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# bandpass

Bandpass-filter signals

## Description

y = bandpass(x,wpass) filters the input signal x using a bandpass filter with a passband frequency range specified by the two-element vector wpass and expressed in normalized units of π rad/sample. bandpass uses a minimum-order filter with a stopband attenuation of 60 dB and compensates for the delay introduced by the filter. If x is a matrix, the function filters each column independently.

example

y = bandpass(x,fpass,fs) specifies that x has been sampled at a rate of fs hertz. The two-element vector fpass specifies the passband frequency range of the filter in hertz.

y = bandpass(xt,fpass) bandpass-filters the data in timetable xt using a filter with a passband frequency range specified in hertz by the two-element vector fpass. The function independently filters all variables in the timetable and all columns inside each variable.

example

y = bandpass(___,Name=Value) specifies additional options for any of the previous syntaxes using name-value arguments. You can change the stopband attenuation, the Bandpass Filter Steepness, and the type of impulse response of the filter.

example

[y,d] = bandpass(___) also returns the digitalFilter object d used to filter the input.

bandpass(___) with no output arguments plots the input signal and overlays the filtered signal.

## Examples

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Create a signal sampled at 1 kHz for 1 second. The signal contains three tones, one at 50 Hz, another at 150 Hz, and a third at 250 Hz. The high-frequency and low-frequency tones both have twice the amplitude of the intermediate tone. The signal is embedded in Gaussian white noise of variance 1/100.

fs = 1e3;
t = 0:1/fs:1;
x = [2 1 2]*sin(2*pi*[50 150 250]'.*t) + randn(size(t))/10;

Bandpass-filter the signal to remove the low-frequency and high-frequency tones. Specify passband frequencies of 100 Hz and 200 Hz. Display the original and filtered signals, and also their spectra.

bandpass(x,[100 200],fs)

Implement a basic digital music synthesizer and use it to play a traditional song. Specify a sample rate of 2 kHz. Plot the spectrogram of the song.

fs = 2e3;
t = 0:1/fs:0.3-1/fs; fq = [-Inf -9:2]/12;
note = @(f,g) [1 1 1]*sin(2*pi*440*2.^[fq(g)-1 fq(g) fq(f)+1]'.*t);

mel = [5 3 1 3 5 5 5 0 3 3 3 0 5 8 8 0 5 3 1 3 5 5 5 5 3 3 5 3 1]+1;
acc = [5 0 8 0 5 0 5 5 3 0 3 3 5 0 8 8 5 0 8 0 5 5 5 0 3 3 5 0 1]+1;

song = [];
for kj = 1:length(mel)
song = [song note(mel(kj),acc(kj)) zeros(1,0.01*fs)];
end
song = song/(max(abs(song))+0.1);

% To hear, type sound(song,fs)

pspectrum(song,fs,"spectrogram",TimeResolution=0.31, ...
OverlapPercent=0,MinThreshold=-60)

Bandpass-filter the signal to separate the middle register from the other two. Specify passband frequencies of 230 Hz and 450 Hz. Plot the original and filtered signals in the time and frequency domains.

pong = bandpass(song,[230 450],fs);

% To hear, type sound(pong,fs)

bandpass(song,[230 450],fs)

Plot the spectrogram of the middle register.

figure
pspectrum(pong,fs,"spectrogram",TimeResolution=0.31, ...
OverlapPercent=0,MinThreshold=-60)

Filter white noise sampled at 1 kHz using an infinite impulse response bandpass filter with a passband width of 100 Hz. Use different steepness values. Plot the spectra of the filtered signals.

fs = 1000;
x = randn(20000,1);

[y1,d1] = bandpass(x,[ 50 150],fs,ImpulseResponse="iir",Steepness=0.5);
[y2,d2] = bandpass(x,[200 300],fs,ImpulseResponse="iir",Steepness=0.8);
[y3,d3] = bandpass(x,[350 450],fs,ImpulseResponse="iir",Steepness=0.95);

pspectrum([y1 y2 y3],fs)
legend("Steepness = " + [0.5 0.8 0.95],Location="south")

Compute and plot the frequency responses of the filters.

[h1,f] = freqz(d1,1024,fs);
[h2,~] = freqz(d2,1024,fs);
[h3,~] = freqz(d3,1024,fs);

plot(f,mag2db(abs([h1 h2 h3])))
legend("Steepness = " + [0.5 0.8 0.95],Location="south")
ylim([-100 10])

Make the filters asymmetric by specifying different values of steepness at the lower and higher passband frequencies.

[y1,d1] = bandpass(x,[ 50 150],fs,ImpulseResponse="iir",Steepness=[0.5 0.8]);
[y2,d2] = bandpass(x,[200 300],fs,ImpulseResponse="iir",Steepness=[0.5 0.8]);
[y3,d3] = bandpass(x,[350 450],fs,ImpulseResponse="iir",Steepness=[0.5 0.8]);

pspectrum([y1 y2 y3],fs)

Compute and plot the frequency responses of the filters.

[h1,f] = freqz(d1,1024,fs);
[h2,~] = freqz(d2,1024,fs);
[h3,~] = freqz(d3,1024,fs);

plot(f,mag2db(abs([h1 h2 h3])))
ylim([-100 10])

## Input Arguments

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Input signal, specified as a vector or matrix.

Example: sin(2*pi*(0:127)/16)+randn(1,128)/100 specifies a noisy sinusoid

Example: [2 1].*sin(2*pi*(0:127)'./[16 64]) specifies a two-channel sinusoid.

Data Types: single | double
Complex Number Support: Yes

Normalized passband frequency range, specified as a two-element vector with elements in the interval (0, 1).

Passband frequency range, specified as a two-element vector with elements in the interval (0, fs/2).

Sample rate, specified as a positive real scalar.

Input timetable. xt must contain increasing, finite, and equally spaced row times of type duration in seconds.

If a timetable has missing or duplicate time points, you can fix it using the tips in Clean Timetable with Missing, Duplicate, or Nonuniform Times.

Example: timetable(seconds(0:4)',randn(5,1),randn(5,2)) contains a single-channel random signal and a two-channel random signal, sampled at 1 Hz for 4 seconds.

Example: timetable(randn(5,1),randn(5,2),SampleRate=1) contains a single-channel random signal and a two-channel random signal, sampled at 1 Hz for 4 seconds.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: ImpulseResponse="iir",StopbandAttenuation=30 filters the input using a minimum-order IIR filter that attenuates by 30 dB the frequencies smaller than fpass(1) and the frequencies larger than fpass(2).

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'ImpulseResponse','iir','StopbandAttenuation',30 filters the input using a minimum-order IIR filter that attenuates by 30 dB the frequencies smaller than fpass(1) and the frequencies larger than fpass(2).

Type of impulse response of the filter, specified as "fir", "iir", or "auto".

• "fir" — The function designs a minimum-order, linear-phase, finite impulse response (FIR) filter. To compensate for the delay, the function appends to the input signal N/2 zeros, where N is the filter order. The function then filters the signal and removes the first N/2 samples of the output.

In this case, the input signal must be at least twice as long as the filter that meets the specifications.

• "iir" — The function designs a minimum-order infinite impulse response (IIR) filter and uses the filtfilt function to perform zero-phase filtering and compensate for the filter delay.

If the signal is not at least three times as long as the filter that meets the specifications, the function designs a filter with smaller order and thus smaller steepness.

• "auto" — The function designs a minimum-order FIR filter if the input signal is long enough, and a minimum-order IIR filter otherwise. Specifically, the function follows these steps:

• Compute the minimum order that an FIR filter must have to meet the specifications. If the signal is at least twice as long as the required filter order, design and use that filter.

• If the signal is not long enough, compute the minimum order that an IIR filter must have to meet the specifications. If the signal is at least three times as long as the required filter order, design and use that filter.

• If the signal is not long enough, truncate the order to one-third the signal length and design an IIR filter of that order. The reduction in order comes at the expense of transition band steepness.

• Filter the signal and compensate for the delay.

Transition band steepness, specified as a scalar or two-element vector with elements in the interval [0.5, 1). As the steepness increases, the filter response approaches the ideal bandpass response, but the resulting filter length and the computational cost of the filtering operation also increase. See Bandpass Filter Steepness for more information.

Filter stopband attenuation, specified as a positive scalar in dB.

## Output Arguments

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Filtered signal, returned as a vector, a matrix, or a timetable with the same dimensions as the input.

Bandpass filter used in the filtering operation, returned as a digitalFilter object.

• Use filter(d,x) to filter a signal x using d. Unlike bandpass, the filter function does not compensate for filter delay. You can also use the filtfilt and fftfilt functions with digitalFilter objects.

• Use Filter Analyzer to visualize the filter response.

• Use designfilt to edit or generate a digital filter based on frequency-response specifications.

## More About

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### Bandpass Filter Steepness

The Steepness argument controls the width of a filter's transition regions. The lower the steepness, the wider the transition region. The higher the steepness, the narrower the transition region.

To interpret the filter steepness, consider the following definitions:

• The Nyquist frequency, fNyquist, is the highest frequency component of a signal that can be sampled at a given rate without aliasing. fNyquist is 1 (×π rad/sample) when the input signal has no time information, and fs/2 hertz when the input signal is a timetable or when you specify a sample rate.

• The lower and upper stopband frequencies of the filter, fstoplower and fstopupper, are the frequencies below which and above which the attenuation is equal to or greater than the value specified using StopbandAttenuation.

• The lower transition width of the filter, Wlower, is fpasslowerfstoplower, where the lower passband frequency fpasslower is the first element of the specified fpass.

• The upper transition width of the filter, Wupper, is fstopupperfpassupper, where the upper passband frequencyfpassupper is the second element of fpass.

• Most nonideal filters also attenuate the input signal across the passband. The maximum value of this frequency-dependent attenuation is called the passband ripple. Every filter used by bandpass has a passband ripple of 0.1 dB.

To control the width of the transition bands, you can specify Steepness as either a two-element vector, [slower,supper], or a scalar. When you specify Steepness as a vector, the function:

• Computes the lower transition width as

Wlower = (1 – slower) × fpasslower.

• When the first element of Steepness is equal to 0.5, the transition width is 50% of fpasslower.

• As the first element of Steepness approaches 1, the transition width becomes progressively narrower until it reaches a minimum value of 1% of fpasslower.

• Computes the upper transition width as

Wupper = (1 – supper) × (fNyquistfpassupper).

• When the second element of Steepness is equal to 0.5, the transition width is 50% of (fNyquistfpassupper).

• As the second element of Steepness approaches 1, the transition width becomes progressively narrower until it reaches a minimum value of 1% of (fNyquistfpassupper).

When you specify Steepness as a scalar, the function designs a filter with equal lower and upper transition widths. The default value of Steepness is 0.85.

## Version History

Introduced in R2018a