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16.36 Communication Systems Engineering
Spring 2009
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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)
r(t) = S(t) + n(t)
•
r(t)
Channel
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 defined as
– Rxx(t1,t2) = E[x(t1)x(t2)]
•
A random process is Wide-sense-stationary (WSS) if its mean and
auto-correlation are not a function of time. That is
– mx(t) = E[x(t)] = m
– Rxx(t1,t2) = Rx(τ), where τ = t1-t2
•
If x(t) is WSS then:
– Rx(τ) = Rx(-τ)
– | Rx(τ)| <= |Rx(0)| (max is achieved at τ = 0)
•
The power content of a WSS process is:
1
Px = E[ lim
t! " T
Eytan Modiano
Slide 3
$
T /2
#T / 2
1
x (t)dt = lim
t !" T
2
$
T /2
Rx (0)dt =Rx (0)
#T / 2
Power Spectrum of a random process
•
If x(t) is WSS then the power spectral density function is given by:
Sx(f) = F[Rx(τ)]
•
The total power in the process is also given by:
"%"
"
Px =
# S ( f )df = # ''& # R (t)e
x
x
!"
=
!" !"
"%"
# ''& # R (t)e
x
!" !"
! j2 $ ft
! j2 $ ft
(
dt * df
*)
(
df * dt
*)
"
%"
(
= Rx (t) ' e ! j2$ft df * dt = Rx (t)+ (t)dt = Rx (0)
'&!"
*)
!"
!"
"
#
Eytan Modiano
Slide 4
#
#
White noise
•
The noise spectrum is flat over all relevant frequencies
–
White light contains all frequencies
Sn(f)
No/2
• Notice that the total power over the entire frequency range is infinite
–
But in practice we only care about the noise content within the signal
bandwidth, as the rest can be filtered out
• After filtering the only remaining noise power is that contained within the
filter bandwidth (B)
SBP(f)
-fc
Eytan Modiano
Slide 5
B
No/2
fc
B
No/2
AWGN
• The effective noise content of bandpass noise is BNo
–
Experimental measurements show that the pdf of the noise samples can be
modeled as zero mean gaussian random variable
fx (x) =
•
–
AKA Normal r.v., N(0,σ2)
–
σ2 = Px = BNo
The CDF of a Gaussian R.V.,
Fx (! ) = P[X " ! ] =
•
!
!
x
#$
#$
% f (x)dx = %
This integral requires numerical evaluation
–
Eytan Modiano
Slide 6
1
# x 2 / 2" 2
e
2! "
Available in tables
1
#x 2 / 2' 2
e
dx
2& '
AWGN, continued
•
X(t) ~ N(0,σ2)
•
X(t1), X(t2) are independent unless t1 = t2
•
#E[X(t + ! )]E[X(t )] ! " 0
Rx (! ) = E[X(t + ! )X(t)] = $
E[X 2 (t)]
! =0
%
#0 !"0
=$ 2
%& ! = 0
•
Eytan Modiano
Slide 7
Rx(0) = σ2 = Px = BNo
Detection of signals in AWGN
Observe: r(t) = S(t) + n(t), t ∈ [0,T]
Decide which of S1, …, Sm was sent
•
Receiver filter
–
r(t)
•
Eytan Modiano
Slide 8
Designed to maximize signal-to-noise power ratio (SNR)
filter
h(t)
Goal: find h(t) that maximized SNR
y(t)
“sample at t=T”
decide
Receiver filter
t
#
y(t) = r(t) * h(t) = r(! )h(t " ! )d!
0
T
#
Sampling at t = T $ y(T )= r(! )h(T " ! )d!
0
r (!) = s(! ) + n(! ) $
T
T
#
y(T) = s(! )h(T " !) d! +
0
SNR=
Eytan Modiano
Slide 9
Ys2 (T)
E[Yn2 (T )]
# n(! )h(T " ! )d! = Y (T) + Y (T )
s
0
=
%T
(
' s(! )h(T " ! )d! *
'& 0
*)
#
N0
2
T
#
0
h2 (T " t)dt
2
=
n
%T
(
' h(! )s(T " !) d! *
'& 0
*)
#
N0
2
T
#
0
h2 (T " t )dt
2
Matched filter: maximizes SNR
Caushy - Schwartz Inequality :
2
"
"
$ "
'
2
2
g
(t)g
(t)dt
*
(g
(t)
)
(g
(t)
)
2
1
2
&% !"1
)(
!"
!"
#
#
#
Above holds with equality iff: g1 (t) = cg2 (t) for arbitrary constant c
$T
'
& s(+ )h(T ! + )d+ )
&% 0
)(
#
SNR=
N0
2
Eytan Modiano
Slide 10
T
#
0
h2 (T ! t )dt
2
T
*
#
T
#
(s(+ )) 2 d+ h 2 (T ! + )d+
0
0
N0
2
T
#
h 2 (T ! t)dt
2
=
N0
T
#
0
2Es
(s(+ )) d+ =
N0
0
Above maximum is obtained iff: h(T-τ) = cS(τ)
=> h(t) = cS(T-t) = S(T-t)
h(t) is said to be “matched” to the signal S(t)
2
Example: PAM
Sm (t) = Amg(t), t ∈ [0,T]
Am is a constant: Binary PAM Am ∈ {0,1} Matched filter is matched to g(t)
g(t)
g(T-t)
A
A
T
Eytan Modiano
Slide 11
T
“matched filter”
Example, continued
t
Ys (t) =
$ S(! )h(t " ! )d! ,
h(t' ) = g(T " t' ) # h(t " ! ) = g(T +! " t )
0
Ys (t) =
t
t
0
0
$ g(!)g(T + ! " t )d! = $ g(! )g(T " t + ! )d!
T
Ys (T ) =
$
g2 (! )d!
0
Ys(t)
A2T
•
Sample at t=T to obtain
maximum value
t
T
Eytan Modiano
Slide 12
Matched filter receiver
Sample at t=kT
U(t)
rx(t)
rx(kT)
g(T-t)
2Cos(2πfct)
Sample at t=kT
U(t)
ry(t)
2Sin(2πfct)
Eytan Modiano
Slide 13
g(T-t)
ry(kT)
Binary PAM example, continued
g(t)
0 => S1 = g(t)
1 => S2 = -g(t)
S(t)
A
T
“S1(t)”
“S2(t)”
Y(t)
“Y1(t)”
2T
T
3T
T
T
Eytan Modiano
Slide 14
2T
“Y2(t)”
2T
Alternative implementation: correlator receiver
r(t) = S(t) + n(t)
Sample at t=kT
! ()
T
r(t)
Y(kT)
0
S(t)
!
T
Y(T) = r (t)S(t) =
0
!
T
0
!
T
S (t) + n(t)S(t) =Ys (T ) + Yn (T )
2
0
Notice resemblance to matched filter
Eytan Modiano
Slide 15