Signals & Systems
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.
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.
Unit Step Function
u(t) definition, properties, relation to other signals.
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.
Rectangular Pulse
rect(t/T), gate function, spectral properties.
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.
Signal Addition and Multiplication
Sum and product of signals, modulation.
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.
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 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 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 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.
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.
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.
Real and Complex Signals
Complex exponential, Euler formula in signals.
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.
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 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 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 Theorem
Nyquist rate fs >= 2fm, band-limited signals.
Aliasing
Folding, frequency ambiguity when undersampled.
Ideal Reconstruction
Sinc interpolation, ideal low pass filter.
Anti-Aliasing Filter
Pre-sampling LPF, guard band, practical design.
Aperiodic Signals
Non-repeating signals, transient signals.
Signum Function
sgn(t), relation to step function.
Triangular Pulse
tri(t), convolution of two rect functions.
Amplitude Scaling
a*x(t) amplification and attenuation.
Feedback Systems
Closed loop, transfer function H/(1+GH).
Invertibility
Inverse system h_inv(t)*h(t) = delta(t).
Fourier Series of Triangle Wave
Odd harmonics only, 1/n² decay.
FT of Signum Function
sgn(t) <-> 2/jw, distribution sense.
Time Scaling Property FT
x(at) <-> (1/|a|)F(w/a), bandwidth-duration tradeoff.
Hilbert Transform
90 degree phase shift, analytic signal, envelope.
Inverse LT of Repeated Poles
Higher order poles, partial fractions with repeated roots.
Z-Transform of Standard Sequences
Unit step, exponential, sinusoidal z-transforms.
Practical Reconstruction
Zero order hold, first order hold, practical filters.
Sampling of Discrete Time Signals
Decimation, upsampling in discrete domain.
Unit Impulse Function
Delta function, sifting property, area interpretation.
Time Reversal
x(-t) reflection about t=0.