Signals & Systems
Time Reversal
x(-t) reflection about t=0.
Unit Impulse Function
Delta function, sifting property, area interpretation.
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.
Time Scaling Property FT
x(at) <-> (1/|a|)F(w/a), bandwidth-duration tradeoff.
FT of Signum Function
sgn(t) <-> 2/jw, distribution sense.
Fourier Series of Triangle Wave
Odd harmonics only, 1/n² decay.
Invertibility
Inverse system h_inv(t)*h(t) = delta(t).
Feedback Systems
Closed loop, transfer function H/(1+GH).
Amplitude Scaling
a*x(t) amplification and attenuation.
Triangular Pulse
tri(t), convolution of two rect functions.
Signum Function
sgn(t), relation to step function.
Causality from Impulse Response
h(t)=0 for t<0 condition for causal systems.
Step Response
s(t) = integral of h(t), relation to impulse response.
Convolution Properties
Commutative, associative, distributive properties.
Linearity Test
Superposition test, additivity and homogeneity.
System Analysis Using Laplace
Solving differential equations, circuit analysis.
Transfer Function
H(s) = Y(s)/X(s), poles and zeros, system characterization.
Inverse Laplace Transform
Partial fraction expansion, residue method.
Final Value Theorem
lim s->0 sF(s) = f(inf), steady state value.
Initial Value Theorem
lim s->inf sF(s) = f(0+), application examples.
LT of Standard Signals
Step, ramp, exponential, sine, cosine transforms.
Region of Convergence
ROC rules, causal vs anti-causal, relationship to stability.
Parseval Theorem
Energy = (1/2pi) integral |F(w)|² dw, energy spectral density.
Multiplication Property FT
x*y in time <-> (1/2pi)X*Y convolution in frequency.
Power Spectrum of Periodic Signals
Line spectrum, power in harmonics, Parseval relation.
Gibbs Phenomenon
9% overshoot at discontinuities, non-uniform convergence.
Fourier Series Properties
Linearity, time shift, frequency shift, Parseval.
Fourier Series of Sawtooth Wave
All harmonics present, alternating sign coefficients.
Fourier Series of Square Wave
Odd harmonics, 1/n decay, spectral analysis.
Fourier Series Symmetry
Even function: bn=0, odd function: an=0, half wave.
Exponential Fourier Series
Complex coefficients cn, compact notation.
Aperiodic Signals
Non-repeating signals, transient signals.
Anti-Aliasing Filter
Pre-sampling LPF, guard band, practical design.
Ideal Reconstruction
Sinc interpolation, ideal low pass filter.
Aliasing
Folding, frequency ambiguity when undersampled.
Sampling Theorem
Nyquist rate fs >= 2fm, band-limited signals.
Stability from Z-Transform
All poles inside unit circle for BIBO stability.
Transfer Function H(z)
Discrete system function, FIR vs IIR characterization.
Real and Complex Signals
Complex exponential, Euler formula in 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.
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.
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 Sum
y[n] = x[n]*h[n], tabular method for DT.
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.
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.
Laplace Transform Definition
F(s) = integral x(t)e^(-st)dt, bilateral and unilateral.
Laplace Transform Properties
Linearity, time shift, s-domain shift, differentiation.
Pole Zero Plot Laplace
s-plane representation, stability from pole locations.
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.