# medfreq

Median frequency

## Syntax

freq = medfreq(x)
freq = medfreq(x,fs)
freq = medfreq(pxx,f)
freq = medfreq(sxx,f,rbw)
freq = medfreq(___,freqrange)
[freq,power] = medfreq(___)
medfreq(___)

## Description

freq = medfreq(x) estimates the median normalized frequency, freq, of the power spectrum of a time-domain signal, x.

example

freq = medfreq(x,fs) estimates the median frequency in terms of the sample rate, fs.

example

freq = medfreq(pxx,f) returns the median frequency of a power spectral density (PSD) estimate, pxx. The frequencies, f, correspond to the estimates in pxx.
freq = medfreq(sxx,f,rbw) returns the median frequency of a power spectrum estimate, sxx, with resolution bandwidth rbw.
freq = medfreq(___,freqrange) specifies the frequency interval over which to compute the median frequency. This syntax can include any combination of input arguments from previous syntaxes, as long as the second input argument is either fs or f. If the second input is passed as empty, normalized frequency will be assumed. The default value for freqrange is the entire bandwidth of the input signal.

example

[freq,power] = medfreq(___) also returns the band power, power, of the spectrum. If you specify freqrange, then power contains the band power within freqrange.
medfreq(___) with no output arguments plots the PSD or power spectrum and annotates the median frequency.

## Examples

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Generate 1024 samples of a chirp sampled at 1024 kHz. Specify the chirp so that it has an initial frequency of 50 kHz and reaches 100 kHz at the end of the sampling. Add white Gaussian noise such that the signal-to-noise ratio is 40 dB. Reset the random number generator for reproducible results.

nSamp = 1024; Fs = 1024e3; SNR = 40; rng default t = (0:nSamp-1)'/Fs; x = chirp(t,50e3,nSamp/Fs,100e3); x = x+randn(size(x))*std(x)/db2mag(SNR);

Estimate the median frequency of the chirp. Plot the power spectral density (PSD) and annotate the median frequency.

medfreq(x,Fs)

ans = 7.4998e+04 

Generate another chirp. Specify an initial frequency of 200 kHz, a final frequency of 300 kHz, and an amplitude that is twice that of the first signal. Add white Gaussian noise.

x2 = 2*chirp(t,200e3,nSamp/Fs,300e3); x2 = x2+randn(size(x2))*std(x2)/db2mag(SNR);

Concatenate the chirps to produce a two-channel signal. Estimate the median frequency of each channel.

y = medfreq([x x2],Fs)
y = 1×2 105 × 0.7500 2.4999 

Plot the PSDs of the two channels and annotate their median frequencies.

medfreq([x x2],Fs);

Add the two channels to form a new signal. Plot the PSD and annotate the median frequency.

medfreq(x+x2,Fs)

ans = 2.3756e+05 

Generate 1024 samples of a 100.123 kHz sinusoid sampled at 1024 kHz. Add white Gaussian noise such that the signal-to-noise ratio is 40 dB. Reset the random number generator for reproducible results.

nSamp = 1024; Fs = 1024e3; SNR = 40; rng default t = (0:nSamp-1)'/Fs; x = sin(2*pi*t*100.123e3); x = x + randn(size(x))*std(x)/db2mag(SNR);

Use the periodogram function to compute the power spectral density (PSD) of the signal. Specify a Kaiser window with the same length as the signal and a shape factor of 38. Estimate the median frequency of the signal and annotate it on a plot of the PSD.

[Pxx,f] = periodogram(x,kaiser(nSamp,38),[],Fs); medfreq(Pxx,f);

Generate another sinusoid, this one with a frequency of 257.321 kHz and an amplitude that is twice that of the first sinusoid. Add white noise.

x2 = 2*sin(2*pi*t*257.321e3); x2 = x2 + randn(size(x2))*std(x2)/db2mag(SNR);

Concatenate the sinusoids to produce a two-channel signal. Estimate the PSD of each channel and use the result to determine the median frequency.

[Pyy,f] = periodogram([x x2],kaiser(nSamp,38),[],Fs); y = medfreq(Pyy,f)
y = 1×2 105 × 1.0012 2.5731 

Annotate the median frequencies of the two channels on a plot of the PSDs.

medfreq(Pyy,f);

Add the two channels to form a new signal. Estimate the PSD and annotate the median frequency.

[Pzz,f] = periodogram(x+x2,kaiser(nSamp,38),[],Fs); medfreq(Pzz,f);

Generate a signal whose PSD resembles the frequency response of an 88th-order bandpass FIR filter with normalized cutoff frequencies $0.25\pi$ rad/sample and $0.45\pi$ rad/sample.

d = fir1(88,[0.25 0.45]);

Compute the median frequency of the signal between $0.3\pi$ rad/sample and $0.6\pi$ rad/sample. Plot the PSD and annotate the median frequency and measurement interval.

medfreq(d,[],[0.3 0.6]*pi);

Output the median frequency and the band power of the measurement interval. Specifying a sample rate of $2\pi$ is equivalent to leaving the rate unset.

[mdf,power] = medfreq(d,2*pi,[0.3 0.6]*pi); fprintf('Mean = %.3f*pi, power = %.1f%% of total \n', ... mdf/pi,power/bandpower(d)*100)
Mean = 0.371*pi, power = 77.4% of total 

Add a second channel with normalized cutoff frequencies $0.5\pi$ rad/sample and $0.8\pi$ rad/sample and an amplitude that is one-tenth that of the first channel.

d = [d;fir1(88,[0.5 0.8])/10]';

Compute the median frequency of the signal between $0.3\pi$ rad/sample and $0.9\pi$ rad/sample. Plot the PSD and annotate the median frequency of each channel and the measurement interval.

medfreq(d,[],[0.3 0.9]*pi);

Output the median frequency of each channel. Divide by $\pi$.

mdf = medfreq(d,[],[0.3 0.9]*pi)/pi
mdf = 1×2 0.3706 0.6500 

## Input Arguments

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Input signal, specified as a vector or matrix. If x is a vector, it is treated as a single channel. If x is a matrix, then medfreq computes the median frequency of each column of x independently. x must be finite-valued.

Example: cos(pi/4*(0:159))+randn(1,160) is a single-channel row-vector signal.

Example: cos(pi./[4;2]*(0:159))'+randn(160,2) is a two-channel signal.

Data Types: single | double

Sample rate, specified as a positive real scalar. The sample rate is the number of samples per unit time. If the time is measured in seconds, then the sample rate is in hertz.

Data Types: single | double

Power spectral density (PSD), specified as a vector or matrix. If pxx is a matrix, then medfreq computes the median frequency of each column of pxx independently.

The power spectral density must be expressed in linear units, not decibels. Use db2pow to convert decibel values to power values.

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.

Data Types: single | double

Frequencies, specified as a vector.

Data Types: single | double

Power spectrum estimate, specified as a vector or matrix. If sxx is a matrix, then medfreq computes the median frequency of each column of sxx independently.

The power spectrum must be expressed in linear units, not decibels. Use db2pow to convert decibel values to power values.

Example: [sxx,w] = periodogram(cos(pi./[4;2]*(0:159))'+randn(160,2),'power') specifies the periodogram power spectrum estimate of a two-channel sinusoid embedded in white Gaussian noise and the normalized frequencies at which it is computed.

Data Types: single | double

Resolution bandwidth, specified as a positive scalar. The resolution bandwidth is the product of two values: the frequency resolution of the discrete Fourier transform and the equivalent noise bandwidth of the window used to compute the PSD.

Data Types: single | double

Frequency range, specified as a two-element vector of real values. If you do not specify freqrange, then medfreq uses the entire bandwidth of the input signal.

Data Types: single | double

## Output Arguments

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Median frequency, specified as a scalar or vector.

• If you specify a sample rate, then freq has the same units as fs.

• If you do not specify a sample rate, then freq has units of rad/sample.

Band power, returned as a scalar or vector.

## References

[1] Phinyomark, Angkoon, Sirinee Thongpanja, Huosheng Hu, Pornchai Phukpattaranont, and Chusak Limsakul. "The Usefulness of Mean and Median Frequencies in Electromyography Analysis." In Computational Intelligence in Electromyography Analysis – A Perspective on Current Applications and Future Challenges, edited by Ganesh R. Naik. InTech, 2012. https://doi.org/10.5772/50639.