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. To compute the power spectrum, medfreq uses the periodogram function with a rectangular window and a number of DFT points equal to the length of x.

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

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, medfreq assumes a normalized frequency. The default value for freqRange is the entire bandwidth of the input signal.

[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

Median Frequency of Chirps

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

Median Frequency of Sinusoids

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);

Median Frequency of Bandlimited Signals

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

x — Input signal

vector | matrix

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

Fs — Sample rate

positive real scalar

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

pxx — Power spectral density

vector | matrix

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

f — Frequencies

vector

Frequencies, specified as a vector.

Data Types: single | double

sxx — Power spectrum estimate

vector | matrix

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

rbw — Resolution bandwidth

positive scalar

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

freqRange — Frequency range

two-element vector

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

freq — Median frequency

scalar | vector

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.

power — Band power

scalar | vector

Band power, returned as a scalar or vector.

Algorithms

To determine the median frequency, medfreq computes a periodogram power spectrum estimate using a rectangular window.

You can obtain the same value of median frequency, medFrq, from a signal x at a sample rate Fs in these three ways.

Directly from the signal	medFrq = medfreq(x,Fs)
From the periodogram of the signal	[P,F] = periodogram(x,[],length(x),Fs); medFrq = medfreq(P,F)
From the power spectral estimate (Welch's PSD) of the signal	[P,F] = pwelch(x,rectwin(length(x)),[],length(x),Fs); medFrq = medfreq(P,F)

Note

Because medfreq uses an intermediary representation to transform the input signal from the time domain to frequency domain, the returned median frequency might vary, depending on the signal transformation method, number of DFT points, and window size.

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. London: IntechOpen, 2012. https://doi.org/10.5772/50639.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

Usage notes and limitations:

• The syntax with no output arguments is not supported.

For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2015a

R2023a: Use gpuArray objects

The medfreq function supports gpuArray objects. You must have Parallel Computing Toolbox™ to use this functionality.