The Chebyshev norm is also called the norm, uniform norm, minimax norm, or simply the maximum absolute value.
An equivalent formulation is to minimize main-lobe width subject to a side-lobe specification:
|Side-Lobe Level in dB||(4.45)|
Thus, gives side-lobes which are dB below the main-lobe peak. Since the side lobes of the Dolph-Chebyshev window transform are equal height, they are often called ``ripple in the stop-band'' (thinking now of the window transform as a lowpass filter frequency response). The smaller the ripple specification, the larger has to become to satisfy it, for a given window length .
The Chebyshev window can be regarded as the impulse response of an optimal Chebyshev lowpass filter having a zero-width pass-band (i.e., the main lobe consists of two ``transition bands''--see Chapter 4 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 3.31 shows the Dolph-Chebyshev window and its transform as designed by chebwin(31,40) in Matlab, and Fig.3.32 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 3.33 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 3.34 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.
Dolph-Chebyshev Window Theory
In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.
The th Chebyshev polynomial may be defined by
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, and . Using the double-angle trig formula , it can be verified that
for . The following properties of the Chebyshev polynomials are well known:
- 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
where is defined by the desired ripple specification:
where is the ``main lobe edge frequency'' defined by
Expanding in terms of complex exponentials yields
where . Thus, the coefficients give the length Dolph-Chebyshev window in zero-phase form.
Dolph-Chebyshev Window Main-Lobe Width
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.
Dolph-Chebyshev Window Length Computation
For (the typical case), the denominator is close to , and we have
Thus, half the time-bandwidth product in radians is approximately
where is the parameter often used to design Kaiser windows (§3.9).
Gaussian Window and Transform