cheb1ord calculates the minimum order of a digital
or analog Chebyshev Type I filter required to meet
a set of filter design specifications.

Digital Domain

[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs) returns the lowest order n of
the Chebyshev Type I filter that loses no more than Rp dB in the passband and has at least Rs dB of attenuation in the stopband. The scalar (or vector)
of corresponding cutoff frequencies Wp, is also
returned. Use the output arguments n and Wp with
the cheby1 function.

Choose the input arguments to specify the stopband and passband
according to the following table.

Description of Stopband and Passband Filter
Parameters

Parameter

Description

Wp

Passband corner frequency Wp, the
cutoff frequency, is a scalar or a two-element vector with values between 0 and 1, with 1 corresponding to the
normalized Nyquist frequency, π radians per sample.

Ws

Stopband corner frequency Ws, is a
scalar or a two-element vector with values between 0 and 1, with 1
corresponding to the normalized Nyquist frequency.

Rp

Passband ripple, in decibels. This value is the maximum
permissible passband loss in decibels.

Rs

Stopband attenuation, in decibels. This value is the
number of decibels the stopband is down from the passband.

Use the following guide to specify filters of different types.

Filter Type Stopband and Passband Specifications

Filter Type

Stopband
and Passband Conditions

Stopband

Passband

Lowpass

Wp < Ws, both
scalars

(Ws,1)

(0,Wp)

Highpass

Wp > Ws, both
scalars

(0,Ws)

(Wp,1)

Bandpass

The interval specified by Ws contains
the one specified by Wp (Ws(1) < Wp(1)
< Wp(2) < Ws(2)).

(0,Ws(1)) and (Ws(2),1)

(Wp(1),Wp(2))

Bandstop

The interval specified by Wp contains
the one specified by Ws (Wp(1) < Ws(1)
< Ws(2) < Wp(2)).

(0,Wp(1)) and (Wp(2),1)

(Ws(1),Ws(2))

If your filter specifications call for a bandpass or bandstop
filter with unequal ripple in each of the passbands or stopbands,
design separate lowpass and highpass filters according to the specifications
in this table, and cascade the two filters together.

Analog Domain

[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs,'s')finds the minimum order n and
cutoff frequencies Wp for an analog Chebyshev Type
I filter. You specify the frequencies Wp and Ws similar
to those described in the Description of Stopband and Passband Filter
Parameters table
above, only in this case you specify the frequency in radians per
second, and the passband or the stopband can be infinite.

For data sampled at 1000 Hz, design a lowpass filter with less than 3 dB of ripple in the passband defined from 0 to 40 Hz and at least 60 dB of ripple in the stopband defined from 150 Hz to the Nyquist frequency.

Wp = 40/500;
Ws = 150/500;
Rp = 3;
Rs = 60;
[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs)
[b,a] = cheby1(n,Rp,Wp);
freqz(b,a,512,1000)
title('n = 4 Chebyshev Type I Lowpass Filter')

n =
4
Wp =
0.0800

Design a bandpass filter with a passband of 60 Hz to 200 Hz, with less than 3 dB of ripple in the passband, and 40 dB attenuation in the stopbands that are 50 Hz wide on both sides of the passband.

Wp = [60 200]/500;
Ws = [50 250]/500;
Rp = 3;
Rs = 40;
[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs)
[b,a] = cheby1(n,Rp,Wp);
freqz(b,a,512,1000)
title('n = 7 Chebyshev Type I Bandpass Filter')

cheb1ord uses the Chebyshev lowpass filter
order prediction formula described in [1]. The function performs its calculations in the analog
domain for both analog and digital cases. For the digital case, it
converts the frequency parameters to the s-domain
before the order and natural frequency estimation process, and then
converts them back to the z-domain.

cheb1ord initially develops a lowpass filter
prototype by transforming the passband frequencies of the desired
filter to 1 rad/s (for low- or highpass filters) or to -1 and 1 rad/s
(for bandpass or bandstop filters). It then computes the minimum order
required for a lowpass filter to meet the stopband specification.

References

[1] Rabiner, Lawrence R., and Bernard Gold. Theory
and Application of Digital Signal Processing. Englewood
Cliffs, NJ: Prentice-Hall, 1975.