Causality from Impulse Response

Causality is the property that a system's output at time t depends only on the present and past values of the input, never on future values. Checking causality from the impulse response is direct: examine where h(t) or h[n] is nonzero.

Core Concept

Causality is a physical requirement for real-time systems. A microprocessor filter, a motor controller, and a radio demodulator must all produce output using only current and past data. The impulse response h(t) encodes this completely: if h(t) is nonzero for any t less than zero, the system would need future input values to compute the current output.

For continuous-time LTI systems, the causality test is: h(t) = 0 for all t < 0. For discrete-time LTI systems, it is: h[n] = 0 for all n < 0. Any system whose impulse response starts before the origin is non-causal.

In the Laplace domain, a causal system has a transfer function H(s) whose Region of Convergence (ROC) is a right-half plane of the form Re(s) > sigma_0. If the ROC is a left-half plane or a vertical strip, the system is non-causal. This ROC criterion is essential for GATE problems involving inverse Laplace transforms.

Key Formula

Continuous-time causality: h(t) = 0 for t < 0, equivalently h(t) = h(t)*u(t).

Discrete-time causality: h[n] = 0 for n < 0.

Laplace domain ROC for causal system: Re(s) > sigma_max, where sigma_max is the real part of the rightmost pole. The ROC extends to the right of all poles.

Problem: Is the system with H(s) = 1/(s+2) - 1/(s-1) causal?
          Assume the ROC is Re(s) > 1.

Step 1 — Identify poles:
  Pole at s = -2 and pole at s = 1.

Step 2 — Check ROC:
  ROC given as Re(s) > 1 (right of the rightmost pole at s=1).

Step 3 — Inverse Laplace for each term with this ROC:
  1/(s+2) with Re(s)>-2 -> e^(-2t)*u(t)  [causal term]
  1/(s-1) with Re(s)>1  -> e^(t)*u(t)    [causal term]

Step 4 — Combine:
  h(t) = [e^(-2t) - e^(t)] * u(t)

Step 5 — Causality check:
  h(t) = 0 for t < 0 because both terms are multiplied by u(t).

Final Answer: System IS causal with this ROC.
Note: If ROC were Re(s)<1, the 1/(s-1) term inverts to -e^(t)*u(-t), making it non-causal.

Exam Tip: In GATE problems, the ROC is not always stated explicitly. When a signal is described as causal (right-sided), assume the ROC is Re(s) > rightmost pole. When described as anti-causal (left-sided), assume ROC is Re(s) < leftmost pole. The same H(s) expression can represent different time-domain signals depending on which ROC you choose, so always state the ROC when writing the answer.

Properties Summary

• Causality test in time domain: h(t) = 0 for all t < 0.
• Causality test in Laplace domain: ROC is Re(s) > sigma for some finite sigma.
• Causality test in Z-domain: ROC is |z| > r for some r; ROC extends outside the outermost pole.
• Anti-causal system: h(t) = 0 for t > 0; ROC is Re(s) < sigma (left-half plane).
• BIBO stability + causality: all poles must have Re(s) < 0 for continuous-time, or |z| < 1 for discrete-time.
• Non-causal FIR filters are realizable offline (audio post-processing) but not in real time.

Quick Revision

• Causal: output at t depends only on x(tau) for tau <= t.
• Impulse response starts at or after t = 0 for a causal system.
• ROC for causal system in s-domain: right-half plane, Re(s) > real part of rightmost pole.
• ROC for causal system in z-domain: exterior of a circle, |z| > radius of outermost pole.
• A non-causal system can still be LTI; it just cannot operate in real time.
• Bilateral Laplace pairs like e^(-|t|) produce an ROC that is a vertical strip, indicating a non-causal, two-sided signal.
• Exam trap: assuming every stable system is causal; a system can be BIBO stable and non-causal if its poles are in the left half-plane but the ROC is chosen as a left-half plane.