# ellip

Elliptic filter design

## Syntax

[b,a] = ellip(n,Rp,Rs,Wp)
[b,a] = ellip(n,Rp,Rs,Wp,ftype)
[z,p,k] = ellip(___)
[A,B,C,D] = ellip(___)
[___] = ellip(___,'s')

## Description

example

[b,a] = ellip(n,Rp,Rs,Wp) returns the transfer function coefficients of an nth-order lowpass digital elliptic filter with normalized passband edge frequency Wp. The resulting filter has Rp decibels of peak-to-peak passband ripple and Rs decibels of stopband attenuation down from the peak passband value.

example

[b,a] = ellip(n,Rp,Rs,Wp,ftype) designs a lowpass, highpass, bandpass, or bandstop elliptic filter, depending on the value of ftype and the number of elements of Wp. The resulting bandpass and bandstop designs are of order 2n.Note:   See Limitations for information about numerical issues that affect forming the transfer function.

example

[z,p,k] = ellip(___) designs a lowpass, highpass, bandpass, or bandstop digital elliptic filter and returns its zeros, poles, and gain. This syntax can include any of the input arguments in previous syntaxes.

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[A,B,C,D] = ellip(___) designs a lowpass, highpass, bandpass, or bandstop digital elliptic filter and returns the matrices that specify its state-space representation.

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[___] = ellip(___,'s') designs a lowpass, highpass, bandpass, or bandstop analog elliptic filter with passband edge angular frequency Wp, Rp decibels of passband ripple, and Rs decibels of stopband attenuation.

## Examples

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Design a 6th-order lowpass elliptic filter with 10 dB of passband ripple, 50 dB of stopband attenuation, and a passband edge frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to $0.6\pi$ rad/sample. Plot its magnitude and phase responses. Use it to filter a 1000-sample random signal.

fc = 300; fs = 1000; [b,a] = ellip(6,10,50,fc/(fs/2)); freqz(b,a,[],fs) subplot(2,1,1) ylim([-100 20])

dataIn = randn(1000,1); dataOut = filter(b,a,dataIn);

Design a 6th-order elliptic bandstop filter with normalized edge frequencies of $0.2\pi$ and $0.6\pi$ rad/sample, 5 dB of passband ripple, and 50 dB of stopband attenuation. Plot its magnitude and phase responses. Use it to filter random data.

[b,a] = ellip(3,5,50,[0.2 0.6],'stop'); freqz(b,a)

dataIn = randn(1000,1); dataOut = filter(b,a,dataIn);

Design a 6th-order highpass elliptic filter with a passband edge frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to $0.6\pi$ rad/sample. Specify 3 dB of passband ripple and 50 dB of stopband attenuation. Convert the zeros, poles, and gain to second-order sections. Plot the magnitude and phase responses.

[z,p,k] = ellip(6,3,50,300/500,"high"); sos = zp2sos(z,p,k); freqz(sos)

Design a 20th-order elliptic bandpass filter with a lower passband frequency of 500 Hz and a higher passband frequency of 560 Hz. Specify a passband ripple of 3 dB, a stopband attenuation of 40 dB, and a sample rate of 1500 Hz. Use the state-space representation. Convert the state-space representation to second-order sections. Visualize the frequency responses.

fs = 1500; [A,B,C,D] = ellip(10,3,40,[500 560]/(fs/2)); sos = ss2sos(A,B,C,D); freqz(sos,[],fs)

Design an identical filter using designfilt. Visualize the frequency responses.

d = designfilt("bandpassiir",FilterOrder=20, ... PassbandFrequency1=500,PassbandFrequency2=560, ... PassbandRipple=3, ... StopbandAttenuation1=40,StopbandAttenuation2=40, ... SampleRate=fs); freqz(d,[],fs)

Design a 5th-order analog Butterworth lowpass filter with a cutoff frequency of 2 GHz. Multiply by $2\pi$ to convert the frequency to radians per second. Compute the frequency response of the filter at 4096 points.

n = 5; fc = 2e9; [zb,pb,kb] = butter(n,2*pi*fc,"s"); [bb,ab] = zp2tf(zb,pb,kb); [hb,wb] = freqs(bb,ab,4096);

Design a 5th-order Chebyshev Type I filter with the same edge frequency and 3 dB of passband ripple. Compute its frequency response.

[z1,p1,k1] = cheby1(n,3,2*pi*fc,"s"); [b1,a1] = zp2tf(z1,p1,k1); [h1,w1] = freqs(b1,a1,4096);

Design a 5th-order Chebyshev Type II filter with the same edge frequency and 30 dB of stopband attenuation. Compute its frequency response.

[z2,p2,k2] = cheby2(n,30,2*pi*fc,"s"); [b2,a2] = zp2tf(z2,p2,k2); [h2,w2] = freqs(b2,a2,4096);

Design a 5th-order elliptic filter with the same edge frequency, 3 dB of passband ripple, and 30 dB of stopband attenuation. Compute its frequency response.

[ze,pe,ke] = ellip(n,3,30,2*pi*fc,"s"); [be,ae] = zp2tf(ze,pe,ke); [he,we] = freqs(be,ae,4096);

Design a 5th-order Bessel filter with the same edge frequency. Compute its frequency response.

[zf,pf,kf] = besself(n,2*pi*fc); [bf,af] = zp2tf(zf,pf,kf); [hf,wf] = freqs(bf,af,4096);

Plot the attenuation in decibels. Express the frequency in gigahertz. Compare the filters.

plot([wb w1 w2 we wf]/(2e9*pi), ... mag2db(abs([hb h1 h2 he hf]))) axis([0 5 -45 5]) grid xlabel("Frequency (GHz)") ylabel("Attenuation (dB)") legend(["butter" "cheby1" "cheby2" "ellip" "besself"])

The Butterworth and Chebyshev Type II filters have flat passbands and wide transition bands. The Chebyshev Type I and elliptic filters roll off faster but have passband ripple. The frequency input to the Chebyshev Type II design function sets the beginning of the stopband rather than the end of the passband. The Bessel filter has approximately constant group delay along the passband.

## Input Arguments

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Filter order, specified as an integer scalar. For bandpass and bandstop designs, n represents one-half the filter order.

Data Types: double

Peak-to-peak passband ripple, specified as a positive scalar expressed in decibels.

If your specification, ℓ, is in linear units, you can convert it to decibels using Rp = 40 log10((1+ℓ)/(1–ℓ)).

Data Types: double

Stopband attenuation down from the peak passband value, specified as a positive scalar expressed in decibels.

If your specification, ℓ, is in linear units, you can convert it to decibels using Rs = –20 log10ℓ.

Data Types: double

Passband edge frequency, specified as a scalar or a two-element vector. The passband edge frequency is the frequency at which the magnitude response of the filter is  decibels. Smaller values of passband ripple, Rp, and larger values of stopband attenuation, Rs, both result in wider transition bands.

• If Wp is a scalar, then ellip designs a lowpass or highpass filter with edge frequency Wp.

If Wp is the two-element vector [w1 w2], where w1 < w2, then ellip designs a bandpass or bandstop filter with lower edge frequency w1 and higher edge frequency w2.

• For digital filters, the passband edge frequencies must lie between 0 and 1, where 1 corresponds to the Nyquist rate—half the sample rate or π rad/sample.

For analog filters, the passband edge frequencies must be expressed in radians per second and can take on any positive value.

Data Types: double

Filter type, specified as one of the following:

• 'low' specifies a lowpass filter with passband edge frequency Wp. 'low' is the default for scalar Wp.

• 'high' specifies a highpass filter with passband edge frequency Wp.

• 'bandpass' specifies a bandpass filter of order 2n if Wp is a two-element vector. 'bandpass' is the default when Wp has two elements.

• 'stop' specifies a bandstop filter of order 2n if Wp is a two-element vector.

## Output Arguments

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Transfer function coefficients of the filter, returned as row vectors of length n + 1 for lowpass and highpass filters and 2n + 1 for bandpass and bandstop filters.

• For digital filters, the transfer function is expressed in terms of b and a as

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{\text{b(1)}+\text{b(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{b(n+1)}\text{\hspace{0.17em}}{z}^{-n}}{\text{a(1)}+\text{a(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{a(n+1)}\text{\hspace{0.17em}}{z}^{-n}}.$

• For analog filters, the transfer function is expressed in terms of b and a as

$H\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}=\frac{\text{b(1)}\text{\hspace{0.17em}}{s}^{n}+\text{b(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{b(n+1)}}{\text{a(1)}\text{\hspace{0.17em}}{s}^{n}+\text{a(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{a(n+1)}}.$

Data Types: double

Zeros, poles, and gain of the filter, returned as two column vectors of length n (2n for bandpass and bandstop designs) and a scalar.

• For digital filters, the transfer function is expressed in terms of z, p, and k as

$H\left(z\right)=\text{k}\frac{\left(1-\text{z(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{z(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{z(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}{\left(1-\text{p(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{p(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{p(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}.$
• For analog filters, the transfer function is expressed in terms of z, p, and k as

$H\left(s\right)=\text{k}\frac{\left(s-\text{z(1)}\right)\text{\hspace{0.17em}}\left(s-\text{z(2)}\right)\cdots \left(s-\text{z(n)}\right)}{\left(s-\text{p(1)}\right)\text{\hspace{0.17em}}\left(s-\text{p(2)}\right)\cdots \left(s-\text{p(n)}\right)}.$

Data Types: double

State-space representation of the filter, returned as matrices. If m = n for lowpass and highpass designs and m = 2n for bandpass and bandstop filters, then A is m × m, B is m × 1, C is 1 × m, and D is 1 × 1.

• For digital filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{c}x\left(k+1\right)=\text{A}\text{\hspace{0.17em}}x\left(k\right)+\text{B}\text{\hspace{0.17em}}u\left(k\right)\\ y\left(k\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{C}\text{\hspace{0.17em}}x\left(k\right)+\text{D}\text{\hspace{0.17em}}u\left(k\right).\end{array}$

• For analog filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{l}\stackrel{˙}{x}=\text{A}\text{\hspace{0.17em}}x+\text{B}\text{\hspace{0.17em}}u\\ y=\text{C}\text{\hspace{0.17em}}x+\text{D}\text{\hspace{0.17em}}u.\end{array}$

Data Types: double

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### Limitations

Numerical Instability of Transfer Function Syntax

In general, use the [z,p,k] syntax to design IIR filters. To analyze or implement your filter, you can then use the [z,p,k] output with zp2sos. If you design the filter using the [b,a] syntax, you might encounter numerical problems. These problems are due to round-off errors and can occur for n as low as 4. The following example illustrates this limitation.

n = 6; Rp = 0.1; Rs = 80; Wn = [2.5e6 29e6]/500e6; ftype = 'bandpass'; % Transfer Function design [b,a] = ellip(n,Rp,Rs,Wn,ftype); % This filter is unstable % Zero-Pole-Gain design [z,p,k] = ellip(n,Rp,Rs,Wn,ftype); sos = zp2sos(z,p,k); % Plot and compare the results [h,f] = freqz(b,a,8192); hsos = freqz(sos,8192); semilogx(f/pi,mag2db(abs([h hsos]))) grid legend('TF Design','ZPK Design') xlabel('Normalized Frequency (\times\pi rad/sample)') ylabel('Magnitude (dB)')

## Algorithms

Elliptic filters offer steeper rolloff characteristics than Butterworth or Chebyshev filters, but are equiripple in both the passband and the stopband. In general, elliptic filters meet given performance specifications with the lowest order of any filter type.

ellip uses a five-step algorithm:

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

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

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

4. 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.

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

## Version History

Introduced before R2006a