Signals & Systems
The language of signal processing. Learn continuous and discrete signals, classification (periodic, energy/power, even/odd, deterministic/random), basic signals (step, impulse, sinusoid, exponential), system properties (linearity, time-invariance, causality, stability), LTI systems, impulse response and convolution.
Audio, images, radio waves, sensor readings and stock prices are all signals. Systems — filters, amplifiers, channels, controllers — transform them. Linear time-invariant (LTI) systems are fully described by their impulse response, which makes them analysable with convolution and transforms.
An analogy
A system's impulse response is like the echo of a single hand-clap in a hall: once you know it, you can predict how the hall will sound for any music by adding up echoes (convolution).
Signal types
Continuous-time (analog) vs discrete-time (samples); periodic (repeats with period T) vs aperiodic; energy signals (finite energy, e.g. pulses) vs power signals (finite average power, e.g. sinusoids); deterministic vs random (noise).
System properties
Linearity (superposition), time-invariance (delay input → delayed output), causality (output depends only on past/present), memory, stability (bounded input → bounded output), invertibility.
LTI and convolution
For LTI systems, output = input convolved with impulse response. Convolution flips, shifts, multiplies and sums. Cascaded systems convolve their impulse responses; parallel systems add them.
Towards transforms
Complex exponentials are eigenfunctions of LTI systems: they pass through scaled by the frequency response. This insight leads to Fourier, Laplace and Z transforms, which turn convolution into multiplication.
Formulas
- Convolution (discrete):
y[n] = Σ_k x[k] h[n − k] - Convolution (continuous):
y(t) = ∫ x(τ) h(t − τ) dτ - Energy / power:
E = Σ|x[n]|²; P = lim (1/N) Σ|x[n]|²; sinusoid P = A²/2 - BIBO stability:
Σ |h[n]| < ∞
Worked example: Discrete convolution
Given: x = [1, 2, 3], h = [1, 1]
Find: y = x * h.
- y[0] = 1
- y[1] = 2 + 1 = 3
- y[2] = 3 + 2 = 5
- y[3] = 3
Answer: y = [1, 3, 5, 3]
Worked example: Signal power
Given: x(t) = 5 cos(2π·50t)
Find: Average power (1 Ω).
- P = A²/2 = 12.5 W
Answer: 12.5 W (RMS = 3.54 V)
Common mistakes
- Wrong: Forgetting to flip h in convolution. — Right: y[n] = Σ x[k]h[n − k].
- Wrong: Assuming all systems are LTI. — Right: Check linearity and time-invariance first.
Applications
- Audio effects: Reverb by convolution with room responses.
- Communications: Channels modelled as LTI systems.
- Control: System responses and stability.
Key takeaways
- Classify signals and systems.
- LTI → impulse response + convolution.
- Stable if Σ|h| < ∞.
- Transforms simplify convolution.
Going further
- Correlation: Similarity measure related to convolution.
- Random signals: Autocorrelation and power spectral density.
- State-space: Multi-input multi-output systems.