Signals and Systems
Signals and Systems tests whether you can move fluently between the time domain, the transform domain, and the sampled domain. GATE EE asks about LTI properties, convolution, Fourier series and transform, Laplace and Z transforms with regions of convergence, and the sampling theorem.
LTI systems
- y[n] = x[n] * h[n]; continuous time uses ∫x(τ)h(t − τ)dτ.
- Causal ⇔ h(t) = 0 for t < 0. BIBO stable ⇔ ∫|h(t)|dt < ∞.
Transforms
- Laplace ROC lies to the right of the rightmost pole for a right-sided signal.
- Z transform: aⁿu[n] ↔ 1/(1 − az⁻¹) with ROC |z| > |a|. A system is stable exactly when the ROC includes the unit circle.
- Periodic signals use Fourier series; finite-energy signals use the Fourier transform.
Worked example: x(t) = sin(2π·3000t) is sampled at f_s = 4 kHz. The Nyquist limit is 2 kHz, so 3 kHz is above it. The alias lands at |3000 − 4000| = 1000 Hz, and the reconstructed signal is a 1 kHz sinusoid indistinguishable from a genuine 1 kHz tone. No post-sampling filter can recover the original frequency, which is precisely why an anti-aliasing filter precedes the sampler.
Traps that cost marks
- Treating stability and causality as the same property; a system can be stable and non-causal.
- Omitting the ROC when inverting a transform, which loses the distinction between different signals sharing one algebraic expression.
- DFT periodicity X[k + N] = X[k] and conjugate symmetry for real sequences - often the whole trick of a numerical question.