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Contents

Signals & Systems

Signal Classification

Continuous Time Signals Discrete Time Signals Periodic Signals Aperiodic Signals Even and Odd Signals Energy Signals Power Signals Deterministic vs Random Signals Real and Complex Signals

Signal Operations

Unit Step Function Unit Impulse Function Ramp Signal Exponential Signals Sinusoidal Signals Signum Function Rectangular Pulse Triangular Pulse Sinc Function Time Shifting Time Scaling Time Reversal Amplitude Scaling Signal Addition and Multiplication

LTI Systems

System Classification Linearity Test Time Invariance Test Impulse Response Convolution Integral Convolution Properties Convolution Sum Step Response Causality from Impulse Response BIBO Stability Cascade and Parallel Systems Feedback Systems Invertibility

Fourier Series

Trigonometric Fourier Series Exponential Fourier Series Fourier Series Symmetry Fourier Series of Square Wave Fourier Series of Sawtooth Wave Fourier Series of Triangle Wave Fourier Series Properties Gibbs Phenomenon Power Spectrum of Periodic Signals

Fourier Transform

Fourier Transform Definition Inverse Fourier Transform FT of Rectangular Pulse FT of Impulse Function FT of Exponential Signal FT of Signum Function FT of Step Function FT of Cosine and Sine Fourier Transform Properties Time Scaling Property FT Convolution Theorem FT Multiplication Property FT Parseval Theorem Hilbert Transform

Laplace Transform

Laplace Transform Definition Region of Convergence LT of Standard Signals Laplace Transform Properties Initial Value Theorem Final Value Theorem Inverse Laplace Transform Inverse LT of Repeated Poles Transfer Function Pole Zero Plot Laplace System Analysis Using Laplace

Z-Transform

Z-Transform Definition Z-Transform ROC Z-Transform of Standard Sequences Z-Transform Properties Inverse Z-Transform Transfer Function H(z)Stability from Z-Transform

Sampling & Reconstruction

Sampling Theorem Aliasing Ideal Reconstruction Practical Reconstruction Anti-Aliasing Filter Sampling of Discrete Time Signals

Other Topics

Other Subjects

Analog Electronics Digital Electronics Control Systems Electromagnetic Theory Digital Communication Analog Communication Network Analysis Microprocessors VLSI Design Verilog HDL Power Electronics Embedded Systems DSP Electronic Devices

Section Progress79%

11 of 14 articles

Time Scaling

x(at) compression and expansion, effect on frequency.

Updated: 7 April 2026

4 min read

Time scaling compresses or expands a signal along the time axis. Audio engineers use this principle to speed up or slow down recorded voice tracks.

Time scaling by a factor of 2 compresses the signal horizontally.

Core Concept

Multiplying the time variable by a constant changes how fast the signal evolves. We refer to this numerical constant as the scaling factor.

When the scaling factor exceeds one, the signal compresses. The events happen more quickly. A one second pulse might shrink to half a second.

When the factor falls between zero and one, the signal expands. It stretches out along the horizontal axis. This resembles watching a video in slow motion.

Key Formula

The scaled signal is written as y(t) = x(a*t). The scalar a must be a positive real number to avoid mixing scaling with reversal.

Example

Given: A rectangular pulse x(t) exists from t = -4 to t = 4. Find the duration of y(t) = x(2t).\nFormula: The new boundaries satisfy -4 <= 2t <= 4.\nSteps:\n1. Set the lower bound: 2t = -4, yielding t = -2.\n2. Set the upper bound: 2t = 4, yielding t = 2.\n3. Calculate the new total duration: 2 - (-2).\nFinal Answer: The new duration is exactly 4 units.

Exam Tip: Time compression widens the frequency spectrum, while time expansion narrows it. This inverse relationship heavily tests your grasp of fundamental Fourier properties.

Properties Summary

Compression: Factor a > 1 shrinks the time duration.
Expansion: Factor 0 < a < 1 stretches the time duration.
Zero point: The value at t = 0 remains firmly anchored.
Frequency scaling: y(t) = x(at) transforms to (1/a) * X(w/a).
Energy scaling: The energy of x(at) becomes 1/a times the original energy.

Quick Revision

Multiplies the independent variable.
Compresses or stretches the shape.
Origin stays completely fixed.
Inverse effect on frequency spread.
Alters total signal energy.
Exam trap: Assuming that x(t/2) means the signal shrinks by half, when it actually stretches the signal to twice its length.

#Signals & Systems

Time Scaling Quiz

Evaluate your command of time compression, expansion, and their frequency effects.

Question 1 of 3

Q1.If FT{x(t)} = X(f), what is the Fourier transform of x(a*t) for a > 0?

Related Articles

Amplitude Scaling

a*x(t) amplification and attenuation.

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Time Scaling Property FT

x(at) <-> (1/|a|)F(w/a), bandwidth-duration tradeoff.

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Discrete Time Signals

Signals defined at integer time indices, sequences.

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Sinusoidal Signals

Amplitude, frequency, phase, representation forms.

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Continuous Time Signals

Signals defined for all time, analog signals.

4 min read