1 Compute the Hilbert transform xH(t) for the following signals:

(a) x(t) = cos(2πt).

(b) x(t) = sin(−3πt).

(c) x(t) = cos(5πt) + 2sin(2πt).

(d) x(t) = δ(t), the Dirac delta.

(e) x(t) = C0, a constant signal.

(f) x(t) = u(t), the unit step signal.

(g) x(t) = u(t + 1) − u(t − 1).

2 Let x(t) = cos(Ω0t) and y(t) = sin(Ω0t).

(a) Show that yH(t) = −x(t), where yH(t) is the Hilbert transform of y(t).

(b) Compute the analytic signal xA(t).

(c) Compute the signal envelope for x(t).

(d) Compute the instantaneous phase φ(t).

(e) Compute the instantaneous frequency ω(t).