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
[#!Dolph46!#,#!Harris78!#],
[#!RabinerAndGold!#, p. 94]:

(4.43) |

The Chebyshev norm is also called the norm,

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 [#!Dolph46!#,#!Harris78!#,#!Helms71!#,#!Lyons01!#]:

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 [#!Lyons01!#]

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
- Example Chebyshev Windows and Transforms
- Chebyshev and Hamming Windows Compared
- Dolph-Chebyshev Window Theory

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