Fundamentals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 9

Another type of time dependence is defined when the considered characteristics
recur periodically on the time axis: we then speak of cyclostationarity. For
example, if:

( ) ( ( ) (

k k k k

() () ) ( ) )

Ext

1

… xt

n

= Ext + T

1

.. xt + T

n

∀ t , … , t

1 n 1 n 1 n

the signal will be called cyclostationary for the moment of order M (see for
example [GAR 89]).

1.2. Representations of signals

The various classes of signals defined above can be represented in many ways,
i.e. forming of the information present in the signal, which does not have any initial
motive other than to highlight certain characteristics of the signal. These
representations may be complete, if they carry all the information of the signal, and
partial otherwise.

1.2.1. Representations of deterministic signals

1.2.1.1. Complete representations
As the postulate of the deterministic signal is that any information that affects it
is contained in the function x ( t ) itself, it is clear that any bijective transformation
applied to x ( t ) will preserve this information.

From a general viewpoint, such a transformation may be written with the help of
integral transformations on the signal, such as:

⎡ ⎤

( )

∫∫

( ) ( ) ( )

… … … … [1.11]

Ru , … u = H G xt ⎡ , … , xt ⎤ ⋅ ψ t , , t u , , , u dt dt

⎣ ⎦

1 m

⎢

1 n 1 n 1 m 1 n

⎥

⎣ ⎦

where the operators H [.] and G [.] are not necessarily linear. The kernel
( )

e π

_2 j ft

ψ = has a fundamental role to play in the characteristics that we wish
to display. This transformation will not necessarily retain the dimensions, and in
general we will use representations such as

tf ,

mn We will expect it to be
inversible (i.e. there exists an exact inverse transformation), and that it helps
highlight certain properties of the signal better than the function x ( t ) itself. It must
also be understood that the demonstration of a characteristic of the signal may
considerably mask other characteristics: the choice of H [.], G [.] and ψ (.) governs
the choice of what will be highlighted in x ( t ). In the case of discrete time signals,
discrete expressions of the same general form exist.

.