📈 Signal Processing

Signal Processing#

Analysing and transforming signals (audio, sensor, time series) to extract or clean information.

Important

✨ AI-generated content. This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it.

What it is#

Signal processing is the discipline of representing, transforming and pulling information out of signals — any quantity that carries information about a phenomenon, usually as a function of time or space. A signal can be continuous-time, written \(x(t)\), or discrete-time (sampled), written \(x[n]\). Almost any sequential or spatial measurement is a signal: audio waveforms, image pixel grids (2-D signals), biomedical traces (ECG, EEG, EMG), radio / Wi-Fi / 5G transmissions, and sensor streams (accelerometer, gyroscope, temperature).

What you do with a signal#

The recurring goals are:

Filtering — remove noise or isolate a band of interest.
Transformation — move the signal to a domain where the task is easier (most often the frequency domain).
Feature extraction — derive compact descriptors (e.g. spectral features for speech recognition).
Compression — shrink the data while preserving what matters (MP3, JPEG).
Restoration — undo degradation (denoising, deblurring).

Three ways to look at a signal#

time domain#

Work directly on the samples in order — smoothing with a moving average, detecting peaks, computing energy. Intuitive, but it hides periodic structure.

frequency domain#

Decompose the signal into the sinusoids that compose it using the Fourier transform. For sampled data this is the Discrete Fourier Transform, computed efficiently by the FFT:

\[X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-j 2\pi k n / N}.\]

This is where “mostly 5 Hz with a little 50 Hz noise” becomes visible and you can filter accordingly.

time-frequency domain#

When the frequency content changes over time (speech, music), one global spectrum is not enough. The Short-Time Fourier Transform (STFT) and the wavelet transform produce a spectrogram — frequency content as a function of time.

Other standard tools: the Laplace transform (continuous-time systems), the Z-transform (discrete-time systems), and FIR / IIR digital filters.

The digital pipeline (DSP)#

Modern signal processing is mostly digital, in four stages:

Sampling — analog to discrete samples. The Nyquist-Shannon theorem requires the sampling rate to exceed twice the highest frequency, \(f_s > 2 f_{\max}\), or information is lost.
Quantization — round continuous amplitudes to discrete levels (adding small quantization noise).
Processing — filter, transform, compress, analyse.
Reconstruction — convert back to continuous form if needed.

Pitfalls and edge cases#

Aliasing — sampling too slowly makes high frequencies masquerade as low ones; always low-pass filter before downsampling.
Spectral leakage — the FFT assumes the segment repeats exactly, so a non-integer number of cycles smears energy across bins. Apply a window (Hann, Hamming) to reduce it.
Edge effects — filters need warm-up samples; the very start and end of a filtered signal are unreliable (use zero-phase filtfilt or drop the transient).
Non-stationarity — a single global spectrum misleads when the statistics drift; prefer time-frequency methods.

Worked example — find the dominant frequency#

import numpy as np

fs = 500                                   # sampling rate (Hz)
t = np.arange(0, 2, 1 / fs)
x = np.sin(2 * np.pi * 5 * t) + 0.3 * np.random.randn(t.size)

spectrum = np.fft.rfft(x)                  # FFT of a real signal
freqs = np.fft.rfftfreq(t.size, d=1 / fs)
dominant = freqs[np.argmax(np.abs(spectrum))]
print(f"dominant frequency ~ {dominant:.1f} Hz")   # ~5.0 Hz

Connection to data science and ML#

Signal processing is the front end of many ML pipelines: speech becomes MFCC features, images are filtered by convolutions (a CNN layer is a learned bank of filters), and ECG / EEG become time-frequency features for classification. Seen this way, CNNs, RNNs and Transformers are advanced, learned signal-processing pipelines.

Theme: Signal Processing & Time Series · All terminology

Hint

Mind map — connected ideas

Low-pass Filtering · Subsampling · Downsampling · Time Series · Time Series Forecasting

Hint

More in Signal Processing & Time Series

ARIMA (AutoRegressive Integrated Moving Average) · Bayesian Time Series · Forecast Error · Forecasting Benchmarks · Forecasting Competitions · Log-Space · Low-pass Filtering · LSTM — Long Short-Term Memory Networks · M-Competitions (Makridakis Competitions) · Naïve Baseline Forecast · Prophet — Time Series Forecasting by Facebook (Meta) · Seasonal Lag · Seasonality · Simple Baseline Methods