A particular DSP system is sampled at 48kHz, and requires a highpass filter with

Question

A particular DSP system is sampled at 48kHz, and requires a highpass filter with a passband ripple of 0.12 dB, a transition band of 400Hz, stopband attenuation of 60dB, and a cutoff frequency of 1400Hz. Specify the filter parameter values. Sketch the appropriate design constraints that the filter must satisfy. Design the filter using Impulse invariance method with both Butterworth and Chebyshev platforms. Increase the filter order N found in 3 and re-plot the filter characteristics and compare the results. Design the filter using Bilinear transformation method with both Butterworth and Chebyshev platforms and compare the results. Design the filter using Kaiser window method. Increase the filter order M and/or the filter parameter beta found in 6 and re-plot the filter characteristics and compare the results.

Explanation / Answer

A high-pass filter (also known as a bass-cut filter) attenuates signals below a cutoff frequency (the stopband) and allows signals above the cutoff frequency (the passband). The output of this filter is directly proportional to rate of change of the input signal.

High-pass filters are often used to clean up low-frequency noise, remove humming sounds in audio signals, redirect higher frequency signals to appropriate speakers in sound systems, and remove low-frequency trends from time series data thereby highlighting the high-frequency trends.

Syntax

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

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

[z,p,k] = butter(___)

[A,B,C,D] = butter(___)

[___] = butter(___,'s')

Description

[b,a] = butter(n,Wn) returns the transfer function coefficients of an nth-order lowpass digital Butterworth filter with normalized cutoff frequency Wn.

[b,a] = butter(n,Wn,ftype) designs a lowpass, highpass, bandpass, or bandstop Butterworth filter, depending on the value of ftype and the number of elements of Wn. 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] = butter(___) designs a lowpass, highpass, bandpass, or bandstop digital Butterworth filter and returns its zeros, poles, and gain. This syntax can include any of the input arguments in previous syntaxes.

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

[___] = butter(___,'s') designs a lowpass, highpass, bandpass, or bandstop analog Butterworth filter with cutoff angular frequency Wn.

Chebyshev Type I filter design

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Syntax

[b,a] = cheby1(n,Rp,Wp)

[b,a] = cheby1(n,Rp,Wp,ftype)

[z,p,k] = cheby1(___)

[A,B,C,D] = cheby1(___)

[___] = cheby1(___,'s')

Description

[b,a] = cheby1(n,Rp,Wp) returns the transfer function coefficients of an nth-order lowpass digital Chebyshev Type I filter with normalized passband edge frequency Wp and Rp decibels of peak-to-peak passband ripple.

[b,a] = cheby1(n,Rp,Wp,ftype) designs a lowpass, highpass, bandpass, or bandstop Chebyshev Type I 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] = cheby1(___) designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type I filter and returns its zeros, poles, and gain. This syntax can include any of the input arguments in previous syntaxes.

[A,B,C,D] = cheby1(___) designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type I filter and returns the matrices that specify its state-space representation.

[___] = cheby1(___,'s') designs a lowpass, highpass, bandpass, or bandstop analog Chebyshev Type I filter with passband edge angular frequency Wp and Rp decibels of passband ripple

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

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