
Signal processing is a method for analyzing, modifying, and synthesizing signals such as sound, light, or microwaves.Signal processing is a mathematical tool for working with information encoded in changes of a physical quantity (sound pressure, voltage, brightness). An analog signal is digitized: instantaneous values are measured at a given frequency, yielding a stream of samples. Then algorithms are applied for filtering (removing interference), compression (MP3 uses psychoacoustics to discard inaudible frequencies), and restoration (removing blur in photos). The central idea is the Fourier transform, which moves a signal from the time domain to the frequency domain: a graph of oscillations becomes a spectrum where the amplitude of each frequency component is visible. The entire telecommunications, audio recording, and medical imaging industries are built on this.
Its roots go back to the harmonic analysis of Jean Fourier (1822), though decomposition into trigonometric series appeared even earlier. In the 20th century, the works of Nyquist and Shannon on sampling gave an impetus, and then the digital revolution of the 1960s-1970s produced the Fast Fourier Transform algorithms (Cooley and Tukey, 1965), making spectral analysis fast. The development of integrated circuits allowed real-time processing — from CD players to cellular communications. In the 21st century, compression methods (JPEG, MPEG), wavelet transforms, and neural network approaches have expanded the capabilities of signal processing beyond recognition.
The basic cycle: an ADC converts the signal into a sequence of samples x[n]. Then a digital filter described by a difference equation is applied, e.g., y[n] = Σ b_k x[n-k] – Σ a_k y[n-k] (recursive filter). To analyze the frequency composition, the Discrete Fourier Transform is computed: X[k] = Σ x[n] e^{-j2πkn/N}. In practice, the Fast Fourier Transform (FFT) is used, reducing the complexity from O(N²) to O(N log N). For non-stationary signals (where frequencies change over time), the Short-Time Fourier Transform or wavelets, localized in both time and frequency, are used. Modern communication systems are built on Orthogonal Frequency-Division Multiplexing (OFDM), which divides the channel into thousands of subcarriers to combat multipath fading.
Noise cancellation in phones, music compression into MP3, medical tomographs, radars, data transmission over Wi-Fi — all of this works thanks to signal processing algorithms.
The MP3 format discards about 90% of the data, yet to our ear the difference is almost imperceptible — the algorithm relies on how the brain masks weak sounds with loud ones at nearby frequencies.
| z | variable | Analytic signal — analytic signal |
| x | variable | Real signal — signal |
| a | variable | Instantaneous amplitude — instantaneous amplitude |
| φ | variable | Instantaneous phase — instantaneous phase radian |
| t | variable | Time variable — time second |
| H | operator | Hilbert transform, shifts phase by -90° — hilbert transform operator |
| i | operator | Imaginary unit, √(-1) — imaginary unit |
| W | variable | Wavelet transform coefficient — wavelet transform coefficient |
| s | variable | Scale parameter — scale parameter |
| τ | variable | Translation parameter — translation parameter second |
| x | variable | Input signal — signal |
| t | variable | Time variable — time second |
| ψ* | variable | Complex conjugate of wavelet function — wavelet function |
| ∫ | operator | Integration over time — integral operator |
























