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

Time Scaling Property FT

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

Darshan N

Updated: 19 March 2026

10 min read

#Signals & Systems

Time Scaling FT

Test your understanding of the time scaling property of the Fourier transform and the bandwidth-duration tradeoff.

Question 1 of 3

Q1.If F{x(t)} = X(w), the Fourier transform of x(at) for a nonzero real constant a is:

Multiplication Property FT

x*y in time <-> (1/2pi)X*Y convolution in frequency.

9 min read

Time Scaling

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

4 min read

FT of Exponential Signal

e^(-at)u(t) <-> 1/(a+jw), causal exponential.

11 min read

FT of Step Function

u(t) <-> pi*delta(w) + 1/jw.

12 min read

FT of Impulse Function

delta(t) <-> 1, flat spectrum.

9 min read

Fourier Transform Properties

Convolution Theorem FT

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