Signal Processing Cheat Sheet

The core ideas of Signal Processing distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/signal-processing/cheatsheet

Quick Reference

Fourier Transform

A mathematical operation that decomposes a time-domain signal into its constituent frequency components, revealing the signal's spectral content. The Discrete Fourier Transform (DFT) and its efficient implementation, the FFT, are used for digital signals.

Sampling Theorem (Nyquist-Shannon)

States that a continuous signal can be perfectly reconstructed from its discrete samples if the sampling rate is at least twice the highest frequency present in the signal. This minimum rate is called the Nyquist rate.

Convolution

A mathematical operation that combines two signals to produce a third, describing how the shape of one signal is modified by the other. In signal processing, convolution expresses the output of a linear time-invariant system given an input and the system's impulse response.

Digital Filtering

The process of selectively modifying a digital signal's frequency content using algorithms. Filters are classified as low-pass, high-pass, band-pass, or band-stop depending on which frequencies they allow through.

Z-Transform

A mathematical tool that converts a discrete-time signal into a complex frequency-domain representation, analogous to the Laplace transform for continuous signals. It is essential for analyzing and designing digital filters and discrete-time systems.

Aliasing

A distortion that occurs when a signal is sampled below the Nyquist rate, causing higher-frequency components to be incorrectly represented as lower frequencies in the sampled data. Once aliased, the original signal cannot be recovered.

Impulse Response

The output of a linear time-invariant (LTI) system when the input is an idealized instantaneous pulse (impulse). The impulse response fully characterizes the system's behavior, since any input can be decomposed into scaled and shifted impulses.

Signal-to-Noise Ratio (SNR)

A measure comparing the level of a desired signal to the level of background noise, typically expressed in decibels. Higher SNR means the signal is clearer relative to the noise.

Windowing

The practice of multiplying a signal segment by a window function before performing spectral analysis, in order to reduce spectral leakage caused by analyzing a finite-length portion of an otherwise infinite signal.

Adaptive Filtering

A filtering technique where the filter coefficients are automatically adjusted in real time based on an optimization algorithm, allowing the system to track changes in signal characteristics or environment.

Key Terms at a Glance

Aliasing:Distortion resulting from sampling a signal at a rate below twice its highest frequency, causing high-frequency components to appear as lower frequencies.

Analog Signal:A continuous signal that varies smoothly over time, representing physical quantities like voltage, pressure, or temperature.

Autocorrelation:A function measuring the similarity between a signal and a time-shifted copy of itself, used to detect periodicity and estimate pitch.

Bandwidth:The range of frequencies occupied by a signal or passed by a system, typically measured as the difference between the upper and lower frequency limits.

Convolution:A mathematical operation that combines two signals to produce a third, expressing the output of an LTI system given its input and impulse response.

Decibel (dB):A logarithmic unit used to express the ratio of two quantities, commonly signal power or amplitude, defined as 10 * log10(P1/P2) for power.

Digital Signal:A signal represented as a sequence of discrete numerical values, typically obtained by sampling and quantizing an analog signal.

Discrete Fourier Transform (DFT):A transform that converts a finite sequence of equally spaced samples into a finite sequence of equally spaced frequency components.

Fast Fourier Transform (FFT):An efficient algorithm for computing the DFT that reduces the computational complexity from O(N^2) to O(N log N).

FIR Filter:A digital filter with a finite-duration impulse response, implemented using only feedforward operations with no feedback loops.

Frequency Domain:A representation of a signal in terms of its frequency components rather than its time-varying amplitude.

IIR Filter:A digital filter with a theoretically infinite-duration impulse response, implemented using both feedforward and feedback (recursive) operations.

Impulse Response:The output of a linear time-invariant system when the input is a unit impulse, fully characterizing the system's behavior.

Laplace Transform:An integral transform used to convert continuous-time signals from the time domain to the complex frequency domain for system analysis.

Linear Time-Invariant (LTI) System:A system satisfying both linearity (superposition) and time-invariance (a time shift in input produces an equal shift in output).

Get study tips in your inbox

We'll send you evidence-based study strategies and new cheat sheets as they're published.

We'll notify you about updates. No spam, unsubscribe anytime.