# cheby1

Chebyshev Type I filter design

## Syntax

## Description

`[`

designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type I
filter, depending on the value of `b,a`

] = cheby1(`n`

,`Rp`

,`Wp`

,`fType`

)`fType`

and the number of
elements of `Wp`

. The resulting bandpass and bandstop designs
are of order 2`n`

.

**Note**

You might encounter numerical instabilities when designing IIR filters with transfer functions for orders as low as 4. See Transfer Functions and CTF for more information about numerical issues that affect forming the transfer function.

`[`

designs a digital Chebyshev Type I filter and returns its zeros, poles, and
gain. This syntax can include any of the input arguments in previous
syntaxes.`z,p,k`

] = cheby1(___)

`[___] = cheby1(___,"s")`

designs
an analog Chebyshev Type I filter using any of the input or output
arguments in previous syntaxes.

`[`

designs a lowpass digital Chebyshev Type I filter using second-order Cascaded Transfer Functions
(CTF). The function returns matrices that list the denominator and numerator
polynomial coefficients of the filter transfer function, represented as a
cascade of filter sections. This approach generates IIR filters with improved
numerical stability compared to single-section transfer functions.`B,A`

] = cheby1(`n`

,`Rp`

,`Wp`

,"ctf")* (since R2024b)*

`[___] = cheby1(`

designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type I
filter, and returns the filter representation using the CTF format. The
resulting design sections are of order 2 (lowpass and highpass filters) or 4
(bandpass and bandstop filters).`n`

,`Rp`

,`Wp`

,`fType`

,"ctf")* (since R2024b)*

`[___,`

also returns the overall gain of the system. You must specify
`gS`

] = cheby1(___)`"ctf"`

to return `gS`

.* (since R2024b)*

## Examples

## Input Arguments

## Output Arguments

## More About

## Algorithms

Chebyshev Type I filters are equiripple in the passband and monotonic in the stopband. Type I filters roll off faster than Type II filters, but at the expense of greater deviation from unity in the passband.

`cheby1`

uses a five-step algorithm:

It finds the lowpass analog prototype poles, zeros, and gain using the function

`cheb1ap`

.It converts the poles, zeros, and gain into state-space form.

If required, it uses a state-space transformation to convert the lowpass filter to a highpass, bandpass, or bandstop filter with the desired frequency constraints.

For digital filter design, it uses

`bilinear`

to convert the analog filter into a digital filter through a bilinear transformation with frequency prewarping. Careful frequency adjustment enables the analog filters and the digital filters to have the same frequency response magnitude at`Wp`

or`w1`

and`w2`

.It converts the state-space filter back to transfer function or zero-pole-gain form, as required.

## References

[1] Lyons, Richard
G. *Understanding Digital Signal Processing*. Upper Saddle River, NJ:
Prentice Hall, 2004.

## Extended Capabilities

## Version History

**Introduced before R2006a**