EE 354
Modern Communication Systems
Optimal Digital Receiver Design
Spring 2014
Instructor: C. R. Anderson
1
Digital Baseband Reception
Consider a bipolar NRZ PCM Signal
+V
0
1
1
0
0
0
1
0
1
-V
Add a little noise:
Add a lot of noise:
+V
+V
0
0
1
-V
1
0
0
0
1
0
`
1
`
1
1
0
0
0
1
0
1
-V
How can I recover my original signal values?
2
The ML Receiver Concept
Threshold Receiver
Formally Know as: Maximum Likelihood (ML) Receiver
yint ( t )
yout ( t )
r=
(t ) m (t ) + n (t )
Noise Limiting Filter
fco = Rb or 2Rb
Sample &
Hold
>
<
m̂ ( t )
Threshold
Decision
Operation
Synchronize TX/RX.
+V
Sample in the middle of
the bit period (blue dot).
0
1
-V
1
0
0
0
1
0
`
1
Compare sample value to
Threshold (gold line).
Decide “1” or “0”.
3
The MAP Receiver Concept
Correlation Receiver
Formally Know as: Maximum A Posteriori (MAP) Receiver
Also Known as: Matched Filter Receiver
∫
Tb
0
Noise Limiting Filter
fco = Rb or 2Rb
-
Integrate &
Dump
yout ( t )
r=
(t ) s (t ) + n (t )
•dt
−
yint
(t )
- Template
Pulse Shape
∫
Tb
0
•dt
>
<
Σ
+
yint
(t )
+
ŝ ( t )
Threshold
Decision
Integrate &
Dump
+ Template
Pulse Shape
Operation
+V
Synchronize TX/RX.
0
Correlate (multiply and integrate)
each bit with a stored template
waveform.
`
-V
Compare correlated value to
Threshold (gold line).
1
1
0
0
0
1
0
1
Decide “1” or “0”.
4
Simplified Digital Receiver Analysis – Problem Setup
{
}
Transmit a binary signal, s ( t ) ∈ s0 ( t ) , s1 (t ) , where s ( t ) is nonzero only on [ 0, Tb ] .
=
p0 Pr
=
The two bits are transmitted with probability:
 s0 ( t )  , p1 Pr  s1 ( t ) 
Assume the bits are equally probable such that:
p=
p=
0.5
0
1
The received signal will be corrupted by AWGN:
r=
(t ) s (t ) + n (t )
For a binary communication system, the Probability of Bit Error is given by:
Pe = Pr [sˆ ≠ s ] = Pr sˆ ≠ s0 s = s0  Pr [s = s0 ] + Pr sˆ ≠ s1 s = s1  Pr [s = s1 ]
Pe = ( 12 ) Pr sˆ ≠ s0 s = s0  + ( 12 ) Pr sˆ ≠ s1 s = s1 
The key to understanding how these two receivers work lies in (a) understanding
how they impact the probabilities and (b) computing these probabilities.
5
Digital Receiver Performance
f y ( y | s0 ) =
1
2πσ n2
Pr sˆ ≠ s1 s = s1 
2
y + A)
(
−
e
2σ n2
Threshold
f y ( y | s1 ) =
1
2πσ n2
2
y − A)
(
−
e
2σ n2
Pr sˆ ≠ s0 s = s0 
6