cheb2ord

Chebyshev Type II filter order

Syntax

[n,Ws] = cheb2ord(Wp,Ws,Rp,Rs)

[n,Ws] = cheb2ord(Wp,Ws,Rp,Rs,'s')

Description

[n,Ws] = cheb2ord(Wp,Ws,Rp,Rs) returns the lowest order n of the Chebyshev Type II 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 Ws is also returned.

[n,Ws] = cheb2ord(Wp,Ws,Rp,Rs,'s') designs a lowpass, highpass, bandpass, or bandstop analog Chebyshev Type II filter with cutoff angular frequencies Ws.

Examples

Chebyshev Type II 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 attenuation in the stopband defined from 150 Hz to the Nyquist frequency.

Wp = 40/500;
Ws = 150/500;
Rp = 3;
Rs = 60;

[n,Ws] = cheb2ord(Wp,Ws,Rp,Rs)

n = 4

Ws = 0.3000

[b,a] = cheby2(n,Rs,Ws);

freqz(b,a,512,1000)
title('n = 4 Chebyshev Type II Lowpass Filter')

Chebyshev Type II 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,Ws] = cheb2ord(Wp,Ws,Rp,Rs)

n = 7

Ws = 1×2

    0.1000    0.5000

[b,a] = cheby2(n,Rs,Ws);

freqz(b,a,512,1000)
title('n = 7 Chebyshev Type II 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 cheb2ord returns:

• If Wp and Ws are both scalars and Wp < Ws, then cheb2ord 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 cheb2ord 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 cheb2ord 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 cheb2ord 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 ripple 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.

Ws — Stopband corner frequency

scalar | two-element vector

Stopband corner frequency, returned as a scalar or a two-element vector. Use the output arguments n and Ws with the cheby2 function.

Algorithms

cheb2ord 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.

cheb2ord initially develops a lowpass filter prototype by transforming the stopband frequencies of the desired filter to 1 rad/s (for low- and highpass filters) and to -1 and 1 rad/s (for bandpass and bandstop filters). It then computes the minimum order and natural frequency required for a lowpass filter to match the stopband specification exactly when using the values in the cheby2 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.