Question

An ideal low pass filter H(s) with zero phase and magnitude response:

1 -π ≤ Ω ≤ π
|H(jΩ)| = {
0 otherwise
a) Find the impulse response h(t) of the low-pass filter. Plot it and indicate whether this filter is causal system or not.

Answer 1

The impulse response of an ideal low-pass filter is described by the sinc function, which is non-causal since it is not zero for t < 0.

Step-by-step explanation:

The student asked for the impulse response h(t) of an ideal low-pass filter with a rectangular magnitude response and zero phase. To find h(t), we apply the inverse Fourier transform to the given magnitude response. Since the filter is ideal with a rectangular magnitude response between -π and π, its impulse response can be determined using the sinc function, given by h(t) = sinc(t). The sinc function is defined as sinc(t) = sin(πt)/(πt), and it represents the inverse Fourier transform of the ideal low-pass filter. The plot of this impulse response will be a sinc function centered at the origin, which oscillates and decays as it moves away from the origin.

Regarding causality, this system is non-causal because the impulse response is not zero for t < 0. A causal system requires that the impulse response h(t) be zero for all t < 0, which is not the case for the sinc function representing an ideal low-pass filter.