Signals and Systems

Syllabus

Continuous and discrete time signals, shifting and scaling; linear, time-invariant and causal systems; Fourier series of periodic signals; sampling theorem; Fourier transform for continuous and discrete time signals; Laplace and z-transforms; RMS and average values of periodic waveforms.

Key concepts and formulas

Systems. A system is linear if it obeys superposition, time-invariant if a shifted input gives a shifted output, causal if the output depends only on present and past inputs, and BIBO stable if every bounded input gives a bounded output (for an LTI system: ∫|h(t)| dt < ∞, or all poles in the left half-plane / inside the unit circle).

Convolution. For an LTI system, y(t) = x(t) ∗ h(t) = ∫ x(τ)h(t − τ) dτ, and in the transform domain Y = XH. The length of the convolution of discrete sequences of lengths N and M is N + M − 1.

Fourier series. A periodic signal x(t) = Σ c_k e^jkω₀t. Parseval: average power = Σ |c_k|². Even signals have only cosine terms; odd signals only sine terms; half-wave symmetry gives only odd harmonics.

Fourier transform pairs and properties. A rectangular pulse ↔ a sinc; time shift multiplies by e^−jωt₀; time scaling x(at) ↔ (1/|a|)X(ω/a); differentiation ↔ jω; convolution ↔ multiplication; duality.

Sampling theorem. A signal band-limited to f_m is recovered from samples taken at f_s ≥ 2f_m (the Nyquist rate is 2f_m). Sampling below it causes aliasing: a tone at f appears at |f − kf_s|.

Laplace and z-transforms. L{e^−atu(t)} = 1/(s + a), L{u(t)} = 1/s; Z{aⁿu[n]} = z/(z − a), with ROC |z| > |a|. Initial value theorem x(0⁺) = lim_s→∞ sX(s); final value theorem x(∞) = lim_s→0 sX(s) (if the poles of sX(s) are in the left half-plane).

RMS and average values.

The RMS of a sum of a DC value and sinusoids at different frequencies is √(V_dc² + Σ V_k,rms²).

Worked example

Compute the discrete convolution of [1, 2, 3] and [1, 1], check the RMS of a full-wave rectified sine numerically, and find the alias of a 7 kHz tone sampled at 10 kHz.

import math

x, h = [1, 2, 3], [1, 1]
y = [sum(x[k] * h[n - k] for k in range(len(x)) if 0 <= n - k < len(h)) for n in range(len(x) + len(h) - 1)]
print("convolution:", y)
N = 100000
vals = [abs(math.sin(2 * math.pi * i / N)) for i in range(N)]
print(f"full-wave rectified sine: RMS = {math.sqrt(sum(v * v for v in vals) / N):.4f}, average = {sum(vals) / N:.4f}")
print(f"1/sqrt(2) = {1 / math.sqrt(2):.4f}, 2/pi = {2 / math.pi:.4f}")
f, fs = 7000, 10000
print("alias of 7 kHz at fs = 10 kHz:", min(abs(f - k * fs) for k in range(3)), "Hz")
v = (5 ** 2 + (10 / math.sqrt(2)) ** 2) ** 0.5
print(f"RMS of 5 + 10 sin(wt) = {v:.4f}")

convolution: [1, 3, 5, 3]
full-wave rectified sine: RMS = 0.7071, average = 0.6366
1/sqrt(2) = 0.7071, 2/pi = 0.6366
alias of 7 kHz at fs = 10 kHz: 3000 Hz
RMS of 5 + 10 sin(wt) = 8.6603

Practice questions