Signals and Systems
Signals and Systems tests LTI properties, convolution, transform methods and sampling. Expect a mix of a short computation and a conceptual ROC or stability question.
Core rules
- An LTI system is characterised by h(t); output y = x * h. Convolution in time is multiplication in frequency.
- Causality: h(t)=0 for t<0. BIBO stability: ∫|h(t)|dt < ∞; in the Laplace domain the ROC must include the jω axis.
- Laplace ROC is a vertical strip or half-plane; for a right-sided signal it lies to the right of the rightmost pole.
- Fourier series applies to periodic signals; Fourier transform applies to finite-energy aperiodic signals.
- Sampling theorem: fs > 2fmax, otherwise aliasing occurs. Reconstruction uses an ideal low-pass filter with cutoff fs/2.
- Z-transform ROC is an annulus; unit-circle inclusion means the DTFT exists.
Worked example
Let x(t)=u(t) and h(t)=e^−2t u(t). For t≥0, y(t)=∫0^t e^−2τ dτ = [−e^−2τ/2]0^t = (1−e^−2t)/2. For t<0 the integral is zero. So y(t)=0.5(1−e^−2t)u(t). As t→∞, y→0.5, consistent with the DC gain H(0)=1/2. The system is stable because the pole at −2 is in the left half-plane.
Traps
- Ignoring the ROC when inverting a transform.
- Using the final-value theorem when poles are not in the left half-plane.
- Confusing time scaling with frequency scaling.
- Forgetting that sampling a sinusoid above Nyquist produces a lower alias frequency.
- Treating a marginally stable system as BIBO stable.