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Question

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Explanation / Answer

Syntax

[z,p,k] = butter(n,Wn)

[z,p,k] = butter(n,Wn,'ftype')

[b,a] = butter(n,Wn)

[b,a] = butter(n,Wn,'ftype')

[A,B,C,D] = butter(n,Wn)

[A,B,C,D] = butter(n,Wn,'ftype')

[z,p,k] = butter(n,Wn,'s')

[z,p,k] = butter(n,Wn,'ftype','s')

[b,a] = butter(n,Wn,'s')

[b,a] = butter(n,Wn,'ftype','s')

[A,B,C,D] = butter(n,Wn,'s')

[A,B,C,D] = butter(n,Wn,'ftype','s')

Description

butter designs lowpass, bandpass, highpass, and bandstop digital and analog Butterworth filters. Butterworth filters are characterized by a magnitude response that is maximally flat in the passband and monotonic overall.

Butterworth filters sacrifice rolloff steepness for monotonicity in the pass- and stopbands. Unless the smoothness of the Butterworth filter is needed, an elliptic or Chebyshev filter can generally provide steeper rolloff characteristics with a lower filter order.

Digital Domain

[z,p,k] = butter(n,Wn) designs an order n lowpass digital Butterworth filter with normalized cutoff frequency Wn. It returns the zeros and poles in length n column vectors z and p, and the gain in the scalar k.

[z,p,k] = butter(n,Wn,'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 cutoff frequency Wn

'low' for a lowpass digital filter with normalized cutoff frequency Wn

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

'bandpass' for an order 2*n bandpass filter if Wn is a two-element vector, Wn = [w1 w2]. The passband is w1 < ? < w2. Specifying a two-element vector, Wn, without an explicit 'ftype' defaults to a bandpass filter.

Cutoff frequency is that frequency where the magnitude response of the filter is . For butter, the normalized cutoff frequency Wn must be a number between 0 and 1, where 1 corresponds to the Nyquist frequency, ? radians per sample.

If Wn is a two-element vector, Wn = [w1 w2], butter returns an order 2*n digital bandpass filter with passband w1 < ? < w2.

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

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

[b,a] = butter(n,Wn) designs an order n lowpass digital Butterworth filter with normalized cutoff frequency Wn. It returns the filter coefficients in length n+1 row vectors b and a, with coefficients in descending powers of z.

[b,a] = butter(n,Wn,'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] = butter(n,Wn) or

[A,B,C,D] = butter(n,Wn,'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] = butter(n,Wn,'s') designs an order n lowpass analog Butterworth filter with angular cutoff frequency Wn rad/s. It returns the zeros and poles in length n or 2*n column vectors z and p and the gain in the scalar k. butter's angular cutoff frequency Wn must be greater than 0 rad/s.

If Wn is a two-element vector with w1 < w2, butter(n,Wn,'s') returns an order 2*n bandpass analog filter with passband w1 < ? < w2.

[z,p,k] = butter(n,Wn,'ftype','s') designs a highpass, lowpass, or bandstop filter using the ftype values described above.

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

[b,a] = butter(n,Wn,'s') designs an order n lowpass analog Butterworth filter with angular cutoff frequency Wn rad/s. It 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] = butter(n,Wn,'ftype','s') designs a highpass, lowpass, or bandstop filter using the ftype values described above.

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

[A,B,C,D] = butter(n,Wn,'s') or

[A,B,C,D] = butter(n,Wn,'ftype','s') where A, B, C, and D are

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

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