Signals and Systems
Signals and Systems tests LTI system properties, convolution, Fourier series and transforms, Laplace and z-transforms, sampling and DFT. ESE numericals are often property-based rather than long computations.
Core rules
- LTI systems are fully described by impulse response h(t); output is convolution y=x*h.
- Convolution in time is multiplication in frequency; time shift adds a linear phase factor.
- Fourier series applies to periodic signals; Fourier transform to finite-energy aperiodic signals. Gibbs ripple appears at discontinuities.
- Laplace ROC determines causality and stability; a causal LTI system is stable if all poles lie in the left half-plane.
- Sampling: fs>2fmax for a bandlimited signal; otherwise aliasing folds frequencies.
- z-transform ROC is an annulus; unit-circle inclusion means stability for causal systems.
Worked example
Let x(t)=u(t) and h(t)=e^(-2t)u(t). Then y(t)=integral from 0 to t of e^(-2tau) dtau for t>=0, so y(t)=0.5(1-e^(-2t))u(t). The reasoning uses causal convolution limits: both signals are zero for negative time, so the integral runs from 0 to t.
Traps
Forgetting the ROC when inverting a Laplace or z-transform; using the sampling theorem on a non-bandlimited signal; confusing Fourier-series coefficients with the Fourier transform; and treating a time-varying system as LTI. In convolution, check the integration limits before integrating.