- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# Signals and Systems β 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(π‘) must be causal.In the

*frequency domain*, this criterion denotes that a necessary and sufficient condition for a magnitude function π»(π) 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 |π»(π)| 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 |π»(π)| 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 |π»(π)| 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.

- Related Articles
- Signals and Systems: BIBO Stability Criterion
- Signals and Systems: Even and Odd Signals
- Signals and Systems: Periodic and Aperiodic Signals
- Signals and Systems: Energy and Power Signals
- Signals and Systems β Classification of Signals
- Signals and Systems: Multiplication of Signals
- Signals and Systems: Classification of Systems
- Signals and Systems: Addition and Subtraction of Signals
- Signals and Systems: Real and Complex Exponential Signals
- Signals and Systems: Linear and Non-Linear Systems
- Signals and Systems: Invertible and Non-Invertible Systems
- Signals and Systems: Amplitude Scaling of Signals
- Signals and Systems β Time Scaling of Signals
- Signals and Systems β Parsevalβs Theorem for Laplace Transform
- Signals and Systems β Properties of Even and Odd Signals