Transcript ELCT 332 Fall 2004

Chapter 6
Random Processes and LTI
 Power Spectral Density
 White Noise Process
 Random Processes in LTI Systems
Huseyin Bilgekul
EEE 461 Communication Systems II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
EEE 461 1
Homework Assignments
• Return date: November 14, 2005.
• Assignments:
Problem 6-15
Problem 6-17
Problem 6-22
Problem 6-25
Problem 6-26
EEE 461 2
Power Spectral Density
• Definition: The PSD Px(f) of a random process is defined by, where
the subscript T denotes the truncated version of the signal
 X f 2
T   

Px  f   lim


T


T
XT  f  
2

T
x(t )e j 2 ft dt
2
T 
• Relationship to Time Autocorrelation, Wiener-Khintchine Theorem:
When x(t) is a wide sense stationary process, the PSD is defined as:

Rx    Px  f 
• Average Power of a Random Process
Px  Rx  0  




Px  f  df  2 Px  f  df  x2
0
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Properties of PSD
Some properties of PSD are:
• Px(f ) is always real
• Px(f ) > 0
• When x(t) is real, Px(-f )= Px(f )
• If x(t) is WSS,





Px  f  df  P  Total Normalized Power
____
2

Px  f  df  P  x  Rx  0 
• PSD at zero frequency is:

Px  0    Rx   d

EEE 461 4
General Expression PSD of a Digital Signal
• General expression for the PSD of a Digital Signal:
Px  f  
F f 
Ts
2


R  k  e j kTs
k 
• F(f ) is the Fourier Transform of the Pulse Shape f(t)
– Ts is the sampling interval
– R(k) is the autocorrelation of the data:
I
R  k     an ak  n  Pi
i 1
– an and ak+n are the levels of the nth and (n+k)th symbol positions
– Pi is the probability of having the ith anan+k product
• PSD only depends on the
– Pulse shape f(t)
– Statistical properties of the data
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Example: PSD of Unipolar NRZ Pulses
•
•
•
Possible levels are +A and 0
Square pulses of width Tb
Find the PSD:
2
F f  
j kT
Px  f  
R
k
e



s
Ts
k 
1) Find the spectrum of pulse:
 t 
sin  fTb
f  t       F  f   Tb
 fTb
 Tb 
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PSD of Unipolar NRZ Pulses
2) Evaluate the autocorrelation function
• For k = 0: there are 2 possibilities, an=A or an=0:
2
R  0     an an i Pi  A2 P 1  02 P  0   A2 / 2 (if P 1  0.5)
i 1
• For k >0: there are 4 possibilities, an=0 or A and
an+k=0 or A:
4
R  k     an ank i Pi  A2 P  an  1 and ank  1  0  A  P  an  0 and an k  1
i 1
 A  0  P  an  1 and ank  1  02  P  an  0 and an k  0   A2 / 4
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PSD of Unipolar NRZ Pulses
A Tb  sin  f Tb 
P f  


4   f Tb 
2
2


j 2 kf Tb 
1   e

 k 

A Tb  sin  f Tb 



4   f Tb 
2
• Simplify using:
– Poisson Sum Formula
– and
2
 1

1    f  
 Tb


e
k 
j 2 kf Tb
1

Tb

   f n T 
k 
b
sin  f Tb
 0 at f  n / Tb for n  0
 f Tb
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Example: PSD for Bipolar NRZ Signalling
1. Find the spectrum of pulse:
 t 
sin  fTb
f  t       F  f   Tb
 fTb
 Tb 
2. Find the Autocorrelation
• For k = 0: an=A or an= -A:
2
R  0     an an i Pi  A2 P 1    A P  0   A2
2
i 1
• For k >0: an=-A or A and an+k=-A or A:
4

R  k     an an k i Pi  A2  A   A    A A    A
i 1
2

1
4
0
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White Noise Process
• A random process is said to be a white noise process
if the PSD is constant over all frequencies:
Px  f  
N0
2
R()
Rx   
N0
  
2
P(f)
N0 /2
N0 /2

f
EEE 461 10
Linear Systems
• Recall that for LTI systems:
y t   h t   x t   Y  f   H  f  X  f 
• This is still valid if x and y are random processes, x
might be signal plus noise or just noise
• What is the autocorrelation and PSD for y(t) when x(t)
is known?
x(t)
X(f )
Rx()
Px(f )
Linear Network
h(t)
H(f )
y t   h t   x t 
Y f H f X  f 
Ry ( )  h     h    Rx ( )
Py ( f )  H ( f ) Px ( f )
2
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Output of an LTI System
• Theorem: If a WSS random process x(t) is applied to a LTI
system with impulse response h(t), the output
autocorrelation is:




Ry    y (t ) y (t   )   h  1  x  t  1 d 1  h  2  x  t    2 d 2



 h   h    R   
 
1
2
x
2
 1  d 1d 2
 h     h    Rx  
• And the output PSD is:
Py  f   H  f  Px  f 
2
• The power transfer function is:
Py  f 
2
Gh  f  
 Hf
Px  f 
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Example RC Low Pass Filter
R
• Input is thermal white noise.
No
Px  f  
2
Py  f   H  f  Px  f  
2
H( f ) 
1
C
x(t)=n(t)
y(t)
No 2
 f 
1 

B
 3dB 
1
, B3dB 

2 RC

2
 f
1 j 
 B3 dB 
N o    RC 
Ry      Px ( f ) 
e
4 RC
No
2
Py  y  Ry  0  
,
YDC  m y  0,
4 RC
1
y
2
No
 y  my 
4 RC
2
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SNR at the Output of a RC LPF
s t 
2
SNR 
n2  t 
• Input SNR is ratio of the input signal to input
noise
• Output SNR is ratio of the output signal to
output noise
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SNR at the Output of a RC LPF
• Same RC LPF as before, assume:
x(t)=si(t)+ni(t)
– si(t) =A cos(0t  q0, deterministic.
– ni(t) is white noise, flat PSD over all frequencies.
– ergodic noise (time avg=statistical avg).
• Input SNR (SNRi) is zero:
– Signal Power: A2/2
– Noise Power is infinity.
ni
2
____
2
i


No
 n   Sn  f df  
df  

 2
EEE 461 15
SNR at the Output of a RC LPF
• Output is y(t)=so(t)+no(t)
so  t   si  t   h  t   A H  f o  cos 0t  q o  H  f o  
• Output Signal Power
s t 
2
o
2
A2

H  f0 
2
• Output Noise Power (from previous example)
____
2
n  t   y  Ry  0  
2
o
so  t 
2
SNR0 
no  t 
2
No
4 RC
2 A H  f o  RC
2
2

No
2 A2 RC

No 1  (2 fo RC )2 
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Noise Equivalent Bandwidth
• For a WSS process x(t), the equivalent bandwidth is:

Rx (0)
1
B
Px ( f )df 

Px ( fo ) 0
2 Px ( fo )
• Input: white noise with a PSD of No/2 to a low pass filter:
no 2  t 
___
 No
2
2
 no 
H  f  df
 2

 No
2
2
H  f  df
0 2

• The Noise Equivalent Bandwidth is the filter bandwidth of
H(f ) that gives the same average noise power as an ideal
low pass filter of DC gain H(0)
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Noise Equivalent Bandwidth
H(f)
Ideal LP Filter
LP Filter
H(0)
B
B
no 2  t 
___
 No
2
2
2
no  t   no  2
H  f  df
0 2

2
 No
H  f  df
0
___
 no2  N o B H 2  0 



2
H  f  df
0
No BH 2  0   No 

2
H  f  df
B 0
H2 0

 
Noise Equivalent
Bandwidth
EEE 461 18