APPENDIX A

VECTOR OPERATIONS

Suppose a, b, c, d are column vectors in a coordinate system with column unit vectors UJ, U2, and U3.

We write, e.g., vector a in the form of a = alul + a2U2 + a3U3 with vector components aJ, a2, a3.

Variables a and fJ are scalars.

Then the following laws and rules apply.

Commutative law for addition:

a+b=b+a (A-I)

Associative law for addition:

a+b +c =(a+ b)+c = a+ (b+c) (A-2)

Associative law for scalar multiplication:

a fJa = a(fJa) = (a fJ)a = fJ(aa) (A-3)

Distributive laws:

a(a+b)=aa+ab (A-4)

(a + fJ)a = a a + fJ a (A-5)

187

188 Appendix A: Vector Operations

Dot (= scalar) product:

a . b = lallbl cos( La,b) , 0:0; La, b :0; Jl" (=angle between a and b ) (A -6)

a·b=b·a (A-7)

a· (b + c) = a· b + a· c (A-8)

Cross (= vector) product:

a x b = lallbl sin(La, b)u ~ , (u ~ = unit vector perpendicular to plane of a and b.

a, b, U ~ form a right-handed system.) (A-lO)

UI Uz "3

axb = al aZ a3

q bz b3

= (az~ - a3bZ)uI + (a3q - alb3)uz + (a1bz - azbr)u3

axb=-bxa

ax (b + c) = axb + ax c

Mixed dot and vector products:

ax(bxc)=b(a·c)-c(a·b)

(A-ll)

(A-12)

(A-13)

(A-14)

Appendix A. Vector Operations

(a x b) x e = b (a . c) - a (b· c)

(a x b) x (ex d) = e(a· (b x d)) -d(a· (b xc))

=b~·~xdll-~b·~xdll

a1 a2 a3 a·(bxe)=q b2 b3

c1 c2 c3

Derivatives of vectors a(a) = a1 (a)u1 + a2(a)u2 + a3(a)u3' b(a) , e(a) , ... :

d db da -(a·b)=a·-+-·b da da da

d db da -(axb) =ax-+-xb da da da

d da db de -(a· (bx c)) = -. (b x c) + a·(-x c) + a· (b x-) da da da da

189

(A-15)

(A-16)

(A-17)

(A-18)

(A-19)

(A-20)

(A-21)

APPENDIX B

MATRIX ALGEBRA

A matrix A of order N by M is a rectangular array of NM quantities organized in N rows and M columns. If the number of rows is equal to the number of columns (N = M), A is called a square matrix. an•m is the (n,m)th element of A where n = 1,2, ... , N denotes the nth row and m = 1,2, ... , Mis the mth column.

If A has a single row (N = 1), A is a row vector.

If A has a single column (M = 1), A is a column vector.

Interchanging of rows and columns in A yield the transpose of A. The transpose of A is denoted by AT.

If all non-diagonal elements of A are zero (i.e., an,m = 0, n * m ), A is

called a diagonal matrix.

If all elements of a square diagonal matrix A are unity, A is an identity matrix. Throughout the text, letter I is used for identity matrices.

Transpose of product of matrices A and B:

(B-1)

A-I, B-1, ... are the inverses of nons in gular square matrix A, B, ... :

(B-2)

191

192 Appendix B: Matrix Algebra

Product of matrix inverses:

(B-3)

Symmetric matrix:

(B-4)

Skew-symmetric matrix:

(B-5)

Orthogonal matrix:

(B-6)

Differentiation of matrix A(a) with respect to scalar variable a:

da!,! daM,!

dA da da (B-7)

da daN,! daN,M

da da

Similarity transform: Two square matrices A and B of same size (N by N) are similar, if there

exists a nonsingular matrix C such that

(B-8)

Appendix B. Matrix Algebra

Eigenvalues A n and associated eigenvectors Xn of square N-by-N matrix A:

nonzero solutions (eigenvalues) determined

Characteristic Nth-degree polynomial of N-by-N matrix A:

det(A - AI)

Diagonal matrix of eigenvalues:

=[~1 ..1,02 ~ : A. . . . . .

° ° AN

193

(B-9)

(B-IO)

(B-11 )

U sing a symmetric and orthogonal transformation matrix C, a real symmetric matrix A can always be transformed into a diagonal matrix of its eigenvalues:

Matrix exponential:

00 Ak exp(A) =1+ L­

k=l k!

(B-12)

(B-13)

194 Appendix B: Matrix Algebra

Hermetian (= conjugate transpose) of complex matrix A:

[ ~11 f!1,2

giN I A~ :~II

f!2,2 f!2,M

f!N,2 f!N,M

n (B-l4)

* * * f!1,l f!2,1 f!N,1 * * *

AH = f!1,2 f!2,2 f!N,2

* * * f!M,1 f!M,2 f!M ,N

Properties of Hermetian:

(B-15)

(B-16)

(B-17)

Frobenius (or Euclidean) norm of N x N matrix A:

(B-18)

where the trace of A, tr(A), is the sum of all diagonal elements of matrix A.

APPENDIX C

LIST OF SCALED-ORTHOGONAL SPACE-TIME CODE (STC) MATRICES

A space-time encoder with K input symbols and f! transmit periods is characterized by its space-time code rate (rsrc) given by the ratio

K rSTC =-.

f! (C-l)

A number of K real or complex symbols is transmitted via N antennas over f! symbol time intervals.

For real-valued symbols (superscript "r") and complex-valued symbols (superscript "e") code matrices are listed below.

2 Antennas:

3 Antennas:

[ x, Xr) = -X2

-X3

-X4

!;) (see example of Alamouti scheme) !I

X2

X3 : xI -X4

X4 xI

-x3 X2

(C-2)

(C-3)

(C-4)

195

196 Appendix C: Scaled-Orthogonal STC Matrices

~l ~2 ~3

-~2 ~l -~4

-~3 ~4 ~l

X(c) --~4 -~3 ~2

Note: rSTC = Y2 (C-5) 3 - * * * ~l ~2 ~3

* * * -~2 ~l -~4

* * * -~3 ~4 ~l

* * * -~4 -~3 ~2

Here, blocks of K = 4 symbols are taken and transmitted in parallel using N=3 antennas over £ = 8 symbol periods. Therefore, the code rate is Yz.

With the following two codes, blocks of K = 3 symbols are taken and transmitted in parallel using N = 3 antennas over £ = 4 symbol periods. Therefore, the code rate is rSTC = 3/4.

~l ~2 ~J2x3 2 -

* * ~J2x3 X(c) -

-~2 ~l 2 -3 -

~J2x3 ~J2x3 I * * 2(-~1 -~l +~2 -~2) 2 - 2 -

(C-6)

~J2x3 -~J2x3 I * * 2(~1-~1 +~2 +~2) 2 - 2 -

[ !J -~2

-~J I X(c) - ~; * ~l

Note: rSTC = %. 3 - * ~3 0 ~l

* * 0 -~3 ~2

(C-7)

Appendix C: Scaled-Orthogonal STC Matrices 197

4 Antennas:

[ x, x2 x3 x4

X~) = -X2 Xl -x4 X3 I (C-8)

-X3 X4 Xl -x2

-X4 -x3 x2 Xl

:!l :!2 :!3 :!4

-:!2 :!l -:!4 :!3

- :!3 :!4 :!l -:!2

X(c) - -:!4 - :!3 :!2 :!l Note: rSTC = Yz (C-9) 4 - * * * *

:!l * -:!2

* - :!3

* -:!4

:!l

(c)_ -:!2 X4 - 1 r;:; *

-'\/ 2x3 2 -1 r;:; * -'\/ 2x3 2 -

Note: rSTC = %

:!2 :!3 * *

:!I -:!4

* * :!4 :!l

* * -:!3 :!2

1 r;:; * -'\/ 2x3 2 -1 r;:; * --'\/ 2x3 2 -

:!4 *

:!3 * -:!2

* :!l

~JiX3 2 -

~Jix3 2 -1 * * 2(-:!1 -:!l +:!2 -:!2)

1 * * 2(:!1 -:!l +:!2 + :!2)

More block STC matrix designs in:

~JiX3 2 -

~Jix3 2 -1 * * 2~1 -:!l -:!2 - :!2)

1 * * 2(-:!1-~1 -~2 +~2)

(C-10)

[Cll H.-F. Lu, P. V. Kumar, and H. Chung. "On Orthogonal Designs and Space-Time Codes," IEEE Communications Letters, Vol. 8, No.4, April 2004, pp. 220 - 222. [C2l W. Zhao, G. Leus, and G. B. Giannakis. "Orthogonal Design of Unitary Constellations and Trellis-Coded Noncoherent Space-Time Systems," IEEE Transactions on Information Theory, Vol. 50, No.6, June 2004, pp. 1319 - 1327.

Additive multi-port system, 27

additive white Gaussian noise (AWGN),182

admittance matrix, 3, 20

Alamouti scheme, 178

INDEX

associated Legendre functions of the first kind, 80

associated Legendre functions of the second kind, 81

attenuation constant, 60

augmented admittance matrix, 9

average power density vector, 95

average radiated power, 110

axial ratio of ellipse, 91

Baseband encoder, 175

beam solid angle, 128

Bessel's equation, 75

Bessel function of the first kind,75

Bessel function of the second kind (= Neumann function), 75

Bessel functions of the third kind (= Hankel functions), 75

Canonic ladder network, 84

cartesian coordinate system, 50,64

causal multi-port system, 29

199

200

Cayley-Hamilton theorem, 35

channel estimator, 182

channel identification algorithm, 164, 166

channel state information, 182

characteristic impedance, 12

circularly polarized wave, 94

coherence time of channel, 182

complex channel coefficients, 181

complex exponential, 148

conical angle, 122

constitutive vector equations, 57

continuous-time multi-port system, 29

convolution integral, 30

conversion formulas for coordinate systems, 52

curl of vector-valued function, 53

cylinder functions, 76, 83

cylindrical coordinate system,

Index

50, 73 Decision error matrix, 174

decision statistics, 182

dipole moment, 106

directional antenna, 106

direction cosines, 87

directivity (of antenna), 122

discrete signal vector, 150

discrete-time multi-port system, 30, 40

dispersionless medium, 73

distance vector, 113

distributive laws of inner products, l38

divergence of vector-valued function, 53

dot product, l3 7

Eccentricity of ellipse, 91

effective aperture (of antenna), 127

effective isotropic radiated power (EIRP), 127, l30

efficiency (of antenna), 122, 124

Appendix: Scaled-Orthogonal STC Matrices

eigenvalues of matrix, 35

electric displacement vector, 56

electric vector potential, 77

elemental instantaneous signals, 67

error metric, 142

error signal energy, 142

Euler's constant, 115

even tesseral harmonics, 81

exterior node, 6

Far field (radiation) zone, 108, 109

first-order difference equations of multi -port system, 40

focus of ellipse, 90

forward insertion loss, power gain, 15

Fraunhofer distance, 131

free space propagation, 130

Friis formula, 132

Frobenius norm of a matrix, 163

201

Gauss's law for magnetism, 58

generalized Fourier coefficient, 141

generalized Fourier series, 141

generalized reflection coeffi­cient, 19

generalized scattering vana­bles, 19

generalized S parameter, 18

Gram-Schmidt orthogonalization procedure, 156

gradient of scalar, 53

group delay, 72

group velocity, 72

Half-wavelength dipole, 116

Hankel functions, 75

Helmholtz equations, 60

Hermetian symmetry of inner products, 138

homogeneous multi-port system, 28

Impedance matrix, 3, 20

202

incident voltage wave, 11

incremental-length dipole, 106

index of refraction, 73

inner product, 13 7, 151

inphase component, 176

internal node, 6

intermediate (induction) zone, 108

intrinsic impedance, 70, 83

impulse response of SISO system, 31

impulse response matrix of multi-port system, 31, 36

isotropic radiator, 106, 123

Kronecker delta, 22

Laplacian of vector function, 57

latera recta of ellipse, 90

linear combinations, 11, 139

linear medium, 55 , 57

linear multi-port system, 27, 28

Index

linearly polarized wave, 94

linear, time-variant (LTV) multi-port system, 29, 32

linear wire antenna, 111 lossless multi-port network, 21

loss tangent, 56

Magnetic induction vector, 56

magnetic polarization vector, 56

magnetic vector potential, 77

main beam, 122

major axis of ellipse, 90, 92

maximum change rate of MIMO channel, 162

maximum directivity (of antenna), 124

maximum likelihood detector (MLD), 183

maximum power gain (of antenna), 124

Maxwell's equations, 54

MIMO channel problem, 163

minor axis of ellipse, 90, 92

mismatch loss, 13

Appendix: Scaled-Orthogonal STC Matrices

modes of free space, 82

monopole antenna, 111

Moore-Penrose pseudoinverse, 165

multiple-input multiple-output (MIMO) system, 27

multiple signal matrix, 150

Natural norm of signal, 140, 151

near field (static) zone, 108, 109

Neumann function, 75, 80

Node Reduction Algorithm (NRA), 5

noise-free channel, 162

non-autonomous system, 33

nonlinear medium, 57

nonlinear multi-port system, 28

nonlinear time-variant (NLTV) multi-port system, 28

normalized scattering variab­les, 14

normal modes, 59

Odd tesseral harmonics, 81

omnidirectionally radiating antenna, 128

open circuit voltage of unloaded antenna, 132

203

orthogonality of signals, 139, 151

orthonormality of signals, 140

orthonormalization, 141, 152

orthonormal signal vector, 142

output equations, 32, 33, 34, 36,40

output load admittance, 7

Partial fraction expansion, 84

path loss, free space, 132

permeability, 55

permittivity, 55

phase constant, 60

phase-shift keying, 176

phase velocity, 72

planar antenna array, 129

plane wave, 70

204

point of gradual cutoff, 84

polar coordinate system, 50

polarization, 89

power density vector, 95, 109

power gain (of antenna), 122

power transfer function, 7

Poynting vector, 95, 109

principal submatrix, 6

product form of electrical field,64

projection theorem, 155

propagation constant, 60

pseudoinverse of matrix, 165

Quadrature amplitude modulation, 176

quadrature phase component, 176

QR-decomposition algorithm, 157

Radiation field pattern, 116

radiation power pattern, 118

radiation resistance of small electric dipole, 110

radiation resistance of magnetic dipole, 111

reciprocal multi-port network,21

reciprocity theorem, 106

Index

rectangular coordinate system, 50

reduced admittance matrix,S

reference admittance matrix, 21

reference impedance, 16

reflection coefficient, 13

reverse power gain, 15

return loss, 13

reflected voltage wave, 11

reflection coefficient, 17

resonance (of antenna), 125

rotational parametric representation of ellipse, 91

Sampling property, 30

scaled-orthogonal code matrix, 179

Appendix: Scaled-Orthogonal STC Matrices

scaling laws of inner products, 138

scaling property, 30

scattering matrix, 14

scattering variables, 12, 15

Schwarz's inequality, 149

semi-axes of ellipse, 90

separation equation, 65

separation of variables (SoV),63

sifting property, 30

signal combiner, 182

simple medium, 55

single-input single-output (SISO) system, 26

solid angle, 122

source-free medium, 58

space-time code rate, 177

space-time coding (STC), 171

space-time signal, 49

spatial amplitude profile, 59

spectral efficiency, 175, 178

205

spherical Bessel functions of the first kind, 80

spherical Bessel functions of the second kind, 80

spherical coordinate system, 50

spherical Hankel functions of the first kind, 82

spherical Hankel functions of the second kind, 82

state equations, 32, 33, 35

state transition matrix, 34

state variables, 31

sub-matrix (of S), 24

superposition principle, 28

symbol detection, 164

symbol duration, 176

system loss factor, 132

symbol rate, 176

T esseral harmonics, 81

Thevenin impedance, 18

time-harmonic field, 59

time-invariant multi-port

206

system, 29

time-varying multi-port system, 33, 161

transfer function, 36

transfer matrix of multi -port system, 36, 40

transmission matrix, 177

transverse electrical (TE) wave, 63, 77

transverse electromagnetic (TEM) wave, 71, 77

transverse magnetic (TM) wave, 63

two-port network, 2

Index

Unloaded antenna, 132

upper triangular matrix, 157

Vector signal space, 140

vector wave equations, 59 voltage transfer function, 7

~aveimpedance, 70,82

wavenumbers, 66

wavenumber-frequency spectrum, 68

wavenumber vector, 61, 67, 87

wave potential, 76

weight vector, 142