Notes : Communication-Systems-Engineering Notes | EduRev

: Notes : Communication-Systems-Engineering Notes | EduRev

 Page 1


MIT OpenCourseWare
http://ocw.mit.edu 
16.36 Communication Systems Engineering
Spring 2009 
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 
Page 2


MIT OpenCourseWare
http://ocw.mit.edu 
16.36 Communication Systems Engineering
Spring 2009 
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 
 
 
Lectures 8 - 9: Signal Detection in Noise
and the Matched Filter 
Eytan Modiano 
Aero-Astro Dept. 
Eytan Modiano
Slide 1 
Page 3


MIT OpenCourseWare
http://ocw.mit.edu 
16.36 Communication Systems Engineering
Spring 2009 
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 
 
 
Lectures 8 - 9: Signal Detection in Noise
and the Matched Filter 
Eytan Modiano 
Aero-Astro Dept. 
Eytan Modiano
Slide 1 
 
 
 
   
       
 
   
  
Noise in communication systems
 
S(t)
Channel	 
r(t)
 r(t) = S(t) + n(t)	 
n(t)
•	 Noise is additional “unwanted” signal that interferes with the
transmitted signal 
–	 Generated by electronic devices 
•	 The noise is a random process 
–	 Each “sample” of n(t) is a random variable 
•	 Typically, the noise process is modeled as “Additive White 
Gaussian Noise” (AWGN) 
–	 White: Flat frequency spectrum 
–	 Gaussian: noise distribution 
Eytan Modiano
Slide 2 
Page 4


MIT OpenCourseWare
http://ocw.mit.edu 
16.36 Communication Systems Engineering
Spring 2009 
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 
 
 
Lectures 8 - 9: Signal Detection in Noise
and the Matched Filter 
Eytan Modiano 
Aero-Astro Dept. 
Eytan Modiano
Slide 1 
 
 
 
   
       
 
   
  
Noise in communication systems
 
S(t)
Channel	 
r(t)
 r(t) = S(t) + n(t)	 
n(t)
•	 Noise is additional “unwanted” signal that interferes with the
transmitted signal 
–	 Generated by electronic devices 
•	 The noise is a random process 
–	 Each “sample” of n(t) is a random variable 
•	 Typically, the noise process is modeled as “Additive White 
Gaussian Noise” (AWGN) 
–	 White: Flat frequency spectrum 
–	 Gaussian: noise distribution 
Eytan Modiano
Slide 2 
 
 
  
     
      
 
           
Random Processes
 
•	 The auto-correlation of a random process x(t) is de?ned as 
–	 R
xx
(t
1
,t
2
) = E[x(t
1
)x(t
2
)] 
•	 A random process is Wide-sense-stationary (WSS) if its mean and
auto-correlation are not a function of time. That is 
– m
x
(t) = E[x(t)] = m 
–	 R
xx
(t
1
,t
2
) = R
x
(t), where t = t
1
-t
2 
•	 If x(t) is WSS then: 
–	 R
x
(t) = R
x
(-t) 
–	 | R
x
(t)| <= |R
x
(0)| (max is achieved at t = 0) 
•	 The power content of a WSS process is: 
P
x
= E[ lim
t!"
1
T
x
2
(t)dt =
#T / 2
T / 2
$
lim
t!"
1
T
R
x
(0)dt =
#T / 2
T / 2
$
R
x
(0)
Eytan Modiano

Slide 3
 
Page 5


MIT OpenCourseWare
http://ocw.mit.edu 
16.36 Communication Systems Engineering
Spring 2009 
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 
 
 
Lectures 8 - 9: Signal Detection in Noise
and the Matched Filter 
Eytan Modiano 
Aero-Astro Dept. 
Eytan Modiano
Slide 1 
 
 
 
   
       
 
   
  
Noise in communication systems
 
S(t)
Channel	 
r(t)
 r(t) = S(t) + n(t)	 
n(t)
•	 Noise is additional “unwanted” signal that interferes with the
transmitted signal 
–	 Generated by electronic devices 
•	 The noise is a random process 
–	 Each “sample” of n(t) is a random variable 
•	 Typically, the noise process is modeled as “Additive White 
Gaussian Noise” (AWGN) 
–	 White: Flat frequency spectrum 
–	 Gaussian: noise distribution 
Eytan Modiano
Slide 2 
 
 
  
     
      
 
           
Random Processes
 
•	 The auto-correlation of a random process x(t) is de?ned as 
–	 R
xx
(t
1
,t
2
) = E[x(t
1
)x(t
2
)] 
•	 A random process is Wide-sense-stationary (WSS) if its mean and
auto-correlation are not a function of time. That is 
– m
x
(t) = E[x(t)] = m 
–	 R
xx
(t
1
,t
2
) = R
x
(t), where t = t
1
-t
2 
•	 If x(t) is WSS then: 
–	 R
x
(t) = R
x
(-t) 
–	 | R
x
(t)| <= |R
x
(0)| (max is achieved at t = 0) 
•	 The power content of a WSS process is: 
P
x
= E[ lim
t!"
1
T
x
2
(t)dt =
#T / 2
T / 2
$
lim
t!"
1
T
R
x
(0)dt =
#T / 2
T / 2
$
R
x
(0)
Eytan Modiano

Slide 3
 
 
 
  
Power Spectrum of a random process
 
•	 If x(t) is WSS then the power spectral density function is given by: 
S
x
(f) = F[R
x
(t)] 
• The total power in the process is also given by: 
P
x
= S
x
( f)df =
!"
"
#
R
x
(t)e
! j2$ft
dt
!"
"
#
% 
& 
' 
' 
( 
) 
* 
* 
!"
"
#
df
= R
x
(t)e
! j2$ft
df
!"
"
#
% 
& 
' 
' 
( 
) 
* 
* 
!"
"
#
dt
= R
x
(t) e
! j2$ft
df
!"
"
#
% 
& 
' 
' 
( 
) 
* 
* 
!"
"
#
dt = R
x
(t)+(t)
!"
"
#
dt = R
x
(0)
Eytan Modiano

Slide 4
 
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