spectralEntropy

Spectral entropy for signals and spectrograms

Syntax

entropy = spectralEntropy(x,f)

entropy = spectralEntropy(x,f,Name=Value)

spectralEntropy(___)

Description

entropy = spectralEntropy(x,f) returns the spectral entropy of the signal, x, over time. How the function interprets x depends on the shape of f.

entropy = spectralEntropy(x,f,Name=Value) specifies options using one or more name-value arguments.

spectralEntropy(___) with no output arguments plots the spectral entropy. You can specify an input combination from any of the previous syntaxes.

• If the input is in the time domain, the spectral entropy is plotted against time.

• If the input is in the frequency domain, the spectral entropy is plotted against frame number.

Examples

Spectral Entropy of Time-Domain Signal

Create a chirp signal with white Gaussian noise and calculate the entropy using default parameters.

fs = 1000;
t = (0:1/fs:10)';
f1 = 300;
f2 = 400;
x = chirp(t,f1,10,f2) + randn(length(t),1);

entropy = spectralEntropy(x,fs);

Plot the spectral entropy against time.

spectralEntropy(x,fs)

Spectral Entropy of Frequency-Domain Signal

Create a chirp signal with white Gaussian noise and then calculate the spectrogram using the stft function.

fs = 1000;
t = (0:1/fs:10)';
f1 = 300;
f2 = 400;
x = chirp(t,f1,10,f2) + randn(length(t),1);

[s,f] = stft(x,fs,FrequencyRange="onesided");
s = abs(s).^2;

Calculate the entropy of the spectrogram over time.

entropy = spectralEntropy(s,f);

Plot the spectral entropy against the frame number.

spectralEntropy(s,f)

Specify Nondefault Parameters

Create a chirp signal with white Gaussian noise.

fs = 1000;
t = (0:1/fs:10)';
f1 = 300;
f2 = 400;
x = chirp(t,f1,10,f2) + randn(length(t),1);

Calculate the entropy of the power spectrum over time. Calculate the entropy for 50 ms Hamming windows of data with 25 ms overlap. Use the range from 62.5 Hz to fs/2 for the entropy calculation.

entropy = spectralEntropy(x,fs, ...
                      Window=hamming(round(0.05*fs)), ...
                      OverlapLength=round(0.025*fs), ...
                      Range=[62.5,fs/2]);

Plot the entropy against time.

spectralEntropy(x,fs, ...
              Window=hamming(round(0.05*fs)), ...
              OverlapLength=round(0.025*fs), ...
              Range=[62.5,fs/2])

Input Arguments

x — Input signal

column vector | matrix | 3-D array | timetable

Input signal, specified as a vector, matrix, 3-D array, or timetable. How the function interprets x depends on the shape of f.

If x is a timetable, it can have one or more variables, and each variable can have one or more columns. The output entropy is a timetable. In that case, do not specify f.

Data Types: single | double

f — Sample rate or frequency vector (Hz)

1 (default) | scalar | vector

Sample rate or frequency vector in Hz, specified as a scalar or vector, respectively. How the function interprets x depends on the shape of f:

• If f is not specified and x is a numeric vector or matrix, spectralEntropy assumes x is sampled at a rate equal to 1 Hz. If f is not specified and x is a timetable, spectralEntropy infers the sample rate from x.

• If f is a scalar, x is interpreted as a time-domain signal, and f is interpreted as the sample rate. In this case, x must be a real vector or matrix. If x is specified as a matrix, the columns are interpreted as individual channels.

• If f is a vector, x is interpreted as a frequency-domain signal, and f is interpreted as the frequencies, in Hz, corresponding to the rows of x. In this case, x must be a real L-by-M-by-N array, where L is the number of spectral values at given frequencies of f, M is the number of individual spectra, and N is the number of channels.

  The number of rows of x, L, must be equal to the number of elements of f.

Data Types: single | double

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.

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

Example: Window=hamming(256)

Note

The following name-value arguments apply if x is a time-domain signal. If x is a frequency-domain signal, only the Instantaneous and Scaled arguments apply.

Window — Window applied in time domain

rectwin(round(f*0.03)) (default) | vector

Window applied in the time domain, specified as a real vector. The number of elements in the vector must be in the range [1, size(x,1)]. The number of elements in the vector must also be greater than OverlapLength. If you do not specify Window, spectralEntropy uses a window length that splits x into eight overlapping segments.

Data Types: single | double

OverlapLength — Number of samples overlapped between adjacent windows

round(f*0.02) (default) | non-negative scalar

Number of samples overlapped between adjacent windows, specified as an integer in the range [0, size(Window,1)). If you do not specify OverlapLength, spectralEntropy uses a value that results in 50% overlap between segments.

Data Types: single | double

FFTLength — Number of bins in DFT

numel(Window) (default) | positive scalar integer

Number of bins used to calculate the DFT of windowed input samples, specified as a positive scalar integer. If unspecified, FFTLength defaults to the number of elements in the Window.

Data Types: single | double

Range — Frequency range (Hz)

[0,f/2] (default) | two-element row vector

Frequency range in Hz, specified as a two-element row vector of increasing real values in the range [0, f/2].

Data Types: single | double

SpectrumType — Spectrum type

"power" (default) | "magnitude"

Spectrum type, specified as "power" or "magnitude":

• "power" –– The spectral entropy is calculated for the one-sided power spectrum.

• "magnitude" –– The spectral entropy is calculated for the one-sided magnitude spectrum.

Data Types: char | string

Instantaneous — Instantaneous time series option

true (default) | false

Since R2024b

Instantaneous time series option, specified as a logical.

• If Instantaneous is true, then spectralEntropy returns the instantaneous spectral entropy as a time-series vector.

• If Instantaneous is false, then spectralEntropy returns the spectral entropy value of the whole signal or spectrum as a scalar.

This argument applies if x is a time-domain signal or if x is a frequency-domain signal.

Data Types: logical

Scaled — Scale by white noise option

true (default) | false

Since R2024b

Scale by white noise option, specified as a logical. Scaling by white noise — or log₂n, where n is the number of frequency points — is equivalent to normalizing the spectral entropy. Scaling allows you to perform a direct comparison on signals of different length.

• If Scaled is true, then spectralEntropy returns the spectral entropy scaled by the spectral entropy of the corresponding white noise.

• If Scaled is false, then spectralEntropy does not scale the spectral entropy.

This argument applies if x is a time-domain signal or if x is a frequency-domain signal.

Data Types: logical

Output Arguments

entropy — Spectral entropy

scalar | vector | matrix | timetable

Spectral entropy, returned as a scalar, vector, matrix, or timetable. Each row of entropy corresponds to the spectral entropy of a window of x. Each column of entropy corresponds to an independent channel.

More About

Spectral Entropy

The spectral entropy (SE) of a signal is a measure of its spectral power distribution. The concept is based on the Shannon entropy, or information entropy, in information theory. The SE treats the signal's normalized power distribution in the frequency domain as a probability distribution, and calculates the Shannon entropy of it. The Shannon entropy in this context is the spectral entropy of the signal. This property can be useful for feature extraction in fault detection and diagnosis [2], [1]. SE is also widely used as a feature in speech recognition [3] and biomedical signal processing [4].

The equations for spectral entropy arise from the equations for the power spectrum and probability distribution for a signal. For a signal x(n), the power spectrum is S(m) = |X(m)|², where X(m) is the discrete Fourier transform of x(n). The probability distribution P(m) is then:

$$P\left(m\right)=\frac{S\left(m\right)}{{\displaystyle {\sum }_{i}S\left(i\right)}}\text{.}$$

The spectral entropy H follows as:

$$H=-{\displaystyle \sum _{m=1}^{N}P\left(m\right){\log }_2}P\left(m\right)\text{.}$$

Normalizing:

$$H_n=-\frac{{\displaystyle \sum _{m=1}^{N}P\left(m\right){\log }_2}P\left(m\right)}{{\log }_{2}N}\text{,}$$

where N is the total frequency points. The denominator, log₂N represents the maximal spectral entropy of white noise, uniformly distributed in the frequency domain.

If a time-frequency power spectrogram S(t,f) is known, then the probability distribution becomes:

$$P\left(m\right)=\frac{{\displaystyle {\sum }_{t}S(t,m)}}{{\displaystyle {\sum }_f{\displaystyle {\sum }_{t}S\left(t,f\right)}}}.$$

Spectral entropy is still:

$$H=-{\displaystyle \sum _{m=1}^{N}P\left(m\right){\log }_2}P\left(m\right)\text{.}$$

To compute the instantaneous spectral entropy given a time-frequency power spectrogram S(t,f), the probability distribution at time t is:

$$P\left(t,m\right)=\frac{S\left(t,m\right)}{{\displaystyle {\sum }_{f}S\left(t,f\right)}}.$$

Then the spectral entropy at time t is:

$$H\left(t\right)=-{\displaystyle \sum _{m=1}^{N}P\left(t,m\right)}{\log }_{2}P\left(t,m\right).$$

Algorithms

The spectral entropy is calculated as described in [5]:

$$\text{entropy}=\frac{-{\displaystyle \sum _{k=b_1}^{b_2}{s}_k\log \left(s_k\right)}}{\log \left(b_2-b_1\right)}$$

where

• s_k is the spectral value at bin k.

• b₁ and b₂ are the band edges, in bins, over which to calculate the spectral entropy.

References

[1] Pan, Y. N., J. Chen, and X. L. Li. "Spectral Entropy: A Complementary Index for Rolling Element Bearing Performance Degradation Assessment." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. Vol. 223, Issue 5, 2009, pp. 1223–1231.

[2] Sharma, V., and A. Parey. "A Review of Gear Fault Diagnosis Using Various Condition Indicators." Procedia Engineering. Vol. 144, 2016, pp. 253–263.

[3] Shen, J., J. Hung, and L. Lee. "Robust Entropy-Based Endpoint Detection for Speech Recognition in Noisy Environments." ICSLP. Vol. 98, November 1998.

[4] Vakkuri, A., A. Yli‐Hankala, P. Talja, S. Mustola, H. Tolvanen‐Laakso, T. Sampson, and H. Viertiö‐Oja. "Time‐Frequency Balanced Spectral Entropy as a Measure of Anesthetic Drug Effect in Central Nervous System during Sevoflurane, Propofol, and Thiopental Anesthesia." Acta Anaesthesiologica Scandinavica. Vol. 48, Number 2, 2004, pp. 145–153.

[5] Misra, H., S. Ikbal, H. Bourlard, and H. Hermansky. "Spectral Entropy Based Feature for Robust ASR." 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

Extended Capabilities

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

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

Version History

Introduced in R2019a

R2024b: Recommended over pentropy

The pentropy function will be removed in a future release. Use spectralEntropy instead.