# mfcc

Extract MFCC, log energy, delta, and delta-delta of audio signal

## Syntax

## Description

specifies options using one or more name-value arguments.`coeffs`

= mfcc(___,`Name=Value`

)

**Example: **`coeffs = mfcc(audioIn,fs,LogEnergy="replace")`

returns
mel-frequency cepstral coefficients for the audio input signal sampled at
`fs`

Hz. The first coefficient in the `coeffs`

vector is replaced with the log energy value.

`[`

also returns the delta, delta-delta, and location of samples corresponding to each
window of data. You can specify an input combination from any of the previous
syntaxes.`coeffs`

,`delta`

,`deltaDelta`

,`loc`

] = mfcc(___)

`mfcc(___)`

with no output arguments plots the
mel-frequency cepstral coefficients. Before plotting, the coefficients are
normalized to have mean 0 and standard deviation 1.

If the input is in the time domain, the coefficients are plotted against time.

If the input is in the frequency domain, the coefficients are plotted against frame number.

If the log energy is extracted, then it is also plotted.

## Examples

### Compute Mel Frequency Cepstral Coefficients

Compute the mel frequency cepstral coefficients of a speech signal using the `mfcc`

function. The function returns `delta`

, the change in coefficients, and `deltaDelta`

, the change in delta values. The log energy value that the function computes can prepend the coefficients vector or replace the first element of the coefficients vector. This is done based on whether you set the `LogEnergy`

argument to "`append"`

or "`replace"`

.

Read an audio signal from the `Counting-16-44p1-mono-15secs.wav`

file using the `audioread`

function. The `mfcc`

function processes the entire speech data in a batch. Based on the number of input rows, the window length, and the overlap length, `mfcc`

partitions the speech into 1551 frames and computes the cepstral features for each frame. Each row in the `coeffs`

matrix corresponds to the log-energy value followed by the 13 mel-frequency cepstral coefficients for the corresponding frame of the speech file. The function also computes `loc`

, the location of the last sample in each input frame.

```
[audioIn,fs] = audioread("Counting-16-44p1-mono-15secs.wav");
[coeffs,delta,deltaDelta,loc] = mfcc(audioIn,fs);
```

Plot the normalized coefficients.

mfcc(audioIn,fs)

### Extract MFCC from Frequency-Domain Audio

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

[audioIn,fs] = audioread("Rainbow-16-8-mono-114secs.wav"); win = hann(1024,"periodic"); S = stft(audioIn,"Window",win,"OverlapLength",512,"Centered",false);

To extract the mel-frequency cepstral coefficients, call `mfcc`

with the frequency-domain audio. Ignore the log-energy.

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

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

nbins = 60; coefficientToAnalyze = 4; histogram(coeffs(:,coefficientToAnalyze+1),nbins,"Normalization","pdf") title(sprintf("Coefficient %d",coefficientToAnalyze))

## Input Arguments

`audioIn`

— Input signal

vector | matrix | 3-D array

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

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 spectra, and*N*is the number of individual channels.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`fs`

— Sample rate (Hz)

positive scalar

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

**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: **```
[coeffs,delta,deltaDelta,loc] =
mfcc(audioIn,fs,LogEnergy="replace",DeltaWindowLength=5)
```

returns mel
frequency cepstral coefficients for the audio input signal sampled at
`fs`

Hz. The first coefficient in the `coeffs`

vector is replaced with the log energy value. A set of 5 cepstral coefficients is
used to compute the delta and the delta-delta values.

`Window`

— Window applied in time domain

`hamming(round(0.03*fs),"periodic")`

(default) | vector

Window applied in time domain, specified as a real vector. The number
of elements in the vector must be in the range
[1,`size(`

]. The
number of elements in the vector must also be greater than
`audioIn`

,1)`OverlapLength`

.

**Data Types: **`single`

| `double`

`OverlapLength`

— Number of overlapping samples between adjacent windows

`round(``fs`

*0.02)

(default) | integer

`fs`

*0.02)`NumCoeffs`

— Number of coefficients returned

`13`

(default) | positive scalar integer

Number of coefficients returned for each window of data, specified as
an integer in the range [2 *v*], where
*v* is the number of valid passbands.

The number of valid passbands is defined as ```
sum(BandEdges
<= floor(fs/2))-2
```

. A passband is valid if its edges
fall below `fs/2`

, where `fs`

is the
sample rate of the input audio signal, specified as the second argument,
`fs`

.

**Data Types: **`single`

| `double`

`BandEdges`

— Band edges of filter bank (Hz)

row vector

Band edges of the filter bank in Hz, specified as a nonnegative
monotonically increasing row vector in the range [0,
`fs`

/2]. The number of band edges must be in the
range [4, 160]. The `mfcc`

function designs
half-overlapped triangular filters based on
`BandEdges`

. This means that all band edges,
except for the first and last, are also center frequencies of the
designed bandpass filters.

By default, `BandEdges`

is a 42-element vector,
which results in a 40-band filter bank that spans approximately 133 Hz
to 6864 Hz. The default bands are spaced as described in [2].

**Data Types: **`single`

| `double`

`FFTLength`

— Number of bins for calculating DFT

`numel(``Window`

)

(default) | positive scalar integer

`Window`

)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`

`Rectification`

— Type of non-linear rectification

`"log"`

(default) | `"cubic-root"`

Type of nonlinear rectification applied prior to the discrete cosine
transform, specified as `"log"`

or
`"cubic-root"`

.

**Data Types: **`char`

| `string`

`DeltaWindowLength`

— Number of coefficients for calculating delta and delta-delta

`9`

(default) | odd integer greater than 2

Number of coefficients used to calculate the delta and the delta-delta
values, specified as an odd integer greater than two. If unspecified,
`DeltaWindowLength`

defaults to
`9`

.

Deltas are computed using the `audioDelta`

function.

**Data Types: **`single`

| `double`

`LogEnergy`

— Specify how the log energy is shown

`"append"`

(default) | `"replace"`

| `"ignore"`

Specify how the log energy is shown in the coefficients vector output, specified as:

`"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 object does not calculate or return the log energy.

**Data Types: **`char`

| `string`

## Output Arguments

`coeffs`

— Mel-frequency cepstral coefficients (MFCCs)

matrix | 3-D array

Mel-frequency 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`

`delta`

— Change in coefficients

matrix | array

Change in coefficients from one frame of data 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.

**Data Types: **`single`

| `double`

`loc`

— Location of the last sample in each input frame

vector

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

### MFCC

Mel-frequency 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 mel-frequency 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 mel filter bank linearly spaces the first 10 triangular filters and logarithmically spaces the remaining filters.

### Log Energy

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

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

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

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

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

## References

[1] Rabiner, Lawrence R., and
Ronald W. Schafer. *Theory and Applications of Digital Speech
Processing*. Upper Saddle River, NJ: Pearson, 2010.

[2] Auditory Toolbox. https://engineering.purdue.edu/~malcolm/interval/1998-010/AuditoryToolboxTechReport.pdf

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

### GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

### GPU Arrays

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

## Version History

**Introduced in R2018a**

### R2020b: Delta and delta-delta computation

The delta and delta-delta calculations are now computed using the `audioDelta`

function, which has a different startup behavior than the
previous algorithm. The default value of the `DeltaWindowLength`

parameter has changed from `2`

to `9`

. A delta
window length of `2`

is no longer supported.

### R2020b: `WindowLength`

will be removed in a future release

The `WindowLength`

parameter will be removed from the
`mfcc`

function in a future release. Use the
`Window`

parameter instead.

In releases prior to R2020b, you could only specify the length of a time-domain window. The window was always designed as a periodic Hamming window. You can replace instances of the code

coeffs = mfcc(audioin,fs,WindowLength=1024);

`coeffs = mfcc(audioIn,fs,Window=hamming(1024,"periodic"));`

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