Signals and Systems

Syllabus

Continuous-time signals: Fourier series and Fourier transform, sampling theorem and applications. Discrete-time signals: DTFT, DFT, z-transform, discrete-time processing of continuous-time signals. LTI systems: properties, causality, stability, impulse response, convolution, poles and zeros, frequency response, group delay, phase delay.

Key concepts and formulas

LTI systems. y = x ∗ h. Causal if h(t) = 0 for t < 0; BIBO stable if ∫|h| < ∞. A causal continuous system is stable if all poles have negative real parts; a causal discrete system if all poles lie inside the unit circle.

Fourier transform. Key pairs: rect ↔ sinc, e^−atu(t) ↔ 1/(a + jω), δ(t) ↔ 1, a constant ↔ 2πδ(ω). Properties: linearity, time shift (e^−jωt₀), frequency shift (modulation), time scaling, convolution ↔ product, Parseval's theorem (energy is the same in both domains).

Sampling. f_s ≥ 2f_max; the spectrum of the sampled signal repeats every f_s. For a bandpass signal, a lower rate can work (bandpass sampling).

DTFT and DFT. The DTFT X(e^jω) is periodic with period 2π. The N-point DFT samples it at ω = 2πk/N: X[k] = Σ x[n] e^−j2πkn/N. DFT multiplication corresponds to circular convolution; to get linear convolution of lengths L and M, use N ≥ L + M − 1. The FFT needs about (N/2) log₂N complex multiplications instead of N².

z-transform. Z{aⁿu[n]} = z/(z − a), ROC |z| > |a|; Z{−aⁿu[−n − 1]} = z/(z − a), ROC |z| < |a|. The ROC never contains poles; a right-sided sequence has an ROC outside its outermost pole.

Delays. For H(e^jω) with phase θ(ω): group delay τ_g = −dθ/dω and phase delay τ_p = −θ(ω)/ω. A linear-phase system (θ = −αω) delays all frequencies equally, so it does not distort the waveform shape. FIR filters with symmetric coefficients have linear phase.

Worked example

Compute the 4-point DFT of x = [1, 2, 0, 1], check Parseval's relation for it, and compute the circular convolution of [1, 2, 3, 0] with [1, 1, 0, 0].

import cmath

x = [1, 2, 0, 1]
N = len(x)
X = [sum(x[n] * cmath.exp(-2j * cmath.pi * k * n / N) for n in range(N)) for k in range(N)]
print("DFT:", [complex(round(v.real, 6) + 0.0, round(v.imag, 6) + 0.0) for v in X])
print("energy in time:", sum(v * v for v in x), " energy from DFT:", round(sum(abs(v) ** 2 for v in X) / N, 6))
a, b = [1, 2, 3, 0], [1, 1, 0, 0]
circ = [sum(a[m] * b[(n - m) % 4] for m in range(4)) for n in range(4)]
print("circular convolution:", circ)

DFT: [(4+0j), (1-1j), (-2+0j), (1+1j)]
energy in time: 6  energy from DFT: 6.0
circular convolution: [1, 3, 5, 3]

Practice questions