Digital Signal Processing Reference
In-Depth Information
17.1.4 Periodogram
The periodogram method is similar to the spectrogram method and exploits the
STFT for spectrum estimation using a window function
g
[
k
] of length
N
w
and
centered at
k
=
b
. The time-windowed sequence
u
b
[
k
], centered at
k
=
b
,is
given by
N
w
2
u
b
[
k
]
=
x
k
+
b
−
g
[
k
]
,
0
≤
k
≤
(
N
w
−
1)
.
(17.10)
The DFT of
u
b
[
k
]isgivenby
N
w
−
1
u
b
[
k
]e
−
j
Ω
k
.
U
b
(
Ω
)
=
(17.11)
k
=
0
The periodogram method estimates the power spectrum
P
xx
(
Ω
) using the fol-
lowing equation:
1
µ
2
U
b
(
Ω
)
2
,
P
xx
(
Ω
)
=
(17.12)
where
µ
is referred to as the norm of the window function
g
[
k
] and is calculated
as follows:
µ =
g
2
[
k
]
.
(17.13)
k
While computing the STFT, different window functions attenuate the original
samples of the signal
x
[
k
] by different amounts. Inclusion of a scaling factor
of 1
/µ
2
in Eq. (17.12) reduces the bias introduced by a particular window
function.
If
g
[
k
] is a rectangular window, the estimate of the power spectrum
P
xx
(
Ω
)
computed with Eq. (17.12) is called the
periodogram
. For all other windows,
the estimate is referred to as the
modified periodogram
.
In its current form, Eq. (17.11) calculates the
N
w
-point DFT that produces
DTFT values for a set of equally spaced
N
w
frequency points within the range
Ω
=
[0
,
2
π
]. As for the spectrogram, we can zero-pad the time-windowed
sequence and increase the DFT length to obtain a denser plot in the frequency
domain.
17.1.5 Average periodogram
To estimate the power spectrum, Eq. (17.12) uses a single window with duration
of 0
≤
k
≤
(
N
w
−
1) within the input signal
x
[
k
]. Improved results are obtained
if several estimates from different locations of the signal are obtained and
the resulting values are averaged. Starting from duration 0
≤
k
≤
(
N
w
−
1),
the first iteration computes the periodogram from
x
[
k
] within the specified
duration. In the second iteration, the window is moved forward by (
N
w
-
N
o
)
samples such that there is an overlap of
N
o
between successive windows. The
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