Digital Signal Processing Reference
In-Depth Information
of the strongest secondary lobe brought to the amplitude of the main lobe. Consider
f
max
as the abscissa of the maximum of the strongest secondary lobe:
⎧
⎫
ˆ
dw
⎪
⎪
0,
T
()
[2.83]
f
=
max
f
≠
0
f
=
0
⎨
⎬
max
df
⎪
⎪
⎩
⎭
The amplitude of the strongest secondary lobe brought to the amplitude of the
main lobe is thus:
(
)
()
ˆ
wf
w
0,
T
max
[2.84]
ˆ
0
0,
T
This quantity if independent of
T.
We generally express it in decibels, that is:
(
)
()
⎛
⎞
wf
ˆ
0,
T
max
(
)
()
⎜
⎟
=
20 log
Wf
−
W
0
[2.85]
10
dB
max
dB
⎜
ˆ
⎟
w
0
0,
T
⎝
⎠
Generally, the amplitude of the lobes reduces as the frequency increases. In the
plane (log
f
,
WdB
(
f
)), the maxima of the lobes pass through a straight line whose
slope can be measured in dB/octave, increase in the value of the ordinate of this line
when the frequency is multiplied by 2. This slope measures the “speed” of
suppression of the secondary lobes when the frequency increases.
A third characteristic is the positivity of the transform of even windows. In Table
2.3, the Fourier transform of the even window
()
T
wt
−
is obtained according to
,
22
e
π
function
(according to the delay theorem). We immediately notice that the Bartlett window
verifies the positivity property.
()
j
f t
the transform of
0,
wt
by multiplying this transform by the
Figure 2.7 graphically represents these indices in the case of the rectangular
window. Table 2.4 gives the value of these different indices for various windows.
Qualitatively speaking, selecting a window representing weak secondary lobes (in
order to detect a sine wave of weak power) at the value of a large main lobe (and
thus has a loss in resolution: adding the main lobes of two sine waves of similar
frequency will give only a single lobe).
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