poctave

Generate octave spectrum

Syntax

p = poctave(x,fs)

p = poctave(xt)

p = poctave(pxx,fs,f)

p = poctave(___,type)

p = poctave(___,Name,Value)

[p,cf] = poctave(___)

[p,cf,t] = poctave(___)

poctave(___)

Description

p = poctave(x,fs) returns the octave spectrum of a signal x sampled at a rate fs. The octave spectrum is the average power over octave bands as defined by the ANSI S1.11 standard [2]. If x is a matrix, then the function estimates the octave spectrum independently for each column and returns the result in the corresponding column of p.

p = poctave(xt) returns the octave spectrum of a signal stored in the MATLAB^® timetable xt.

p = poctave(pxx,fs,f) performs octave smoothing by converting a power spectral density, pxx, to a 1/b octave power spectrum, where b is the number of subbands in the octave band. The frequencies in f correspond to the PSD estimates in pxx.

p = poctave(___,type) specifies the kind of spectral analysis performed by the function. Specify type as 'power' or 'spectrogram'.

p = poctave(___,Name,Value) specifies additional options for any of the previous syntaxes using name-value pair arguments.

[p,cf] = poctave(___) also returns the center frequencies of the octave bands over which the octave spectrum is computed.

[p,cf,t] = poctave(___) additionally returns a time vector, t, corresponding to the center times of the segments used to compute the power spectrum estimates when type is 'spectrogram'.

poctave(___) with no output arguments plots the octave spectrum or spectrogram in the current figure. If type is specified as 'spectrogram', then this function is supported only for single-channel input.

Examples

Octave Spectra of White and Pink Noise

Generate ${10}^5$ samples of white Gaussian noise. Create a signal of pseudopink noise by filtering the white noise with a filter whose zeros and poles are all on the positive x-axis. Visualize the zeros and poles.

N = 1e5;
wn = randn(N,1);

z = [0.982231570015379 0.832656605953720 0.107980893771348]';
p = [0.995168968915815 0.943841773712820 0.555945259371364]';

[b,a] = zp2tf(z,p,1);
pn = filter(b,a,wn);

zplane(z,p)

Create a two-channel signal consisting of white and pink noise. Compute the octave spectrum. Assume a sample rate of 44.1 kHz. Set the frequency band from 30 Hz to the Nyquist frequency.

sg = [wn pn];

fs = 44100;

poctave(sg,fs,'FrequencyLimits',[30 fs/2])
legend('White noise','Pink noise','Location','SouthEast')

The white noise has an octave spectrum that increases with frequency. The octave spectrum of the pink noise is approximately constant throughout the frequency range. The octave spectrum of a signal illustrates how the human ear perceives the signal.

Octave Smoothing of White and Pink Noise

Generate ${10}^5$ samples of white Gaussian noise sampled at 44.1 kHz. Create a signal of pink noise by filtering the white noise with a filter whose zeros and poles are all on the positive x-axis.

N = 1e5;
fs = 44.1e3;
wn = randn(N,1);

z = [0.982231570015379 0.832656605953720 0.107980893771348]';
p = [0.995168968915815 0.943841773712820 0.555945259371364]';
[b,a] = zp2tf(z,p,1);

pn = filter(b,a,wn);

Compute the Welch estimate of the power spectral density for both signals. Divide the signals into 2048-sample segments, specify 50% overlap between adjoining segments, window each segment with a Hamming window, and use 4096 DFT points.

[pxx,f] = pwelch([wn pn],hamming(2048),1024,4096,fs);

Display the spectral densities over a frequency band ranging from 200 Hz to the Nyquist frequency. Use a logarithmic scale for the frequency axis.

pwelch([wn pn],hamming(2048),1024,4096,fs)
ax = gca;
ax.XScale = 'log';
xlim([200 fs/2]/1000)
legend('White','Pink')

Compute and display the octave spectra of the signals. Use the same frequency range as in the previous plot. Specify six bands per octave and compute the spectra using 8th-order filters.

poctave(pxx,fs,f,'BandsPerOctave',6,'FilterOrder',8,'FrequencyLimits',[200 fs/2],'psd')
legend('White','Pink')

Octave Spectrogram of Audio Signal

Read an audio recording of an electronic toothbrush into MATLAB®. The toothbrush turns on at about 1.75 seconds and stays on for approximately 2 seconds.

[y,fs] = audioread('toothbrush.m4a');

Compute the octave spectrogram of the audio signal. Specify 48 bands per octave and 82% overlap. Restrict the total frequency range from 100 Hz to fs/2 Hz and use C-weighting.

poctave(y,fs,'spectrogram','BandsPerOctave',48,'OverlapPercent',82,'FrequencyLimits',[100 fs/2],'Weighting','C')

Octave Spectrum Weighting

Generate ${10}^5$ samples of white Gaussian noise sampled at 44.1 kHz. Create a signal of pink noise by filtering the white noise with a filter whose zeros and poles are all on the positive x-axis.

N = 1e5;
fs = 44.1e3;
wn = randn(N,1);

z = [0.982231570015379 0.832656605953720 0.107980893771348]';
p = [0.995168968915815 0.943841773712820 0.555945259371364]';
[b,a] = zp2tf(z,p,1);

pn = filter(b,a,wn);

Compute the octave spectrum of the signal. Specify three bands per octave and restrict the total frequency range from 200 Hz to 20 kHz. Store the name-value pairs in a cell array for later use. Display the spectrum.

flims = [200 20e3];
bpo = 3;
opts = {'FrequencyLimits',flims,'BandsPerOctave',bpo};

poctave(pn,fs,opts{:});

Compute the octave spectrum of the signal with the same settings, but use C-weighting. The C-weighted spectrum falls off at frequencies above 6 kHz.

hold on
poctave(pn,fs,opts{:},'Weighting','C')

Compute the octave spectrum again, but now use A-weighting. The A-weighted spectrum peaks at about 3 kHz and falls off above 6 kHz and at the lower end of the frequency band.

poctave(pn,fs,opts{:},'Weighting','A')
hold off
legend('Pink noise','C-weighted','A-weighted','Location','SouthWest')

Input Arguments

x — Input signal

vector | matrix

Input signal, specified as a vector or matrix. If x is a vector, then poctave treats it as a single channel. If x is a matrix, then poctave computes the octave spectrum or spectrogram independently for each column and returns the result in the corresponding column of p. If type is set to 'spectrogram', the function concatenates the spectrograms along the third dimension of p.

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

fs — Sample rate

positive scalar

Sample rate, specified as a positive scalar expressed in hertz. The sample rate cannot be lower than 7 Hz.

xt — Input timetable

timetable

Input timetable. xt must contain increasing, finite, uniformly spaced row times. If xt represents a multichannel signal, then it must have either a single variable containing a matrix or multiple variables consisting of vectors.

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)) specifies a random process sampled at 1 Hz for 4 seconds.

pxx — Power spectral density

vector | matrix

Power spectral density (PSD), specified as a vector or matrix with real nonnegative elements. The power spectral density must be expressed in linear units, not decibels. Use db2pow to convert decibel values to power values. If type is 'spectrogram', then each column in pxx is considered to be the PSD for a particular time window or sample.

Example: [pxx,f] = periodogram(cos(pi./[4;2]*(0:159))'+randn(160,2)) specifies the periodogram PSD estimate of a noisy two-channel sinusoid sampled at 2π Hz and the frequencies at which it is computed.

f — PSD frequencies

vector

PSD frequencies, specified as a vector. f must be finite, strictly increasing, and uniformly spaced in the linear scale.

Example: [pxx,f] = periodogram(cos(pi./[4;2]*(0:159))'+randn(160,2)) specifies the periodogram PSD estimate of a noisy two-channel sinusoid sampled at 2π Hz and the frequencies at which it is computed.

type — Type of spectrum to compute

'power' (default) | 'spectrogram'

Type of spectrum to compute, specified as 'power' or 'spectrogram'.

• 'power' — Compute the octave power spectrum of the input.

• 'spectrogram' — Compute the octave spectrogram of the input. The function divides the input into segments and returns the short-time octave power spectrum of each segment.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Weighting','A','FilterOrder',8 computes the octave spectrum using A-weighting and 8th-order filters.

'BandsPerOctave' — Number of subbands in octave band

1 (default) | 3/2 | 2 | 3 | 6 | 12 | 24 | 48 | 96

Number of subbands in the octave band, specified as the comma-separated pair consisting of 'BandsPerOctave' and 1, 3/2, 2, 3, 6, 12, 24, 48, or 96. This parameter dictates the width of a fractional-octave band. In such a frequency band, the upper edge frequency is the lower edge frequency times 2^1/b, where b is the number of subbands.

Data Types: single | double

'FilterOrder' — Order of bandpass filters

6 (default) | positive even integer

Order of bandpass filters, specified as the comma-separated pair consisting of 'FilterOrder' and a positive even integer.

Data Types: single | double

'FrequencyLimits' — Frequency band

[max(3,3*fs/48e3) fs/2] (default) | two-element vector

Frequency band, specified as the comma-separated pair consisting of 'FrequencyLimits' and an increasing two-element vector expressed in Hz. The lower value of the vector must be at least 3 Hz. The upper value of the vector must be smaller than or equal to the Nyquist frequency. If the vector does not contain an octave center, poctave may return a center frequency outside the specified limits. To ensure a stable filter design, the actual minimum achievable frequency limit increases to 3*fs/48e3 if the sample rate exceeds 48 kHz. If this argument is not specified, poctave uses the interval [max(3,3*fs/48e3) fs/2].

Data Types: single | double

'Weighting' — Frequency weighting

'none' (default) | 'A' | 'C' | vector | matrix | 1-by-2 cell array | digitalFilter object

Frequency weighting, specified as the comma-separated pair consisting of 'Weighting' and one of the following:

• 'none' — poctave does not perform any frequency weighting on the input.

• 'A' — poctave performs A-weighting on the input. The ANSI S1.42 standard defines the A-weighting curve. The IEC 61672-1 standard defines the minimum and maximum attenuation limits for an A-weighting filter. The ANSI S1.42.2001 standard defines the weighting curve by specifying analog poles and zeros.

• 'C' — poctave performs C-weighting on the input. The ANSI S1.42 standard defines the C-weighting curve. The IEC 61672-1 standard defines the minimum and maximum attenuation limits for a C-weighting filter. The ANSI S1.42.2001 standard defines the weighting curve by specifying analog poles and zeros.

• Vector — poctave treats the input as a vector of coefficients that specify a finite impulse response (FIR) filter.

• Matrix — poctave treats the input as a matrix of second-order section coefficients that specify an infinite impulse response (IIR) filter. The matrix must have at least two rows and exactly six columns.

• 1-by-2 cell array — poctave treats the input as the numerator and denominator coefficients, in that order, that specify the transfer function of an IIR filter.

• digitalFilter object — poctave treats the input as a filter that was designed using designfilt.

This argument is supported only when the input is a signal. Octave smoothing does not support frequency weighting.

Example: 'Weighting',fir1(30,0.5) specifies a 30th-order FIR filter with a normalized cutoff frequency of 0.5π rad/sample.

Example: 'Weighting',[2 4 2 6 0 2;3 3 0 6 0 0] specifies a third-order Butterworth filter with a normalized 3-dB frequency of 0.5π rad/sample.

Example: 'Weighting',{[1 3 3 1]/6 [3 0 1]/3} specifies a third-order Butterworth filter with a normalized 3-dB frequency of 0.5π rad/sample.

Example: 'Weighting',designfilt('lowpassiir','FilterOrder',3,'HalfPowerFrequency',0.5) specifies a third-order Butterworth filter with a normalized 3-dB frequency of 0.5π rad/sample.

Data Types: single | double | char | string | cell

'MinThreshold' — Lower bound for nonzero values

-Inf (default) | real scalar

Lower bound for nonzero values, specified as the comma-separated pair consisting of 'MinThreshold' and a real scalar. The function sets those elements of p such that 10 log₁₀(p) ≤ 'MinThreshold' to zero. Specify 'MinThreshold' in decibels.

Data Types: single | double

'WindowLength' — Length of data segments

nonnegative integer

Length of data segments, specified as the comma-separated pair consisting of 'WindowLength' and a nonnegative integer. 'WindowLength' must be less than or equal to the length of the input signal. If not specified, the length of data segments is calculated based on the size of the input signal. This input is valid only when type is 'spectrogram'.

Data Types: single | double

'OverlapPercent' — Overlap percent between adjoining segments

real scalar in the interval [0, 100)

Overlap percent between adjoining segments, specified as the comma-separated pair consisting of 'OverlapPercent' and a real scalar in the interval [0, 100). If not specified, 'OverlapPercent' is zero. This input is valid only when type is 'spectrogram'.

Data Types: single | double

Output Arguments

p — Octave spectrum or spectrogram

vector | matrix | 3-D array

Octave spectrum or spectrogram, returned as a vector, matrix, or 3-D array. The third dimension, if present, corresponds to the input channels.

cf — Center frequencies

vector

Center frequencies, returned as a vector. cf contains a list of center frequencies of the octave bands over which poctave estimated the octave spectrum. cf has units of hertz.

t — Center times

vector

Center times, returned as a vector. If the input is a PSD, then t represents the sample indices corresponding to the columns of pxx. This argument applies only when type is 'spectrogram'.

References

[1] Smith, Julius Orion, III. "Example: Synthesis of 1/F Noise (Pink Noise)." In Spectral Audio Signal Processing. https://ccrma.stanford.edu/~jos/sasp/.

[2] Specification for Octave-Band and Fractional-Octave-Band Analog and Digital Filters. ANSI Standard S1.11-2004. Melville, NY: Acoustical Society of America, 2004.