Dolph-Chebyshev Window Theory
In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.
Chebyshev Polynomials
The
th Chebyshev polynomial may be defined by
![]() |
(4.46) |
The first three even-order cases are plotted in Fig.3.35. (We will only need the even orders for making Chebyshev windows, as only they are symmetric about time 0.) Clearly,



![]() |
(4.47) |
for

is an
th-order polynomial in
.
is an even function when
is an even integer, and odd when
is odd.
has
zeros in the open interval
, and
extrema in the closed interval
.
for
.
Dolph-Chebyshev Window Definition
Let
denote the desired window length. Then the zero-phase
Dolph-Chebyshev window is defined in the frequency domain by
[155]
![]() |
(4.48) |
where

![]() |
(4.49) |
where

![]() |
(4.50) |
Expanding

![]() |
(4.51) |
where



Dolph-Chebyshev Window Main-Lobe Width
Given the window length
and ripple magnitude
, the main-lobe
width
may be computed as follows [155]:
![\begin{eqnarray*}
x_0 &=& \cosh\left[\frac{\cosh^{-1}\left(\frac{1}{r}\right)}{M-1}\right]\\
\omega_c &=& 2\cos^{-1}\left(\frac{1}{x_0}\right)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img553.png)
This is the smallest main-lobe width possible for the given window length and side-lobe spec.
Dolph-Chebyshev Window Length Computation
Given a prescribed side-lobe ripple-magnitude
and main-lobe width
, the required window length
is given by [155]
![]() |
(4.52) |
For


![]() |
(4.53) |
Thus, half the time-bandwidth product in radians is approximately
![]() |
(4.54) |
where

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Matlab for the Gaussian Window
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Chebyshev and Hamming Windows Compared