Chapter 5 Optimum Receivers for the Additive White Gaussian Noise Channel
5.1 Optimum receiver for signals corrupted by additive white Gaussian noise
(1). A mathematical model for such a system is as below:
The signal waveforms at the output of transmitter: {sm (t ), m  1,2,3, , M , 0  t  T }
The propagation channel: AWGN
The received signal:
(5.1-1)
r(t )  sm (t )  n(t ), 0  t  T
n(t ) represents noise and is a sample function of the AWGN process with PSD
1
 nn ( f )  N 0 W/Hz.
2
Criterion of receiver design: minimum of probability of error.
(2). Configuration of receiver:
(a). Signal demodulator: the function of signal demodulator is to convert the received
waveform r (t ) into an N-dimensional vector r  [r1 r2 rN ] . Two kinds of
demodulators will be discussed, i.e., signal correlator and matched filter.
(b). Detector: the function of detector is to decide which of the M possible signal
waveforms was transmitted based on the vector r. The optimum detector is
designed to minimize the probability of error.
5.1.1 Correlation demodulator
Suppose a set of N basis functions { f n (t ), n  1,2,..., N } span the signal space and therefore
is enough to decompose the received signal into N-dimensional vector.
The received signal is passed through a parallel bank of N cross correlators, which transfers
the waveform into vector element (the projection of r (t ) onto { f n (t ), n  1,2,..., N } .
Thus, we have at the output of the integrator
T
T
 r  t  f  t  dt    s t   n t  f t  dt
k
m
0
k
0
(5.1-2)
rk  smk  nk , k  1,2,..., N
where
T
smk   sm  t  f k  t  dt ,k  1, 2,..., N
0
(5.1-3)
T
nk   n  t  f k  t  dt ,k  1, 2,..., N
0
Note that smk , k  1,2,..., N are deterministic and form the vector s m , however, nk is a
random variable.
The received signal in the interval 0  t  T can be expressed by
N
N
N
k 1
k 1
k 1
r  t    smk f k (t )   nk f k (t )  n '(t )   rk f k (t )  n '(t )
(5.1-4)
The term n '( t ) defined below is a zero-mean Gaussian process. In the following, it is
shown that n '( t ) is irrelevant to the decision as to which signal was transmitted.
Consequently, the decision may be based entirely on the correlators’ output signal and noise
components rk  smk  nk , k  1,2,..., N .
N
n '(t )  n(t )   nk f k (t )
(5.1-5)
k 1
The means of nk for all n are
T
E (nk )   E  n(t ) f k (t )dt  0
0
Their variances are
(5.1-6)
T T
E  nk nm     E  n  t  n    f k  t  f m   dtd 
0 0
T T
1
N 0   t    f k  t  f m   dtd
2 0 0
(5.1-7)
T
1
1
 N 0  f k  t  f m   dt  N 0 mk
2 0
2
1 m  k
where  mk  
0 m  k
Therefore, the N noise components {nk } are zero-mean uncorrelated Gaussian random
variables with a common variance  n 2  12 N 0 .
Conditioned on sm (t ) , the correlator output is a Gaussian random variable with mean
E  rk   E  smk  nk   smk
(5.1-8)
and equal variance
 r2   n2 
1
N0
2
(5.1-9)
Since {nk } are uncorrelated Gaussian random variables, they are also statistically
independent, therefore, {rk } are statistically independent Gaussian random variables.
The pdf of r  [r1 r2 rN ] conditioned on s m is
N
p  r | sm    p  rk | smk , m  1, 2,..., M
(5.1-10)
  rk  smk 2 
1
p  rk | smk  
exp  
 , k  1, 2,..., N
N0
 N0


(5.1-11)
k 1
where
Hence the joint conditional pdf is
 N  rk  smk 2 
p  r | sm  
exp  
 , m  1, 2,..., M
N /2
N0
 N 0 
 k 1

1
(5.1-12)
Next, the correlator outputs {rk } can be shown to be sufficient statistics, in other words, it
suffices to demonstrate n '( t ) offers no additional information for detection, or to show
n '( t ) is uncorrelated with the N correlator outputs, {rk } .
E  n '(t )rk   E  n '(t )  smk  E  n '(t )nk   E  n '(t )nk 
N
 
 
 E   n ( t )   n j f j ( t )  nk 
j 1
 
 
T
N
0
j 1
  E  n(t )n    f k ( )d   E  n j nk  f j (t )
(5.1-13)
1
1
N 0 f k (t )  N 0 f k (t )  0
2
2
Since n '( t ) and {rk } are Gaussian and uncorrelated, they are statistically independent.

Thus, n '( t ) can be ignored.
Example 5.1-1 Consider an M-ary baseband PAM signal set in which the basic pulse shape
g (t ) is rectangular as shown in Figure 5.1-4. Noise n(t ) is AWGN. Find the basis function
f (t ) and the output of the correlator-type demodulator.
Solution: The energy in the rectangular pulse is
T
T
0
0
Eg   g 2  t  dt   a 2dt  a 2T
Since the PAM signal set is one-dimensional, N=1, the basis function is
f (t ) 
1/ T
g (t )  
a 2T
 0
(0  t  T )
1
(otherwise)
The output of the correlator-type demodulator is
T
T
1
r (t )dt
T 0
0
Here we have found that the correlator becomes a simple integrator when f (t ) is
r   r (t ) f (t )dt 
rectangular.
By substituting for r (t ) , the resultant output is
T
T
T


1 
1 
r
   sm (t )  n(t )   dt 
  sm (t )dt   n(t )dt   sm  n
T 0
T 0

0

where the noise term E ( n )  0 and
1
 1
n(t )n( )dtd  


T 0 0
 T
 n2  E 
T T
T T
  E n(T )n( )dtd 
0 0
N0
2T
T T
1
   (t   )dtd  2 N
0
0 0
The pdf for the sampled output is
  r  sm 2 
1
p( r | sm ) 
exp  

N 0 
 N0

5.1.2 Matched-filter demodulator
Instead of using a bank of N correlators to generate the variables, {rk } , we may use a bank
of N linear filters.
Suppose that the impulse responses of the N linear filters are
 f (T  t ), 0  t  T
hk (t )   k
otherwise
0
(5.1-14)
where { f k (t )} are the N basis functions.
The outputs of these filters are
t
t
0
0
yk  t    r   hk  t    d   r   f k T  t    d , k  1,2,..., N
(5.1-15)
Sampling the outputs of these filters at t  T , we obtain
T
yk (T )   r( ) f k ( )d  rk , k  1,2,..., N
(5.1-16)
0
The result at t  T is the same as that obtained from the linear correlators. However, this
is not to conclude both type demodulators are all the same any time. Indeed, it emphasizes
that the equality holds only at the time instant t  T . The figure below illustrates the two
demodulators’ behaviors. (Whalen, A. D., Detection of signals in noise, p. 170)
A matched filter to the signal s (t ) is that its impulse response is h(t )  s(T  t ) , where
s (t ) is assumed to be confined to the time interval 0  t  T . An example is shown below.
The response of h(t )  s(T  t ) to the signal s (t ) is
t
y (t )   s( ) s(T  t   )d
(5.1-17)
0
It is the time-autocorrelation function of the signal s (t ) , and the illustration is shown as
below.
Note: the autocorrelation function is an even function of t, which attains a peak at t  T .
Figure 5.1-7 demonstrates the matched filter demodulator that generates the observed
variables {rk } .
Property of the matched filter
If a signal s (t ) is corrupted by AWGN, the filter with an impulse response matched to s (t )
maximizes the output signal-to-noise ratio (SNR).
Suppose n(t ) is AWGN of zero-mean and PSD  nn ( f )  12 N 0 W/Hz.
The filter response to the signal and noise components is
t
t
t
y (t )   r( )h(t   )d   s( )h(t   )d   n( )h(t   )d
0
0
0
At the sampling instant t  T , the signal and noise components are
(5.1-18)
T
T
0
0
y (T )   s( )h(T   )d   n( )h(T   )d  ys (T )  yn (T )
(5.1-19)
The problem is to select the filter impulse response that maximizes the output signal-to-noise
(SNR0) defined as
ys 2 (T )
(5.1-20)
SNR 0 
E  yn 2 (T ) 
E  yn 2 (T )  is the variance of the noise term at the output of the filter, and can be evaluated
as
T T
E  yn (T )     E  n ( )n(t )  h(T   )h(T  t )dtd
2
0 0
T T

1
N 0  (t   )h(T   )h(T  t )dtd
2 0 0
(5.1-21)
T
1
 N 0  h 2 (T  t )dt
2 0
Note that the variance depends on the PSD of noise and the energy in the impulse response
h(t ) .
By substituting for ys (T ) and E  yn 2 (T )  , the output SNR as
2
T

T

s
(

)
h
(
T


)
d



  h( ) s(T   )d 
  0

SNR 0   0
T
T
1
1
N 0 h 2 (T  t )dt
N 0 h 2 (T  t )dt
2 0
2 0
2
(5.1-22)
Since the denominator depends on the energy in h(t ) , the maximum output SNR over h(t )
is obtained by maximizing the numerator subject to the constraint that the denominator is
held constant.
By use of the Cauchy-Schwarz inequality, which states if g1 (t ) and g2 (t ) are
finite-energy signals, then
2




2
2
(5.1-23)
  g1  t  g2  2  dt    g1 (t )dt  g 2 (t )dt

 
 
with equality when g1 (t )  Cg2 (t ) for arbitrary constant C.
Therefore, the SNR is maximized when h(t )  Cs(T  t ) (we may arbitrarily choose C=1),
i.e., h(t ) is matched to the signal s (t ) . The output (Maximum) SNR is then
T
SNR 0 
2
2E
s 2 (t )dt 

N0 0
N0
(5.1-24)
It is worthwhile to note that the output SNR depends only on the energy of s (t ) but not on
the detailed characteristics of s (t ) .
Frequency-domain interpretation of the matched filter
Because h(t )  s(T  t ) , the Fourier transform of this relationship is
T
T

H ( f )   s(T  t )e  j 2 ft dt    s( )e j 2 f  d  e  j 2 fT  S * ( f )e 2 fT
0
0

(5.1-25)
Note:
(1). The matched filter has a frequency response that is the complex conjugate of the
transmitted signal spectrum multiplied by the phase factor e  j 2 fT , representing the
sampling delay of T.
(2). | H ( f ) || S ( f ) | , the magnitude response of the matched filter is identical to the
transmitted signal spectrum. In addition, the phase of H ( f ) is the negative of the
phase of S ( f ) .
Henceforth, the filter output has a spectrum Y ( f ) | S ( f ) |2 e j 2 fT . The output waveform is

y s (t ) 
 Y ( f )e
j 2 ft

df 

 | S( f ) |
2
e  j 2 fT e j 2 ft df
(5.1-26)

By sampling the output of the matched filter at t  T , we have

ys (T ) 
T

S ( f ) df   s 2 (t )dt  E
2

(5.1-27)
0
The noise at the output of the matched filter has a PSD
1
2
H ( f ) N0
2
The total noise power at the output of the matched filter is
0 ( f ) 


(5.1-28)

1
1
1
2
2
Pn    0 ( f )df  N 0  H ( f ) df  N 0  S ( f ) df  EN 0
2 
2 
2

The signal power at the output of the matched filter is
Ps  ys 2 (T )
The output SNR is then
P
E2
2E
SNR 0  s 

Pn 1 EN
N0
0
2
which agrees with the result given by (5.1-24).
(5.1-29)
(5.1-30)
(5.1-31)
Example 5.1-2
(a). Signals: M=4 biorthogonal signals constructed from the two orthogonal signals. The
signals are shown in Figure 5.1-8(a).
1
(b). Channel: AWGN of zero-mean and PSD N 0 .
2
Find
(i). The basis functions. (ii). The impulse response of the matched filter demodulators. (iii).
The output waveforms of the matched filter demodulators when the transmitted signal is
s1 (t ) .
Solution:
Apparently, N=2, only two basis functions are needed.

 2/T
f1 ( t )  
 0

1 

0  t  T 
2 

 otherwise 
(5.1-32)

 2/T
f 2 (t )  
 0

1

 t T
2

 otherwise 
The impulse responses of the two matched filters are

 2/T
h1 (t )  f1 (T  t )  
 0

1

 T t T
2

 otherwise 
(5.1-33)

 2/T
h2 (t )  f 2 (T  t )  
 0

and is shown in Figure 5.1-8(b).
1 

0  t  T 
2 

 otherwise 
If s1 (t ) is transmitted, the noise-free responses of the two matched filters are shown in
Figure 5.1-8(c).
A2T  E and y2 s (T )  0 . Accordingly, the
received vector from the output of matched filters at the sampling instant t  T is
(5.1-34)
r   r1 r2    E  n1 n2 
where n1  y1n (T ) and n2  y2 n (T ) are the noise components at the outputs of the
matched filters, given by
It is easy to see at t  T , y1s (T ) 
1
2
T
ykn (T )   n(t ) f k (t )dt, k  1,2
(5.1-35)
0
It is easy to show the means of the noise components are zero, i.e., E ( nk )  E[ ykn 2 (T )] .
Their variance is
T T
 n 2  E  ykn 2 (T )     E  n(t )n( )  f k (t ) f k ( ) dtd
0 0
T T

1
N 0   t    f k ( ) f k (t ) dtd
2 0 0
T
1
 N0 
2 0
f
2
k
(5.1-36)
1
(t ) dt  N 0
2
The SNR0 for the first matched filter is
SNR 0
 E

1
N0
2
2

2E
N0
(5.1-37)
which agrees with the previous result. Note that the four possible outputs corresponding to
the four possible transmitted signals are
(r1, r2 )  ( E  n1, n2 ), (n1, E  n2 ), (  E  n1, n2 ), (n1,  E  n2 )
5.1.3 The optimum detector
The optimal detector is to make an optimum decision rule based on the observation vector, r
at the output of the demodulator.
Three kinds of detectors will be discussed in the sequel. There are symbol-by-symbol
maximum likelihood detector for signals without memory, maximum likelihood sequence
detector, and symbol-by-symbol MAP detector, both for signals with memory.
In this section, we design the symbol-by-symbol maximum likelihood (ML) detector to
make the probability of correct decision maximum.
Define the posterior probabilities as
P( signal sm was transmitted|r)  P( sm | r), m  1,2,..., M
The decision criterion is based on selecting the signal corresponding to the maximum of the
set of posterior probabilities {P( sm | r )} , hence, such a criterion is called as maximum a
posteriori probability (MAP) criterion.
The posterior probabilities can be expressed by
P (r | s m ) P ( s m )
(5.1-38)
P (r )
where P(r | sm ) is the conditional PDF of the observed vector given s m , and P(sm ) is the
P (s m | r ) 
a priori probability of the mth signal being transmitted.
The denominator can be written as
M
P (r )   P(r | s m )P(s m )
(5.1-39)
m 1
Therefore, to compute the posterior probabilities P( sm | r ) requires knowledge of the a
priori probabilities P(sm ) and the conditional PDFs P(r | sm ) for m  1,2,..., M .
We observe that the denominator in (5.1-38) is independent of which signal is transmitted.
Furthermore, suppose the M signals are equally likely, that is, the a priori probability is
P(sm )  1/ M for all M. Consequently, to maximize P(sm | r ) is equivalent to maximizing
P (r | s m ) .
The conditional PDF P(r | sm ) or any monotonic function of it is usually called the
likelihood function. The decision criterion based on the maximum of the likelihood function,
P(r | sm ) over the M signals is called the maximum-likelihood (ML) criterion.
P(r | sm ) is shown in (5.1-12). To simplify the computation, we take natural logarithm both
sides as
1
1 N
(5.1-40)
ln p(r | sm )   N ln( N 0 ) 
( rk  smk )2

2
N 0 k 1
Apparently, the maximum of ln p(r | sm ) over s m is equivalent to finding the signal s m
that minimizes the Euclidean distance
N
D (r, sm )    rk  smk 
2
(5.1-41)
k 1
D(r, sm ), m  1,2, , M is called the distance metrics.
As a result, this decision rule is called as minimum distance detection.
(5.1-41) can be expanded as
N
N
N
n 1
n 1
D(r, sm )   r n  2 rn smn   s 2 mn || r ||2 2r  sm  || sm ||2 , m  1, 2,..., M
n 1
2
(5.1-42)
The term || r ||2 can be ignored in the computation of the metrics since it is common to all
distance metrics. The modified distance metrics are
D '(r, sm )  2r  sm  || sm ||2
(5.1-43)
To minimize D '(r, sm ) is equivalent to maximizing C (r, sm ) , as below
C (r, sm )  2r  sm  || sm ||2
(5.1-44)
Notes:
(1). The term r  sm represents the projection of the received signal vector onto each of the
M possible transmitted signals vectors. The projection is a measure of the correlation
between the received vector and the mth signal, therefore, C (r, sm ), m  1,2,..., M is
called the correlation metrics for deciding which of the M signals was transmitted.
(2). The term || sm ||2  Em , m  1, 2,..., M may be viewed as bias for signal sets that have
unequal energies. Accordingly, for signals with same energy, this term can be ignored.
The correlation metrics C (r, sm ), m  1,2,..., M can be expressed as
T
C (r, sm )  2 r(t ) sm (t )dt  Em , m  1,..., M
(5.1-45)
0
Therefore, (5.1-45) can be generated by a demodulator that cross-correlates the received
signal r (t ) with each of the M possible transmitted signals with the individual energy.
Consequently, the optimum receiver (demodulator and detector) can be implemented as
Figure 5.1-9.
As a result, the MAP criterion can be simplified to the ML criterion when all signals have
same energy. On the other hand, if the energies of signals are unequal, the MAP criterion
should be adopted, and the corresponding metrics are
PM (r, sm )  p(r | sm ) P(sm )
Example 5.1-3. Consider binary PAM signals with two possible points s1   s2  Eb ,
where Eb is the energy per bit. The prior probabilities are P( s1 )  p and P( s2 )  1  p
Find the metrics for the optimum MAP detector in AWGN environment.
Solution:
The received signal vector is
r   Eb  yn (T )
(5.1-46)
where yn (T ) is a zero-mean Gaussian random variable with variance  n 2  12 N 0 .
The conditional PDFs p( r | sm ) for the two signals are
p( r | s1 ) 



 
 r E
b
1
exp  
2

2 n
2 n

 r E
n
p( r | s2 ) 
exp  
2

2 n
2 n

1
Then the metrics PM r, s1  and PM  r, s2 
2




(5.1-47)
2


(5.1-48)

 r E
b
PM (r, s1 )  pp( r | s1 ) 
exp  
2

2 n
2 n

 r E 2
b
1 p

PM (r, s2 ) 
exp  
2


2 n
2 n


p

 
2



(5.1-49)
(5.1-50)
The decision rule can be written as
PM (r, s1 ) s1
1
PM (r, s2 ) s2
But

(5.1-51)
 
 r E 2 r E
b
b
PM (r, s1 )
p

exp 
2

PM (r, s2 ) 1  p
2 n

Therefore, (5.1-51) can be expressed by
r 
  r 
2
Eb
Eb

2
2 n2
 
s1 1  p
 ln
p
s
2


(5.1-52)
(5.1-53)
2
Or equivalently,
s1
Eb r


1 2 1 p 1
1 p
 n ln
 N 0 ln
 h
2
p
4
p
(5.1-54)
s2
Note:
(1). For ML criterion, the probability of a correct decision given that sm (t ) was transmitted
is
P( c | s m ) 
 p(r | s
m
)dr
(5.1-55)
Rm
where Rm denotes the region in the N-dimensional space in which the signal sm (t ) is
thought to be transmitted when the vector r  [r1 r2 rN ] is received.
The average probability of a correct decision is
M
M
1
1
P( c)   P( c | sm )    p(r | sm )dr
Rm
m 1 M
m 1 M
(5.1-56)
Hint: P ( c ) is maximized by selecting the signal s m if p(r | sm ) is larger than p(r | sk )
for all m  k . That is, by choosing the signal s m having the largest p(r | sm ) ,
m  1,2,..., M .
(2). For MAP criterion (when the M signals are not equally probable), the average
probability of a correct decision is
M
P(c)    p(sm | r ) p(r )dr
m 1
Rm
5.1.4 The maximum-likelihood sequence detector
In this and the next sections, we will design the optimum detectors for signals with memory.
When the transmitted signal has memory, i.e., the signals transmitted in successive symbol
intervals are interdependent, the optimum detector is a detector that bases its decisions on
observation of a sequence of received signals over successive signal intervals.
To develop the maximum-likelihood sequence detection algorithm, we take the NRZI signal
in Section 4.3.2 as an example. Its memory is characterized by the trellis shown in Figure
4.3-14 and is drawn here again for convenience.
(i). The signal transmitted in each signal interval is binary PAM.
(ii). Two possible transmitted signal points are s1   s2  Eb , Eb is the energy per bit.
(iii). The output of the matched filter or correlator demodulator for binary PAM in the kth
signal interval is
rk   Eb  nk
(5.1-57)
where nk is a zero-mean Gaussian random variable with variance  n 2  12 N 0 .
Consequently, the conditional PDFs for the two possible transmitted signals are




 r  E
k
b
p ( rk | s1 ) 
exp  
2

2 n
2 n

1
 r  E
k
b
1
p ( rk | s2 ) 
exp  
2

2 n
2 n

2
2




(5.1-58)




Suppose now we observe the sequence of matched-filter outputs r1, r2 ,..., rK .
Since
(1). The channel noise is assumed to be white and Gaussian, and
f (t  iT ) and
f (t  jT ) for i  j are orthogonal, it
(2).
E (nk n j )  0, k  j .
follows
that
The noise sequence n1 , n2 ,..., nK is also white.
Consequently, for any given transmitted sequence s ( m ) , the joint PDF of r1, r2 ,..., rK is
K
p( r1 , r2 ,..., rk | s  )   p( rk | sk  )
m
m
k 1
K

k 1

 r  sm
k
k
1
exp  

2 n2
2 n

 
2

K
 K r  sm
 1 
k
k


 exp   
2 n2
k 1
 2 n 

where sk m  is
(5.1-59)


 
2


Eb or  Eb . Then, given the received sequence r1, r2 ,..., rK at the output
of the matched-filter or correlation demodulator, the detector determines the sequence
s( m )  {s1( m ) , s2( m ) ,..., sK ( m ) } that maximizes the conditional PDF p(r1, r2 ,..., rk | sm ) . Such a
detector is called the maximum-likelihood (ML) sequence detector.
By taking the logarithm of (5.1-59) and neglecting the terms that are independent of
( r1 , r2 ,..., rK ) , the ML detector is equivalent to minimizing the Euclidean distance metric,
K

D (r, s m  )   rk  sk m 
k 1

2
(5.1-60)
In searching through the trellis for the sequence that minimizes the Euclidean distance
D (r, s  ) , it may appear that we must compute the distance D (r, s  ) for every possible
m
m
sequence. Intuitively, if the number of outputs of the demodulator is K, the total number of
sequences is 2K .
Fortunately, by using the Viterbi algorithm to eliminate sequences as new data is received
from the demodulator, the number of sequences to be searched can be tremendously reduced.
The Viterbi algorithm is a sequential trellis search algorithm for performing ML sequence
detection.
Assume that the search process begins initially at state S 0 . The corresponding trellis is
shown in Figure 5.1-11.

D 1,1   r 
D  0,1   r 
D 1, 0    r 
  r   
   r   
   r   
   r   
D  0, 0, 0   D  0, 0    r   
D  0,1,1  D  0,1   r   
D  0, 0,1  D  0, 0    r   
D  0,1, 0   D  0,1   r   
P  s   A | r , r
,..., r 
p  r ,..., r | s    A  P  s    A 
| r ,..., r  
p r , r
,..., r 
D0  0, 0   r1   b
2
2
b
2
0
1
b
1
1
b
1
2
2
2
b
2
b
2
2
2
1
(5.1-61)
2
b
2
b
3
b
(5.1-62)
2
0
0
2
0
1
3
(5.1-63)
b
2
1
0
3
b
2
1
1
3
b
k
m

P s
k D
k  D 1
1
k
k 
s
 Am
~ k 
k D
k D

1
m
1

k  D 1
 
1
 arg max
p rk  D ,..., r1 | s    Am P s    Am
k 
s
k
(5.1-65)
k
m
k D
(5.1-64)
k

(5.1-66)
(5.1-67)



 
s ~1  arg max
p r1 D ,..., r1 | s 1  Am P s 1  Am
1

 arg max

1
 s s1D 
s
 p r
,..., r1 | s 

 arg max

1
 s s1 D 
 p  s
1 D
2
s 
1 D 
 
,..., s   P s 
1
1 D 

,..., s1 


(5.1-68)
2
1 
,..., s   , s   


p  s  ,..., s  , s     p  r ,..., r | s   ,..., s    P  s   ,..., s   
p  r ,..., r | s   ,..., s   

 p  r | s   ,..., s 
 p  r | s  ,..., s  ...
p  r | s  , s   p  r | s  
s    arg max p  r ,..., r | s    A  P  s    A 
1 D
2
1 D 
1
2
s 
1 D
1
1 D
1
1 D
1
1 D
1 D
1 D
1
1 D  L
DL
D
D
2
1
1
~ 2
2

 arg max

 2
 s s  2 D 
2 D
 p r
2 D
3
s 

 p r2 D | s
m
,..., r1 | s

,..., s
2
1
p r2 D ,..., r1 | s
 2 D 
 2 D  L 
 2 D 
2 D 
,..., s
,..., s 
 p r
 2
 P s
 2 D 
1 D 
,..., s
 

p r1 D ,..., r1 | s 
1 D 
  p r1 D ,..., r1 | s 
1 D 

  p s
1
s 
,..., s 

2
 
,..., s   P s 
1
1 D 
1 D 
,..., s  
1
 2
 2

1


 p r2 D | s 
2 D 
,..., s 



(5.1-73)

 P  s    p  s
1 D 
2 D
1
s 

(5.1-74)

2 D  L 
(5.1-71)
(5.1-72)

 2 D 
 3  2  
s ~ 2  arg max
p
s
,...,
s
,s 


2
 2
 s s2D s3

 2 D 
 3  2 
p2 s
,..., s , s





,..., s   , s  
2
,..., s

2
,..., r1 | s
1 D

m
p r1 D ,..., r1 | s 1 D  ,..., s  2 P s 1 D  ,..., s  2

(5.1-70)
1
2
2
s 
(5.1-69)
1
1 D
1
s 
1
1
1
,..., s   , s  
2
1

(5.1-75)
  

s ~ k   arg max
p rk  D ,..., r1 | s  k  P s  k 
k 
s

 arg max

k 
 s s k  D 

pk s 

 p rk  D | s 
k  D
,..., s 
 p s
k 1
s 
k  D
k  DL
k D
k
,..., s 
k 1
,..., s
, s
k
 k 1
,s
k D
k 1


 P  s    p  s
s
k 
k 1
k 1 D 
(5.1-76)



,..., s 
k 1

(5.1-77)
5.2
r  s1  n   b  n
p  r | s1  
1
 r 
e
 N0
p  r | s2  
1
 r 
e
 N0
P  e | s1  
(5.2-1)
b

b

2
/ N0
2
(5.2-2)
/ N0
(5.2-3)
0
 p  r | s  dr
1


 r 
b
1

exp  


N0
 N 0 

0
1

2

1
2
 2 b / N 0


2

dr


e  x / 2 dx
2



e  x / 2 dx
2
2 b / N0
 2 b
 Q 
 N0



(5.2-4)
Pb 
1
1
P  e | s1   P  e | s2 
2
2
 2 b
 Q 
 N0



(5.2-5)
2 

d
12 

Pb  Q
 2 N0 


s1    b 0
(5.2-6)
(5.2-7)
s 2  0
 b 
r    b  n1 n2 
(5.2-8)
P  e | s1   P C  r, s 2   C  r, s1   P n2  n1   b 


P n2  n1   b 
1
2 N 0

e
b
 x 2 / 2 N0
dx
(5.2-9)
1

2


e  x / 2 dx
2
b / N0
  
 Q  b 
 N0 

Pb  Q 

(5.2-10)
b 
Q
N 0 

b

(5.2-11)
M
C  r, s m   r  s m   rk smk ,m  1, 2,..., M
(5.2-12)
k 1
r    s  n1 n2
n3

C  r, s1    s
nM 
 s  n1
(5.2-13)

C  r, s 2    s n2
(5.2-14)
C  r, s M    s nM


 x   2
1
s
1

pr1  x1  
exp  


N0
 N0


2
1
prm  xm  
e xm / N0 , m  2,3 , M
 N0

Pc 
 P n
2
(5.2-15)
(5.2-16)
, nM  r1 | r1  p  r1  dr1
 r1 , n3  r1 ,
(5.2-17)

P  nm  r1 | r1  
r1
 p  x dx

rm
m
m
, m  2,3,
,M
1

2
r1 2 / N 0

e x / 2 dx
2

(5.2-18)
 1
pC   
 2



1
PM 
2
  1
 1   2
 

r1

x /2

e
dx



PM  1  Pc
2 / N0
2

 x2 / 2
e
dx




y
M 1
M 1
p  r1  dr1

 1
2 s
 exp    y 

N0

 2


PM
P
 kM
M 1 2 1
k
k P
2k  1
n  k M  k k
PM

2 1
n 1  n  2  1
(5.2-19)
(5.2-20)



2

 dy


(5.2-21)
(5.2-22)
(5.2-23)
P
2k  1
PM  M , k 1
k
2 1
2
PM   M  1 P2   M  1 Q  s / N 0  MQ
Pb 

Q  / N   e
s
(5.2-24)


 s / N0

 s / 2 N0
(5.2-25)
(5.2-26)
0
PM  Me  s / 2 N0  2k e  kb / 2 N0
b
N0
e
 k b / N0 2ln 2 / 2
 2 ln 2  1.39 1.42dB 
PM  2e
b
N0
(5.2-27)
k

 b / N 0  ln 2

(5.2-28)
2
(5.2-29)
 ln 2  0.693  1.6dB 
r    s  n1 n2
(5.2-30)
nM / 2 
(5.2-31)
M /2
1
, M
2
C  r , sm   r  sm   rk smk , m  1, 2,
k 1
r1
1
P  nm  r1 | r1  0  
 N0
e
 x 2 / N0
 r1
1
dx 
2
r1 / N0 / 2

(5.2-32)
e x / 2 dx
2
 r1 / N0 / 2
(5.2-33)
 1
Pc   
 2
0

r1 / N0 / 2

 r1 / N0 / 2
e
 x2 / 2

dx 


M / 2 1
p  r1  dr1
M / 2 1
  2 s / N 0


2
1
 x2 / 2


Pc  
e
dx
e  / 2 d

 2    2 / N

 2 s / N0

s
0


M
10 log 1     10 log
dB
M 1
sm   sm1 sm2 smN  , m  1, 2, , M



PM   M  1 Pb   M  1 Q 


 
d   
e
min
2
2 N0
 d e 2 
min

 2k exp  
 4 N0 


1
sm 
 g Am , m  1, 2, , M
2
Am   2m 1  M  d , m  1, 2, , M





(5.2-34)
(5.2-35)
(5.2-36)
(5.2-37)
(5.3-38)
1
 av 
M
M
d 2 g
m 1
2M
 m 
M
  2m  1  M 
2
m 1
(5.2-39)
d g 1
 1
2

M
M

1
  M 2  1 d 2 g




2M  3
 6
d 2 g
 av 1
2
Pav 
  M  1
T
6
T
1
r  sm  n 
 g Am  n
2
M 1 
1 
pM 
P  r  sm  d
g 
M
2 

2
(5.2-40)
(5.2-41)
M 1 2

M
 N0

M 1 2
M
2


e x
/ N0
dx
d g / 2


e  x / 2 dx
2
d 2 g / N 0
2  M  1  d 2 g

Q
 N0
M

(5.2-42)
6
d 2 g  2
PavT
M 1

2  M  1 
6 PavT

PM 
Q
  M 2  1 N 0 
M



2  M  1 
6 av

PM 
Q
  M 2  1 N 0 
M


2  M  1   6 log 2 M   bav 

PM 
Q
  M 2  1 N 0 
M


2

s m  t   g  t  cos  2 f ct 
 m  1 ,1  m  M , 0  t  T
M


2




(5.2-43)
(5.2-44)
(5.2-45)
(5.2-46)
(5.2-47)
2
2

s M    S cos
 m  1  S sin  m  1
M
M


C  r, sm   r  sm , m  1, 2, , M
(5.2-48)
(5.2-49)
 r  tan 1
r2
r1
(5.2-50)
s1    s
0
(5.2-51)
r1   s  n1
(5.2-52)
r2  n2


 r   2  r2 
1
s
2 
pr  r1 , r2  
exp  
2
2


2 r
2 r


1
(5.2-53)
V  r12  r22
 r  tan 1
pV ,r V ,   
(5.2-54)
r2
r1
 V 2   s  2  s V cos r
exp


2 r2
2 r2

V





pr   r    pV ,r V ,  r  dV
0

1  s sin 2 r
V 

e
Ve

2
0
 /M

PM  1 
2 s cos  r

(5.2-55)
2
/2
dV
p   r  d  r
(5.2-56)
 / M
 2 b
P2  Q 
 N0
Pc  1  P2 
2



(5.2-57)
2

 2 b
 1  Q 
 N0


 
 
  1  2 b
 1  Q 
  2  N 0

 
 
(5.2-58)
P4  1  Pc
 2 b
 2Q 
 N0
s
cos r e

pr  r  
 /M
PM  1 


 /M

2



s
cos  r e 

s sin
2
s sin
r
2
r
(5.2-59)
(5.2-60)
d r
e  u du
2
(5.2-61)
2 s sin  / M
 
 


 2Q  2 s sin   2Q  2k b sin 
M
M


課本此式第二行之積分下限多了ㄧ個右括號請老師定奪
1
Pb  PM
k
rk    s cos  k     nk1
 s sin  k     nk 2 
rk   s e
j  k  
 nk
(5.2-62)
(5.2-63)
rk 1   s e
 k 1  
rk rk*1   s e
j  k  k 1 
 s e
j  k   *
k 1
 nk 1
 s e
n
(5.2-64)
 j  k 1  
rk rk*1   s   s  nk  nk*1   nk nk*1
nk  nk nk*1
x   s  Re  nk  nk*1 
(5.2-66)
(5.2-67)
y  Im  nk  nk*1 
y
x
 r  tan 1
(5.2-65)
(5.2-68)
Re  rk rk*1  
1
rk rk*1  rk*rk 1 

2
1
Pb  e b / N0
2
1
 1

Pb  Q1  a, b   I 0  ab  exp    a 2  b 2  
2
 2


1
a  2 b  1 

2 

sm  t   Amc g  t  cos 2 fct  Ams g t  sin 2 fct ,0  t  T

1
1 
s m   Amc
 g Ams  g 
2
2 

1
Pav   4  2 A2  2 A2
4
1
Pav   2  3 A2   2 A2   2 A2
2
1 M
2
2
Pav 
Amc
 Ams



M m 1
a
M
2
mc
m 1

Pc  1  P M



PM  1  1  2Q 




 4Q 



(5.2-76)



2
3 av
 M  1 N 0
3k bav 

 M  1 N 0 
(5.2-75)
(5.2-77)
1  
3  av

P M  2 1 
 Q  M  1 N
M  

0

(5.2-74)
2
PM  1  1  P M
(5.2-72)
(5.2-73)

a
2
ms
(5.2-70)
(5.2-71)

1
b  2 b  1 

2 

A2

M
(5.2-69)
(5.2-78)
(5.2-79)




2
(5.2-80)


e  2
PM   M  1 Q   d min
/ 2 N0 



 

PM  2Q  2 s sin 
M

3 /  M  1
M 
2sin 2  / M 
W
R
log 2 M
R
 log 2 M
W
R
 2 log 2 M
W
M
M
M
W


R
2T 2  k / R  2log 2 M
s t  
2
cos 2 f ct    t;  
T
(5.2-81)
(5.2-82)
(5.2-83)
(5.2-84)
(5.2-85)
(5.2-86)
(5.3-1)
5.3
s t  
2
cos 2 f ct    t;  
T
r t   s t   n t 
(5.3-1)
(5.3-2)
n  t   n  t  cos 2 f t  n  t  sin 2 f t
c
c
s
c
n
  t ; I   2 h  I k q  t  kT 
k  
nL
n
  h  I k  2 h
 Ik q  t  kT 
k  
k  n  L 1
  n    t ; I  , nT  t   n  1 T
t
q  t    g   d
0
  m 2 m
  0,
,
,
s  p
p
 m
  0,
,
s  p
  t; I   2 h
n 1

,
,
 p  1 m 
p
p




I k q  t  kT   2 hIn q  t  nT 
, I nL1
 pM L 1  even m 
N 
s 2 pM L 1  odd m 

Sn1  n1 , In , In-1 , , In-L2 
 n1   n   hIn- L1
1
1
3


  0,   ,   ,   ,  
s 
4
2
4

1
1
1
1
 0,1 ,  0, 1 ,  ,1 ,  , 1 ,   ,1 ,   , 1 ,   ,1 ,   , 1 ,
4
 4
 2
 2

3
 3
  1
  1
  1
  1

  ,1 ,   , 1 ,    ,1 ,    , 1 ,    ,1 ,    , 1 ,
4
 4
  4
  4
  2
  2

 3
  3

   ,1 ,    , 1
 4
  4

 n1   n   hIn-1
1
3
      
4
4
(5.3-4)
(5.3-5)
 2 p  1 m  
k  n  L 1
Sn  n , I n1 , I n2 ,
(5.3-3)
(5.3-6)
(5.3-7)
(5.3-8)
(5.3-9)
(5.3-10)
CM
I 
n
 n 1T
 r  t  cos ct    t; I dt

 n 1T
 CM
n 1
I  
nT
n  I; n  
(5.3-11)
r  t  cos  t    t ; I    dt
n
 c
 n 1T
 r  t  cos ct    t; I   
nT
d2 
ij
NT


0
NT

0
(5.3-12)
2
s 2  t  dt 
i
 2 N 
2
T
 dt

 s t  s t  dt
 i   j   
 2 N  2

n
2
T
2
T
NT

0
NT
s 2  t  dt  2  s  t  s  t dt
j
i
j
0
 cos ct    t; Ii  cos ct    t; I j dt

NT


(5.3-13)
0
 cos   t; Ii     t; I j dt

NT


0
 1  cos   t; Ii     t; I j  dt
NT


 

0
d 2  2  2
ij
b ij
log 2 M
2 
ij
T
2 
ij
(5.3-14)
 1  cos   t; Ii     t; I j  dt
NT



 
(5.3-15)
0
log 2 M
T
NT
 1  cos   t; dt
(5.3-16)
0

 b 2 

PM  K
Q



min 
min  No


2
2

 lim min 
min N  i, j ij
 log 2 M NT

1  cos   t ; Ii  I j   dt 
 lim min 



N 
i, j  T
0

I  1, 1, I 2 , I3
j
I  1, 1, I 2 , I3
j
  2, 2,0,0, 
 sin 2 h 
d 2  h   2 1 
, M  2
B
2 h 


 sin 2k h  
d 2  h   min  2log 2 M  1 

1

k

M

1
B
2k h  


(5.3-17)
(5.3-18)
(5.3-19)
(5.3-20)
(5.3-21)
(5.3-22)
 1 
2 t 

1  cos
  0  t  LT 
2 LT 
g  t    2 LT 

0  otherwise 

j 2 f t 

c 
s  t   Re   t  e


n 1


   h t   n  1 T  In

  t   exp  j  
  h Ik  0  
T
k 0

 
 

p  r1 , r2 , , r1 D | I1, I2 , , I1 D 
p  r1 , r2 ,
, r1 D | I1, I2 ,
, I1 D  P  I1, I2 ,
, I1 D 
(5.3-23)
j 2 f t
s  t   Re  g  t  e c 


j 2 f c  t t0 

s  t  t0   Re  g  t  t0  e

 j 2 f c t0 j 2 f c t

 Re  g  t  t0  e
e
  2 f ct0
sm  t   Re  slm  t  e j 2 fct  , m  1, 2, 0  t  T
T
1T
2
   s 2  t  dt   s  t  dt
m
2 lm
0
(5.4-1)
(5.4-2)
0
T
12   

1
sl*1  t  s12  t  dt

2 0
(5.4-3)

n  t   Re nc  t   jns  t  e j 2 fct 
 Re  z  t  e j 2 fct 

r  t   Re  slm  t  e j  z  t  e j 2 fct
rl  t   sm  t  e j  z  t  ,0  t  T
(5.4-4)

h  t   sl* T  t 
T
 s t 
l
2
dt  2
(5.4-5)
(5.4-6)
(5.4-7)
(5.4-8)
0
rm  rmc  jrms , m  1, 2
(5.4-9)
r2  2  cos    0   n2c  j 2  sin    0   n2 s 
p  r | sm  p sm 
P  sm | r  
, m  1, 2
p r
(5.4-10)
r1  2 cos   n1c  j  2 sin   n1s 
(5.4-11)
s1
p  s1 | r 
p  r | s1 
p  r | s2 


p s2 | r 
s2
s1
p s2 

p  s1 

(5.4-12)
s2
 r 
p  r | sm  
p  r | s1 
p  r | s2 
2
 p r | s
0
m
,   p   d 
(5.4-13)
(5.4-14)
r1  r1c  jr1s
 2 cos   n1c  j  2 sin   n1s 
r2  r2 c  jr2 s
(5.4-15)
 n2 c  jn2 s
  r1c  2 cos  2   r1  2 sin  2 
p  r1c , r1s | s1 ,   
exp  

2 2
2 2


1
p  r2 c , r2 s  
1
2
 r r 
exp  

2
2 2 

2
2c
1
2
2
 p r
1c
, r1s | s1 ,   d
0
 2  r1c cos   r1s sin   
d
0 exp 
2

2
 2 r 2  r 2 
 2  r1c cos   r1s sin   
1
1c
1s


exp
d


I


0
2 
2
2


2 0






 r22c  r22s  4 2   2 r12c  r12s 
1

p  r2c , r2 s | s 2  
exp

 I 0 

2 2
2 2
2

 

s1
2
 r  r  4 2  1

exp


2 2 0
2 2

 2
1
2
1c
2
1c

 r 
I  2
2

/ 
I 0 2 r12c  r12s /  2
0
(5.4-16)
2
2s
r22c  r22s


2
s2
p  s2 
p  s1 
(5.4-17)
(5.4-18)
(5.4-19)
(5.4-20)
s1
r12c  r12s


r22c  r22s
(5.4-21)
s2
s1  t   2 b / Tb cos 2 f1t , 0  t  Tb
s2  t   2 b / Tb cos 2 f 2t , 0  t  Tb
sl1  t   2 / Tb , 0  t  Tb
sl 2  t   2 / Tb e j 2
r t  
f1m  t  
ft
, 0  t  Tb
2 b
cos  2 f mt  m   n  t 
Tb
2
cos  2 f1  2 m f  t  , m  0,1
Tb
2
f2m  t  
sin  2 f1  2 m f  t  , m  0,1
Tb
(5.4-22)
(5.4-23)
(5.4-24)
(5.4-25)
 sin  2  k  m  fT 
rkc    
cos m
 2  k  m  fT


cos  2  k  m  fT   1
sin m   nkc , k , m  1, 2
2  k  m  fT

 cos  2  k  m  fT   1
rks   
cos m
2  k  m  fT


(5.4-26)

sin  2  k  m  fT 
sin m   nkc , k , m  1, 2
2  k  m  fT

rmc   b cos m  nmc
(5.4-27)
rms   b sin m  nms
rkc  nkc , rks  nks , k  m
sm  t   Re  slm  t  e
j 2 f c t
 , m  1, 2,
(5.4-28)
, M ,0  t  T
(5.4-29)
T
*
rm  rmc  jrms   rl  t  slm
 t  dt , m  1, 2,
,M
(5.4-29)
,M
(5.4-30)
0
T
*
rm  rmc  jrms   rl  t  slm
 t  dt ,m  1, 2,
0
rm  rmc2  rms2 , m  1, 2,
,M
(5.4-31)
rm  rmc2  rms2 , m  1, 2,
,M
(5.4-32)
r1c   s cos 1  n1c
(5.4-33)
r1s   s sin 1  n1s
rmc  nmc , rms  nms , m  2,3,
,M
2
2 

 r12c  r12s   s    s  r1c  r1s  
pr1   r1c , r1s  
exp  
 I0 

2 2
2 2
2

 


2
2
 r r 
1
prm  rmc , rms  
exp   mc 2 ms  , m  2,3, , M
2
2
2 

1
Rm 
(5.4-34)
(5.4-35)
(5.4-36)
rmc2  rms2

r
 m  tan ms
rmc
(5.4-37)
1
J
 cos m
 sin m
  2 Rm
 Rm sin m  Rm cos m
p  R1 , 1  
 1
    2 s 
R1
exp    R12  2 s   I 0 
R1 
2
2
N
N
0
0



 

(5.4-38)
(5.4-39)
Rm
 1 
exp   Rm2  , m  2,3, , M
2
 2 
Pc  p  R2  R1 , R3  R1 , , RM  R1 
p  Rm , m  

  P  R2  R1 , R3  R1 ,
, RM  R1 | R1  x  pR1  x  dx
(5.4-40)
(5.4-41)
0

Pc    P  R2  R1 | R1  x  
M 1
pR1  x  dx
(5.4-42)
0
x
P  R2  R1 | R1  x    pR2  r2  dr2
0
 1  e x
2
/2
M 1
n  M  1  nx2 / 2
   1 
e
n 0
 n 
M 1
 n s 
n  M  1 1
Pc    1 
exp 


n 0
 n  n 1
  n  1 N 0 

1  e x
M 1
2
/2
PM    1

(5.4-43)
M 1
n 1
n 1

 M  1 1
nk b 
exp  



 n  n 1
  n  1 N 0 
1  b / 2 N0
e
2
2k 1
Pb  k
PM
2 1
R log 2 M

W
M
2
 Rm
 Rm   m2    m Rm 
exp


 I0 
  Rm  0 
4

N
2

N
p  Rm    2 N 0
0
0

 


0  Rm  0 

P2 
(5.4-44)
(5.4-45)
(5.4-46)
(5.4-47)
(5.4-48)
(5.4-49)
(5.4-50)

Pb  P  R2  R1     p  x1 , x2  d x1 d x2
(5.4-51)
0 x1
Pb  P  R2  R1   P  R22  R12   P  R22  R12  0 
(5.4-52)
1  a 2  b 2  / 2
Pb  Q1  a, b   e
I 0  ab 
2
(5.4-53)
a
1
2N 

1
2N 
b
1 
2
1 
2
0
b
b
0



  1
Pb  Q1  0, 0   e b / 2 N0
N0  2

(5.4-54)
(5.4-55)

 
Q1  0, b   e b / 2 N0
N0 

r t    s t   n t 

Pb  Q 


Pb  KQ 


Pb  Q 




2 b 

N 0 
2 b 

KN 0 
 2 b
10 5  100Q 
 N0
(5.5-1)
2 b
N0
 2 b 
10 7  Q 

 N0 
 2 b
105  Q 
 100 N 0
PG A
PR  T T 2 R
4 d
GR  2 2
AR 
m
4
PG G
PR  T T R 2
 4 d /  
(5.5-2)
(5.5-3)






  
Ls  

 4 d 
PR  PT GT GR Ls La
1
AR   D 2
4
(5.5-4)
(5.5-5)
(5.5-6)
2
 D 
GR   

  
10 A
GR  2
(5.5-7)
(5.5-8)
(5.5-9)
2
(5.5-10)
(5.5-11)


 B  70   / D 
(5.5-12)
 PR dB   PT dB  GT dB  GR dB   Ls dB   La dB
(5.5-13)
Ls  195.6dB
 PR dB  20  17  39 196.6
 119.6dBW
PR  1.11012 W
N0  kBT0 W/Hz
 b Tb PR 1 PR


N0
N0
R N0
(5.5-14)
(5.5-15)
 
PR
 R b 
N0
 N 0  req
(5.5-16)
PR  1.11012 W(119.6dBW)
N 0  4.11021 W/Hz
PR
 119.6  203.9  84.3dBHz
N0
RdB  84.3  10
 74.3dB
P 
 
RdB   R 
  b   M dB
 N 0  dB Hz  N 0 req
  PT dBW   GT dB   GR dB
 
  La dB   Ls dB   b   M dB
 N 0 req
(with respect to 1 bit/s)
(5.5-17)