FIR Filters & Convolution

Four complementary methods for filtering and convolution, from educational time-domain approaches to efficient FFT-based processing.

Time-domain convolution

For signals of length N and kernels of length M, computes the full linear convolution. Output length is N + M - 1.

import pyminidsp as md
import numpy as np

signal = md.impulse(100, amplitude=1.0, position=0)
kernel = np.array([1.0, 2.0, 3.0])
out = md.convolution_time(signal, kernel)
# out[:3] == [1.0, 2.0, 3.0]
# len(out) == 102

Moving-average filter

A simple low-pass filter that computes the running mean over a window. Output matches input length with zero-padded startup.

signal = md.sine_wave(1024, freq=440.0, sample_rate=44100.0)
smoothed = md.moving_average(signal, window_len=5)

General FIR filter

Apply a causal FIR filter with arbitrary coefficients:

\[\text{out}[n] = \sum_{k=0}^{T-1} \text{coeffs}[k] \cdot \text{signal}[n-k]\]

Output matches input length.

coeffs = np.array([0.25, 0.5, 0.25])
filtered = md.fir_filter(signal, coeffs)

FFT overlap-add

Same result as time-domain convolution but much faster for long kernels by processing blocks in the frequency domain.

kernel = md.hann_window(256)
out_time = md.convolution_time(signal, kernel)
out_fft = md.convolution_fft_ola(signal, kernel)
np.testing.assert_allclose(out_time, out_fft, atol=1e-10)

Comparison

Method	Complexity	Output length	Best for
convolution_time	O(NM)	N + M − 1	Teaching, short kernels
moving_average	O(N)	N	Simple smoothing
fir_filter	O(NM)	N	Standard FIR design
convolution_fft_ola	O(N log N)	N + M − 1	Long kernels, production

md.shutdown()