Documentation

[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.

[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.

[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.

[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.

[___] = 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

Lowpass Elliptic Transfer Function

Design a 6th-order lowpass elliptic filter with 5 dB of passband ripple, 40 dB of stopband attenuation, and a passband edge frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to $0.6 π$ rad/sample. Plot its magnitude and phase responses. Use it to filter a 1000-sample random signal.

[b,a] = ellip(6,5,40,0.6);
freqz(b,a)

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

Bandstop Elliptic Filter

Design a 6th-order elliptic bandstop filter with normalized edge frequencies of $0.2 π$ and $0.6 π$ 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);

Highpass Elliptic Filter

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

[z,p,k] = ellip(6,3,50,300/500,'high');
sos = zp2sos(z,p,k);
fvtool(sos,'Analysis','freq')

Bandpass Elliptic Filter

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. Design an identical filter using designfilt.

[A,B,C,D] = ellip(10,3,40,[500 560]/750);
d = designfilt('bandpassiir','FilterOrder',20, ...
    'PassbandFrequency1',500,'PassbandFrequency2',560, ...
    'PassbandRipple',3, ...
    'StopbandAttenuation1',40,'StopbandAttenuation2',40, ...
    'SampleRate',1500);

Convert the state-space representation to second-order sections. Visualize the frequency responses using fvtool.

sos = ss2sos(A,B,C,D);
fvt = fvtool(sos,d,'Fs',1500);
legend(fvt,'ellip','designfilt')

Comparison of Analog IIR Lowpass Filters

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

n = 5;
f = 2e9;

[zb,pb,kb] = butter(n,2*pi*f,'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*f,'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*f,'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*f,'s');
[be,ae] = zp2tf(ze,pe,ke);
[he,we] = freqs(be,ae,4096);

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

plot(wb/(2e9*pi),mag2db(abs(hb)))
hold on
plot(w1/(2e9*pi),mag2db(abs(h1)))
plot(w2/(2e9*pi),mag2db(abs(h2)))
plot(we/(2e9*pi),mag2db(abs(he)))
axis([0 4 -40 5])
grid
xlabel('Frequency (GHz)')
ylabel('Attenuation (dB)')
legend('butter','cheby1','cheby2','ellip')

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.

Input Arguments

`n` — Filter order
integer scalar

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

Data Types: double

`Rp` — Peak-to-peak passband ripple
positive scalar

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

`Rs` — Stopband attenuation
positive scalar

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

`Wp` — Passband edge frequency
scalar | two-element vector

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 –Rp 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

`ftype` — Filter type
`'low'` | `'bandpass'` | `'high'` | `'stop'`

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

`b,a` — Transfer function coefficients
row vectors

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

$H (z) = \frac{B (z)}{A (z)} = \frac{b(1) + b(2) z^{- 1} + \dots + b(n+1) z^{- n}}{a(1) + a(2) z^{- 1} + \dots + a(n+1) z^{- n}} .$
For analog filters, the transfer function is expressed in terms of b and a as

$H (s) = \frac{B (s)}{A (s)} = \frac{b(1) s^{n} + b(2) s^{n - 1} + \dots + b(n+1)}{a(1) s^{n} + a(2) s^{n - 1} + \dots + a(n+1)} .$

Data Types: double

`z,p,k` — Zeros, poles, and gain
column vectors, scalar

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
$H (z) = k \frac{(1 - z(1) z^{- 1}) (1 - z(2) z^{- 1}) \dots (1 - z(n) z^{- 1})}{(1 - p(1) z^{- 1}) (1 - p(2) z^{- 1}) \dots (1 - p(n) z^{- 1})} .$
For analog filters, the transfer function is expressed in terms of z, p, and k as
$H (s) = k \frac{(s - z(1)) (s - z(2)) \dots (s - z(n))}{(s - p(1)) (s - p(2)) \dots (s - p(n))} .$

Data Types: double

`A,B,C,D` — State-space matrices
matrices

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

$\begin{matrix} x (k + 1) = A x (k) + B u (k) \\ y (k) = C x (k) + D u (k) . \end{matrix}$
For analog filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{l} \dot{x} = A x + B u \\ y = C x + D u . \end{array}$

Data Types: double

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