Digital Signal Processing Reference
In-Depth Information
This last condition is more or less fulfilled if the frequential step
f
+
−=
f
f
1
1
ˆ
()
⎧
⎫
h
0
2
⎪
⎪
()
is equal to the bandwidth a-3 dB of the filters (interval length
fh f
>
).
⎨
⎬
0
⎪
⎪
⎩
⎭
From this analysis, we obtain an estimator of power spectral density using a
realization
x
(
t
), 0
≤
t
≤
T
from the following sequence of operations:
- filtering.
The signal
x
(
t
), 0
≤
t
≤
T
is filtered by the bank; we obtain the filter
signals
()
y t
, 0
≤
t
≤
T
;
-
quadration.
We calculate the instantaneous power of these filter signals, that is
()
2
, 0
≤
t
≤
T
;
- integration.
The power of each filter signal is estimated, without bias, by time
averaging the instantaneous power:
()
2
yt
1
T
=
∫
P
y
t
dt
[5.34]
0
-
normalization.
The estimation of the spectral density is obtained by
standardizing the estimated power by the energy of the pulse response of the
()
2
+∞
+∞
ˆ
2
()
∫
∫
E
=
h
f
df
=
h
t dt
filters:
0
0
−∞
−∞
P
()
Sf
=
[5.35]
xx
E
h
This estimator is approximately without bias when the spectral density
S
xx
does
not vary much in a frequential sampling step. For a sine wave
(
)
ae
π +
where
φ
is an initial phase uniformly distributed between 0 and 2
π
, the spectrum is equal to
a
2
δ
(
f - f
s
); we obtain from formula [5.31] the average value of the spectrum:
j
2
f
t
s
)
2
ˆ
(
hf
−
f
0
s
(
)
()
2
ES
f
=
a
xx
E
h
The pulse of the exact spectrum leaks to the adjacent analysis frequencies; this
phenomenon is more prominent when the frequential sampling step is large. In a
signal made up of narrow and large band components, the spectral contribution of
the narrow band components can be masked by the spectral contribution of large
band components, and all the more when the centre frequency of these narrow band
components is farther from an analysis frequency.
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