Signals & System Properties
Continuous vs discrete, energy vs power, periodicity; then the four properties every exam asks: linearity, time-invariance, causality, BIBO stability.
- Classify signals: CT/DT, energy/power, periodic/aperiodic, even/odd
- Test a system for linearity and time-invariance with standard inputs
- State causality and BIBO stability and test each on given systems
Every ECE exam paper in signals starts the same way: "Check whether the system y(t) = x(t)cos(t0) is linear, causal and stable." Four tests, ten minutes, full marks — if you know the tests by heart.
Signal vocabulary:
- CT vs DT: x(t) continuous argument vs x[n] integer index.
- Energy vs power: E = ∫|x(t)|²dt (finite energy ⇒ energy signal, zero average power); P = lim(1/T)∫|x|² over T. Power signals have infinite energy (e.g. periodic). A zero signal is both (trivially).
- Periodic: x(t) = x(t+T0), fundamental T0 = smallest such T. DT: x[n] = x[n+N], fundamental N smallest positive integer. Sum of two periodic signals is periodic only if T1/T2 is rational.
- Even/odd: x(t)=x(−t) even; x(t)=−x(−t) odd; any signal = even part + odd part.
System properties (the four exam tests):
- Linearity: T{a·x1 + b·x2} = a·T{x1} + b·T{x2}. Test with weighted sum of two inputs. Breaks on: amplitude nonlinearity (x²), offsets (x + 1), modulations by t·x(t).
- Time-invariance (TI): input delayed by t0 ⇒ output delayed by t0. Test: y(t,t0) = T{x(t−t0)} equals y(t−t0). Breaks on: coefficients that vary with t (t·x(t), cos(t)x(t)).
- Causality: output at time t depends only on present/past inputs, never future. Test: x1 = x2 on (−∞, t0] ⇒ outputs equal there. Any system writing y(t) = x(t+1) is noncausal.
- BIBO stability: every bounded input (|x| ≤ Mx < ∞) gives bounded output (|y| ≤ My < ∞). For LTI: stable iff impulse response absolutely integrable, ∫|h(t)|dt < ∞ (DT: Σ|h[n]| < ∞).
LTI = Linear + Time-Invariant. Only for LTI systems do convolution, transfer functions, frequency response and the whole transform toolkit apply. That is why the first two tests come first in every question.
| One-line test | |
|---|---|
| Linearity | Weighted-sum input gives weighted-sum output |
| Time-invariance | Delay input by t0 ⇒ output delayed by t0 |
| Causality | No dependence on future input values |
| BIBO stability | Bounded input ⇒ bounded output (∫|h| < ∞ for LTI) |
x², |x|, offset (+1) kill linearity; any t multiplying x kills TI.
Without LTI there is no ∫|h| shortcut — argue directly with bounded inputs.
- 1
Linearity
T{a x1 + b x2} = t(a x1 + b x2) = a t x1 + b t x2 — linear (scaling/additivity hold).
- 2
TI test
Input x(t−t0) gives t·x(t−t0); desired output is (t−t0)x(t−t0). Not equal ⇒ NOT time-invariant.
- 3
Causality
y(t) depends only on x(t) at the same instant ⇒ causal.
- 4
Stability
x = u(t) bounded (M=1): y(t) = t → grows unbounded ⇒ NOT BIBO stable.
- 5
Verdict
Linear, causal, not TI, not stable — so no convolution/transfer-function analysis.
Solution
Linear + causal; not TI (t coefficient), not stable (ramp gain).
y(t) = x(t) + 3. Linear? Time-invariant? Why does the offset break linearity?
Hint: Test T{0} — linearity requires the zero input to give zero output.
Testing TI by delaying the output expression and calling it done
Both sides: response to delayed input must equal delayed original response.
Saying causal systems are always stable
Causality and stability are independent properties — t·x(t) is causal but unstable.
State BIBO stability for LTI systems in terms of the impulse response.
The system is BIBO stable iff h(t) is absolutely integrable: ∫|h(t)|dt < ∞ (DT: Σ|h[n]| < ∞).
Source: Standard 4–6 mark question.
- Energy vs power; periodicity requires rational period ratio for sums.
- Linearity, TI, causality, BIBO — four standard tests.
- LTI unlocks convolution and transforms; ∫|h| < ∞ ⇔ stable.
