Glossary of Terms — Signals & Systems

Introduction

This glossary provides concise definitions of the essential terminology used in Signals and Systems. Terms are organized alphabetically for quick reference. Understanding precise definitions is important because many terms have specific technical meanings that differ from their everyday usage. For example, "stable" in signal processing has a precise mathematical definition (BIBO stability) that is more specific than the colloquial meaning of "not changing."

A

Aliasing: The phenomenon where high-frequency components appear as lower frequencies after undersampling (sampling below the Nyquist rate). Irreversible once it occurs.

Amplitude: The instantaneous value or peak value of a signal at a given time.

Analog signal: A continuous-time, continuous-amplitude signal. Defined for all time values with amplitudes from a continuous range.

Anti-aliasing filter: A low-pass filter applied before sampling to remove frequency components above $f_s/2$, preventing aliasing.

Autocorrelation: A measure of how similar a signal is to a delayed version of itself: $R_{xx}(\tau) = \int x(t)x(t+\tau)dt$.

B

Bandwidth: The range of frequencies occupied by a signal or passable by a system. Multiple definitions exist (3-dB, null-to-null, noise equivalent).

BIBO Stability: Bounded-Input Bounded-Output stability — every bounded input produces a bounded output.

Bilateral transform: Transform that integrates/sums over the entire time axis ($-\infty$ to $+\infty$).

Bode plot: A pair of plots (magnitude in dB and phase in degrees) versus log-frequency showing a system\'s frequency response.

C

Causal system: A system whose output at any time depends only on present and past inputs (not future inputs). Physically realizable.

Convolution: The operation $y(t) = \int x(\tau)h(t-\tau)d\tau$ that computes the output of an LTI system from its input and impulse response.

Cross-correlation: Measures similarity between two different signals at various time lags.

Cutoff frequency: The frequency at which a filter\'s response drops to a specified level (typically -3 dB).

D

Decibel (dB): A logarithmic unit: $G_{dB} = 20\log_{10}(V_{out}/V_{in})$ for amplitude, $10\log_{10}(P_{out}/P_{in})$ for power.

DFT (Discrete Fourier Transform): The Fourier Transform for finite-length discrete sequences: $X[k] = \sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$.

Dirac delta function: $\delta(t)$ — an idealized impulse with zero width, infinite height, and unit area. Satisfies $\int f(t)\delta(t-a)dt = f(a)$.

Dynamic range: The ratio between the largest and smallest representable signal levels, typically expressed in dB.

E-F

Eigenfunction: A function that passes through a system unchanged in form (only scaled). For LTI systems, complex exponentials $e^{st}$ are eigenfunctions.

Energy signal: A signal with finite total energy ($0 < E < \infty$) and zero average power.

FFT (Fast Fourier Transform): An efficient algorithm for computing the DFT in $O(N\log N)$ operations instead of $O(N^2)$.

FIR filter: Finite Impulse Response — a filter whose impulse response has finite duration. Always stable.

Frequency response: $H(j\omega)$ — the system\'s gain and phase shift as a function of frequency.

G-I

Group delay: $\tau_g(\omega) = -d\angle H/d\omega$ — the delay experienced by the envelope of a narrowband signal at frequency $\omega$.

Homogeneity: $T\{ax\} = aT\{x\}$ — scaling property required for linearity.

IIR filter: Infinite Impulse Response — a recursive filter whose impulse response theoretically extends to infinity. Can be unstable.

Impulse response: $h(t) = T\{\delta(t)\}$ — the output of a system when excited by a unit impulse. Completely characterizes an LTI system.

L-N

Laplace Transform: $X(s) = \int x(t)e^{-st}dt$ — generalizes the Fourier Transform to complex frequency $s = \sigma + j\omega$.

Linear system: A system satisfying superposition: $T\{ax_1+bx_2\} = aT\{x_1\}+bT\{x_2\}$.

LTI system: Linear Time-Invariant — satisfies both linearity and time-invariance. The most important class for analysis.

Nyquist rate: $2f_{max}$ — the minimum sampling rate for perfect signal reconstruction.

P-R

Parseval\'s theorem: Energy in time equals energy in frequency: $\int|x|^2 dt = \frac{1}{2\pi}\int|X|^2 d\omega$.

Pole: A value of $s$ (or $z$) where the transfer function goes to infinity. Determines natural response modes.

Power spectral density (PSD): Distribution of signal power across frequency: $S_{xx}(\omega) = \mathcal{F}\{R_{xx}(\tau)\}$.

ROC (Region of Convergence): The set of complex values for which a transform converges absolutely.

S-Z

Sampling theorem: A bandlimited signal ($f_{max}$) is perfectly recoverable from samples if $f_s > 2f_{max}$.

Superposition: The defining property of linear systems — response to sum equals sum of responses.

Time-invariant: System behavior does not change over time: delaying input delays output by the same amount.

Transfer function: $H(s) = Y(s)/X(s)$ — the Laplace (or Z) Transform of the impulse response.

Zero: A value of $s$ (or $z$) where the transfer function equals zero. Creates nulls in frequency response.

Z-Transform: $X(z) = \sum x[n]z^{-n}$ — the discrete-time counterpart of the Laplace Transform.

Key Takeaways

• Technical terms have precise mathematical definitions — use them correctly
• Many concepts have parallel continuous-time and discrete-time versions
• Understanding terminology enables clear communication with colleagues and in examinations
• This glossary covers the core vocabulary; specialized topics may introduce additional terms