What is meant by causal and non causal system

In the context of signal processing and system theory, the terms "causal" and "non-causal" refer to different types of systems based on the timing of their outputs relative to their inputs.

Causal System:

A causal system is one where the output at any given time ttt depends only on the values of the input at the current time ttt and at times in the past t′<tt' < tt′<t. In other words, the output does not depend on future values of the input. Mathematically, for a system HHH:

y(t)=H[x(t)]=f(x(t),x(t−1),x(t−2),…)y(t) = H[x(t)] = f(x(t), x(t-1), x(t-2), \ldots)y(t)=H[x(t)]=f(x(t),x(t−1),x(t−2),…)

• Example: A simple moving average filter, where the current output depends on the current input and previous inputs, is a causal system because it does not require future inputs to compute the current output.

Non-Causal System:

A non-causal system is one where the output at a given time ttt depends on future values of the input, in addition to the current and past values. This violates the principle of causality because the system requires knowledge of future inputs to compute the output at the present time.

• Example: A system that predicts future values based on current and past inputs is non-causal because it uses future inputs that are not yet known to compute the current output.

Practical Considerations:

• Real-Time Processing: Causal systems are typically preferred in real-time applications because they can process data as it arrives without needing future data, which may not be available or known.

• Theoretical Analysis: Non-causal systems are sometimes used in theoretical contexts or predictive models where future information is hypothesized or extrapolated, but they are less common in practical implementations due to the requirement for future data.

Understanding whether a system is causal or non-causal is crucial for designing and analyzing signal processing algorithms, control systems, filters, and other applications where the relationship between input and output timing is critical.