Digital Signal Processing Reference
In-Depth Information
For finite
N
, the convolution of the real spectrum with Bartlett's window can
result in a spectral leakage phenomenon. For example, for a cisoid of amplitude
a
,
the frequency
v
0
, of uniformly distributed initial phase between 0 and 2π,
the
periodogram obtained can be written as
2
ˆ
(
)
−
−
.
For an addition of two
cisoids, the average value of the periodogram obtained is the sum of the
contributions of each cisoid. If these two cisoids are of largely different powers, the
secondary lobes of Bartlett's window centered on the frequency of the sine wave of
high power, that much higher as
N
is smaller, can mask the main lobe of Bartlett's
window centered on the frequency of the sine wave of low power. Thus, the
periodogram is less sensitive.
aB
νν
NN
,
0
To calculate and draw the periodogram, the frequency axis must be calibrated
[0, 1[, for example, using a regular calibration of
M
points on this interval
(with
M
≥
N
):
ν
⎧
⎫
∈
,0
≤ ≤
M
−
1
[5.47]
⎨
⎬
M
⎩
⎭
Thus, we have, for all
{
}
…
:
∈
0,
,
M
−
1
2
N
−
1
1
⎛⎞
=
k
M
−
j
2
π
⎜
⎝⎠
∑
()
I
x k e
[5.48]
xx
MN
k
=
0
By adding
M - N
zeros to the series of
N
available points (
zero-padding
)
,
and by
choosing for
M
a power of 2, we calculate this estimator using fast Fourier
transform algorithms (FFT: see Chapter 2).
This aspect provides other interpretations of the periodogram. Let us consider
the series of
M
points (that is (
x
(0), …,
x
(
N -
1)) completed by
M - N
zeros) contain
a period of a periodic signal
x
p
of period
M.
This periodic signal can be broken
varying from 0 to
N
- 1, of
down into a sum of
M
cisoids of frequencies
,
M
(
)
amplitude
()
()
ˆ
p
x
and initial phase
ˆ
arg
x
:
p
M
−
1
k
M
e
π
j
2
()
=
∑
()
ˆ
xk
x
[5.49]
p
p
=
0
where ˆ
p
is the discrete Fourier transform of the series
x
p
,
that is:
x
N
−
1
1
k
M
−
j
2
π
()
=
∑
()
ˆ
x
x k e
[5.50]
p
M
k
=
0
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