:

ceiling operator that returns the smallest integer larger than or equal to

:

the set of integers

:

the set of positive integers (great than 0)

:

the set of real numbers

:

the set of complex numbers

:

arbitrary matrix of size constructed by matrix entrance with where , and

:

identity matrix of size

:

the space of all squares summable discrete functions/sequences

:

the space of all Lesbesgue squares integrable functions

:

real part of a number, matrix, or a function

:

imaginary part of a number, matrix, or a function

:

Sinc function

:

Kronecker delta, or Dirac‐delta function, or unit impulse with infinite size

:

root of and is equal to

:

th root of unity and equals to

:

discrete Fourier transform operator

:

inverse discrete Fourier transform operator

:

discrete Fourier transform matrix of size ; with . The Fourier matrix is of arbitrary size when is missing

:

discrete cosine transform matrix of size ; the cosine matrix is of arbitrary size when is missing

:

convolution operator

:

interval in domain ; the interval domain is arbitrary when is missing

:

angular frequency

:

spatial angular frequency in domain

:

sampling angular frequency with sampling interval in domain ()

:

comb filter impulse response function in domain with being the separation between adjacent impulses in the comb filter;

:

frequency response of the comb filter , i.e.

:

impulse train in analog domain with being the separation between adjacent indices

, with

:

discrete impulse sequence

, with

A word on notations

1. 1. (Indices) We denote continuous variable and discrete variable induced signals as and , respectively.
2. 2. (Vector‐matrix) The blackboard bold is used to represent matrix‐valued signal and function, and is used to represent the vector‐valued signal and function. The normal characters are used to represent signal in scalar form.
3. 3. (Rows versus columns) For vector‐matrix multiplication written as , we may take vector as a row vector.