Subject : Network Theory
System Causality
Introduction:
A causal system is a system where the output y(t) at some specific instant t_{0 }only depends on the input x(t) for values of t less than or equal to t_{0} . Therefore these kinds of systems have outputs and internal states that depends only on the current and previous input values.
Mathematical Expression:
- There are two important considerations causality and stability.
- By causality we mean that a voltage cannot appear between any pair of terminals in the network before a current is impressed, or vice versa.
- In other words, the impulse response of the network must be zero for t < 0, that is,
- The impulse response
- is not causal.
- In certain cases, the impulse response could be made realizable (causal) by delaying it appropriately. For example, the impulse response in Fig. 10.1a is not realizable.
- If we delay the response by T seconds, we find that the delayed response h (t — T) is realizable (Fig. 10.1b).
Paley-Wiener criterion:
- In the frequency domain, causality is implied when the Paley-Wiener Criterion is satisfied for the amplitude function \H(jω)\.
- The Paley-Wiener criterion states that a necessary and sufficient condition for an amplitude function \H(jω)\ to be realizable (causal) is that
- The following conditions must be satisfied before the Paley-Wiener criterion is valid: h(t) must possess a Fourier transform H(jω); the square magnitude function \H(jω)\^{2} must be integrable, that is,
- Another way of looking at the Paley-Wiener criterion is that the amplitude function cannot fall off to zero faster than exponential order.