Why are ideal low-pass filters non-causal and not realizable with a causal, finite-delay system?

• A. Because it requires nonlinear processing to implement.
• B. Because it has a finite impulse response which cannot be realized.
• C. Its impulse response is a sinc function with infinite duration, making it noncausal.
• D. Because it has a pole at z = 1 that prevents causality.

Answer: (C)

The essential idea is that an ideal low-pass filter has perfect frequency selectivity, which in time corresponds to an impulse response that extends forever in both directions. If you take the inverse Fourier transform of the ideal passband (unity gain up to the cutoff and zero beyond), you get h(t) = (1/π) sin(ωc t)/t, a sinc function. This function has nonzero values for all times, including negative t. A causal system, by definition, must have h(t) = 0 for t < 0, so this impulse response cannot be realized by any causal, finite-delay system. You can delay it to start later (multiplying the spectrum by e^{-jωD}), but that only shifts the phase and cannot turn the infinite, nonzero-negative-time sinc into a truly causal, finite-delay realization without altering the exact magnitude response. That’s why an ideal low-pass filter is non-causal and not realizable with a strictly causal, finite-delay system.

The other statements don’t fit: linear processing is sufficient (it's a linear time-invariant operation), the impulse response isn’t finite-length (so it’s not a finite impulse response), and there isn’t a simple pole at z = 1 defining its behavior since the ideal response is not a rational transfer with a finite set of poles.