Signals & Systems
83 articles • Complete guide
Understand signals, transforms (Fourier, Laplace, Z), convolution, and sampling – the heart of communication, control & DSP. Clear visuals and examples make even tough math feel natural and exam ready.
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Curriculum
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Signal Classification
Continuous, discrete, periodic, aperiodic, energy, power signals
Continuous Time Signals
Signals defined for all time, analog signals.
Discrete Time Signals
Signals defined at integer time indices, sequences.
Periodic Signals
Period T, fundamental frequency, conditions for periodicity.
Aperiodic Signals
Non-repeating signals, transient signals.
Even and Odd Signals
Symmetry properties, decomposition into even and odd parts.
Energy Signals
Finite energy, zero average power, square integrability.
Power Signals
Finite average power, infinite energy, periodic signals.
Deterministic vs Random Signals
Predictable vs stochastic, mathematical description.
Real and Complex Signals
Complex exponential, Euler formula in signals.
Signal Operations
Time shifting, scaling, reversal, basic signals
Unit Step Function
u(t) definition, properties, relation to other signals.
Unit Impulse Function
Delta function, sifting property, area interpretation.
Ramp Signal
r(t) = t*u(t), relation to step function by integration.
Exponential Signals
Real and complex exponentials, growth and decay.
Sinusoidal Signals
Amplitude, frequency, phase, representation forms.
Signum Function
sgn(t), relation to step function.
Rectangular Pulse
rect(t/T), gate function, spectral properties.
Triangular Pulse
tri(t), convolution of two rect functions.
Sinc Function
sinc(t) = sin(πt)/(πt), Fourier dual of rect.
Time Shifting
x(t-t0) delay, x(t+t0) advance operations.
Time Scaling
x(at) compression and expansion, effect on frequency.
Time Reversal
x(-t) reflection about t=0.
Amplitude Scaling
a*x(t) amplification and attenuation.
Signal Addition and Multiplication
Sum and product of signals, modulation.
LTI Systems
Convolution, impulse response, transfer function, causality, stability
System Classification
Memory, causality, stability, linearity, time invariance.
Linearity Test
Superposition test, additivity and homogeneity.
Time Invariance Test
Time shift input, compare output shift.
Impulse Response
h(t) complete characterization of LTI systems.
Convolution Integral
y(t) = x(t)*h(t), flip and slide method.
Convolution Properties
Commutative, associative, distributive properties.
Convolution Sum
y[n] = x[n]*h[n], tabular method for DT.
Step Response
s(t) = integral of h(t), relation to impulse response.
Causality from Impulse Response
h(t)=0 for t<0 condition for causal systems.
BIBO Stability
Bounded input bounded output, integral of |h(t)| is finite.
Cascade and Parallel Systems
Series h=h1*h2, parallel h=h1+h2 combinations.
Feedback Systems
Closed loop, transfer function H/(1+GH).
Invertibility
Inverse system h_inv(t)*h(t) = delta(t).
Fourier Series
Trigonometric, exponential Fourier series, spectra
Trigonometric Fourier Series
a0, an, bn coefficients, DC and harmonic terms.
Exponential Fourier Series
Complex coefficients cn, compact notation.
Fourier Series Symmetry
Even function: bn=0, odd function: an=0, half wave.
Fourier Series of Square Wave
Odd harmonics, 1/n decay, spectral analysis.
Fourier Series of Sawtooth Wave
All harmonics present, alternating sign coefficients.
Fourier Series of Triangle Wave
Odd harmonics only, 1/n² decay.
Fourier Series Properties
Linearity, time shift, frequency shift, Parseval.
Gibbs Phenomenon
9% overshoot at discontinuities, non-uniform convergence.
Power Spectrum of Periodic Signals
Line spectrum, power in harmonics, Parseval relation.
Fourier Transform
Properties, pairs, Parseval theorem, applications
Fourier Transform Definition
F(w) = integral x(t)e^(-jwt)dt, existence conditions.
Inverse Fourier Transform
x(t) = (1/2pi) integral F(w)e^(jwt)dw synthesis.
FT of Rectangular Pulse
rect(t/T) <-> T*sinc(wT/2pi), spectral spreading.
FT of Impulse Function
delta(t) <-> 1, flat spectrum.
FT of Exponential Signal
e^(-at)u(t) <-> 1/(a+jw), causal exponential.
FT of Signum Function
sgn(t) <-> 2/jw, distribution sense.
FT of Step Function
u(t) <-> pi*delta(w) + 1/jw.
FT of Cosine and Sine
cos(w0t) <-> pi[delta(w-w0)+delta(w+w0)].
Fourier Transform Properties
Linearity, duality, time shift, frequency shift.
Time Scaling Property FT
x(at) <-> (1/|a|)F(w/a), bandwidth-duration tradeoff.
Convolution Theorem FT
x*h in time <-> X*H multiplication in frequency.
Multiplication Property FT
x*y in time <-> (1/2pi)X*Y convolution in frequency.
Parseval Theorem
Energy = (1/2pi) integral |F(w)|² dw, energy spectral density.
Hilbert Transform
90 degree phase shift, analytic signal, envelope.
Laplace Transform
Properties, ROC, inverse transform, system analysis
Laplace Transform Definition
F(s) = integral x(t)e^(-st)dt, bilateral and unilateral.
Region of Convergence
ROC rules, causal vs anti-causal, relationship to stability.
LT of Standard Signals
Step, ramp, exponential, sine, cosine transforms.
Laplace Transform Properties
Linearity, time shift, s-domain shift, differentiation.
Initial Value Theorem
lim s->inf sF(s) = f(0+), application examples.
Final Value Theorem
lim s->0 sF(s) = f(inf), steady state value.
Inverse Laplace Transform
Partial fraction expansion, residue method.
Inverse LT of Repeated Poles
Higher order poles, partial fractions with repeated roots.
Transfer Function
H(s) = Y(s)/X(s), poles and zeros, system characterization.
Pole Zero Plot Laplace
s-plane representation, stability from pole locations.
System Analysis Using Laplace
Solving differential equations, circuit analysis.
Z-Transform
Properties, ROC, inverse, discrete system analysis
Z-Transform Definition
X(z) = sum x[n]z^(-n), relation to Laplace via z=e^(sT).
Z-Transform ROC
ROC shapes, causal and anti-causal, stability criteria.
Z-Transform of Standard Sequences
Unit step, exponential, sinusoidal z-transforms.
Z-Transform Properties
Linearity, time shift, convolution, initial/final value.
Inverse Z-Transform
Partial fractions, long division, contour integration.
Transfer Function H(z)
Discrete system function, FIR vs IIR characterization.
Stability from Z-Transform
All poles inside unit circle for BIBO stability.
Sampling & Reconstruction
Nyquist theorem, aliasing, interpolation
Sampling Theorem
Nyquist rate fs >= 2fm, band-limited signals.
Aliasing
Folding, frequency ambiguity when undersampled.
Ideal Reconstruction
Sinc interpolation, ideal low pass filter.
Practical Reconstruction
Zero order hold, first order hold, practical filters.
Anti-Aliasing Filter
Pre-sampling LPF, guard band, practical design.
Sampling of Discrete Time Signals
Decimation, upsampling in discrete domain.
Other Topics
State-space, advanced topics
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