Description

1 Definitions
1. Let X(t) be a random process. The autocorrelation function is then defined as
RX (τ) = E [X(t)X∗ (t + τ)] 2. The Fourier transform of g(t) is defined as
Z ∞ (1.1)
G g(t)e−ȷ2πft dt (1.2)
3. The power spectral density of X(t) is defined as
F
RX(τ) ←→ S X(f) (1.3)
4.
F
g(t) ←→ G(f) (1.4)
F
=⇒ G(t) ←→ g(− f) (1.5)
5.
F 1 f !
g(at) ←→ G (1.6)
|a| |a|
6.
 1
rect(t) =  , 2 (1.7)
0 otherwise sin(πf)
sinc(f) = (1.8)
πf
7. If
Y(t) = X(t) ∗ h(t), (1.9)
S Y(f) = |H(f)|2 S X(f) (1.10)
8.

△ t 1 − |t| t ∈ (−1,1) (1.11)
=
2 0 otherwise
9.
F
g1(t) ∗ g2(t) ←→ G1(f)G2(f) (1.12)
10. The Dirac delta function is defined as
δ(t) = 0,t , 0 (1.13)
Z ∞
δ(t)dt = 1 (1.14)
−∞
2 Problems
1. Show that
F
rect(t) ←→ sinc(f) (2.1)
2. Show that
F −ȷ2πfT
g(t − T) ←→ G(f)e (2.2)
3. Let
Z T
Y = X(t) (2.3)
0
Show that
Y = Y(t)|t=T (2.4)
where
Y(t) = X(t) ∗ h(t) (2.5)
h(t) = rect t − T2  (2.6) T
4. Show that
t
rect(t) ∗ rect(t) = △ (2.7)
2
5. Show that
t F 2
△ ←→ sinc (f) (2.8)
2
6. Show that
Z ∞
E Y2 S Y(f)df (2.9)
7. Show that
E Y2 = T Z T △ τ RX(τ)dτ (2.10)
−T T
8. Show that
F
δ(t) ←→ 1 (2.11)
9. Show that
−a|t| F 2a
e ←→ (2.12)
a2 + ȷ2πf2
10. For
T = 10, E [X(t)] = 8,RX (τ) = 64 + e−2|τ|, (2.13)
find the mean and variance of Y.