Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 123

e

If the power spectral density S xx does not vary much in the bandwidth of the

filter, then:

+∞

() 2 ()

∫

ˆ

PS f ≈ hf f

[5.27]

xx

−∞

Thus, the power of the output filter e , normalized by the energy of the pulse
response of this filter, restores the power spectral density in the center frequency, in
the case where this density does not vary much in the bandwidth.

In practice, the frequencies of interest are uniformly distributed along the
frequency axis:

f = f

[5.28]

1

and the filters are obtained by spectral offset of the centre low-pass filter on the zero
frequency:

ˆ ˆ

() ( )

hf hf f = −

[5.29]

0

This condition being fulfilled, the output power y while considering in
addition that the transfer function module of the low-pass filter of pulse response h 0
has an even module can be written as:

2

+∞
−∞

ˆ

( ) ()

∫

P = h f − f S f df

[5.30]

xx

2

⎛ ⎞

ˆ

()

= S ⊗ h f

[5.31]

⎜ ⎟

xx

⎝ ⎠

To ensure that there is no loss of data, we set a filter bank with perfect
reconstruction ()

( )

()

x t = ∑ y t

, which leads to the following condition:

∑

ˆ

()

hf = 1

[5.32]

()

or, something less restrictive, that x ( t ) and

y t

have the same power spectral

∑

density, such as the condition:

() 2

∑

ˆ

hf = 1

[5.33]