,α

)

+ Xαcov(

H(l|l − 1))

XHαP−1

Z,α

−Xαcov(

H(l|l − 1))

XHβ P−1

Z,β− Xβcov

(

H(l|l − 1))

XHαP−1

Z,β+ Xβcov

(

H(l|l − 1))

XHβ P−1

Z,β

]

(7.2)

To obtain insight into code design for the CE/DD scheme, (7.2) is simplified by assuming

that the fading channel is quasi-static. Since the channels are IID and Rayleigh (i.e.

zero-mean and unit variance), cov (H(l)) = I, cov(

H(l|l − 1))

= P(l|l − 1) = σ2I, and

cov(

H(l|l − 1))

= cov(H(l)) − cov(

H(l|l − 1))

= (1 − σ2)I. So, (7.2) becomes

tr[(

XαXHα + R

)(

P−1

Z,β− P−1

Z,α

)

+(1 − σ2

) (

XαXHαP−1

Z,α

−XαXHβ P−1

Z,β− XβX

HαP−1

Z,β+ XβX

Hβ P−1

Z,β

)]

(7.3)


Using the matrix inversion lemma and assuming that the STBC is unitary, (7.3) reduces

to

(1

r2 + rσ2T

)

r(1 − σ2)tr

[(

Xα − Xβ

)(

Xα − Xβ

)H]

+σ2

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

(7.4)

where, with the state-space model, P(l|l− 1) = σ2I can be calculated a priori in an open-

loop fashion by the matrix KF. Using the matrix identity ln(det(M)) = tr(ln(M)), and

the Taylor expansion ln(1 + x) ≈ x [1], it is easy to show that (7.4) resembles the KL

distance criterion in [47]. From (7.4), it is observed that there are two components: the

first part dominates when σ2 → 0 (coherent detection), and the second part dominates

when σ2 → 1 (differential demodulation). In the case of known CSI, i.e. σ2 = 0, (7.4)

becomes

1

rtr

[(

Xα − Xβ

)(

Xα − Xβ

)H]

(7.5)

which is the trace criterion (ds) for designing coherent STBCs [5]. This suggests that

linear unitary STBC should be used when CSI is known [1, 5]. Let λn be the eigenvalues

of the matrix

[(

Xα − Xβ

)(

Xα − Xβ

)H]

and d2min be the minimum of these eigenvalues,

then NTd2min ≤∑NT

n=1 λn. Hence, maximizing (7.5) is equivalent to maximizing d2min.

When the channel is ignored, i.e. σ2 = 1, (7.4) becomes

1

r2 + rT

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

(7.6)

In (7.6), XHαXβ resembles the differential demodulator which suggests that differential

demodulation and group code should be used when CSI is neither known nor estimated


[10, 12]. Let λn be the eigenvalues of the matrix

[(

XHαXβ

)(

XHαXβ

)H]

and δ2max be

the maximum of these eigenvalues, then∑NT

n=1 λn ≤ NTδ2max. So, maximizing (7.6) is

equivalent to minimizing δ2max.

From (7.5)–(7.6), one might propose the following scheme for CE/DD: employ group

code and differential demodulation at the beginning while the channel estimation error is

large, then switch to linear unitary code once the channel estimation error is small enough.

However, for a CE/DD scheme, 0 < σ2 < 1 because the channels are estimated. Moreover,

differential demodulation provides only the unitary rotational difference (which can be seen

as the space-time extension of the phase difference) between successive block codes, thus

the sets (XL1 , H

L1 ) and (XL

1 UH,UHL

1 ) remain isometric. So, without pilot training or

asymmetric STBCs, the channel estimates remain ambiguous, and the CE/DD scheme

cannot switch from differential demodulation to coherent detection. Therefore, switching

from differential demodulator and group STBCs to coherent detector and lineary unitary

STBCs are both ineffective and complicated. Instead, a single type of STBC and a single

receiver structure that perform well in various 0 < σ2 < 1 and break isometry should be

employed.

7.2.2 Self-Matching Space-Time Block Codes

A STBC that is both asymmetric and adaptive is now being designed, and it is henceforth

called the self-matching STBC. The size of the self-matching STBC is now considered.

Training has been used traditionally to reduce estimation error: pilot matrices are used

in [3], and codes with embedded training are used in [48]. Motivated by better detection


performance and higher capacity as suggested in [48], codes with embedded training are

considered. Furthermore, codes with embedded training mitigate isometry because they

are asymmetric by default. Hence, the self-matching STBC (T ×M) employs embedded

training, and it is divided into a training part (Tt ×M) and a information-bearing part

(Td ×M), where T = Tt + Td. To achieve maximal transmit diversity, Td ≥ M . Thus,

T > M and the self-matching STBC must be rectangular.

Since square linear unitary codes are optimal for coherent detection, they are adopted

for the information-bearing part of the self-matching code [1, 6, 7]. Thus, Td = M . Ac-

cording to [3,49], the training part should be a Tt ×M unitary matrix. If a Tt ×M matrix

is unitary, then all of its M column vectors are linearly independent of each other. In any

set of linearly independent vectors of length Tt, there can be at most Tt vectors. Therefore,

for the training part to be a unitary matrix, Tt ≥ M . In summary, Td = M and Tt ≥ M ,

so T ≥ 2M .

At the heart of the self-matching code is really the square information-bearing part. If

the information-bearing part is a square STBC that achieves MN diversity in quasi-static

fading channels, then it is stated in [41] that it also achieves MN diversity in continu-

ous fading channels. Furthermore, from [41], it is easy to show that if the information-

bearing part achieves MN diversity in continuous fading channels, then the rectangular

self-matching code also achieves MN diversity in continuous fading channels. Therefore,

even though the code design criterion in (7.4) is derived assuming the quasi-static fading

model. The use of (7.4) to design rectangular codes for continuous fading model does not

reduce achievable diversity, and is thus justified.


It is stated in [41] that if a square STBC achieves MN diversity in quasi-static fading

channels, then it also achieves MN diversity in continuous fading channels. At the heart

of the proposed rectangular self-matching code, there exists a square STBC information-

bearing part which achieves MN diversity. Therefore, it can be argued that even though

the design criterion in (7.4) is derived assuming the quasi-static fading model, the use

of (7.4) to design rectangular codes for continuous fading model does not reduce the

achievable diversity.

From (7.4), it is obvious that the self-matching code should adapt to the channel

estimation error. When the estimation error is negligible, the training part of the code

is not as important, and more energy should be allocated to the information-bearing

part. As the amount of channel knowledge decreases, training becomes more important,

and the code energy should be distributed appropriately between the training part and

the information-bearing part. The distribution of energy between the training part and

the information-bearing part can be controlled by a weighting factor. From (7.4), the

weighting factor should be P = σ2I. To minimize (4.2), (7.4) should be maximized by

emphasizing the maximization of d2min when P = σ2I is small and the minimization of δ2

max

when P = σ2I is large.

However, training reduces transmission rate. Since it has been proven in [43] that

training is the limiting case of asymmetry, the training part can be replaced by information-

bearing asymmetric code. Both embedded training or asymmetric code break isometry,

provide the detector with a better channel estimate and thus assist in minimizing (4.2).

A limited version of the rectangular self-matching code has been proposed heuristically


in [18]. The modulation scheme in [18] is inefficient: extra pilot training matrices are used

for channel estimation, and the embedded training property of its self-matching code is

never exploited. The embedded training property of the self-matching code in the present

report is explicitly stated in the design and effectively utilized. Moreover, the CE/DD

scheme in [18] is not practical: the channel estimation is a function of the pilot training

matrix which must precede the information-bearing codewords, and the ML detection is a

function of this previous channel estimation. Therefore, the channel estimates used in the

ML detection are always outdated, and the scheme can be used for very slow fading only.

The CE/DD scheme proposed in this report is derived based on the continuous fading

model, and the channel estimates used for detection are kept up-to-date recursively using

the matrix KF. Therefore, the CE/DD scheme in this report can be used for various fading

rates. Furthermore, the CE/DD scheme in this report provides an easy way to calculate

σ2 a priori for every l, but the scheme in [18] provides a static estimation of σ2 and does

not account for the fact that the theoretical performance of an adaptive channel estimator

improves over time as more information become available.

Using the design process discussed above, the following T × M rectangular unitary

self-matching (SM) code, which reduces (4.2) by adapting its d2min and δ2

max according to

σ2, results:

Cself-matching

(

bpPp=1 , cq

Qq=1

)

=

√u+ v

1 + σ2

[σ√u

(

U(

bpPp=1

))H∣∣∣∣

1√v

(

V(

cqQq=1

))H]H

(7.7)

where U(·) is an (T −M)×M asymmetric unitary code (i.e. unitary code that is variant

to unitary rotation) or an arbitrarily assigned constant unitary matrix (i.e. training), V(·)


is an M ×M unitary code, T ≥ 2M , bpPp=1 and cqQ

q=1 are independent information

symbols, u = tr(UH(·)U(·)) = (T −M)M , and v = tr(VH(·)V(·)) = M2.

Remark—Difference between the self-matching code in (7.7) and the one in [18]: The

self-matching STBC in (7.7) is different than the one in [18] because the training part

has been replaced by asymmetric STBC. The self-matching STBC in [18] uses a fixed U.

Therefore, even though it improves detection performance in CE/DD system, the data

rate is reduced by at least half. The self-matching STBC in (7.7) uses information-bearing

asymmetric STBC for U(·). Hence, it breaks isometry and improves detection performance

without having to reduce the code rate substantially.

Let d2v be the minimum eigenvalue of

[

(Vα − Vβ) (Vα − Vβ)H]

, and assuming that

either U(·) is known (training) or detected correctly by breaking isometry (asymmetry),

the d2min and δ2

max of the self-matching STBC in (7.7) are:

d2min =

T

M

(d2

v

1 + σ2

)

(7.8)

δ2max = T 2

(

1 − σ2

(1 + σ2)2

d2v

M

)

(7.9)

Thus, the properties of the self-matching code, d2min and δ2

max, reduce to functions of the

distance-square property of the square code V(·), d2v. From (7.8)–(7.9), the STBC in (7.7)

adjusts its TMd2

v ≥ d2min ≥ T

Md2

v

2≥ d2

v (since T ≥ 2M) and T 2 ≤ δ2max ≤ T 2

(

1 − 14

d2v

M

)

according to 0 ≤ σ2 ≤ 1. Since maximizing d2min and minimizing δ2

max are competing

criteria, when σ2 → 0, the self-matching code emphasizes on increasing d2min → T

Md2

v

and compromises on reducing δ2max by increasing δ2

max → T 2; and when σ2 → 1, the

self-matching code emphasizes on reducing δ2max → T 2

(

1 − 14

d2v

M

)

and compromises on

increasing d2min by reducing d2

min → TM

d2v

2. Hence, the self-matching code in (7.7) satisfies


the design criterion in (7.4) by adapting its code properties (d2min and δ2

max) according to

the estimation performance.

Since U(·) is either asymmetric or fixed a priori, the self-matching STBC is asymmetric.

Because training is the limiting case of asymmetry, the asymmetric self-matching STBC in

(7.7) generalizes the training-based self-matching code proposed in [18]. If U(·) is known

a priori and doesn’t carry any information, it acts like embedded training for the channel

estimator. The embedded training reduces estimation MSE and thus improves detection

performance in (4.2). The unitarity of U(·) is consistent with what has been shown as

optimal training [3, 49].

However, since the embedded training of the self-matching STBC does not carry any

information and T is at least doubled when compared to the lineary unitary STBCs, the

code rate of the self-matching STBC is reduced at least by half. Because asymmetry is a

generalized solution to break isometry and training is merely a special case of asymmetry,

the code rate reduction of the self-matching STBC can be compensated by replacing the

embedded training with information bearing asymmetric STBC [15, 45]. The asymmetry

of the self-matching STBC breaks isometry according to (7.1) which improves estimation

performance and hence also improves detection performance in (4.2). Thus, in general,

the U(·) can be an information-bearing asymmetric STBC.

In summary, the three characteristics of the self-matching STBC that improve detection

performance are: self-matching, asymmetry, and embedded training.

Remark—Open-loop design of the self-matching code: In realistic applications, H(l) =

H(l)− H(l) is difficult to calculate because H(l) is unknown. Given the correct fDTs, the


matrix state-space model will be accurate, and the matrix KF can recursively calculate

a priori the theoretical covariance P(l|l) = cov(

H(l))

for every l. Then, P(l|l) can be

used to scale the self-matching code. Since P(l|l) can be calculated a priori, the need for

a feedback loop or other elaborate procedures to calculate σ2 is eliminated. Moreover, the

CE/DD system in Fig. 2.1 with self-matching STBC resembles the closed-loop systems

in [50] and [51] where the feedback is replaced by the theoretical estimation error covari-

ance P(l|l) which scales the codewords. So, the knowledge of the channel statistics, the

modelling of the system using the matrix state-space model, the use of the matrix KF to

track and estimate the channels and the employment of the self-matching STBCs allow

the simpler open-loop system in this report to perform equivalent operations of a more

elaborate closed-loop system and to take advantage of the performance gains inaccessible

to other ordinary open-loop systems.

7.2.3 Detection Performance of Self-Matching Space-Time Block

Codes

Approximating the fading with the quasi-static fading model and assuming that self-

matching code is used, the ratio −λαβ+

λαβ−

in (4.2) and (4.4) also becomes (5.4) where (Ap-

pendix H)

T1 =TN

M

σ4T + r − rσ4

(r2 + rσ2T )(1 + σ2)2tr[

(Vα − Vβ) (Vα −Vβ)H]

Remark—Parameters affecting the performance of rectangular unitary self-matching

codes: The detection performance of a self-matching STBC is related to the distance

property of its square unitary code U(·). This means the design of a self-matching STBC


is distilled down to the design of a square unitary STBC. Hence, the design of self-matching

STBCs is made very easy because the design of square unitary STBCs have been very well

studied.

Chapter 8

Results

Table 8.1 compares the properties of some of the STBCs used in this report: Alamouti

code, self-matching code (U is assumed to be known or detected perfectly), and 16-ary

cyclic group code. Unless otherwise stated, it is assumed that QPSK constellation, M =

N = 2, T = 2 for unitary code in (5.3), and T = 4 for self-matching code in (7.7) are

used. For the simulations, the continuous fading matrix state-space model in (2.3) and

(3.3) is used to model the system, the matrix KF in (3.4) is used to track and estimate

the channel, and the ML detector in (3.5) is used to detect the data. For each experiment,

5000 independent iterations are performed, and the simulation setup in Table 8.2 is used.

Fig. 8.1 compares the theoretical detection performance, at 20 dB over different esti-

mation MSE, of various receiver and modulation schemes: receiver with known CSI and

Alamouti with QPSK, differential demodulation and 16-ary cyclic group code, CE/DD

and Alamouti with QPSK, CE/DD and self-matching STBC with QPSK. Square unitary

STBC works well only when the CSI is known or when the estimation MSE is near zero,

48

CHAPTER 8. RESULTS 49

Table 8.1: Various linear space-time block codes and their corresponding properties.

∆αβ = Cα − Cβ where Cα,Cβ ∈ ΩC : Cα 6= Cβ. N/A indicates that these proper-

ties are not defined for the STBC.

Alamouti Self-Matching Cyclic Group

T ×M 2 × 2 4 × 2 2 × 2

size of constellation 16 16 16

code rate 1 0.5 1

linearity Yes No No

unitarity Yes Yes Yes

symmetry Yes No Yes

traceless self-interference (SI) [1, 6] Yes N/A N/A

minimum SI [1, 6] 0 N/A N/A

symbol homogeneity (SH) [1, 6] Yes N/A N/A

maximal symbolwise diversity (MSD) [1, 6] maximum N/A N/A

rank = rank(∆αβ) [3, 41] 2 2 2

5 dB Alamouti Self-matching Cyclic Group

det = det(∆Hαβ∆αβ) [4, 5] 4 8.4016 1.3726

ds = tr(∆Hαβ∆αβ) [5] 4 5.7971 2.3431

dp = det(∆Hαβ∆αβ)1/T [4, 5] 2 1.7025 1.1716

d2min = min eig

(∆H

αβ∆αβ

)[18] 2 2.8986 0.3045

δ2max = max eig

((CH

αCβ

) (CH

αCβ

)H)

[18] 4 12.8074 4

15 dB Alamouti Self-matching Cyclic Group

det = det(∆Hαβ∆αβ) 4 14.7929 1.3726


Table 8.2: Summary of simulation parameters.

Parameter Value

center frequency fc = 1.8 × 109 Hz (high tier IS-136) [52]

symbol period Ts = 4.12 × 10−5 s [52]

pulse shaping square root raised cosine with a roll-off factor of 0.35 [52]

frame length 162 symbols [52]

Rayleigh fading model Jakes model [53]

hypermodel for KF AR-2 model [33–35]

normalized fading rate fDTs = 0.00637 ≤ 0.03 (i.e., slow fading) [29, 33, 54]

vehicular speeds 92.8 km/h

signal constellation QPSK

# transmit antennas M = 2

# receive antennas N = 2

pilot training unitary matrices

unitary STBC Alamouti’s STBC

differential STBC Cyclic group STBC

asymmetric STBC Asymmetric STBC in the example of Algorithm 1

self-matching STBC Self-Matching STBC with T = 4

# Monte Carlo runs 5000


as discussed in Section 5.3. When the estimation MSE is no longer negligible, the per-

formance of square unitary can be worse than that of differential STBC. At 20 dB, the

threshold is when MSE is 0.05. As discussed in Section 7.2.2, self-matching STBC, how-

ever, always performs better than differential STBC and square unitary STBC because of

its self-matching, isometry-breaking asymmetric and embedded training properties.

Fig. 8.2 and 8.3 show the experimental BER and estimation MSE of the CE/DD scheme

using various STBCs and isometry breaking solutions. These results are compared against

the BER curve with known CSI and the MSE curve with known data.

In Fig. 8.2, the BER in the case of known CSI and the BER in the absence of any CSI

are compared. For lower SNR (< 10 dB), the absence of any CSI degrades the detection

performance by an order of magnitude. For higher SNR (> 10 dB), isometry induces an

error floor of about 6 × 10−3 in the absence of any CSI. If a pilot block is introduced to

break isometry in differential demodulation, then while the BER at lower SNR (< 10 dB)

remains unchanged, the error floor in the higher SNR (> 10 dB) is mitigated. However,

the BER of differential demodulation with a pilot block is still two orders of magnitude

higher than the BER in the case of known CSI. These results confirm the discussion in

Section 5 that differential demodulation offers poorer performance. Hence, it is clear that

a CE/DD scheme which offers some CSI should improve the detection performance.

As discussed in Sections 6 and 7 and shown in Fig. 8.2 and 8.3, when neither train-

ing nor known CSI is available, and symmetric STBC (Alamouti’s with QPSK) is used,

isometry causes severe irreducible error floors in both estimation and detection (BER ≈

0.5, MSE ≈ 2), rendering the CE/DD inoperable. Training breaks isometry and mitigates


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Steady State Estimation MSE P(∞|∞)

BE

R

Known CSI with Alamouti Differential with Cyclic GroupCE/DD with Alamouti CE/DD with Self-Matching (T=4)

Figure 8.1: Detection performance comparison among the different receivers and modula-

tion schemes.


5 10 15 2010

-6

10-5

10-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

Known CSI Differential (No CSI) Differential (Pilot) Alamouti (without Training) Alamouti (Training) Alamouti (Asymmetric STBC) Self-Matching (Embedded Training)Self-Matching (Asymmetric STBC)

Figure 8.2: Bit error rate (BER) of the joint estimation and detection scheme for various

space time block codes and isometry breaking solutions.


5 10 15 2010

-3

10-2

10-1

100

SNR (dB)

MS

E

Known Data Alamouti (without Training) Alamouti (Training) Alamouti (Asymmetric STBC) Self Matching (Embedded Training)Self-Matching (Asymmetric STBC)

Figure 8.3: Estimation mean square error (MSE) of the joint estimation and detection

scheme for various space time block codes and isometry breaking solutions.


the estimation and detection error floors (at 15 dB, BER and MSE are reduced by 3

and 2 orders of magnitude respectively). Also, asymmetric Alamouti’s STBC constructed

from Alamouti’s STBC and ASC (asymmetric QPSK) as described in Algorithm 1, breaks

isometry as well as training does. These results show that symmetry-induced isometry

causes severe problem in CE/DD, illustrate how asymmetry can mitigate isometry, and

confirm that training is a special case of asymmetry. As SNR increases, the estimation

MSE decreases, and the BER of the Alamouti’s code approaches that of the BER in the

case of known CSI. However, at lower SNR (< 10 dB), the performance of Alamouti’s

STBC in CE/DD, with either training or asymmetric STBC, is only marginally better

than that of differential demodulation. This is because Alamouti’s code, although optimal

when CSI is known, is sub-optimal when the CSI is estimated.

As discussed in Section 7, the self-matching STBC was designed for CE/DD. From Fig.

8.2 and 8.3, it is shown that the self-matching STBC performs better than Alamouti’s code

in a CE/DD system (at 15 dB, BER is reduced by an order of magnitude), Furthermore,

the self-matching code performs near optimally in a CE/DD system: the BER performance

of the self-matching code is only about 1 dB worse than that of the Alamouti’s code with

known CSI. Thus, it is confirmed that the three characteristics of the self-matching STBC

improve the detection and estimation performance in a CE/DD system: (a) the self-

matching property adjusts the code’s d2min and δ2

max to reduce (4.2); (b) the asymmetry

breaks isometry and eliminates detection and estimation error floors; and (c) the embedded

training improves KF estimation performance, and the more accurate channel estimate

provides better detection performance.


As discussed in Section 7, since the embedded training reduces the code rate of the

self-matching STBC and is a special case of asymmetry, it can be replaced by information-

bearing asymmetric code as suggested in Section 7. As shown in Fig. 8.2 and 8.3, self-

matching STBC with asymmetric STBC (asymmetric binary code in Fig. 7.1 constructed

using Algorithm 2) performs as well as self-matching STBC with embedded training. This

confirms that the self-matching code in (7.7) with Usm being an asymmetric STBC gen-

eralizes the self-matching code in [18] where Usm is a constant unitary matrix. In fact,

in a CE/DD system, the self-matching code with asymmetric STBC is shown to be the

best choice of the STBCs examined because it offers near-optimal performance without

drastically reducing the code rate.

Table 8.3 compares the theoretical and experimental detection performance of various

STBCs for different M ’s and with differernt constellations. The theoretical BER bound

calculated from (4.1) agree with the experimental BER. Comparing the BERs among the

square unitary STBCs, as discussed in Section 5.3, Alamouti with QPSK consistently

performs better than Alamouti with 16QAM does because the former has a much larger

d2min. Also, increasing the number of antennas from M = 2 to M = 4 improves detection

performance because of the larger d2min as well. Comparing the BERs among the rectan-

gular unitary STBCs, as discussed in Section 7.2.3, because of the larger d2v, the 8 × 4

code performs better than the 4×2 code with QPSK does, and the 4×2 code with QPSK

performs better than the 4×2 code with 16QAM does. Comparing the BERs between the

square unitary STBCs and their rectangular self-matching counterparts, as discussed in

Section 7.2, the self-matching codes perform better than the square unitary codes because


Table 8.3: Comparison between theoretical and experimental detection performance of

various space-time block codes. (An experimental BER of 0 indicates that no error resulted

in the 5000 independent iterations.)

T ×M : 2 × 2 2 × 2 4 × 4 4 × 2 4 × 2 8

Code: Alamouti Alamouti Unitary Self-Matching Self-Matching Self-Matc

Sub-code: Alamouti Alamouti 4 × 4

Constellation: QPSK 16QAM QPSK QPSK 16QAM QPSK

d2min or d2

v 2 0.4 2.6667 2 0.4 2.6667

10 dB experimental BER 2 × 10−2 2 × 10−1 6 × 10−3 2 × 10−3 2 × 10−1 2 ×

theoretical BER 2 × 10−2 1 × 10−1 5 × 10−3 1 × 10−3 1 × 10−1 1 ×

15 dB experimental BER 6 × 10−4 1 × 10−1 3 × 10−5 4 × 10−5 3 × 10−2 3 ×

theoretical BER 8 × 10−4 7 × 10−2 2 × 10−5 2 × 10−5 2 × 10−2 3 ×

20 dB experimental BER 7 × 10−7 1 × 10−2 0 4 × 10−7 3 × 10−4

theoretical BER 8 × 10−7 1 × 10−2 2 × 10−8 4 × 10−7 4 × 10−4 4 ×


of their self-matching, isometry-breaking asymmetric and embedded training properties.

Chapter 9

Conclusions

MIMO systems are used in many high data rate applications. Since CSI is realistically un-

available, CE/DD is chosen over differential demodulation for good capacity and detection

performance. In slow flat Rayleigh fading, a matrix state-space model based on continuous

fading channel model is introduced, and the corresponding MIMO CE/DD system using

matrix KF and matrix normalized innovations-based detector is derived. The matrix KF is

derived using the vectorization technique developed in and the definition of the covariance

of a matrix developed in [20, 21] [22]. The matrix normalized innovations-based detector

is shown to be a weighted Euclidean distance-square (i.e. Mahalanobis distance-square)

detector, where the weighting covariance matrix is calculated recursively by the matrix

KF and represents the effective SNR.

In a MIMO CE/DD system affected by multiplicative fading, analysis and simulation

have shown that a large class of STBCs classified as symmetric STBCs induce isometry,

which leads to irreducible error floors. This can be mitigated by the use of asymmetric

59

CHAPTER 9. CONCLUSIONS 60

STBC, which breaks isometry. Pilot training is shown to be a special case of asymmetry-

based solution used to break isometry. Two algorithms to construct asymmetric STBCs

have been introduced and tested, and it has been shown that information-bearing asym-

metric STBCs mitigate isometry as well as data-rate reducing pilots do.

Through the probability of error equation, detection performance is shown to be in-

tricately linked to the STBC design, the channel model, the theoretical estimation per-

formance and the past detection performance. In addition to breaking isometry, the SM-

STBC in (7.7) is introduced to further improve the detection performance of the CE/DD

scheme. The SM-STBC is asymmetric, has embedded training, and adapts its code prop-

erties to the estimation MSE. With the matrix KF, the scaling factor of the SM-STBC

can be calculated a priori to form an simple open-loop system that takes advantage of

performance gains available for closed-loop systems without the need of a feedback loop.

Due to these features, the SM-STBC performs better than other STBCs investigated.

The SM-STBC in this report differs from that in [18] because the fixed unitary matrix

is replaced by an asymmetric STBC. The fixed unitary matrix of the SM-STBC in [18]

is shown in this report to be a form of embedded training. However, embedded training

using known matrices reduces code rate. Since training is a special case of asymmetry,

the embedded training can be replaced by information-bearing asymmetric STBC. Thus,

the SM-STBC in this report is more efficient than that in [18]. It has been shown that

the SM-STBC with an asymmetric STBC performs as well as the SM-STBC with a fixed

matrix.

The novel asymmetric and self-matching STBCs have been shown both analytically

CHAPTER 9. CONCLUSIONS 61

and experimentally to be effective, robust and simple solutions to mitigate error floors

and to improve detection performance. This report has contributed to wireless MIMO

transceiver and STBC design by introducing the matrix state-space model, the matrix

KF, the matrix normalized innovations-based ML detector, and the asymmetric STBCs,

both self-matching and non-self-matching. They form a complete, consistent and unified

open-loop MIMO transceiver and STBC design framework package that has been shown

both analytically and experimentally to improve detection performance.

Appendix A

Matrix KF Recursion Algorithm for

MIMO Matrix State-Space Model

The ordinary KF recursion algorithm only works with scalar and vector state-space model.

Hence, to employ KF, a matrix KF recursion algorithm needs to be derived to be used with

the following matrix state-space model (repeated from (2.3) and (3.3) for convenience):

H(l) = AH(l− 1) + W(l) (A.1)

Y(l) = X(l)H(l) + V(l) (A.2)

62

APPENDIX A. MATRIX KF RECURSION ALGORITHM FOR MIMO MATRIX STATE-SPACE MODEL

Employing the following vectorization technique in [21]:

vec

m1,1 m1,2 . . . m1,C

m2,1 m2,2 . . . m2,C

......

. . ....

mR,1 mR,2 . . . mR,C

=

m1,1

m2,1

...

mR,1

m1,2

m2,2

...

mR,2

...

...

m1,C

m2,C

...

mR,C

(A.3)

and the following equality in [21]:

vec (ABC) =(CH ⊗A

)vec (B) (A.4)

for any matrices A, B and C, the matrix state-space model in (A.1)–(A.2) can be vector-

ized as:

vec (H(l)) = vec (AH(l − 1) + W(l))

= vec (AH(l − 1)) + vec (W(l))


= vec (AH(l − 1)I) + vec (W(l))

= (I ⊗A) vec (H(l)) + vec (W(l)) (A.5)

vec (Y(l)) = vec (X(l)H(l) + V(l))

= vec (X(l)H(l)) + vec (V(l))

= vec (X(l)H(l)I) + vec (V(l))

= (I ⊗X(l)) vec (H(l)) + vec (V(l)) (A.6)

Applying the regular KF recursion to the vectorized state-space model in (A.5)–(A.6),

the following KF recursion is obtained:

vec(

H(l|l − 1))

= (I⊗ A) vec(

H(l − 1|l − 1))

(A.7)

P(l|l − 1) = (I⊗ A)P(l − 1|l − 1) (I ⊗ A)H + Q (A.8)

vec(Z(l|l − 1)) = vec (Y(l)) − (I ⊗X(l)) vec(

H(l|l − 1))

(A.9)

PZ(l|l − 1) = (I ⊗X(l))P(l|l − 1) (I ⊗ X(l))H + R (A.10)

K(l) = P(l|l− 1) (I ⊗ X(l))H P−1

Z(l|l − 1) (A.11)

vec(

H(l|l))

= vec(

H(l|l − 1))

+ K(l)vec(

Z(l|l − 1))

(A.12)

P(l|l) = [I− K(l) (I ⊗ X(l))]P(l|l − 1) (A.13)

where

Q = Evec(W(l))vec(W(l))H

= cov (W(l))

R = E

vec (V(l)) vec (V(l))H

= cov (V(l))


The vectorized KF recursion is then transformed into the following matrix form by

reversing the vectorization technique in (A.3) and (A.4) as follows1:

Z(l|l − 1) = Y(l) − X(l)H(l|l − 1) (A.14)

PZ(l|l − 1) = (I ⊗X(l))P(l|l − 1) (I ⊗ X(l))H + R (A.15)

K(l) = P(l|l− 1) (I ⊗ X(l))H P−1

Z(l|l − 1) (A.16)

H(l + 1|l) = vec−1

(I ⊗ A)[

vec(

H(l|l − 1))

+ K(l)vec(

Z(l|l − 1))]

(A.17)

P(l + 1|l) = (I ⊗A) [I− K(l) (I ⊗ X(l))]P(l|l − 1) (I ⊗A)H + Q (A.18)

where the operator vec−1(·) is the inverse of the vectorization operator in (A.3), and it

turns the input vector back into a matrix.

It is clear that the appearance of Kronecker product in the matrix KF recursion algo-

rithm is due to the vectorization of the matrix state-space model. Thus, the KF recursion

formulation for a MIMO CE/DD scheme depends intimately upon the form of the MIMO

state-space model. For example, if the original MIMO state-space model is written in

vector form or if it is written in a different matrix form, then the resulting KF recursion

algorithm may have a different formulation.

1In addition, since matrix manipulations are more time consuming and only the prediction equations

are needed in the data detection, the estimation equations are absorbed into the prediction equations in

an attempt to minimize computational complexity.

Appendix B

Derivation of the ML Space-Time

Block Code Detector

Given H(l|l − 1) and X(l), Y(l) is Gaussian distributed with the mean and covariance:

E

Y(l)|H(l|l − 1),X(l)

= X(l)H(l|l − 1)

E

∣∣∣Y(l) −X(l)H(l|l − 1)

∣∣∣

2∣∣∣∣H(l|l − 1),X(l)

= (I ⊗ X(l))P(l|l − 1) (I ⊗X(l))H + R

Given the current observation Y(l), the a posteriori probability of X(l) is P (X(l)|Yl−11 ,Y(l),Xl−1

1 ) =

P (Y(l)|Yl−11 ,Xl−1

1 ,X(l))P (X(l))P (Yl−11 ,Xl−1

1 )/P (Yl−11 ,Y(l),Xl−1

1 ). Hence, given Y(l),

Yl−11 , Xl−1

1 , and quantities (H(l|l − 1) and PZ(l|l − 1)) from the matrix KF, the MAP

detector is

X(l) = arg maxΞ∈ΩX

P (Y(l)|Yl−1

1 ,Xl−11 ,Ξ)P (Ξ)

= arg maxΞ∈ΩX

exp[

−12ZP

(

Y(l),Ξ, H(l|l − 1),P(l|l− 1))]

√

(2π)NT det (PZ(l|l − 1)Ξ)

P (Ξ)

(B.1)

66

APPENDIX B. DERIVATION OF THE ML SPACE-TIME BLOCK CODE DETECTOR67

When all the codewords are equiprobable, i.e. P (Ξ) = 1/‖ΩX‖, (B.1) reduces to

X(l) = arg maxΞ∈ΩX

exp[

−12ZP

(

Y(l),Ξ, H(l|l − 1),P(l|l − 1))]

√

(2π)NT det (PZ(l|l − 1)Ξ)

= arg minΞ∈ΩX

1

2ZP

(

Y(l),Ξ, H(l|l − 1),P(l|l− 1))

+ ln

(√

(2π)NTdet (PZ(l|l − 1)Ξ)

)

(B.2)

In (B.2), the second term is usually insignificant compared to the first term. Hence, it can

be ignored, and (B.2) reduces to the ML block code detector in (3.5).

Appendix C

Derivation of the Detection Pairwise

Error Probability

Using the technique in [3], given H(l|l− 1) and P(l|l− 1), the probability of detecting Xβ

when X(l) = Xα is transmitted [3]

Pα→β = P

ln[

P(

Y(l)|H(l|l − 1),Xβ

)]

> ln[

P(

Y(l)|H(l|l − 1),Xα

)]

= P

ZP

(

Y(l),Xβ, H(l|l − 1),P(l|l− 1))

+ ln det(P

Z(l|l − 1)Xβ

)<

ZP

(

Y(l),Xα, H(l|l − 1),P(l|l− 1))

+ ln det (PZ(l|l − 1)Xα

)

= P

vec(

Y(l) − XβH(l|l − 1))H

P−1

Z(l|l − 1)Xβ

vec(

Y(l) − XβH(l|l − 1))

−

vec(

Y(l) − XαH(l|l − 1))H

P−1

Z(l|l − 1)Xα

vec(

Y(l) −XαH(l|l − 1))

<

ln

(

det (PZ(l|l − 1)Xα

)

det(P

Z(l|l − 1)Xβ

)

)

= PΥHΘα,βΥ < ψα,β

68

APPENDIX C. DERIVATION OF THE DETECTION PAIRWISE ERROR PROBABILITY69

=

−∑Residue [φ(s) exp(sψα,β)/s]right plane poles if ψα,β ≤ 0

∑Residue [φ(s) exp(sψα,β)/s]left plane poles ∪ 0 if ψα,β > 0

(C.1)

where

φ(s) = (det (I + scov (Υ)Θα,β))−1

Υ =

[

vec (Y(l))T vec(

H(l|l − 1))T]T

Θα,β =

P−1

Z,β−P−1

Z,αP−1

Z,αXα −P−1

Z,βXβ

XHαP−1

Z,α− XH

β P−1

Z,βXH

β P−1

Z,βXβ − XH

αP−1

Z,αXα

ψα,β = ln

(

det (PZ(l|l − 1)Xα

)

det(P

Z(l|l − 1)Xβ

)

)

Using the technique in [39], φ(s) =(

1(1+λαβ+s)(1+λαβ−

s)

)NT

where λαβ+ and λαβ− are

respectively the positive and negative eigenvalues of cov(Υ)Θα,β, and Pα→β is found to be

(4.2).

Appendix D

Derivation of Detection Performance

for Square Unitary Codes

If the fading model is assumed to be quasi-static, then cov(H(l|l)) = I, P(l|l − 1) = σ2I,

cov(

H(l|l − 1))

= cov(H(l|l))−P(l|l−1) = (1−σ2)I, R = rI, P−1

Z,α= 1

rI− σ2

r2+rσ2TXαX

Hα ,

P−1

Z,β= 1

rI − σ2

r2+rσ2TXβX

Hβ , and (4.3) is simplified to the following:

XαXHα + rI (1 − σ2)Xα

(1 − σ2)XHα (1 − σ2)I

σ2

r2+rσ2T

(

XαXHα − XβX

Hβ

)r

r2+rσ2T

(

XαXβ

)

rr2+rσ2T

(

XHαXH

β

)

0

So, T1 = tr (cov(Υ)Θαβ) is

T1 = tr

(

XαXHα + rI

)( σ2

r2 + rσ2T

)(

XαXHα − XβX

Hβ

)

+(1 − σ2)Xα

(r

r2 + rσ2T

)(

XHα − XH

β

)

+ (1 − σ2)XHα

(r

r2 + rσ2T

)(

Xα − Xβ

)

=r2

r2 + rσ2Ttr

T XαXHα − XαX

HαXβX

Hβ + rXαX

Hα − rXβX

Hβ

+(1 − σ2)

(r

r2 + rσ2T

)

tr

XαXHα − XαX

Hβ + XαX

Hα − XβX

Hα

70

APPENDIX D. DERIVATION OF DETECTION PERFORMANCE FOR SQUARE UNITARY CODES

=σ2

r2 + rσ2T

T 2MN − tr(

XαXHαXβX

Hβ

)

+(1 − σ2)

(r

r2 + rσ2T

)

2TMN − tr(

XαXHβ

)

− tr(

XβXHα

)

(D.1)

and T2 = tr (cov(Υ)Θαβcov(Υ)Θαβ) is

T2 =

(σ2

r2 + rσ2T

)2

T 2MN(T 2 + 2Tr + 2r2) + tr(

XαXHαXβX

Hβ XαX

HαXβX

Hβ

)

−2(T 2 + Tr + r2)tr(

XαXHαXβX

Hβ

)

+rσ2(1 − σ2)

(1

r2 + rσ2T

)2

4T 2MN(T + r) − 4(T + r)tr(

XαXHαXβX

Hβ

)

−2T 2tr(

XαXHβ

)

− 2T 2tr(

XβXHα

)

+2tr(

XαXHβ XαX

HαXβX

Hβ

)

+ 2tr(

XβXHαXβX

Hβ XαX

Hα

)

+(1 − σ2)2

(r

r2 + rσ2T

)2

2T 2MN − 2T tr(

XαXHβ

)

− 2T tr(

XβXHα

)

+tr(

XαXHβ XαX

Hβ

)

+ tr(

XβXHαXβX

Hα

)

+(1 − σ2)

(r

r2 + rσ2T

)2

2T 2MN + 4TMNr + 2tr(

XαXHαXβX

Hβ

)

−2(T + r)tr(

XαXHβ

)

− 2(T + r)tr(

XβXHα

)

(D.2)

We observe that λαβ+ and λαβ− are respectively the positive and negative roots of the

characteristic equation det (cov(Υ)Θαβ − λI) = 0 [55]. Assuming that the fading is quasi-

static, the dimension of cov(Υ)Θαβ is N(M +T )×N(M +T ), rank(cov(Υ)Θαβ) = 2MN ,

and the multiplicity of λαβ+ and λαβ− is MN [56, 57]. Hence,

det (cov(Υ)Θαβ − λI) = 0

λN(M+T )−2MN (λ− λαβ+)MN(λ− λαβ−)MN = 0

λN(T−M)(λ− λαβ+)MN(λ− λαβ−)MN = 0

(λ− λαβ+)MN(λ− λαβ−)MN = 0


(λ− λαβ+)(λ− λαβ−) = 0

λ2 − (λαβ+ + λαβ−)λ+ λαβ+λαβ− = 0 (D.3)

Since T1 = MN(λαβ+ + λαβ−) and T2 = MN(λ2αβ+ + λ2

αβ−) [55],

λαβ+λαβ− = −1

2

(−2λαβ+λαβ− + λ2

αβ+ − λ2αβ+ + λ2

αβ− − λ2αβ−

)

= −1

2

[−(λ2

αβ+ + 2λαβ+λαβ− + λ2αβ−

)+ λ2

αβ+ + λ2αβ−

]

= −1

2

[− (λαβ+ + λαβ−)2 + λ2

αβ+ + λ2αβ−

]

= −1

2

[

−(

T1

MN

)2

+T2

MN

]

(D.4)

Hence, (D.3) becomes

λ2 − T1

MNλ− 1

2

[

−(

T1

MN

)2

+T2

MN

]

= 0 (D.5)

Solving (D.5), we obtain

λαβ+ =

T1

MN+

√(

T1

MN

)2 − 4(−1

2

) [

−(

T1

MN

)2+ T2

MN

]

2(D.6)

λαβ− =

T1

MN−√(

T1

MN

)2 − 4(−1

2

) [

−(

T1

MN

)2+ T2

MN

]

2(D.7)

Therefore, (5.4) becomes

−λαβ+

λαβ−= −

T1

MN+√

2T2

MN−(

T1

MN

)2

T1

MN−√

2T2

MN−(

T1

MN

)2(D.8)

Assuming that the STBCs are square and unitary, (D.1) and (D.2) simplify to

T1 = (1 − σ2)

(r

r2 + rσ2T

)

tr

[(

Xα − Xβ

)(

Xα − Xβ

)H]

(D.9)


T2 = (1 − σ2)2

(r

r2 + rσ2T

)2

−2T tr(

XαXHβ

)

− 2T tr(

XβXHα

)

+1

MN

[

tr(

XαXHβ + XβX

Hα

)]2

+(1 − σ2)

(r

r2 + rσ2T

)24T 2MN + 4TMNr

−2(T + r)tr(

XαXHβ

)

− 2(T + r)tr(

XβXHα

)

(D.10)

From (D.9) and (D.10), it can be shown that T2 =T 21

MN+ 2T1. Thus, (D.8) becomes

−λαβ+

λαβ−

= −T1

MN+√(

T1

MN

)2+ 4T1

MN

T1

MN−√(

T1

MN

)2+ 4T1

MN

= −T1

MN+√(

T1

MN

)2+ 4T1

MN

T1

MN−√(

T1

MN

)2+ 4T1

MN

·T1

MN+√(

T1

MN

)2+ 4T1

MN

T1

MN+√(

T1

MN

)2+ 4T1

MN

= −(

T1

MN

)2+(

T1

MN

)2+ 4 T1

MN+ 2T1

MN

√(

T1

MN

)2+(

4T1

MN

)

(T1

MN

)2 −(

T1

MN

)2+ 4 T1

MN

= −2(

T1

MN

)2+ 4 T1

MN+√

4(

T1

MN

)4+ 16

(T1

MN

)3

4 T1

MN

= 1 +T1

2MN+

√(

T1

2MN

)2

+T1

MN

Appendix E

Proofs for Lemmas, Propositions,

and Corollaries

Proof: [Lemma 1] Since A = I⊗A and Q = I⊗Q, then AU = UA and UQUH = Q,

where U is a unitary transform such that UHU = I and X(l) ∈ ΩX implies X(l)UH ∈ ΩX.

Based on this result, (6.2) can be easily proven by substituting X(l)UH and UH(l|l − 1)

for X(l) and H(l|l − 1), respectively, in (A.14)-(A.18).

Proof: [Lemma 2] From (3.5), it is observed that

ZP

(

Y(l), X(l), H(l|l − 1),P(l|l− 1))

= ZP

(

Y(l), X(l)UH,UH(l|l − 1),UP(l|l− 1)UH)

Thus, if X(l) = arg minΞ∈ΩX

ZP

(

Y(l),Ξ, H(l|l − 1),P(l|l− 1))

,

then X(l)UH = arg minΞ∈ΩX

ZP

(

Y(l),ΞUH,UH(l|l − 1),UP(l|l− 1)UH)

.

Proof: [Proposition 3] For L = 1, Proposition 3 is reduced to Lemma 2, and it has

already been proven. Assume that Proposition 3 is true for L = l. For L = l + 1, given

Y(l + 1) and H(l + 1|l), X(l + 1) is the detected code, and H(l + 2|l + 1) is the channel

74

APPENDIX E. PROOFS FOR LEMMAS, PROPOSITIONS, AND COROLLARIES 75

prediction. By Lemma 2, given Y(l + 1) and UH(l + 1|l), X(l + 1)UH is the detected

code. Then, by Lemma 1, UH(n+2|n+1) is the channel prediction. So, by mathematical

induction, given YL1 , the CE/DD scheme obtains XL

1 and HL1 with H(0|0) and P(0|0), but

it obtains XL1 U

H and UHL1 with UH(0|0) and UP(0|0)UH.

Proof: [Corollary 4] Given H(0|0) = UH(0|0) = 0 and P(0|0) = (I ⊗ U)P(0|0) (I ⊗U)H =

I, Proposition 3 states that the CE/DD scheme obtains either(

XL1 , H

L1

)

, or(

XL1 U,UHL

1

)

.

The detection criterion for X(1) in (3.5) is reduced to

X(1) = arg minΞ∈ΩX

vec(Y(1))H

(I⊗

(ΞΞH

)+ R

)vec(Y(1))

making X(1) and(

X(1)U)

equally likely detected codes, and hence, XL1 and

(

XL1 U)

equally likely detected sequences.

Appendix F

Derivation of Code Design Criterion

The matrix inversion lemma states that if a positive definite M ×M matrix A has the

form

A = B + CDCH (F.1)

where B is a positive definite M ×M matrix, C is a M ×N matrix, and D is a positive

definite N ×N matrix, then

A−1 = B−1 − B−1C(D−1 + CHB−1C

)−1CHB−1 (F.2)

From (A.15), PZ(l|l − 1) has the form shown in (F.1). Therefore, by (F.2)

P−1

Z(l|l − 1) = R−1 − R−1 (I ⊗ X(l))

(

P−1(l|l − 1) + (I ⊗ X(l))H R−1 (I⊗ X(l)))−1

(I ⊗ X(l))H R−1 (F.3)

where R = rI. Since block fading model is assumed, P(l|l−1) = σ2I. Thus, (F.3) becomes

P−1

Z(l|l − 1) =

1

rI −

(1

rI

)

(I ⊗ X(l))

(1

σ2I + (I⊗ X(l))H

(1

rI

)

(I ⊗ X(l))

)−1

76

APPENDIX F. DERIVATION OF CODE DESIGN CRITERION 77

(I ⊗ X(l))H

(1

rI

)

=1

rI − 1

r2(I ⊗ X(l))

(1

σ2I +

1

r(I ⊗X(l))H (I⊗ X(l))

)−1

(I ⊗X(l))H(F.4)

Assuming that the STBC is unitary, i.e. XH(l)X(l) = T IM×M , then

(IN×N ⊗ X(l))H (IN×N ⊗ X(l)) = T IMN×MN (F.5)

Substituing (F.5) into (F.4),

P−1

Z(l|l − 1) =

1

rI − 1

r2(I ⊗ X(l))

(1

σ2I +

T

rI

)−1

(I ⊗ X(l))H

=1

rI − 1

r2(I ⊗ X(l))

(r + σ2T

rσ2I

)−1

(I ⊗ X(l))H

=1

rI − σ2

r2 + rσ2T(I ⊗ X(l)) (I ⊗X(l))H (F.6)

Hence, using (F.6), (7.2) can be simplified as follows:

tr[(

XαXHα + R

)(

P−1

Z,β−P−1

Z,α

)

+(1 − σ2

) (

XαXHαP−1

Z,α

−XαXHβ P−1

Z,β− XβX

HαP−1

Z,β+ XβX

Hβ P−1

Z,β

)]

= tr

[(

XαXHα + rI

)((1

rI − σ2

r2 + rσ2TXβX

Hβ

)

−(

1

rI − σ2

r2 + rσ2TXαX

Hα

))

+1 − σ2

r

(

XαXHα − XαX

Hβ − XβX

Hα + XβX

Hβ

)

− σ2 − σ4

r2 + rσ2T

(

XαXHαXαX

Hα − XαX

Hβ XβX

Hβ − XβX

HαXβX

Hβ + XβX

Hβ XβX

Hβ

)]

= tr

[(

XαXHα + rI

)( σ2

r2 + rσ2TXαX

Hα − σ2

r2 + rσ2TXβX

Hβ

)

+1 − σ2

r

(

Xα − Xβ

)(

Xα − Xβ

)H

− σ2 − σ4

r2 + rσ2T

(

T XαXHα − T XαX

Hβ − XβX

HαXβX

Hβ + T XβX

Hβ

)]

= tr

[σ2

r2 + rσ2TXαX

HαXαX

Hα − σ2

r2 + rσ2TXαX

HαXβX

Hβ


+σ2

r + σ2TXαX

Hα − σ2

r + σ2TXβX

Hβ +

1 − σ2

r

(

Xα − Xβ

)(

Xα − Xβ

)H

− σ2 − σ4

r2 + rσ2T

(


Hβ − XβX

HαXβX

Hβ + T XβX

Hβ

)]

= tr

[σ2T

r2 + rσ2TXαX

Hα − σ2

r2 + rσ2TXαX

HαXβX

Hβ

+σ2

r + σ2TXαX

Hα − σ2

r + σ2TXβX

Hβ +

1 − σ2

r

(

Xα − Xβ

)(

Xα − Xβ

)H

− σ2 − σ4

r2 + rσ2T

(


Hβ − XβX

HαXβX

Hβ + T XβX

Hβ

)]

(F.7)

Since tr(A + B) = tr(A) + tr(B) for any matrices A and B of the same dimension, and

tr(CD) = tr(DC) for any square matrices C and D, (F.7) becomes

σ2T

r2 + rσ2Ttr

XHαXα

− σ2

r2 + rσ2Ttr

XHαXβX

Hβ Xα

σ2

r + σ2Ttr

XHαXα

− σ2

r + σ2Ttr

XHβ Xβ

+1 − σ2

rtr

(

Xα − Xβ

)H (

Xα − Xβ

)

− σ2 − σ4

r2 + rσ2T

(

T tr

XHαXα

− T tr

XHβ Xα

− tr

XHαXβX

Hβ Xβ

+ T tr

XHβ Xβ

)

=σ2MNT 2

r2 + rσ2T− σ2

r2 + rσ2Ttr(

XHαXβ

)(

XHβ Xα

)

σ2MNT

r + σ2T− σ2MNT

r + σ2T+

1 − σ2

rtr

(

Xα − Xβ

)H (

Xα − Xβ

)

− σ2 − σ4

r2 + rσ2T

(

T tr

XHαXα

− T tr

XHβ Xα

− T tr

XHαXβ

+ T tr

XHβ Xβ

)

=σ2MNT 2

r2 + rσ2T− σ2

r2 + rσ2Ttr

(

XHαXβ

)(

XHαXβ

)H

+1 − σ2

rtr

(

Xα − Xβ

)H (

Xα − Xβ

)

−(1 − σ2) σ2T

r2 + rσ2Ttr

XHαXα − XH

β Xα − XHαXβ + XH

β Xβ

=σ2

r2 + rσ2T

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

+1 − σ2

rtr

(

Xα − Xβ

)H (

Xα − Xβ

)

−(1 − σ2) σ2T

r2 + rσ2Ttr

(

Xα − Xβ

)H (

Xα − Xβ

)


=σ2

r2 + rσ2T

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

+

(1 − σ2

r− (1 − σ2) σ2T

r2 + rσ2T

)

tr

[(

Xα − Xβ

)H (

Xα − Xβ

)]

=σ2

r2 + rσ2T

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

+(1 − σ2

)(r + σ2T − σ2T

r2 + rσ2T

)

tr

[(

Xα − Xβ

)H (

Xα − Xβ

)]

=σ2

r2 + rσ2T

MNT 2 − tr

[(

XHαXβ

)(

XHαXβ

)H]

+r (1 − σ2)

r2 + rσ2Ttr

[(

Xα − Xβ

)H (

Xα − Xβ

)]

which reduces to (7.3).

Appendix G

Derivation of Distance Properties

Equation (7.8) is derived as follows:

d2min = min eig

(Xα − Xβ)H(Xα −Xβ)

=u+ v

1 + σ2min eig

([σ√uUH

∣∣∣∣

1√vVH

α

]

−[σ√uUH

∣∣∣∣

1√vVH

β

])

([σ√uUH

∣∣∣∣

1√vVH

α

]H

−[σ√uUH

∣∣∣∣

1√vVH

β

]H)

=u+ v

1 + σ2min eig

[

0

∣∣∣∣

1√v

(Vα − Vβ)H

] [

0

∣∣∣∣

1√v

(Vα − Vβ)H

]H

=u+ v

1 + σ2

1

vmin eig

(Vα − Vβ)H (Vα − Vβ)

=u+ v

v

(d2

v

1 + σ2

)

=MT

M2

(d2

v

1 + σ2

)

=T

M

(d2

v

1 + σ2

)

And equation (7.9) is derived as follows:

δ2max = max eig

(XH

αXβ

) (XH

αXβ

)H

80

APPENDIX G. DERIVATION OF DISTANCE PROPERTIES 81

= max eig

(u+ v

1 + σ2

)2([

σ√uUH

∣∣∣∣

1√vVH

α

] [σ√uUH

∣∣∣∣

1√vVH

β

]H)

([σ√uUH

∣∣∣∣

1√vVH

α

] [σ√uUH

∣∣∣∣

1√vVH

β

]H)H

= max eig

(u+ v

1 + σ2

)2(σ2

uUHU +

1

vVH

αVβ

)(σ2

uUHU +

1

vVH

αVβ

)H

= max eig

(u+ v

1 + σ2

)2(σ2

u(T −M)I +

1

vVH

αVβ

)(σ2

u(T −M)I +

1

vVH

αVβ

)H

= max eig

(u+ v

1 + σ2

)2(σ2

MI +

1

vVH

αVβ

)(σ2

MI +

1

vVH

αVβ

)H

= max eig

(u+ v

1 + σ2

)2(σ4

M2I +

σ2

Mv

(VH

αVβ + VHβ Vα

)+

1

v2VH

αVβVHβ Vα

)

= max eig

(u+ v

1 + σ2

)2(σ4

M2I +

σ2

M3

(VH

α Vβ + VHβ Vα

)+

1

M4M2I

)

= max eig

(u+ v

1 + σ2

)2(1 + σ4

M2I +

σ2

M3

(VH

αVβ + VHβ Vα

))

= max eig

(u+ v

1 + σ2

)2(1 + σ4

M2I +

σ2

M3

(−VH

αVα + VHαVβ + VH

β Vα − VHβ Vβ

)+

σ2

M3

(VH

αVα + VHβ Vβ

))

= max eig

(u+ v

1 + σ2

)2(1 + σ4

M2I − σ2

M3

(VH

αVα −VHαVβ −VH

β Vα + VHβ Vβ

)+

σ2

M3(MI +MI)

)

= max eig

(u+ v

1 + σ2

)2(1 + σ4

M2I +

2σ2

M2I − σ2

M3(Vα − Vβ)

H (Vα − Vβ)

)

= max eig

(u+ v

1 + σ2

)2(1 + 2σ2 + σ4

M2I − σ2

M3(Vα − Vβ)H (Vα − Vβ)

)

= max eig

(u+ v

1 + σ2

)2((

1 + σ

M

)2

I − σ2

M3(Vα −Vβ)H (Vα −Vβ)

)

(G.1)

Since eig (aI − A) = a−eig (A) for any real number a, max (eig (aI −A)) = max (a− eig (A)) =

APPENDIX G. DERIVATION OF DISTANCE PROPERTIES 82

a− min (eig (A)). Hence, (G.1) becomes

δ2max =

(u+ v

1 + σ2

)2(1 + σ2

M

)2

−(u+ v

1 + σ2

)2σ2

M3min eig

(

(Vα −Vβ)H (Vα −Vβ))

=

(u+ v

1 + σ2

)2(1 + σ2

M

)2

−(u+ v

1 + σ2

)2σ2

M3d2

v

=(MT )2

(1 + σ2)2

(1 + σ2)2

M2− σ2

(1 + σ2)2

(MT )2

M3d2

v

= T 2 − σ2

(1 + σ2)2

T 2

Md2

v

= T 2

(

1 − σ2

(1 + σ2)2

d2v

M

)

Appendix H

Derivation of Detection Performance

for Self-Matching Codes

For now, the proof is done experimentally. Calculations of T1 and T2 are performed

repeatedly over different R’s, P(l|l − 1)’s, Xα’s and Xβ’s. The results consistently show

that the relationship between T1 and T2 stated in (5.4) holds. An analytical proof will be

provided soon.

83

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