ESECE: Signals and Systems

Signals and Systems rewards fluency with transforms and their regions of convergence. ESECE tests LTI properties, convolution, Fourier series and transform pairs, Laplace and z-transform analysis of circuits and difference equations, sampling and aliasing, and DFT properties.

LTI fundamentals

• Convolution in time is multiplication in frequency: y(t) = x(t)*h(t). The eigenfunction result holds only for LTI systems.
• Stability means the impulse response is absolutely summable or integrable, i.e. the ROC contains the unit circle (DT) or the jω axis (CT). Causality requires the ROC to be the outside of a circle or a right half-plane.
• Energy signals use Parseval; power signals use time averages.

Transforms

• Laplace: differentiation becomes multiplication by s, so circuit analysis turns into algebra with initial conditions entering as sources through 1/sC and sL.
• z-transform: H(z) = Y(z)/X(z); all poles inside the unit circle means stable and causal.
• Periodicity: the CTFT is not periodic; the DTFT repeats every 2π; the DFT is finite and periodic in N.
• Sampling: alias-free when f_s > 2f_max. A sinusoid above f_s/2 folds to |f − k·f_s|.

Worked example

Let h(t) = e^(−2t)u(t) and x(t) = u(t). Then y(t) = ∫₀ᵗ e^(−2τ)dτ = 0.5(1 − e^(−2t))u(t). Check with Laplace: H(s) = 1/(s+2), X(s) = 1/s, so Y(s) = 1/[s(s+2)] = 0.5/s − 0.5/(s+2), which is the same answer. Now sample a 1 kHz tone at 1.5 kHz: the fold is |1000 − 1500| = 500 Hz, so a spurious 500 Hz tone appears where none existed.

Traps

Omitting the ROC so causal and anticausal answers look identical; assuming a DFT's amplitude equals the time-domain amplitude without the 1/N or 2/N scale; treating f_s = 2f_max as safe when practical reconstruction also needs an anti-alias filter and oversampling; calling a system LTI merely because it is linear.