Examples of Causal systems

Assume y[n] and y(t) are respectively the outputs corresponding to input signals x[n] and x(t)

1. System with description y[n] = x[n-1] + x[n] is clearly causal, as output "at" n depends on only values of the input "at instants" less than or equal to n ( in this case n and n-1 ).

2. Similarly, the continuous-time system with description is causal, as value of output at any time t₀ depends on only value of the input at t₀and before.

3. But system with description y[n] = x[n+1] is not causal as output at n depends on input one instant later.

If you think the idea of non-causal systems is counter intuitive, i.e: if you think no system can "anticipate the future", remember the independent variable need not be time. Visualizing non-causal systems with , say one-dimensional space as the independent variable is not difficult at all ! Even if the independent variable is time, we need not always be dealing with real-time, i.e. with the time axes of the input and output signals synchronized. The input signal may be a recorded audio signal and the output may be the same signal played backwards. This is clearly not causal !