Signals, LTI Systems and Convolution
Signal classification, energy and power signals, basic operations, system properties (linearity, time invariance, causality, stability, memory), impulse response and convolution.
A signal carries information as a function of time; a system transforms an input signal into an output. Linear time-invariant (LTI) systems are fully described by one function: the impulse response.
Classifying signals
- Continuous / discrete time; analog / digital amplitude.
- Periodic: x(t + T) = x(t). The sum of two periodic signals is periodic only if the ratio of their periods is rational; the period is the LCM.
- Even x(−t) = x(t); odd x(−t) = −x(t). Every signal = even part + odd part.
- Energy signal: finite energy E = ∫|x|² dt (power 0). Power signal: finite non-zero average power (energy infinite), like periodic signals.
- Operations: time shift x(t − t₀), scaling x(at), reversal x(−t). For x(at − b), shift by b first, then scale by a.
Standard signals: unit impulse δ(t) (∫δ = 1; sifting: ∫x(t)δ(t − t₀)dt = x(t₀)), unit step u(t), ramp, exponential, sinc.
System properties
| Property | Test |
|---|---|
| Linear | Additive and homogeneous: a x₁ + b x₂ → a y₁ + b y₂ |
| Time-invariant | Shifted input gives the same output, shifted |
| Causal | Output depends only on present and past input |
| Memoryless | Output depends only on the present input |
| BIBO stable | Every bounded input gives a bounded output |
| Invertible | Distinct inputs give distinct outputs |
Examples: y(t) = x(2t) is linear but time-varying; y(t) = x(t) + 1 is not linear; y(t) = x(t + 1) is not causal.
Convolution
For an LTI system with impulse response h(t):
y(t) = x(t) * h(t) = ∫ x(τ) h(t − τ) dτ; discrete: y[n] = Σ x[k] h[n − k].
- Causal if h(t) = 0 for t < 0. Stable if ∫|h(t)| dt < ∞ (Σ|h[n]| < ∞).
- Convolution is commutative, associative and distributive.
- Length of the discrete convolution of sequences with lengths N and M: N + M − 1.
- Convolving with δ(t − t₀) shifts the signal by t₀.
Worked example
x[n] = {1, 2} and h[n] = {1, 1, 1} (both starting at n = 0).
- y[n] = {1, 3, 3, 2}, length 2 + 3 − 1 = 4.
Practice
What is the energy of x(t) = 3 for 0 ≤ t ≤ 2 (and zero elsewhere)?
Show answer and explanation
Answer: 18E = ∫|x|²dt = 9 × 2 = 18.
What is the sum of the output samples when {1, 2, 3} is convolved with {1, −1, 2}?
Show answer and explanation
Answer: 12The sum of a convolution equals the product of the sums: 6 × 2 = 12.
x(t) = cos(2πt/3) + sin(2πt/4). What is its fundamental period?
Show answer and explanation
Answer: 12The periods are 3 and 4; their LCM is 12.
The system y(t) = t·x(t) is:
Show answer and explanation
Answer: AScaling and adding inputs works (linear), but the gain t changes with time, so a shifted input does not give a shifted output.