ECE 310 Digital Signal Processing
Homework 5
1. Evaluate the following integrals:
(a) R ∞
−∞(t
2 + 5t − 1)δ(t)dt =
(b) R ∞
1
(t
2 + 5t − 1)δ(t)dt =
(c) [e
−tu(t)] ∗ δ(5t − 15) =, where u(t) is a unit step function.
2. Determine the Fourier transform of the following functions:
(a) δ(2t − 3)
(b) sin(Ω0t + φ0), where Ω0 and φ0 are known real numbers.
(c) u(t) − u(t − T), where T is a constant.
3. Compute the discrete-time Fourier transform (DTFT) of the following sequence. x[n] = α
n
sin (ω0n)u[n],
where α and ω0 are real constants with |α| < 1.
4. Let Xd(ω) denote the DTFT of the complex valued signal x[n], where the real and imaginary parts
of x[n] are given below. Perform the following calculations without explicitly evaluating Xd(ω).
a) Evaluate Xd(0)
b) Evaluate Xd(π)
c) Evaluate Z π
−π
Xd(ω)dω
d) Determine and sketch the signal whose DTFT is X∗
d
(−ω)
5. Let x[n] be an arbitrary sequence, not necessarily real-valued, with DTFT Xd(ω). Express the DTFT
of the following sequences in terms of Xd(ω)
a) x
∗
[n]
b) x
∗
[−n]
6. Consider the complex sequence x[n] = (u[n] − u[n − N])/N.
a) Find closed-form expressions for |Xd(ω)| and ∠Xd(ω).
b) For N = 5, plot |Xd(ω)|; How will the shape of |Xd(ω)| change if N increases.
c) For N = 5, plot ∠Xd(ω); How will the shape of ∠Xd(ω) change if N increases.
