Introduction - Signal Processing and Optimization for Transceiver Systems - page 11

mization.

Part 4: Appendices

There are eight appendices at the end of the topic. They contain short discus-
sions on useful topics from inequalities, matrix theory, singular value decompo-
sitions, random processes, Wiener filtering, sampling theory, and so forth. In
addition, there are appendices at the ends of some individual chapters, which
contain useful material relevant to those chapters. Topic appendices are num-
bered as Appendix A, Appendix B, and so forth. Chapter appendices are num-
bered as Appendix 2.A (App. A at the end of Chap. 2), and so forth. Appendix
I at the end of the topic gives a summary of the main optimization results in
Part 2 of the topic, with each major result summarized in one page.

1.6 Commonly used notations

Bold-faced letters, such as A and v , indicate matrices and vectors. Superscript
T,

T , A ∗ , and A † denote, respectively, the transpose, conjugate,
and transpose-conjugate of a matrix. The determinant of a square matrix A
is denoted as det ( A ), and the trace as Tr ( A ) , with brackets omitted when
redundant. Given two Hermitian matrices A and B , the notation A ≥ B means
that A − B is positive semidefinite, and A > B means that A − B is positive
definite (Appendix B). For a continuous-time function h ( t )the Laplace transform
is denoted as H ( s )andthe Fourier transform as H ( jω ) . The frequency variable
f = ω/ 2 π is also sometimes used. For a discrete-time function g ( n )the z-
transform is denoted as G ( z ) and the Fourier transform as G ( e jω ) . The tilde
notation on a function of z is defined as follows:

∗ , and † , as in A

H ( z )= H † (1 /z ∗ ) .

Thus,

H ( z )=
n

⇒ H ( z )=
n

h ( n ) z −n h † ( n ) z n ,

so that the tilde notation effectively replaces all coe cients with the transpose
conjugates, and replaces z with 1 /z. For example,

H ( z )= h (0) + h (1) z − 1 ⇒ H ( z )= h ∗ (0) + h ∗ (1) z,

and

H ( z )= a 0 + a 1 z − 1

1+ b 1 z − 1 ⇒ H ( z )= a 0

+ a 1

z
1+ b 1 z
Note that H ( e jω )= H † ( e jω ). That is, the tilde notation reduces to transpose
conjugation on the unit circle.