## 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 [60,97],
[212, p. 94]:

*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:

The optimal Dolph-Chebyshev window *transform* can be written in
closed form [60,97,100,147]:

The zero-phase Dolph-Chebyshev window, , is then computed as the
inverse DFT of
.^{4.9} The
parameter controls the side-lobe level via the formula [147]

**Side-Lobe Level in dB**

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).

### 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 3.25 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.

### 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

- 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 [146]

#### Dolph-Chebyshev Window Main-Lobe Width

Given the window length and ripple magnitude , the main-lobe width may be computed as follows [146]:

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 [146]

**Next Section:**

Gaussian Window and Transform

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Kaiser Window