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# 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
Page 2

MIT OpenCourseWare
http://ocw.mit.edu
16.36 Communication Systems Engineering
Spring 2009

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

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

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

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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