Filtering and Convolution

The linear convolution of two sequences $x[n]$ (length $N$) and $h[n]$ (length $M$) produces a sequence of length $N + M - 1$:

$$(x * h)[n] = \sum_{m=0}^{M-1} h[m]\, x[n - m]$$

In SciPy:

from scipy.signal import convolve, fftconvolve
y = convolve(x, h, mode='full')       # direct method
y = fftconvolve(x, h, mode='full')    # FFT-based (faster for large arrays)

An FIR filter computes the output as a weighted sum of the current and past input samples:

$$y[n] = \sum_{k=0}^{M-1} b_k\, x[n-k]$$

where $\{b_k\}$ are the filter coefficients. The output is simply the convolution $y = x * b$. FIR filters are always stable and can have exactly linear phase (constant group delay).

from scipy.signal import firwin
b = firwin(numtaps=51, cutoff=0.3)  # lowpass, cutoff = 0.3 * Nyquist

An IIR filter uses feedback from past outputs, implementing a rational transfer function:

$$H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}$$

IIR filters achieve steep rolloff with fewer coefficients than FIR filters but can be unstable and have nonlinear phase.

from scipy.signal import butter, cheby1, sosfilt
sos = butter(N=5, Wn=0.3, btype='low', output='sos')
y = sosfilt(sos, x)

A second-order section (biquad) is a 2nd-order IIR filter:

$$H_i(z) = \frac{b_{i0} + b_{i1}z^{-1} + b_{i2}z^{-2}}{1 + a_{i1}z^{-1} + a_{i2}z^{-2}}$$

Any IIR filter of order $N$ can be factored into $\lceil N/2 \rceil$ cascaded second-order sections. This representation avoids the numerical instability of high-order polynomials:

sos = butter(10, 0.2, output='sos')   # 5 second-order sections
y = sosfilt(sos, x)                    # cascade filtering

The matched filter for a known signal $s[n]$ in additive white Gaussian noise is the filter whose impulse response is the time-reversed conjugate of $s$:

$$h_\text{MF}[n] = s^*[M - 1 - n]$$

The matched filter maximizes the output signal-to-noise ratio (SNR). In practice, it is implemented as cross-correlation:

from scipy.signal import correlate
y = correlate(received, template, mode='full')

The Remez exchange algorithm (Parks-McClellan) designs equiripple FIR filters that minimize the maximum error in specified frequency bands:

from scipy.signal import remez
b = remez(
    numtaps=65,
    bands=[0, 0.2, 0.3, 0.5],   # passband: 0-0.2, stopband: 0.3-0.5
    desired=[1, 0],               # gain in each band
    fs=1.0,
)

The result has equiripple behavior: the approximation error oscillates with equal amplitude across each band (Chebyshev optimality).