machine_learning.mfcc

Mel Frequency Cepstral Coefficients (MFCC) Calculation

MFCC is an algorithm widely used in audio and speech processing to represent the short-term power spectrum of a sound signal in a more compact and discriminative way. It is particularly popular in speech and audio processing tasks such as speech recognition and speaker identification.

How Mel Frequency Cepstral Coefficients are Calculated: 1. Preprocessing:

• Load an audio signal and normalize it to ensure that the values fall within a specific range (e.g., between -1 and 1).

• Frame the audio signal into overlapping, fixed-length segments, typically using a technique like windowing to reduce spectral leakage.

2. Fourier Transform: - Apply a Fast Fourier Transform (FFT) to each audio frame to convert it

   from the time domain to the frequency domain. This results in a representation of the audio frame as a sequence of frequency components.

3. Power Spectrum: - Calculate the power spectrum by taking the squared magnitude of each

   frequency component obtained from the FFT. This step measures the energy distribution across different frequency bands.

4. Mel Filterbank: - Apply a set of triangular filterbanks spaced in the Mel frequency scale

   to the power spectrum. These filters mimic the human auditory system’s frequency response. Each filterbank sums the power spectrum values within its band.

5. Logarithmic Compression: - Take the logarithm (typically base 10) of the filterbank values to

   compress the dynamic range. This step mimics the logarithmic response of the human ear to sound intensity.

6. Discrete Cosine Transform (DCT): - Apply the Discrete Cosine Transform to the log filterbank energies to

   obtain the MFCC coefficients. This transformation helps decorrelate the filterbank energies and captures the most important features of the audio signal.

7. Feature Extraction: - Select a subset of the DCT coefficients to form the feature vector.

   Often, the first few coefficients (e.g., 12-13) are used for most applications.

References: - Mel-Frequency Cepstral Coefficients (MFCCs):

https://en.wikipedia.org/wiki/Mel-frequency_cepstrum

• Speech and Language Processing by Daniel Jurafsky & James H. Martin: https://web.stanford.edu/~jurafsky/slp3/

• Mel Frequency Cepstral Coefficient (MFCC) tutorial http://practicalcryptography.com/miscellaneous/machine-learning /guide-mel-frequency-cepstral-coefficients-mfccs/

Author: Amir Lavasani

Functions

audio_frames(→ numpy.ndarray)	Split an audio signal into overlapping frames.
calculate_fft(→ numpy.ndarray)	Calculate the Fast Fourier Transform (FFT) of windowed audio data.
calculate_signal_power(→ numpy.ndarray)	Calculate the power of the audio signal from its FFT.
discrete_cosine_transform(→ numpy.ndarray)	Compute the Discrete Cosine Transform (DCT) basis matrix.
example(→ numpy.ndarray)	Example function to calculate Mel Frequency Cepstral Coefficients
freq_to_mel(→ float)	Convert a frequency in Hertz to the mel scale.
get_filter_points(→ tuple[numpy.ndarray, numpy.ndarray])	Calculate the filter points and frequencies for mel frequency filters.
get_filters(→ numpy.ndarray)	Generate filters for audio processing.
mel_spaced_filterbank(→ numpy.ndarray)	Create a Mel-spaced filter bank for audio processing.
mel_to_freq(→ float)	Convert a frequency in the mel scale to Hertz.
mfcc(→ numpy.ndarray)	Calculate Mel Frequency Cepstral Coefficients (MFCCs) from an audio signal.
normalize(→ numpy.ndarray)	Normalize an audio signal by scaling it to have values between -1 and 1.

Module Contents

machine_learning.mfcc.audio_frames(audio: numpy.ndarray, sample_rate: int, hop_length: int = 20, ftt_size: int = 1024) → numpy.ndarray

Split an audio signal into overlapping frames.

Args:

audio: The input audio signal. sample_rate: The sample rate of the audio signal. hop_length: The length of the hopping (default is 20ms). ftt_size: The size of the FFT window (default is 1024).

Returns:

An array of overlapping frames.

Examples: >>> audio = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]*1000) >>> sample_rate = 8000 >>> frames = audio_frames(audio, sample_rate, hop_length=10, ftt_size=512) >>> frames.shape (126, 512)

machine_learning.mfcc.calculate_fft(audio_windowed: numpy.ndarray, ftt_size: int = 1024) → numpy.ndarray

Calculate the Fast Fourier Transform (FFT) of windowed audio data.

Args:

audio_windowed: The windowed audio signal. ftt_size: The size of the FFT (default is 1024).

Returns:

The FFT of the audio data.

Examples: >>> audio_windowed = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]]) >>> audio_fft = calculate_fft(audio_windowed, ftt_size=4) >>> bool(np.allclose(audio_fft[0], np.array([6.0+0.j, -1.5+0.8660254j, … -1.5-0.8660254j]))) True

machine_learning.mfcc.calculate_signal_power(audio_fft: numpy.ndarray) → numpy.ndarray

Calculate the power of the audio signal from its FFT.

Args:

audio_fft: The FFT of the audio signal.

Returns:

The power of the audio signal.

Examples: >>> audio_fft = np.array([1+2j, 2+3j, 3+4j, 4+5j]) >>> power = calculate_signal_power(audio_fft) >>> np.allclose(power, np.array([5, 13, 25, 41])) True

machine_learning.mfcc.discrete_cosine_transform(dct_filter_num: int, filter_num: int) → numpy.ndarray

Compute the Discrete Cosine Transform (DCT) basis matrix.

Args:

dct_filter_num: The number of DCT filters to generate. filter_num: The number of the fbank filters.

Returns:

The DCT basis matrix.

Examples: >>> float(round(discrete_cosine_transform(3, 5)[0][0], 5)) 0.44721

machine_learning.mfcc.example(wav_file_path: str = './path-to-file/sample.wav') → numpy.ndarray

Example function to calculate Mel Frequency Cepstral Coefficients (MFCCs) from an audio file.

Args:

wav_file_path: The path to the WAV audio file.

Returns:

np.ndarray: The computed MFCCs for the audio.

machine_learning.mfcc.freq_to_mel(freq: float) → float

Convert a frequency in Hertz to the mel scale.

Args:

freq: The frequency in Hertz.

Returns:

The frequency in mel scale.

Examples: >>> float(round(freq_to_mel(1000), 2)) 999.99

machine_learning.mfcc.get_filter_points(sample_rate: int, freq_min: int, freq_high: int, mel_filter_num: int = 10, ftt_size: int = 1024) → tuple[numpy.ndarray, numpy.ndarray]

Calculate the filter points and frequencies for mel frequency filters.

Args:

sample_rate: The sample rate of the audio. freq_min: The minimum frequency in Hertz. freq_high: The maximum frequency in Hertz. mel_filter_num: The number of mel filters (default is 10). ftt_size: The size of the FFT (default is 1024).

Returns:

Filter points and corresponding frequencies.

Examples: >>> filter_points = get_filter_points(8000, 0, 4000, mel_filter_num=4, ftt_size=512) >>> filter_points[0] array([ 0, 20, 51, 95, 161, 256]) >>> filter_points[1] array([ 0. , 324.46707094, 799.33254207, 1494.30973963,

2511.42581671, 4000. ])

machine_learning.mfcc.get_filters(filter_points: numpy.ndarray, ftt_size: int) → numpy.ndarray

Generate filters for audio processing.

Args:

filter_points: A list of filter points. ftt_size: The size of the FFT.

Returns:

A matrix of filters.

Examples: >>> get_filters(np.array([0, 20, 51, 95, 161, 256], dtype=int), 512).shape (4, 257)

machine_learning.mfcc.mel_spaced_filterbank(sample_rate: int, mel_filter_num: int = 10, ftt_size: int = 1024) → numpy.ndarray

Create a Mel-spaced filter bank for audio processing.

Args:

sample_rate: The sample rate of the audio. mel_filter_num: The number of mel filters (default is 10). ftt_size: The size of the FFT (default is 1024).

Returns:

Mel-spaced filter bank.

Examples: >>> float(round(mel_spaced_filterbank(8000, 10, 1024)[0][1], 10)) 0.0004603981

machine_learning.mfcc.mel_to_freq(mels: float) → float

Convert a frequency in the mel scale to Hertz.

Args:

mels: The frequency in mel scale.

Returns:

The frequency in Hertz.

Examples: >>> round(mel_to_freq(999.99), 2) 1000.01

machine_learning.mfcc.mfcc(audio: numpy.ndarray, sample_rate: int, ftt_size: int = 1024, hop_length: int = 20, mel_filter_num: int = 10, dct_filter_num: int = 40) → numpy.ndarray

Calculate Mel Frequency Cepstral Coefficients (MFCCs) from an audio signal.

Args:

audio: The input audio signal. sample_rate: The sample rate of the audio signal (in Hz). ftt_size: The size of the FFT window (default is 1024). hop_length: The hop length for frame creation (default is 20ms). mel_filter_num: The number of Mel filters (default is 10). dct_filter_num: The number of DCT filters (default is 40).

Returns:

A matrix of MFCCs for the input audio.

Raises:

ValueError: If the input audio is empty.

Example: >>> sample_rate = 44100 # Sample rate of 44.1 kHz >>> duration = 2.0 # Duration of 1 second >>> t = np.linspace(0, duration, int(sample_rate * duration), endpoint=False) >>> audio = 0.5 * np.sin(2 * np.pi * 440.0 * t) # Generate a 440 Hz sine wave >>> mfccs = mfcc(audio, sample_rate) >>> mfccs.shape (40, 101)

machine_learning.mfcc.normalize(audio: numpy.ndarray) → numpy.ndarray

Normalize an audio signal by scaling it to have values between -1 and 1.

Args:

audio: The input audio signal.

Returns:

The normalized audio signal.

Examples: >>> audio = np.array([1, 2, 3, 4, 5]) >>> normalized_audio = normalize(audio) >>> float(np.max(normalized_audio)) 1.0 >>> float(np.min(normalized_audio)) 0.2