Fundamentals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 13

This function gives the frequency distribution of the power, such that its sum
along the axis f once again gives the total power:

∫

() ()

Sf f = γ 0 = P

[1.24]

x xx

ℜ

All this is greatly detailed, for example in [DUV 91], and taken up in section 2.2
of this topic.

The reader will easily understand that the previously mentioned definitions of
energy (or power) spectral densities are based on the Fourier transform, only for the
isometry represented by equation [1.19]. We can thus easily generalize these
concepts through other orthogonal transforms (Hadamard-Walsh, Haar, etc.) that
have the same property of preserving the inner product. The axis along which the
ESD will give the energy distribution will no longer be a frequency axis, but an axis
corresponding to the subscripting of the family of orthogonal functions retained (see
[HAR 69], for example).

1.2.2. Representations of random signals

1.2.2.1. General approach
As specified in section 1.1.2, the complete knowledge of the information present
in the signal requires the knowledge of the probability laws. This occurs in practice
only when we have a theoretical model establishing the law, and only its parameters
remain to be determined: this is a problem that therefore deals with the theory of
parametric estimation (see Chapter 3).

Except for this slightly ideal case, the representations of random signals will only
be partial. Among these representations, the moments and cumulants are the most
used (if not the only ones). We will talk of knowledge of the order K of the signal if
all moments up to the K order are known, i.e. the set of:

)

(

k k

() ()

Ext

1

… xt

n

for M = 1, 2, … , K

1 n

= ∑

with Mk

i

i

Each moment is a function of n variables, and it is obvious that for a high value
of K , the representation will be very complex. The practical ambitions are largely
limited, and we practically limit ourselves to K = 2, and in a few cases to K = 3 or 4.
For a definition and presentation of cumulants, refer to [LAC 97] and [NIK 93].