Mel Filterbanks & MFCCs¶

Two essential features for speech and audio machine learning:

Mel filterbank energies — triangular spectral bands spaced on the mel scale, which compresses frequency representation to match human hearing.
MFCCs — decorrelated coefficients derived from the log mel energies via a DCT, widely used in speech recognition and audio classification.

Mel scale¶

The HTK mapping:

\[\text{mel}(f) = 2595 \cdot \log_{10}\!\left(1 + \frac{f}{700}\right)\]

This densifies low frequencies and coarsens high frequencies, reflecting how humans perceive pitch.

Building a mel filterbank¶

import pyminidsp as md

fb = md.mel_filterbank(512, sample_rate=16000.0, num_mels=26)
# fb.shape == (26, 257)  — 26 triangular filters over 257 FFT bins

Computing mel energies¶

From a single frame:

signal = md.sine_wave(512, freq=440.0, sample_rate=16000.0)
mel = md.mel_energies(signal, sample_rate=16000.0, num_mels=26)
# mel.shape == (26,)

Processing steps (internally):

Apply a Hann window.
Compute one-sided PSD bins via FFT: |X(k)|² / N.
Apply mel filterbank weights and sum per band.

Computing MFCCs¶

coeffs = md.mfcc(signal, sample_rate=16000.0, num_mels=26, num_coeffs=13)
# coeffs.shape == (13,)

Conventions:

HTK mel mapping for filter placement.
Natural-log compression: log(max(E_mel, 1e-12)).
DCT-II with HTK-C0 normalisation.
Coefficient C0 is in coeffs[0].

Processing a full utterance¶

To extract MFCCs from a longer signal, use the STFT to break it into frames first:

import numpy as np

sr = 16000.0
frame_size = 512
hop = 128

# Load or generate a signal
signal = md.chirp_linear(int(sr), f_start=200.0, f_end=4000.0, sample_rate=sr)

num_frames = md.stft_num_frames(len(signal), frame_size, hop)
all_mfcc = np.zeros((num_frames, 13))
for f in range(num_frames):
    start = f * hop
    frame = signal[start:start + frame_size]
    all_mfcc[f] = md.mfcc(frame, sample_rate=sr, num_mels=26, num_coeffs=13)

md.shutdown()