Biomedical Engineering Reference
In-Depth Information
From trigonometric identities, an alternate method for expressing the cosine win-
dow is to write the expression in terms of the “Sine” function, when the data are not
symmetrical about the origin. For example if
α
=
2 the cosine model becomes as shown
in by (16.5).
sin
2
(
x
)
cos
2
(
x
)
=
1
−
(16.5)
Hence, the basic cosine model could be expressed as (16.6).
sin
alpha
n
for alpha values of 2 to 4
N
w
(
n
)
=
(16.6)
16.3.6 Welch Window
This window is a modification of the Parzen window with some better characteristics
especially in the coherent gain; however, the Welch window produces greater main lobe
leakage due to a wider spreading. The mathematical model for the Welch window is given
by (16.7). Notice the polynomial similarity between Parzen and Welch expressions.
i
2
−
0
.
5(
N
−
1)
w
(
i
)
=
1
−
(16.7)
0
.
5(
N
−
1)
This type of window is not good for signals with closely related spectral compo-
nents. The main side-lobe effect can be worse than the leakage from the rectangular
window.
16.3.7 Blackman Window
The name “Blackman Window” was given to honor Dr. Blackman, a pioneer in the
harmonic signal detection. There are several windows under his name, differing in the
amount of variables used. For simplicity only the “Exact Blackman Window” will be
described, since this window has superb tone discrimination characteristics (used in
acoustics).
The expression for the Exact Blackman Window is given by (16.8).
50 cos
2
n
08
2
2
n
N
N
w
(
n
)
=
0
.
42
+
0
.
+
0
.
(16.8)
where
n
=−
N
/
2,.... ,
−
1, 0,
+
1,.... ,
+
N
/
2
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