Signals and Systems
Signals and Systems tests signal classification, LTI properties, convolution, Fourier series and transforms, Laplace and z-transforms, and sampling and reconstruction.
Core rules
- An LTI system is fully characterised by h(t). BIBO stability needs ∫|h(t)|dt < ∞ (or Σ|h[n]| < ∞); causality needs h(t) = 0 for t < 0.
- Convolution in time becomes multiplication in the transform domain - use this to avoid integrals.
- Laplace ROC: a right-sided signal has its ROC to the right of the rightmost pole; a stable causal system needs every pole in the left half-plane.
- Sampling: fs ≥ 2fm for reconstruction; aliasing folds frequencies to |f − kfs|, so an anti-aliasing filter must precede the sampler.
Worked example
For h(t) = e^(−2t)u(t) and x(t) = u(t): y(t) = ∫₀ᵗ e^(−2τ)dτ = ½(1 − e^(−2t))u(t). Cross-check with transforms: H(s) = 1/(s + 2) and X(s) = 1/s, so Y(s) = 1/[s(s + 2)] = ½[1/s − 1/(s + 2)], whose inverse Laplace is identical. The pole at s = −2 lies in the left half-plane, confirming a stable, causal system.
Traps
Dropping u(t) or the ROC; confusing a time shift with a frequency shift; applying the Nyquist rate to an already-sampled sequence; and calling a system with a pole on the jω axis stable - it is marginally stable at best.