cheb1ord

Chebyshev Type I filter order

Syntax

[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs)

[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs,'s')

Description

[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.

[n,Wp] = cheb1ord(Wp,Ws,Rp,Rs,'s') designs a lowpass, highpass, bandpass, or bandstop analog Chebyshev Type I filter with cutoff angular frequencies Wp.

Examples

Chebyshev Type I Filter Design

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)

n = 4

Wp = 0.0800

[b,a] = cheby1(n,Rp,Wp);
freqz(b,a,512,1000) 
title('n = 4 Chebyshev Type I Lowpass Filter')

Chebyshev Type I Bandpass Filter Design

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)

n = 7

Wp = 1×2

    0.1200    0.4000

[b,a] = cheby1(n,Rp,Wp);
freqz(b,a,512,1000)
title('n = 7 Chebyshev Type I Bandpass Filter')

Input Arguments

Wp — Passband corner (cutoff) frequency

scalar | two-element vector

Passband corner (cutoff) frequency, specified as a scalar or a two-element vector with values between 0 and 1 inclusive, with 1 corresponding to the normalized Nyquist frequency, π rad/sample. For digital filters, the unit of passband corner frequency is in radians per sample. For analog filters, passband corner frequency is in radians per second, and the passband can be infinite. The values of Wp and Ws determine the type of filter cheb1ord returns:

• If Wp and Ws are both scalars and Wp < Ws, then cheb1ord returns the order and cutoff frequency of a lowpass filter. The stopband of the filter ranges from Ws to 1, and the passband ranges from 0 to Wp.

• If Wp and Ws are both scalars and Wp > Ws, then cheb1ord returns the order and cutoff frequency of a highpass filter. The stopband of the filter ranges from 0 to Ws, and the passband ranges from Wp to 1.

• If Wp and Ws are both vectors and the interval specified by Ws contains the interval specified by Wp (Ws(1) < Wp(1) < Wp(2) < Ws(2)), then cheb1ord returns the order and cutoff frequencies of a bandpass filter. The stopband of the filter ranges from 0 to Ws(1) and from Ws(2) to 1. The passband ranges from Wp(1) to Wp(2).

• If Wp and Ws are both vectors and the interval specified by Wp contains the interval specified by Ws (Wp(1) < Ws(1) < Ws(2) < Wp(2)), then cheb1ord returns the order and cutoff frequencies of a bandstop filter. The stopband of the filter ranges from Ws(1) to Ws(2). The passband ranges from 0 to Wp(1) and from Wp(2) to 1.

  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))

Data Types: single | double

Note

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

Ws — Stopband corner frequency

scalar | two-element vector

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

• For digital filters, stopband corner frequency is in radians per sample.

• For analog filters, stopband corner frequency is in radians per second and the stopband can be infinite.

Note

The values of Wp and Ws determine the filter type.

Rp — Passband ripple

scalar

Passband ripple, specified as a scalar in dB.

Data Types: single | double

Rs — Stopband attenuation

scalar

Stopband attenuation, specified as a scalar in dB.

Data Types: single | double

Output Arguments

n — Lowest filter order

integer scalar

Lowest filter order, returned as an integer scalar.

Wp — Passband corner frequency

scalar | two-element vector

Passband corner frequency, returned as a scalar or a two-element vector. Use the output arguments n and Wp with the cheby1 function.

Algorithms

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 lowpass or highpass filters) or to -1 and 1 rad/s (for bandpass or bandstop filters). It then computes the order and natural frequency required for a lowpass filter to match the passband specification exactly when using the values in the cheby1 function.

References

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

All inputs must be constants. Expressions or variables are allowed if their values do not change.