Signals and Systems – 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.,

$$\mathrm{\mathit{h\left ( t \right )\mathrm{=}\mathrm{0};\; \; \mathrm{for}\: \: t< 0}}$$

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 β„Ž(𝑑) must be causal.

  • In the frequency domain, this criterion denotes that a necessary and sufficient condition for a magnitude function 𝐻(πœ”) to be physically realizable is given by,

$$\mathrm{\mathit{\int_{-\infty }^{\infty }\frac{\mathrm{ln}\left | H\left ( \omega \right ) \right |}{\left ( \mathrm{1}\mathrm{+}\omega ^{\mathrm{2}} \right )}d\omega < \infty }}$$

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

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

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(πœ”)| 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(πœ”)| 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.