## Dolph-Chebyshev Window

The*Dolph-Chebyshev Window*(or

*Chebyshev window*, or

*Dolph window*) minimizes the

*Chebyshev norm*of the side lobes for a given main-lobe width [61,101], [224, p. 94]:

(4.43) |

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:

(4.44) |

The optimal Dolph-Chebyshev window

*transform*can be written in closed form [61,101,105,156]:

^{4.14}The parameter controls the side-lobe level via the formula [156]

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).

### Matlab for the Dolph-Chebyshev Window

In Matlab, the function`chebwin(M,ripple)`computes a length Dolph-Chebyshev window having a side-lobe level

`ripple`dB below that of the main-lobe peak. For example,

w = chebwin(31,60);designs a length window with side lobes at dB (when the main-lobe peak is normalized to 0 dB).

### Example Chebyshev Windows and Transforms

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.#### 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, and . Using the double-angle trig formula , it can be verified that

(4.47) |

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

Let denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by [155](4.48) |

where is defined by the desired ripple specification:

(4.49) |

where is the ``main lobe edge frequency'' defined by

(4.50) |

Expanding in terms of complex exponentials yields

(4.51) |

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 [155]:#### 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 (the typical case), the denominator is close to , and we have

(4.53) |

Thus, half the time-bandwidth product in radians is approximately

(4.54) |

where is the parameter often used to design Kaiser windows (§3.9).

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Gaussian Window and Transform

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Kaiser Window