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ellipord

Minimum order for elliptic filters

Syntax

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

Description

ellipord calculates the minimum order of a digital or analog elliptic filter required to meet a set of filter design specifications.

Digital Domain

[n,Wp] = ellipord(Wp,Ws,Rp,Rs) returns the lowest order, n, of the elliptic 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 in ellip.

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 attenuated with respect to the passband response.

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] = ellipord(Wp,Ws,Rp,Rs,'s') finds the minimum order n and cutoff frequencies Wp for an analog 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.

Use ellipord for lowpass, highpass, bandpass, and bandstop filters as described in the Filter Type Stopband and Passband Specifications table above.

Examples

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Lowpass Elliptic Filter Order

For 1000 Hz data, 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, 500 Hz. Find the filter order and cutoff frequency.

Wp = 40/500; Ws = 150/500;
Rp = 3; Rs = 60;
[n,Wp] = ellipord(Wp,Ws,Rp,Rs)
n =

     4


Wp =

    0.0800

Specify the filter in terms of second-order sections and plot the frequency response.

[z,p,k] = ellip(n,Rp,Rs,Wp);
sos = zp2sos(z,p,k);
freqz(sos,512,1000)
title(sprintf('n = %d Elliptic Lowpass Filter',n))

Bandpass Elliptic Filter Order

Design a bandpass filter with a passband from 60 Hz to 200 Hz with at most 3 dB of ripple and at least 40 dB attenuation in the stopbands. Specify a sampling rate of 1 kHz. Have the stopbands be 50 Hz wide on both sides of the passband. Find the filter order and cutoff frequencies.

Wp = [60 200]/500; Ws = [50 250]/500;
Rp = 3; Rs = 40;
[n,Wp] = ellipord(Wp,Ws,Rp,Rs)
n =

     5


Wp =

    0.1200    0.4000

Specify the filter in terms of second-order sections and plot the frequency response.

[z,p,k] = ellip(n,Rp,Rs,Wp);
sos = zp2sos(z,p,k);
freqz(sos,512,1000)
title(sprintf('n = %d Elliptic Bandpass Filter',n))

More About

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Algorithms

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

ellipord initially develops a lowpass filter prototype by transforming the passband 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 required for a lowpass filter to meet the stopband specification.

References

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

See Also

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