What is the difference between a causal vs noncausal impulse response and how does it affect realizability?

What is the difference between a causal vs noncausal impulse response and how does it affect realizability?

• A. If h[n] is nonzero for negative indices, the system is noncausal and cannot be realized by a causal filter.
• B. A causal system can have nonzero impulse response for negative indices.
• C. Noncausal systems can be realized only with future input.
• D. Realizability does not depend on causality.

Answer: (A)

In discrete-time LTI systems, causality means the output at time n can depend only on inputs at times up to n (no future inputs). This is tied to the impulse response: for a causal system, h[n] must be zero for all n < 0. If the impulse response is nonzero for negative indices, the system would respond at times before the impulse is even applied, which is a hallmark of noncausal (acausal) behavior. In real-time processing, you can’t rely on future inputs, so a filter with such an impulse response cannot be realized as a causal filter. That’s why the statement is correct: a nonzero response for negative indices means the system is noncausal and cannot be realized by a causal filter in real time. If you relax the requirement to nonreal-time or allow buffering and processing that uses future samples, you can realize such a system, but it won’t be causal. Realizability relies on causality because causal systems only use present and past data; noncausal systems necessitate looking ahead.

In discrete-time LTI systems, causality means the output at time n can depend only on inputs at times up to n (no future inputs). This is tied to the impulse response: for a causal system, h[n] must be zero for all n < 0. If the impulse response is nonzero for negative indices, the system would respond at times before the impulse is even applied, which is a hallmark of noncausal (acausal) behavior. In real-time processing, you can’t rely on future inputs, so a filter with such an impulse response cannot be realized as a causal filter.

That’s why the statement is correct: a nonzero response for negative indices means the system is noncausal and cannot be realized by a causal filter in real time. If you relax the requirement to nonreal-time or allow buffering and processing that uses future samples, you can realize such a system, but it won’t be causal. Realizability relies on causality because causal systems only use present and past data; noncausal systems necessitate looking ahead.