Introduction to Spectral Analysis of Non-Stationary Random Signals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 233

() ( ) ()

x t = 100 0 y t

with:

() () ()

⎛ − at − a t − a t ⎞

1 2 n

⎜ ⎟

1 0 0 T

⎜ ⎟

() ()

⎡ ⎤

A = and b = bt b − 1 t

⎣

0 p

⎦

⎜ ⎟

⎜ ⎟

⎜ ⎟

0 1 0

⎝ ⎠

Thus, the evolutive spectrum is rational and defined by:

() ( )
() ( )

− 1

btzbtz , ,

1

( )

[9.3]

st f ,

2 π

− 1

atzatz , ,

j 2 π f

ze =

where () () are written:

atz , d , btz

() ()

− 1

()

− n

atz , =+ 1 a t z + + a t z

1 n

− 1 − + n 1

( ) () () ()

btz , = b tb tz + + b tz

0 1 − n

Grenier [GRE 81a, GRE 81b] shows that the rational evolutive spectrum behaves
in a more satisfactory way than the evolutive spectrum of equation [9.2].

9.1.2. Evolutive spectrum properties

The evolutive spectrum defined this way observes a large number of the
properties desired by Loynes [LOY 68]:
- the variance of the process can be written:

+∞
−∞

2

⎡ ⎤ =

()

2

() ( )

∫

Ext σ t = stfdf ,

⎢ ⎣ ⎦

x

- s (t, f ) coincides in the stationary case with the power spectral density of the
process;
- the evolutive spectrum is a real and positive function of t and f ;
- if the process is real, ( ) (

)

s tf , =−; st f ,