1
R
Set of Real numbers
C
Set of complex numbers
Rm
Real Euclidean space of dimension m
H
Denotes a Hilbert space
·T
Matrix transpose
·H
Hermitian operation for a matrix
·∗
Complex conjucation
x
Bold letter denotes a vector
x
Letter in normal font denotes a scalar
X
Capital bold letter denotes a matrix
xi or x(i)
The ith component of vector x
X(n) or x(n) or xi (n) or x(n)
Time dependence on the time index n,
depending if it is a matrix, vector, vector component or scalar.
The gradient of a function
∇f (x) or ∇x f (x) or ∂f∂(xx)
X(i, j) or Xij
The (i, j) element of matrix X
Xij (n)
Time dependence on the time index n of the i, j element of a matrix
X
E[·]
Denotes expectation
x̂
Denotes the estimate of a variable, x
In case that authors wish to distinguish the random variable from its corrsponding values that ocur from the experiments, the following notation is proposed.
Very often, authors use bold fonts to denote the random variabe and non-bold
for the values that result from the experiment. However, we have adopted the
use of bold fonts for the vectors.
X
x
x
Denotes a random matrix
Denotes a random vector
Denotes a random scalar
x⊥y
Denotes orthogonal vectors
|x| and |z|
Absolute values for a real number x and a complex number z
||x|| or ||x||2
Denotes Euclidean norm
||A||F
Denotes Frobenious norm
||x||p
Denotes lp norm
Re(x)
Denotes the real part of a complex number
Im(x)
Denotes the imaginary part of a complex number
Im or I
Denotes the identity matrix of dimension n × n
vec(A)
Collumn vector formed by stacking the collumns of A
diag(A)
Collumn vector with the diagonal entries of A
rank(A)
Rank of matrix A
A⊗B
The Kronecker product of two matrices
det(·)
The determinant of a matrix
Tr(·)
The trace of a matrix
λi , i = 1, 2, . . . , m
The eigenvalues of an m × m matrix.
2
R(A)
Range space of A
N (A)
Null space of A
P (·)
Probability of a discrete event
p(·)
Probability density function of a random variable
R or Rx
The autocorrelation matrix of a random vector x
Σ or Σx
The covariance matrix of a random vector x
σ 2 or σx2
The variance of the random variable x
log a
The logarithm of a relative to base 10
ln a
The natural logarithm of a
exp (·)
The exponential function
A>B
Means that A − B is positive definite
A≥B
Means that A − B is positive semidefinite
X(z)
The z-transform of sequence x(n)
X(ejω )
The Fourier transform of a sequence x(n)
X(ejΩT )
The Fourier transform of a sampled sequence x(nT ), with sampling
period T
X(jΩ)
The Fourier transform of a function x(t)
X(m) or Xm
The DFT of x(n)
F (Ωx , Ωy )
The Fourier transform of f (x, y)
F (ωx , ωy ) or F (ω1 , ω2 ), etc.
The Fourier transform of f (n, m)
F (k, l)
The DFT of the sequence f (n, m)
X(s)
The Laplase transform of a function x(t)
Notation of a matrix A


A(1, 1) A(1, 2) A(1, 3)
A =  A(2, 1) A(2, 2) A(2, 3) 
A(3, 1) A(3, 2) A(3, 3)
Notation of a vector x



x = [x1 , x2 , . . . , xm ]T = 

x1
x2
..
.





xm
It will also be nice to make an effort to use the following symbols:
• For input signals: u(n) or x(n)
• For the output signal y(n) and d(n) for the desired response
• For the implulse response of a system: h(n) or w(n) or g(n) and similarly
for the point spread function, h(n, m), or w(n, m) or g(n, m). Similar
notation for analogue systems, i.e., h(t) or h(x, y).
• e(n) for an error estimation sequence, ea (n) for an a-posteriori error sequence and ep (n) for an a-priori error sequence