Digital signal processing with scipy.signal: filter design, spectral analysis, peak detection, PSD estimation, and adaptive filtering for engineering workflows.
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Digital signal processing with scipy.signal: filter design, spectral analysis, peak detection, PSD estimation, and adaptive filtering for engineering workflows.
A comprehensive skill for digital signal processing (DSP) using scipy.signal.
Covers FIR/IIR filter design, time-series filtering, spectral analysis, peak
detection, power spectral density estimation, cross-correlation, matched filters,
and adaptive filtering. Designed for real-world applications such as biomedical
signal processing and vibration/fault analysis.
Core Functions
1. Filter Design
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
defdesign_bandpass_filter(
lowcut: float,
highcut: float,
fs: float,
order: int = 4,
filter_type: str = "butter",
rp: float = 1.0,
rs: float = 60.0,
) -> tuple[np.ndarray, np.ndarray]:
"""
Design a bandpass IIR or FIR filter.
Parameters
----------
lowcut : float
Lower cutoff frequency in Hz.
highcut : float
Upper cutoff frequency in Hz.
fs : float
Sampling frequency in Hz.
order : int
Filter order (default 4).
filter_type : str
One of 'butter', 'cheby1', 'cheby2', 'ellip', 'firwin'.
rp : float
Maximum ripple in the passband (dB), used by cheby1/ellip.
rs : float
Minimum attenuation in the stopband (dB), used by cheby2/ellip.
Returns
-------
b, a : array_like
Numerator and denominator polynomials of the IIR filter.
For FIR (firwin), a = [1.0].
"""
nyq = 0.5 * fs
low = lowcut / nyq
high = highcut / nyq
if filter_type == "butter":
b, a = signal.butter(order, [low, high], btype="bandpass")
elif filter_type == "cheby1":
b, a = signal.cheby1(order, rp, [low, high], btype="bandpass")
filter_type == :
b, a = signal.cheby2(order, rs, [low, high], btype=)
filter_type == :
b, a = signal.ellip(order, rp, rs, [low, high], btype=)
filter_type == :
order % != :
order +=
b = signal.firwin(order + , [low, high], pass_zero=)
a = np.array([])
:
ValueError()
b, a
() -> [np.ndarray, np.ndarray]:
w0 = freq / ( * fs)
b, a = signal.iirnotch(w0, quality)
b, a
() -> [np.ndarray, np.ndarray]:
nyq = * fs
normal_cutoff = cutoff / nyq
filter_type == :
b, a = signal.butter(order, normal_cutoff, btype=)
filter_type == :
order % == :
order +=
b = signal.firwin(order, normal_cutoff)
a = np.array([])
:
ValueError()
b, a
elif
"cheby2"
"bandpass"
elif
"ellip"
"bandpass"
elif
"firwin"
# FIR bandpass via window method; order must be even for bandpass
if
2
0
1
1
False
1.0
else
raise
f"Unsupported filter_type: {filter_type!r}"
return
def
design_notch_filter
freq: float, fs: float, quality: float = 30.0
tuple
"""
Design a notch (band-stop) filter to remove a single frequency.
Parameters
----------
freq : float
Frequency to attenuate in Hz.
fs : float
Sampling frequency in Hz.
quality : float
Quality factor Q = freq / bandwidth.
Returns
-------
b, a : array_like
"""
defapply_filter(
data: np.ndarray,
b: np.ndarray,
a: np.ndarray,
method: str = "sosfilt",
) -> np.ndarray:
"""
Apply a digital filter to a 1-D signal.
Parameters
----------
data : np.ndarray
Input signal array (1-D).
b : np.ndarray
Numerator coefficients.
a : np.ndarray
Denominator coefficients.
method : str
'sosfilt' (recommended, numerically stable) or 'filtfilt' (zero-phase).
Returns
-------
np.ndarray
Filtered signal with the same length as `data`.
"""if method == "sosfilt":
sos = signal.tf2sos(b, a)
return signal.sosfiltfilt(sos, data)
elif method == "filtfilt":
return signal.filtfilt(b, a, data)
elif method == "lfilter":
return signal.lfilter(b, a, data)
else:
raise ValueError(f"Unknown method: {method!r}")
defapply_envelope_detection(
data: np.ndarray, fs: float, lowpass_cutoff: float = 10.0) -> np.ndarray:
"""
Compute the amplitude envelope using the Hilbert transform followed by
a lowpass filter.
Parameters
----------
data : np.ndarray
Bandpass-filtered signal.
fs : float
Sampling frequency in Hz.
lowpass_cutoff : float
Cutoff for the smoothing lowpass filter (Hz).
Returns
-------
np.ndarray
Smoothed amplitude envelope.
"""
analytic = signal.hilbert(data)
envelope = np.abs(analytic)
b, a = design_lowpass_filter(lowpass_cutoff, fs, order=4)
return apply_filter(envelope, b, a)
3. Spectral Analysis
defcompute_spectrogram(
data: np.ndarray,
fs: float,
window: str = "hann",
nperseg: int = 256,
noverlap: int | None = None,
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""
Compute the Short-Time Fourier Transform (STFT) spectrogram.
Parameters
----------
data : np.ndarray
Input time-series signal.
fs : float
Sampling frequency in Hz.
window : str
Window function name (e.g. 'hann', 'hamming', 'blackman').
nperseg : int
Length of each STFT segment.
noverlap : int or None
Number of overlapping samples. Defaults to nperseg // 2.
Returns
-------
f : np.ndarray
Frequency bins (Hz).
t : np.ndarray
Time bins (seconds).
Sxx : np.ndarray
Power spectrogram in dB (shape: [freq_bins, time_bins]).
"""if noverlap isNone:
noverlap = nperseg // 2
f, t, Zxx = signal.stft(data, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap)
Sxx = 20 * np.log10(np.abs(Zxx) + 1e-12)
return f, t, Sxx
defcompute_psd_welch(
data: np.ndarray,
fs: float,
nperseg: int = 512,
window: str = "hann",
) -> tuple[np.ndarray, np.ndarray]:
"""
Estimate power spectral density using Welch's method.
Parameters
----------
data : np.ndarray
Input signal.
fs : float
Sampling frequency in Hz.
nperseg : int
Length of each Welch segment.
window : str
Window function name.
Returns
-------
freqs : np.ndarray
Frequency bins (Hz).
psd : np.ndarray
Power spectral density (V²/Hz).
"""
freqs, psd = signal.welch(data, fs=fs, window=window, nperseg=nperseg)
return freqs, psd
defplot_frequency_response(
b: np.ndarray,
a: np.ndarray,
fs: float,
title: str = "Filter Frequency Response",
) -> None:
"""Plot magnitude and phase response using freqz."""
w, h = signal.freqz(b, a, worN=8000, fs=fs)
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 6))
ax1.plot(w, 20 * np.log10(np.abs(h) + 1e-12))
ax1.set_xlabel("Frequency (Hz)")
ax1.set_ylabel("Magnitude (dB)")
ax1.set_title(title)
ax1.grid(True)
ax2.plot(w, np.angle(h, deg=True))
ax2.set_xlabel("Frequency (Hz)")
ax2.set_ylabel("Phase (degrees)")
ax2.grid(True)
plt.tight_layout()
plt.show()
4. Peak Detection and Correlation
defdetect_peaks(
data: np.ndarray,
min_height: float | None = None,
min_distance: int = 1,
prominence: float | None = None,
) -> tuple[np.ndarray, dict]:
"""
Detect peaks in a 1-D signal using scipy.signal.find_peaks.
Parameters
----------
data : np.ndarray
Input signal.
min_height : float or None
Minimum peak height.
min_distance : int
Minimum number of samples between peaks.
prominence : float or None
Minimum peak prominence.
Returns
-------
peaks : np.ndarray
Indices of detected peaks.
properties : dict
Peak properties (heights, prominences, etc.).
"""
kwargs: dict = {"distance": min_distance}
if min_height isnotNone:
kwargs["height"] = min_height
if prominence isnotNone:
kwargs["prominence"] = prominence
peaks, properties = signal.find_peaks(data, **kwargs)
return peaks, properties
defcompute_cross_correlation(
x: np.ndarray, y: np.ndarray, normalize: bool = True) -> tuple[np.ndarray, np.ndarray]:
"""
Compute full cross-correlation between two signals.
Returns
-------
lags : np.ndarray
Lag values in samples.
corr : np.ndarray
Cross-correlation coefficients.
"""
corr = signal.correlate(x, y, mode="full")
lags = signal.correlation_lags(len(x), len(y), mode="full")
if normalize:
corr = corr / (np.std(x) * np.std(y) * len(x))
return lags, corr
defmatched_filter(template: np.ndarray, noisy_signal: np.ndarray) -> np.ndarray:
"""
Apply a matched filter (cross-correlate signal with time-reversed template).
Returns
-------
np.ndarray
Matched filter output (same length as noisy_signal).
"""
h = template[::-1]
return signal.convolve(noisy_signal, h, mode="same")
5. Adaptive Filtering (LMS)
deflms_adaptive_filter(
desired: np.ndarray,
input_signal: np.ndarray,
filter_order: int = 32,
mu: float = 0.01,
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""
Least Mean Squares (LMS) adaptive filter.
Parameters
----------
desired : np.ndarray
Desired (reference) signal.
input_signal : np.ndarray
Input (noisy) signal to filter.
filter_order : int
Number of filter taps.
mu : float
Step size (learning rate). Must be small for stability (< 1 / (filter_order * P_x)).
Returns
-------
output : np.ndarray
Filtered output signal.
error : np.ndarray
Error signal (desired - output).
weights : np.ndarray
Final filter weight vector.
"""
n = len(input_signal)
weights = np.zeros(filter_order)
output = np.zeros(n)
error = np.zeros(n)
for i inrange(filter_order, n):
x = input_signal[i - filter_order : i][::-1]
y = np.dot(weights, x)
e = desired[i] - y
weights += 2 * mu * e * x
output[i] = y
error[i] = e
return output, error, weights
Example 1: EMG Signal Processing Pipeline
This example simulates an electromyography (EMG) recording and runs a full
processing pipeline: bandpass filtering, power-line notch removal, and
amplitude envelope extraction.
Example 2: Vibration Analysis for Rotating Machinery
This example performs fault-frequency detection on a vibration signal from a
rotating machine. It uses FFT, spectrogram, and automated peak detection to
identify bearing defect frequencies.
Filter stability: Always use sosfilt / sosfiltfilt for high-order
filters. Direct form II (lfilter) can suffer from numerical instability for
orders above 6–8.
Zero-phase filtering: filtfilt applies the filter twice (forward and
backward), eliminating phase distortion but doubling the effective order.
Avoid for online/real-time applications.
Nyquist constraint: Cutoff frequencies must be strictly between 0 and
fs/2. Always validate inputs before calling design functions.
FIR vs IIR: FIR filters (firwin) are inherently stable and have linear
phase, but require higher orders to achieve sharp transitions. IIR filters
(butter, ellip) achieve steeper roll-off at lower order but introduce phase
distortion.
LMS convergence: The step size mu must satisfy 0 < mu < 1 / (M * Px)
where M is filter order and Px is the input signal power. Start small
(e.g., mu = 0.001) and increase cautiously.
Window selection for STFT: Hann window is a good default. Blackman-Harris
offers lower sidelobes for detecting weak tones near strong ones.
Welch PSD segments: Larger nperseg gives finer frequency resolution but
fewer averages (higher variance). Typical values: nperseg = fs (1-second
segments) for stationary signals.
