## 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]:

The zero-phase Dolph-Chebyshev window,
, is then computed as the
inverse DFT of
.^{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]:

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 (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).

**Next Section:**

Gaussian Window and Transform

**Previous Section:**

Kaiser Window