Chapter 10
Analysis and Processing of
Random Signals
ENCS6161 - Probability and Stochastic
Processes
Concordia University
Power Spectral Density
For WSS r.p. X(t), the power spectral density (PSD)
Z +∞
SX (f ) , F [RX (τ )] =
RX (τ )e−j2πf τ dτ
−∞
R +∞
So, RX (τ ) = −∞ SX (f )ej2πf τ df
The average power of X(t)
E[X(t)2 ] = RX (0) =
Z
+∞
SX (f )df
−∞
Note: SX (f ) ≥ 0 (see pg. 412 of textbook for proof)
Cross-Power Spectral Density:
Fro two WSS r.p. X(t), Y (t)
SX,Y (f ) = F [RX,Y (τ )]
where RX,Y (τ ) = E[X(t + τ )Y (t)].
ENCS6161 – p.1/11
Power Spectral Density
Example: a WSS r.p. with SX (f ) =
N0
2
for |f | < w
SX (f )
N0
2
−w
w
f
RX (τ )
1
2w
1
w
w
N0
df = N0 w
E[X (t)] =
−w 2
Z w
N0 j2πf τ
RX (τ ) =
e
df
−w 2
N0 sin(2πwτ )
=
2πτ
2
Z
τ
k
, then X(t) and X(t + τ ) are uncorrelated.
If τ = ± 2w
ENCS6161 – p.2/11
Power Spectral Density
White Noise: SX (f ) =
N0
2 ,
for all f
N0
δ(τ )
⇒ RX (τ ) =
2
Discrete-Time Random Processes
∞
X
RX (k)e−j2πf k
SX (f ) = F [RX (k)] =
k=−∞
RX (k) =
Z
1
2
− 21
SX (f )ej2πf k df
We only need to consider − 12 < f < 12 , since SX (f ) is
periodic in f with period of 1.
ENCS6161 – p.3/11
Linear Time-Invariant Systems
x(t)
T (·)
y(t) = T [x(t)]
A system is linear if,
T [αx1 (t) + βx2 (t)] = αT [x1 (t)] + βT [x2 (t)]
A system is time-invariant if,
y(t) = T [x(t)] ⇒ y(t − τ ) = T [x(t − τ )]
ENCS6161 – p.4/11
Response of LTI Systems
h(t) = T [δ(t)] is called the Impulse Response of the
system T (·).
The response of T (·) to x(t)Z will then be
∞
y(t) = T [x(t)] =
x(s)h(t − s)ds
−∞
Z ∞
=
h(s)x(t − s)ds
−∞
= x(t) ∗ h(t)
ENCS6161 – p.5/11
Response of LTI Systems
Let’s now consider the output of an LTI to a random
WSS signal X(t).
X(t)
Y (t)
h(·)
Z
mY (t) = E[Y (t)] = E
h(s)X(t − s)ds
−∞
Z ∞
Z ∞
=
h(s)E[X(t − s)]ds = mX
h(s)ds
−∞
−∞
R∞
Let H(f ) = F [h(t)] = −∞ h(t)e−j2πf t dt, then
mY (t) = mX H(0)
∞
ENCS6161 – p.6/11
Response of LTI Systems
The autocorrelation of the output Y (t)
RY (τ ) = E[Y (t)Y (t + τ )]
Z ∞
Z ∞
= E
h(s)X(t − s)ds
h(r)X(t + τ − r)dr
−∞
−∞
Z ∞Z ∞
=
h(s)h(r)E[X(t − s)X(t + τ − r)]dsdr
−∞ −∞
Z ∞Z ∞
=
h(s)h(r)RX (τ + s − r)dsdr
−∞
−∞
ENCS6161 – p.7/11
Response of LTI Systems
The power
Z spectral density
∞
SY (f ) =
RY (τ )e−j2πf τ dτ
−∞
Z ∞Z ∞Z ∞
=
h(s)h(r)RX (τ + s − r)e−j2πf τ dsdrdτ
−∞
−∞
−∞
(let u = τ + s − r)
Z ∞Z ∞Z ∞
=
h(s)h(r)RX (u)e−j2πf (u−s+r) dsdrdu
−∞ −∞ −∞
Z ∞
Z ∞
Z ∞
=
h(s)ej2πf s ds
h(r)e−j2πf r dr
RX (u)e−j2πf u du
−∞
∗
−∞
−∞
= H (f )H(f )SX (f ) = |H(f )|2 SX (f )
ENCS6161 – p.8/11
Response of LTI Systems
Similarly,
RY,X (τ ) = E[Y (t + τ )X(t)] = RX (τ ) ∗ h(τ )
SY,X (f ) = H(f )SX (f )
also,
∗
SX,Y (f ) = SY,X
(f ) = H ∗ (f )SX (f )
since RX,Y (τ ) = RY,X (−τ )
ENCS6161 – p.9/11
Response of LTI Systems
Example: A white Gaussian Signal X(t) is applied to
an RC circuit. Find the average power and
autocorrelation of the output Y (t).
R
+
X(t)
C
Y(t)
-
dy
+ y(t)
x(t) = RC
dt
⇒ x(f ) = j2πf RCy(f ) + y(f )
y(f )
1
⇒ H(f ) =
=
x(f )
1 + j2πf RC
ENCS6161 – p.10/11
Response of LTI Systems
2
N0
1
2
SY (f ) = |H(f )| SX (f ) = 1 + j2πf RC 2
N0 /2
=
1 + 4π 2 f 2 R2 C 2
N0 − |τ |
−1
RY (τ ) = F [SY (f )] =
e RC
4RC
N0
Average Power: E[Y 2 (t)] = RY (0) = 4RC
ENCS6161 – p.11/11