In sound processing, the mel-frequency cepstrum (MFC) is a representation of the short-term power spectrum of a sound, based on a linear cosine transform of a log power spectrum on a nonlinear mel scale of frequency. Mel-frequency cepstral coefficients (MFCCs) are coefficients that collectively make up an MFC. They are derived from a type of cepstral representation of the audio clip (a nonlinear "spectrum-of-a-spectrum"). The difference between the cepstrum and the mel-frequency cepstrum is that in the MFC, the frequency bands are equally spaced on the mel scale, which approximates the human auditory system's response more closely than the linearly-spaced frequency bands used in the normal spectrum. This frequency warping can allow for better representation of sound, for example, in audio compression that might potentially reduce the transmission bandwidth and the storage requirements of audio signals. MFCCs are commonly derived as follows:
Take the Fourier transform of (a windowed excerpt of) a signal. Map the powers of the spectrum obtained above onto the mel scale, using triangular overlapping windows or alternatively, cosine overlapping windows. Take the logs of the powers at each of the mel frequencies. Take the discrete cosine transform of the list of mel log powers, as if it were a signal. The MFCCs are the amplitudes of the resulting spectrum. There can be variations on this process, for example: differences in the shape or spacing of the windows used to map the scale, or addition of dynamics features such as "delta" and "delta-delta" (first- and second-order frame-to-frame difference) coefficients. The European Telecommunications Standards Institute in the early 2000s defined a standardised MFCC algorithm to be used in mobile phones.
Applications MFCCs are commonly used as features in speech recognition systems, such as the systems which can automatically recognize numbers spoken into a telephone. MFCCs are also increasingly finding uses in music information retrieval applications such as genre classification, audio similarity measures, etc.
MFCC for speaker recognition
Since Mel-frequency bands are distributed evenly in MFCC, and they are very similar to the voice system of a human, MFCC can efficiently be used to characterize speakers. For instance, it can be used to recognize the speaker's cell phone model characteristics, and further the details of the speaker's voice. This type of mobile device recognition is possible because the production of electronic components in a phone have tolerances, because different electronic circuit realizations do not have exact same transfer functions. The dissimilarities in the transfer function from one realization to another becomes more prominent if the task performing circuits are from different manufacturers. Hence, each cell phone introduces a convolutional distortion on input speech that leaves its unique impact on the recordings from the cell phone. Therefore, a particular phone can be identified from the recorded speech by multiplying the original frequency spectrum with further multiplications of transfer functions specific to each phone followed by signal processing techniques. Thus, by using MFCC one can characterize cell phone recordings to identify the brand and model of the phone. Considering recording section of a cellphone as Linear time-invariant (LTI) filter: Impulse response- h(n), recorded speech signal y(n) as output of filter in response to input x(n). Hence, y ( n ) = x ( n ) ∗ h ( n ) {\displaystyle y(n)=x(n)*h(n)} (convolution) As speech is not stationary signal, it is divided into overlapped frames within which the signal is assumed to be stationary. So, the p t h {\displaystyle p^{th}} short-term segment (frame) of recorded input speech is:
y p w ( n ) = [ x ( n ) w ( p W − n ) ] ∗ h ( n ) {\displaystyle y_{p}w(n)=[x(n)w(pW-n)]*h(n)} , where w(n): windowed function of length W. Hence, as specified the footprint of mobile phone of the recorded speech is the convolution distortion that helps to identify the recording phone. The embedded identity of the cell phone requires a conversion to a better identifiable form, hence, taking short-time Fourier transform:
Y p w ( f ) = X p w ( f ) H ( f ) {\displaystyle Y_{p}w(f)=X_{p}w(f)H(f)}
H ( f ) {\displaystyle H(f)} can be considered as a concatenated transfer function that produced input speech, and the recorded speech Y p w ( f ) {\displaystyle Y_{p}w(f)} can be perceived as original speech from cell phone. So, equivalent transfer function of vocal tract and cell phone recorder is considered as original source of recorded speech. Therefore,
X p w ( f ) = X e p w ( f ) X v ( f ) , H ′ ( f ) = H ( f ) X v ( f ) , {\displaystyle X_{p}w(f)=Xe_{p}w(f)X_{v}(f),H'(f)=H(f)X_{v}(f),}
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