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This chapter is from the book

1.6 Symbols and Common Definitions

We use the notation in Table 1.2 and assign specific definitions to the variables in Table 1.3 throughout this text.

Table 1.2 Generic notation used in this text.

by definition

convolution operator

a

bold lowercase is used to denote column vectors

A

bold uppercase is used to denote matrices

α A

non-bold letters are used to denote scalar values

|α|

magnitude of scalar α

a

vector 2-norm of a

AF

Frobenius norm of A

A

calligraphic letters denote sets

|A|

cardinality of set A

AT

matrix transpose

A*

conjugate transpose

Ac

conjugate

A1/2

matrix square root

A-1

matrix inverse

A

Moore-Penrose pseudo inverse

ak

kth entry of vector a

[A]k,l

scalar entry of A in kth row lth column

[A]:,k

kth column of matrix A

[A]:,k:m

column consisting of rows k, k + 1,..., m of matrix A

(·)

used to index a continuous signal

a(t)

continuous scalar signal and value at t

a(t)

continuous vector signal and value at t

A(t)

continuous matrix signal and value at t

[·]

used to index a discrete-time signal

a[n]

denotes discrete-time scalar signal and value at n

a[n]

discrete-time vector signal and value at n

A[n]

discrete-time matrix signal and value at n

a[n]

denotes discrete-time vector signal in frequency domain at subcarrier n

A[n]

discrete-time matrix signal in frequency domain at subcarrier n

log

denotes log2 unless otherwise mentioned

Table 1.3 Common definitions used in this text.

Es

signal energy

No

noise energy

L

channel order

tab1.jpg

discrete-time ISI channel impulse response with (L + 1) taps

tab2.jpg

frequency domain channel transfer function

y[n]

symbol sampled received signal

x[n]

symbol sampled transmitted signal

s[n]

symbol sampled transmitted signal before precoding

IN

N × N identity matrix

0N,M

N × M all zeros matrix

j

imaginary number tab3.jpg

E

expectation operator

x ~ N(m, σ2)

means that x is a Gaussian random variable with mean m and variance σ2

x ~ Nc(m, σ2)

means that x is a circularly symmetric complex Gaussian random variable with complex mean m, total variance σ2, the real and imaginary parts of x are independent, and the variance of the real and imaginary parts are each σ2/2

Aeff

effective aperture of an antenna (square meters)

Amax

maximum effective aperture of an antenna (square meters)

d

transmitter-receiver separation distance (meters)

EIRP

effective isotropic radiated power

λ

wavelength (meters)

c

speed of light in free space = 3 × 108 m/s

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