Demodulation / Detection

7th Dec, 2020

 CHAPTER 3
 Detection of Binary Signal in Gaussian Noise
 Matched Filters and Correlators
 Bayes’ Decision Criterion
 Maximum Likelihood Detector
 Error Performance
 Demodulation and Detection
AWGN

DETECT
DEMODULATE & SAMPLE
SAMPLE
at t = T
RECEIVED
WAVEFORM FREQUENCY
RECEIVING EQUALIZING
DOWN
FILTER FILTER THRESHOLD MESSAGE
TRANSMITTED CONVERSION
WAVEFORM COMPARISON SYMBOL
OR
CHANNEL
FOR COMPENSATION
SYMBOL
BANDPASS FOR CHANNEL
SIGNALS INDUCED ISI

OPTIONAL

ESSENTIAL

Figure 3.1: Two basic steps in the demodulation/detection of digital signals

The digital receiver performs two basic functions:

 Demodulation, to recover a waveform to be sampled at t = nT.
 Detection, decision-making process of selecting possible digital symbol
Detection of Binary Signal in Gaussian Noise
2

-1

-2
0 2 4 6 8 10 12 14 16 18 20

-1

-2
0 2 4 6 8 10 12 14 16 18 20

-1

-2
0 2 4 6 8 10 12 14 16 18 20
Detection of Binary Signal in Gaussian Noise

 For any binary channel, the transmitted signal over a symbol interval
(0,T) is:
s0 (t ) 0  t  T for a binary 0
si (t )  
 s1 (t ) 0  t  T for a binary 1

 The received signal r(t) degraded by noise n(t) and possibly

degraded by the impulse response of the channel hc(t), is

r ( t )  s i ( t ) * hc ( t )  n ( t ) i  1, 2 (3.1)
Where n(t) is assumed to be zero mean AWGN process
 For ideal distortionless channel where hc(t) is an impulse function
and convolution with hc(t) produces no degradation, r(t) can be
represented as:
r (t )  s (t )  n(t ) i  1,2
i 0t T (3.2)
Detection of Binary Signal in Gaussian Noise

 The recovery of signal at the receiver consist of two parts

 Filter
 Reduces the received signal to a single variable z(T)
 z(T) is called the test statistics
 Detector (or decision circuit)
 Compares the z(T) to some threshold level 0 , i.e.,
H 1

z (T ) 
 where H1 and H0 are the two
 0
H 0 possible binary hypothesis
Receiver Functionality
The recovery of signal at the receiver consist of two parts:
1. Waveform-to-sample transformation
 Demodulator followed by a sampler
 At the end of each symbol duration T, predetection point yields a
sample z(T), called test statistic
z(T )  a (t)  n (t ) i  1,2
i 0
(3.3)

Where ai(T) is the desired signal component,

and no(T) is the noise component
2. Detection of symbol
 Assume that input noise is a Gaussian random process and
receiving filter is linear

1  1  n0  
2

p ( n0 )  exp      (3.4)
 0 2  2   0  
  Then output is another Gaussian random process

1  1  z  a0  
2

p(z | s0 )  exp    
 0 2  2   0  

1  1  z  a1  
2

p( z | s1 )  exp    
 0 2  2   0  
Where 0 2 is the noise variance
 The ratio of instantaneous signal power to average noise power ,
(S/N)T, at a time t=T, out of the sampler is:
 S  a i2
   (3.45)
 N T  02

 Need to achieve maximum (S/N)T

Find Filter Transfer Function H0(f)

 Objective: To maximizes (S/N)T

 Expressing signal ai(t) at filter output in terms of filter transfer
function H(f)

a i (t )  

H ( f ) S ( f ) e j 2  ft df (3.46)

where S(f) is the Fourier transform of input signal s(t)

 Output noise power can be expressed as:
N0 
  
2
0 | H ( f ) | 2 df
2  (3.47)
 Expressing (S/N)T as:
 2


j 2  fT
H ( f ) S( f ) e df
 S  
  
 N T N0 
(3.48)
2 

| H ( f ) | 2 df
 For H(f) = H0(f) to maximize (S/N)T, ; use Schwarz’s Inequality:

 2  2  2

 
f1 ( x) f 2 ( x)dx  

f1 ( x) dx  
f 2 ( x) dx (3.49)

 Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and *

indicates complex conjugate
 Associate H(f) with f1(x) and S(f) ej2 fT with f2(x) to get:

 2  2  2


H ( f ) S ( f ) e j 2fT df   H ( f ) df
 
S ( f ) df (3.50)

 Substitute in eq-3.48 to yield:

S  2  2
  
 N T N 0

S ( f ) df (3.51)
  S  2E
 Or max    and energy E of the input signal s(t):
 N T N0
 2
 Thus (S/N)T depends on input signal energy E
and power spectral density of noise and
 

S ( f ) df

NOT on the particular shape of the waveform

S  2E
 Equality for max    holds for optimum filter transfer
 N T N 0
function H0(f)
such that:
H ( f )  H 0 ( f )  kS * ( f ) e  j 2fT (3.54)

h ( t )    1 kS * ( f ) e  j 2  fT  (3.55)

 For real valued s(t):  kS (T  t ) 0  t  T

h (t )   (3.56)
0 else where
 The impulse response of a filter producing maximum output signal-
to-noise ratio is the mirror image of message signal s(t), delayed by
symbol time duration T.
 The filter designed is called a MATCHED FILTER

 kS (T  t ) 0  t  T
h (t )  
0 else where

 Defined as:
a linear filter designed to provide the maximum
signal-to-noise power ratio at its output for a given
transmitted symbol waveform
Correlation realization of Matched filter
 A filter that is matched to the waveform s(t), has an impulse
response
 kS (T  t ) 0tT
h (t )  
0 else where
 h(t) is a delayed version of the mirror image (rotated on the t = 0
axis) of the original signal waveform

Signal Waveform Mirror image of signal Impulse response of

waveform matched filter
Figure 3.7
 This is a causal system
 Recall that a system is causal if before an excitation is applied at

time t = T, the response is zero for -  < t < T

 The signal waveform at the output of the matched filter is
t (3.57)
z (t )  r (t ) * h (t )  0
r ( )h ( t   ) d 

 Substituting h(t) to yield:

r ( ) s T  ( t   ) d 
t
z (t )  0

r ( ) s T  t   d 
t
 0 (3.58)
 When t=T,
T
z (t )   0
r ( ) s ( ) d 
(3.59)
 The function of the correlator and matched filter are the same

 Compare (a) and (b)

T
 From (a)
z (t )  0
r ( t ) s ( t ) dt

T
z (t ) t T  z (T )   r ( ) s( )d
0
 From (b)  t

z' (T )  r(t) *h(t)   r( )h(t  )d   r( )h(t  )d
 0

But
h(t)  s(T  t)  h(t  )  s[T  (t  )]  s(T   t)
t
 z ' (t )   r ( ) s (  T  t ) d
0

 At the sampling instant t = T, we have

T T
z' (t ) t T  z' (t )   r ( )s(  T  T )d   r ( )s( )d
0 0

 This is the same result obtained in (a)

T
z ' (T )  
0
r ( ) s ( ) d 
 Hence
z(T )  z' (T )
Detection
 Matched filter reduces the received signal to a single variable z(T), after
which the detection of symbol is carried out
 The concept of maximum likelihood detector is based on Statistical
Decision Theory
 It allows us to
 formulate the decision rule that operates on the data

 optimize the detection criterion

H 1

z (T ) 

 0
H 0
Probabilities Review

 P[s0], P[s1]  a priori probabilities

 These probabilities are known before transmission
 P[z]
 probability of the received sample
 p(z|s0), p(z|s1)
 conditional pdf of received signal z, conditioned on the class si
 P[s0|z], P[s1|z]  a posteriori probabilities
 After examining the sample, we make a refinement of our
previous knowledge
 P[s1|s0], P[s0|s1]
 wrong decision (error)
 P[s1|s1], P[s0|s0]
 correct decision
How to Choose the threshold?
 Maximum Likelihood Ratio test and Maximum a posteriori (MAP)
criterion:
If

p ( s0 | z )  p ( s1 | z )   H 0
else
p ( s1 | z )  p ( s0 | z )   H 1


 Problem is that a posteriori probability are not known.

 Solution: Use Bay’s theorem:
p( z | s ) p(s )
p(s | z)  i i
i p(z)

H1 H1
p( z | s1 ) P(s1 ) p( z | s0 ) P ( s0 )
 

 p( z | s1) P(s )
1 
p( z | s0 ) P(s0 )
P( z ) H0
P( z) H0
 MAP criterion:
H1
p ( z | s1 ) P (s0 )
L( z)  

 likelihood ratio test ( LRT )
p( z | s0 ) H0
P ( s1 )

 When the two signals, s0(t) and s1(t), are equally likely, i.e., P(s0) =
P(s1) = 0.5, then the decision rule becomes
H1
p ( z | s1 )
L( z)  

1  max likelihood ratio test
p( z | s0 ) H0
 This is known as maximum likelihood ratio test because we are
selecting the hypothesis that corresponds to the signal with the
maximum likelihood.

 In terms of the Bayes criterion, it implies that the cost of both types
of error is the same
 Substituting the pdfs

1  1  z  a0  
2

H0 : p( z | s0 )  exp    
 0 2  2   0  

1  1  z  a1  
2

H1 : p ( z | s1 )  exp     
 0 2  2   0  

H1 1  1 2 H1
exp   z  a1  
p ( z | s1 )   0 2  2 0  
L( z)  1 1
p ( z | s0 )  1  1 2 
exp   z  a 0  
H0  0 2  2 0  H0
 Hence:

 z ( a1  a 0 ) ( a12  a 02 )  
exp    1
 02
2 02
 

 Taking the log of both sides will give

H1
z (a1  a0 ) (a12  a02 ) 
  ln{L( z )}   0
02
2 02

H0

H1
z ( a1  a 0 )  ( a12  a 02 ) ( a1  a 0 )( a1  a 0 )
 
0 2
 2 02
2 02
H0
 Hence

H1 H1
  02 (a1  a0 )(a1  a0 )  ( a1  a0 )
z z 0
 2 02 (a1  a0 )  2
H0 H0

where z is the minimum error criterion and  0 is optimum threshold

 For antipodal signal, s1(t) = - s0 (t)  a1 = - a0

H1

z 0

H0
This means that if received signal was positive, s1 (t) was sent,
else s0 (t) was sent
Probability of Error
 Error will occur if
 s1 is sent  s0 is received

P ( H 0 | s1 )  P (e | s1 )
0
P (e | s1 )   p ( z | s1 ) dz

 s0 is sent  s1 is received
P ( H 1 | s0 )  P (e | s0 )

P (e | s0 )   0
p ( z | s 0 ) dz
 The total probability of error is the sum of the errors
2
PB   P (e, si )  P ( e | s1 ) P ( s1 )  P (e | s0 ) P ( s0 )
i 1

 P ( H 0 | s1 ) P ( s1 )  P ( H 1 | s0 ) P ( s0 )
 If signals are equally probable
PB  P ( H 0 | s1 ) P ( s1 )  P ( H 1 | s0 ) P ( s 0 )
1
 P ( H 0 | s1 )  P ( H 1 | s0 ) 
2
1
PB  P( H 0 | s1 )  P( H1 | s0 ) bySymmetry
 P( H1 | s0 )
2
 Hence, the probability of bit error PB, is the probability that an
incorrect hypothesis is made
 Numerically, PB is the area under the tail of either of the conditional
distributions p(z|s1) or p(z|s2)
 
PB   0
P ( H 1 | s 0 ) dz   0
p ( z | s 0 ) dz

 1  1 za 
2

  0 0 2
exp   
 2   0
0


 dz

  1  1za 
2

PB   exp    0
  dz
0 
0 2  2   0  
( z  a0 )
 u
0
 1  u2 
 ( a1  a 0 )
2 0 2
exp  
 2 
 du

 The above equation cannot be evaluated in closed form (Q-function)

 Hence,
 a1  a 0 
PB  Q    equation B .18
 2 0 
1  x2 
Q ( x)  exp   
x 2  2
Conclusion