Signals and Systems: practice problems and exam questions

Discrete convolution: the output of a linear time-invariant system is y[n] = Σ_k x[k]·h[n − k]: flip h, shift it by n, multiply sample by sample with x and add. If x has L_x samples starting at n_x and h has L_h samples starting at n_h, then y has L_x + L_h − 1 samples starting at n_x + n_h. The sum of all samples of y equals (Σx)·(Σh). An FIR filter y[n] = b₀x[n] + b₁x[n − 1] + … is a convolution with h[n] = b_n.

Laplace transform: F(s) = ∫₀^∞ f(t)·e^−st dt turns derivatives into multiplications by s: ℒ{f′} = sF(s) − f(0), ℒ{f″} = s²F(s) − s·f(0) − f′(0). Standard pairs: 1 ↔ 1/s, e^−at ↔ 1/(s + a), t·e^−at ↔ 1/(s + a)², sin ωt ↔ ω/(s² + ω²), cos ωt ↔ s/(s² + ω²), and a shift s → s + σ multiplies the time function by e^−σt. To invert a rational F(s), split it into partial fractions: the residue of a simple pole p is A = [(s − p)·F(s)] at s = p. Final value theorem: f(∞) = lim_s→0 s·F(s), valid only when all poles of s·F(s) lie in the left half-plane. Initial value theorem: f(0⁺) = lim_s→∞ s·F(s).