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# ellip

Elliptic filter design

## Syntax

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

## Description

ellip designs lowpass, bandpass, highpass, and bandstop digital and analog elliptic filters. Elliptic filters offer steeper rolloff characteristics than Butterworth or Chebyshev filters, but are equiripple in both the pass- and stopbands. In general, elliptic filters meet given performance specifications with the lowest order of any filter type.

### Digital Domain

[z,p,k] = ellip(n,Rp,Rs,Wp) designs an order n lowpass digital elliptic filter with normalized passband edge frequency Wp, Rp dB of ripple in the passband, and a stopband Rs dB down from the peak value in the passband. It returns the zeros and poles in length n column vectors z and p and the gain in the scalar k.

The normalized passband edge frequency is the edge of the passband, at which the magnitude response of the filter is -Rp dB. For ellip, the normalized cutoff frequency Wp is a number between 0 and 1, where 1 corresponds to half the sampling frequency (Nyquist frequency). Smaller values of passband ripple Rp and larger values of stopband attenuation Rs both lead to wider transition widths (shallower rolloff characteristics).

If Wp is a two-element vector, Wp = [w1 w2], ellip returns an order 2*n bandpass filter with passband w1 < ω < w2.

[z,p,k] = ellip(n,Rp,Rs,Wp,'ftype') designs a highpass, lowpass, or bandstop filter, where the string 'ftype' is one of the following:

• 'high' for a highpass digital filter with normalized passband edge frequency Wp

• 'low' for a lowpass digital filter with normalized passband edge frequency Wp

• 'stop' for an order 2*n bandstop digital filter if Wp is a two-element vector, Wp = [w1 w2]. The stopband is w1 < ω < w2.

With different numbers of output arguments, ellip directly obtains other realizations of the filter. To obtain the transfer function form, use two output arguments as shown below.

 Note:   See Limitations for information about numerical issues that affect forming the transfer function.

[b,a] = ellip(n,Rp,Rs,Wp) designs an order n lowpass digital elliptic filter with normalized passband edge frequency Wp, Rp dB of ripple in the passband, and a stopband Rs dB down from the peak value in the passband. It returns the filter coefficients in the length n+1 row vectors b and a, with coefficients in descending powers of z.

[b,a] = ellip(n,Rp,Rs,Wp,'ftype') designs a highpass, lowpass, or bandstop filter, where the string 'ftype' is 'high', 'low', or 'stop', as described above.

To obtain state-space form, use four output arguments as shown below:

[A,B,C,D] = ellip(n,Rp,Rs,Wp) or

[A,B,C,D] = ellip(n,Rp,Rs,Wp,'ftype') where A, B, C, and D are

and u is the input, x is the state vector, and y is the output.

### Analog Domain

[z,p,k] = ellip(n,Rp,Rs,Wp,'s') designs an order n lowpass analog elliptic filter with angular passband edge frequency Wp rad/s and returns the zeros and poles in length n or 2*n column vectors z and p and the gain in the scalar k.

The angular passband edge frequency is the edge of the passband, at which the magnitude response of the filter is -Rp dB. For ellip, the angular passband edge frequency Wp must be greater than 0 rad/s.

If Wp is a two-element vector with w1 < w2, then ellip(n,Rp,Rs,Wp,'s') returns an order 2*n bandpass analog filter with passband w1 < ω< w2.

[z,p,k] = ellip(n,Rp,Rs,Wp,'ftype','s') designs a highpass, lowpass, or bandstop filter, where the string 'ftype' is 'high', 'low', or 'stop', as described above.

With different numbers of output arguments, ellip directly obtains other realizations of the analog filter. To obtain the transfer function form, use two output arguments as shown below:

[b,a] = ellip(n,Rp,Rs,Wp,'s') designs an order n lowpass analog elliptic filter with angular passband edge frequency Wp rad/s and returns the filter coefficients in the length n+1 row vectors b and a, in descending powers of s, derived from this transfer function:

[b,a] = ellip(n,Rp,Rs,Wp,'ftype','s') designs a highpass, lowpass, or bandstop filter, where the string 'ftype' is 'high', 'low', or 'stop', as described above.

To obtain state-space form, use four output arguments as shown below:

[A,B,C,D] = ellip(n,Rp,Rs,Wp,'s') or

[A,B,C,D] = ellip(n,Rp,Rs,Wp,'ftype','s') where A, B, C, and D are

and u is the input, x is the state vector, and y is the output.

## Examples

### Lowpass Filter

For data sampled at 1000 Hz, design a sixth-order lowpass elliptic filter with a passband edge frequency of 300 Hz, which corresponds to a normalized value of 0.6, 3 dB of ripple in the passband, and 50 dB of attenuation in the stopband:

```[z,p,k] = ellip(6,3,50,300/500);
[sos,g] = zp2sos(z,p,k);	      % Convert to SOS form
Hd = dfilt.df2tsos(sos,g);    % Create a dfilt object
h = fvtool(Hd)                % Plot magnitude response
set(h,'Analysis','freq')      % Display frequency response
```

## Limitations

In general, you should 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 and an sos dfilt structure. For higher order filters (possibly starting as low as order 8), numerical problems due to roundoff errors may occur when forming the transfer function using the [b,a] syntax. The following example illustrates this limitation:

```n = 6;
Rp = .1; Rs = 80;
Wn = [2.5e6 29e6]/500e6;
ftype = 'bandpass';

% Transfer Function design
[b,a] = ellip(n,Rp,Rs,Wn,ftype);
h1=dfilt.df2(b,a);    % This is an unstable filter.

% Zero-Pole-Gain design
[z, p, k] = ellip(n,Rp,Rs,Wn,ftype);
[sos,g]=zp2sos(z,p,k);
h2=dfilt.df2sos(sos,g);

% Plot and compare the results
hfvt=fvtool(h1,h2,'FrequencyScale','log');
legend(hfvt,'TF Design','ZPK Design')
```

## More About

expand all

### Algorithms

The design of elliptic filters is the most difficult and computationally intensive of the Butterworth, Chebyshev Type I and II, and elliptic designs. ellip uses a five-step algorithm:

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

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

3. It transforms the lowpass filter to a bandpass, highpass, or bandstop filter with the desired cutoff frequencies using a state-space transformation.

4. For digital filter design, ellip uses bilinear to convert the analog filter into a digital filter through a bilinear transformation with frequency prewarping. Careful frequency adjustment guarantees that the analog filters and the digital filters will 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.

## See Also

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