Question

Sketch the Hilbert transforms (frequency domain only) of the following signals

(a) Π(f)
(b) Δ(f)

Answer 1

The Hilbert transforms of a rectangular function Π(f) and a triangular function Δ(f) involve phase shifts of the signal in the frequency domain.

Step-by-step explanation:

The question pertains to the Hilbert transforms of certain signals in the frequency domain. For a rectangular function, denoted Π(f), its Hilbert transform corresponds to a signum function in the frequency domain, shifting phase by -90 degrees for positive frequencies and by 90 degrees for negative frequencies.

For the triangular function, denoted Δ(f), its Hilbert transform is slightly more complicated. The Hilbert transform of a triangular function can be thought of as the convolution of the Hilbert transform of a rectangular function with the original triangular function, applying phase shifts to each frequency component that are dependent on the original signal's characteristics.