An equivalent formulation is to minimize main lobe width subject to a side-lobe specification:
The Chebysev window can be regarded as the impulse response of an optimal Chebyshev lowpass filter having a zero-width passband (i.e., the main lobe consists of two ``transition bands''--see Appendix E regarding FIR filter design more generally).
w = chebwin(31,60);designs a length window with side lobes at dB (when the main-lobe peak is normalized to 0 dB).
Figure 3.25 shows the Dolph-Chebyshev window and its transform as designed by chebwin(31,40) in Matlab, and Fig.3.26 shows the same thing for chebwin(31,200). As can be seen from these examples, higher side-lobe levels are associated with a narrower main lobe and more discontinuous endpoints.
Figure 3.27 shows the Dolph-Chebyshev window and its transform as designed by chebwin(101,40) in Matlab. Note how the endpoints have actually become impulsive for the longer window length. The Hamming window, in contrast, is constrained to be monotonic away from its center in the time domain.
The ``equal ripple'' property in the frequency domain perfectly satisfies worst-case side-lobe specifications. However, it has the potentially unfortunate consequence of introducing ``impulses'' at the window endpoints. Such impulses can be the source of ``pre-echo'' or ``post-echo'' distortion which are time-domain effects not reflected in a simple side-lobe level specification. This is a good lesson in the importance of choosing the right error criterion to minimize. In this case, to avoid impulse endpoints, we might add a continuity or monotonicity constraint in the time domain (see §3.13.2 for examples).
Chebyshev and Hamming Windows Compared
Figure 3.28 shows an overlay of Hamming and Dolph-Chebyshev window transforms, the ripple parameter for chebwin set to dB to make it comparable to the Hamming side-lobe level. We see that the monotonicity constraint inherent in the Hamming window family only costs a few dB of deviation from optimality in the Chebyshev sense at high frequency.
In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.
The th Chebyshev polynomial may be defined by
- 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 .
Given the window length and ripple magnitude , the main-lobe width may be computed as follows :
This is the smallest main-lobe width possible for the given window length and side-lobe spec.
Given a prescribed side-lobe ripple-magnitude and main-lobe width , the required window length is given by 
Gaussian Window and Transform