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## 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]:

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:

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**
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 low-pass

filter
frequency response). The smaller the ripple specification, the larger

has to become to satisfy it, for a given window length

.

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).

**Subsections**

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**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.