Causality and Paley-Wiener Criterion for Physical Realization

Condition of Causality

A causal system is the one which does not produce an output before the input is applied. Therefore, for an LTI (Linear Time-Invariant) system to be causal, the impulse response of the system must be zero for t less than zero, i.e.,

$$\mathrm{h(t) \:=\: 0\:;\: \: for \: \: t\:\lt\: 0}$$

The term physical realization denotes that it is physically possible to construct that system in real time. A system which is physically realizable cannot produce an output before the input is applied. This is called the condition of causality for the system.

• Therefore, the time domain criterion for a physically realizable system is that the unit impulse response h(t) must be causal.
• In the frequency domain, this criterion denotes that a necessary and sufficient condition for a magnitude function H(ω) to be physically realizable is given by,

$$\mathrm{\int_{-\infty }^{\infty }\frac{ln\: |H(\omega)|}{(1 \:+\: \omega^{2})}\:d\omega \:\lt\: \infty}$$

However, the magnitude function |H(ω)| must be square integrable before the Paley-Wiener criterion is valid, i.e.,

$$\mathrm{\int_{-\infty}^{\infty}\:\left | H(\omega) \right |^{2}\:d\omega \:\lt\: \infty}$$

Therefore, a system whose magnitude function violates the Paley-Wiener criterion has an impulse response which is non-causal, i.e., the response of the system exists prior to the application of the input signal.

Conclusions from the Paley-Wiener Criterion

The conclusions drawn from the Paley-Wiener criterion are given as follows −

• The magnitude function |H(ω)| may be zero at some discrete frequencies, but it cannot be zero over a finite band of frequencies because this will cause the integral in the equation of Paley-Wiener criterion to become infinity, which means that the ideal filters are not physically realizable.
• The magnitude function |H(ω)| cannot be reduced to zero faster than a function of exponential order. It denotes that a realizable magnitude characteristic cannot have too great a total attenuation.