# gtcc

Extract gammatone cepstral coefficients, log-energy, delta, and delta-delta

## Description

example

coeffs = gtcc(audioIn,fs) returns the gammatone cepstral coefficients (GTCCs) for the audio input, sampled at a frequency of fs Hz.

example

coeffs = gtcc(___,Name,Value) specifies options using one or more Name,Value pair arguments.

example

[coeffs,delta,deltaDelta,loc] = gtcc(___) also returns the delta, delta-delta, and location in samples corresponding to each window of data.

## Examples

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Get the gammatone cepstral coefficients for an audio file using default settings. Plot the results.

[coeffs,~,~,loc] = gtcc(audioIn,fs);

t = loc./fs;

plot(t,coeffs)
xlabel('Time (s)')
title('Gammatone Cepstral Coefficients')
legend('logE','0','1','2','3','4','5','6','7','8','9','10','11','12', ...
'Location','northeastoutside')

Calculate 20 GTCC using filters equally spaced on the ERB scale between hz2erb(62.5) and hz2erb(12000). Calculate the coefficients using 50 ms periodic Hann windows with 25 ms overlap. Replace the 0th coefficient with the log-energy. Use time-domain filtering.

[coeffs,~,~,loc] = gtcc(audioIn,fs, ...
'NumCoeffs',20, ...
'FrequencyRange',[62.5,12000], ...
'Window',hann(round(0.05*fs),'periodic'), ...
'OverlapLength',round(0.025*fs), ...
'LogEnergy','Replace', ...
'FilterDomain','Time');

Plot the results.

t = loc/fs;

plot(t,coeffs)
xlabel('Time (s)')
title('Gammatone Cepstral Coefficients')
legend('logE','1','2','3','4','5','6','7','8','9','10','11','12','13', ...
'14','15','16','17','18','19','Location','northeastoutside');

Read in an audio file and convert it to a frequency representation.

win = hann(1024,"periodic");
S = stft(audioIn,"Window",win,"OverlapLength",512,"Centered",false);

To extract the gammatone cepstral coefficients, call gtcc with the frequency-domain audio. Ignore the log-energy.

coeffs = gtcc(S,fs,"LogEnergy","Ignore");

In many applications, GTCC observations are converted to summary statistics for use in classification tasks. Plot a probability density function for one of the gammatone cepstral coefficients to observe its distributions.

nbins = 60;
coefficientToAnalyze = 4;

histogram(coeffs(:,coefficientToAnalyze+1),nbins,'Normalization','pdf')
title(sprintf("Coefficient %d",coefficientToAnalyze))

## Input Arguments

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Input signal, specified as a vector, matrix, or 3-D array.

If 'FilterDomain' is set to 'Frequency' (default), then audioIn can be real or complex.

• If audioIn is real, it is interpreted as a time-domain signal and must be a column vector or a matrix. Columns of the matrix are treated as independent audio channels.

• If audioIn is complex, it is interpreted as a frequency-domain signal. In this case, audioIn must be an L-by-M-by-N array, where L is the number of DFT points, M is the number of individual spectrums, and N is the number of individual channels.

If 'FilterDomain' is set to 'Time', then audioIn must be a real column vector or matrix. Columns of the matrix are treated as independent audio channels.

Data Types: single | double
Complex Number Support: Yes

Sample rate of the input signal in Hz, specified as a positive scalar.

Data Types: single | double

### 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: coeffs = gtcc(audioIn,fs,'LogEnergy','Replace') returns gammatone cepstral coefficients for the audio input signal sampled at fs Hz. For each analysis window, the first coefficient in the coeffs vector is replaced with the log energy of the input signal.

Window applied in time domain, specified as the comma-separated pair consisting of 'Window' and a real vector. The number of elements in the vector must be in the range [1,size(audioIn,1)]. The number of elements in the vector must also be greater than OverlapLength.

Data Types: single | double

Number of samples overlapped between adjacent windows, specified as the comma-separated pair consisting of 'OverlapLength' and an integer in the range [0, numel(Window)). If unspecified, OverlapLength defaults to round(0.02*fs).

Data Types: single | double

Number of coefficients returned for each window of data, specified as the comma-separated pair consisting of 'NumCoeffs' and an integer in the range [2, v]. v is the number of valid passbands. If unspecified, NumCoeffs defaults to 13.

The number of valid passbands is defined as the number of ERB steps (ERBN) in the frequency range of the filter bank. The frequency range of the filter bank is specified by FrequencyRange.

Data Types: single | double

Domain in which to apply filtering, specified as the comma-separated pair consisting of 'FilterDomain' and 'Frequency' or 'Time'. If unspecified, FilterDomain defaults to Frequency.

Data Types: string | char

Frequency range of gammatone filter bank in Hz, specified as the comma-separated pair consisting of 'FrequencyRange' and a two-element row vector of increasing values in the range [0, fs/2]. If unspecified, FrequencyRange defaults to [50, fs/2]

Data Types: single | double

Number of bins used to calculate the discrete Fourier transform (DFT) of windowed input samples. The FFT length must be greater than or equal to the number of elements in the Window.

Data Types: single | double

Type of nonlinear rectification applied prior to the discrete cosine transform, specified as 'log' or 'cubic-root'.

Data Types: char | string

Number of coefficients used to calculate the delta and the delta-delta values, specified as the comma-separated pair consisting of 'DeltaWindowLength' and an odd integer greater than two. If unspecified, DeltaWindowLength defaults to 9.

Deltas are computed using the audioDelta function.

Data Types: single | double

Log energy usage, specified as the comma-separated pair consisting of 'LogEnergy' and 'Append', 'Replace', or 'Ignore'. If unspecified, LogEnergy defaults to 'Append'.

• 'Append' –– The function prepends the log energy to the coefficients vector. The length of the coefficients vector is 1 + NumCoeffs.

• 'Replace' –– The function replaces the first coefficient with the log energy of the signal. The length of the coefficients vector is NumCoeffs.

• 'Ignore' –– The function does not calculate or return the log energy.

Data Types: char | string

## Output Arguments

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Gammatone cepstral coefficients, returned as an L-by-M matrix or an L-by-M-by-N array, where:

• L –– Number of analysis windows the audio signal is partitioned into. The input size, Window, and OverlapLength control this dimension: L = floor((size(audioIn,1) − numel(Window)))/(numel(Window)OverlapLength) + 1.

• M –– Number of coefficients returned per frame. This value is determined by NumCoeffs and LogEnergy.

When LogEnergy is set to:

• 'Append' –– The function prepends the log energy value to the coefficients vector. The length of the coefficients vector is 1 + NumCoeffs.

• 'Replace' –– The function replaces the first coefficient with the log energy of the signal. The length of the coefficients vector is NumCoeffs.

• 'Ignore' –– The function does not calculate or return the log energy. The length of the coefficients vector is NumCoeffs.

• N –– Number of input channels (columns). This value is size(audioIn,2).

Data Types: single | double

Change in coefficients from one analysis window to another, returned as an L-by-M matrix or an L-by-M-by-N array. The delta array is the same size and data type as the coeffs array. See coeffs for the definitions of L, M, and N.

Data Types: single | double

Change in delta values, returned as an L-by-M matrix or an L-by-M-by-N array. The deltaDelta array is the same size and data type as the coeffs and delta arrays. See coeffs for the definitions of L, M, and N.

Data Types: single | double

Location of last sample in each analysis window, returned as a column vector with the same number of rows as coeffs.

Data Types: single | double

## Algorithms

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The gtcc function splits the entire data into overlapping segments. The length of each analysis window is determined by Window. The length of overlap between analysis windows is determined by OverlapLength. The algorithm to determine the gammatone cepstral coefficients depends on the filter domain, specified by FilterDomain. The default filter domain is frequency.

### Frequency-Domain Filtering

Gammatone cepstrum coefficients are popular features extracted from speech signals for use in recognition tasks. In the source-filter model of speech, cepstral coefficients are understood to represent the filter (vocal tract). The vocal tract frequency response is relatively smooth, whereas the source of voiced speech can be modeled as an impulse train. As a result, the vocal tract can be estimated by the spectral envelope of a speech segment.

The motivating idea of gammatone cepstral coefficients is to compress information about the vocal tract (smoothed spectrum) into a small number of coefficients based on an understanding of the cochlea. Although there is no hard standard for calculating the coefficients, the basic steps are outlined by the diagram.

The default gammatone filter bank is composed of gammatone filters spaced linearly on the ERB scale between 50 and 8000 Hz. The filter bank is designed by designAuditoryFilterBank.

The information contained in the zeroth gammatone cepstral coefficient is often augmented with or replaced by the log energy. The log energy calculation depends on the input domain.

If the input is a time-domain signal, the log energy is computed using the following equation:

$\mathrm{log}E=\mathrm{log}\left(\text{sum}\left({x}^{2}\right)\right)$

If the input is a frequency-domain signal, the log energy is computed using the following equation:

$\mathrm{log}E=\mathrm{log}\left(\text{sum}\left({|x|}^{2}\right)/FFTLength\right)$

### Time-Domain Filtering

If FilterDomain is specified as 'Time', the gtcc function uses the gammatoneFilterBank to apply time-domain filtering. The basic steps of the gtcc algorithm are outlined by the diagram.

The FrequencyRange and sample rate (fs) parameters are set on the filter bank using the name-value pairs input to the gtcc function. The number of filters in the gammatone filter bank is defined as hz2erb(FrequencyRange(2)) − hz2erb(FrequencyRange(1)).This roughly corresponds to placing a gammatone filter every 0.9 mm in the cochlea.

The output from the gammatone filter bank is a multichannel signal. Each channel output from the gammatone filter bank is buffered into overlapped analysis windows, as specified by the Window and OverlapLength parameters. The energy for each analysis window of data is calculated. The STE of the channels are concatenated. The concatenated signal is then passed through a logarithm function and transformed to the cepstral domain using a discrete cosine transform (DCT).

The log-energy is calculated on the original audio signal using the same buffering scheme applied to the gammatone filter bank output.

## Compatibility Considerations

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Behavior changed in R2020b

Behavior change in future release

## References

[1] Shao, Yang, Zhaozhang Jin, Deliang Wang, and Soundararajan Srinivasan. "An Auditory-Based Feature for Robust Speech Recognition." IEEE International Conference on Acoustics, Speech and Signal Processing. 2009.

[2] Valero, X., and F. Alias. "Gammatone Cepstral Coefficients: Biologically Inspired Features for Non-Speech Audio Classification." IEEE Transactions on Multimedia. Vol. 14, Issue 6, 2012, pp. 1684–1689.