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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 Progress71%

10 of 14 articles

Time Shifting

x(t-t0) delay, x(t+t0) advance operations.

Updated: 7 April 2026

10 min read

Time shifting delays or advances a signal along the horizontal axis. Radar systems rely on this delay to compute the distance to a physical target.

Time delaying a signal moves its waveform to the right.

Core Concept

Shifting a signal moves the entire waveform left or right without altering its basic shape. We simply replace the independent time variable with a shifted version.

A negative shift advances the signal in time. This shifts the mathematical graph toward the left. It implies the event happens earlier than the original reference.

A positive shift delays the signal. The graph shifts completely to the right. Physical systems can only delay signals because advancing requires seeing the future.

Key Formula

The delayed signal is written as y(t) = x(t - t0). Here t0 represents the specific delay time measured in seconds.

Example

Given: Original signal x(t) peaks at t = 2. Find the new peak for y(t) = x(t + 3).\nFormula: y(t) = x(t - (-3)), which represents a left shift by 3 units.\nSteps:\n1. The peak of x(t) occurs when the argument equals 2.\n2. Set the argument of y(t) to 2: t + 3 = 2.\n3. Solve for t to find the new peak location.\nFinal Answer: The new peak occurs at t = -1.

Exam Tip: When performing multiple operations, always apply time shifting first. Performing scaling before shifting often leads to errors in the final shift magnitude.

Properties Summary

Delay: x(t - a) moves the graph right by a units.
Advance: x(t + a) moves the graph left by a units.
Amplitude: The peak and minimum values remain strictly unchanged.
Energy: Total signal energy stays identical before and after the shift.
Fourier domain: A time shift introduces a linear phase shift in frequency.

Quick Revision

Moves waveform horizontally.
Does not stretch or compress.
Left movement means advance.
Right movement means delay.
Preserves signal energy perfectly.
Exam trap: Mixing up the sign convention and shifting a delayed signal to the left instead of the right.

#Signals & Systems

Time Shifting Quiz

Test your understanding of time delay and advance operations on signals.

Question 1 of 3

Q1.If x(t) has a Fourier transform X(f), what is the Fourier transform of the time-shifted signal x(t - t0)?

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Time Reversal

x(-t) reflection about t=0.

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Ramp Signal

r(t) = t*u(t), relation to step function by integration.

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Sinusoidal Signals

Amplitude, frequency, phase, representation forms.

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Discrete Time Signals

Signals defined at integer time indices, sequences.

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Exponential Signals

Real and complex exponentials, growth and decay.

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