The document discusses the optimal receiver structure for digital communication systems over additive white Gaussian noise (AWGN) channels. It describes two types of receivers: 1) Correlation receiver which consists of a bank of correlators that correlate the received signal with locally generated reference signals, followed by a maximum likelihood detector. 2) Matched filter receiver which is optimal for AWGN channels. It describes deriving the matched filter by maximizing the output signal-to-noise ratio. The matched filter has an impulse response that is the time-reversed version of the transmitted signal.

1. Module 6: Bandpass System-II
Non – Coherent FSK

2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
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Amplitude
x=cos(20*pi*t)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
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0.5
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Time
Amplitude
y=cos(40*pi*t)
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200
Time
Amplitude
z=conv(x,y)
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Amplitude
z1=conv(x,x)
0 0.5 1 1.5 2
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Amplitude
z2=conv(y,y)

3. (a)

6. Receivers
Figure 3.1: Mathematical model of a digital communication system
The detector block shown in Figure 3.1 has mainly two receivers; they are Correlation
Receiver and Matched filter receiver.
Correlation Receiver:
We note that for an AWGN channel and for the case when the transmitted signals s1(t),
s2(t),…, sM(t) are equally likely, the optimum receiver consists of two sub systems, detailed in
Figure 3.8 and described next:
1. The detector part of the receiver is shown in Figure 3.8(a). It consists of a bank of M
product integrators or correlators supplied with a corresponding set of coherent
reference signals or orthonormal basis functions ϕ1(t), ϕ2(t),…, ϕN(t) that are generated
locally. This bank of correlators operate on the received signal x(t), 0 ≤ t ≤ T, to
produce the observation vector x.
2. The second part of the receiver, namely, the vector receiver is shown in Figure 3.8(b).
It is implemented in the form of a maximum likelihood detector that operates on the
observation vector x to produce an estimate m̂ of the transmitted symbol mi, i = 1,
2,…, M, in a way that would minimize the average probability of symbol error. The N
elements of the observation vector x are first multiplied by the corresponding N
elements of each of the M signal vectors s1, s2,…,sM, and the resulting products are
successively summed in accumulators to form the corresponding set of inner products
{(x, sk)}, k = 1, 2,…,M. Next, the inner products are corrected for the fact that the
transmitted signal energies may be unequal. Finally, the largest in the resulting set of
numbers is selected, and a corresponding decision on the transmitted message is
made.
Message
Source
Vector
Transmitter
Modulator
Channel
Detector
Vector
Receiver
Estimate
m̂
Noise
{ }
i
m { }
i
s
{ ( )}
i
s t
( )
x t
x
Transmitter
Receiver

7. (a)
(b)
Figure 3.8: (a) Detector (b) Vector Receiver
Matched Filter Receiver:
Since each of the orthonormal basis functions ϕ1(t), ϕ2(t),…, ϕN(t) is assumed to be zero
outside the interval 0 ≤ t ≤ T, the use of multipliers shown in Fig. 3.8(a) may be avoided. This
is desirable because analog multipliers are usually hard to build. Consider, for example, a
linear filter with impulse response hj(t). If the received signal x(t) is used as the filter input,
the resulting filter output, yj(t), is defined by the convolution integral:
( ) ( ) ( )
j j
y t x h t d
  


 
 (3.30)
Suppose we now set the impulse response
( ) ( )
j j
h t T t

  (3.31)
Then the resulting filter output is
1
x

1( )
t

( )
x t
2
x

2 ( )
t

N
x

( )
N t


0
T
dt

0
T
dt

0
T
dt

Observation
vector
x
1
x

1
s
x 


Accumulator
2
s
N
s
1
x
Accumulator
1
x
Accumulator

1
1
2
E
2
1
2
E
1
2
M
E


Select
largest
Estimate
m̂
1
( , )
x s
2
( , )
x s
( , )
M
x s







8. ( ) ( ) ( )
j j
y t x T t d
   


  
 (3.32)
Sampling this output at time t = T, we get
( ) ( ) ( )
j j
y T x d
   


  (3.33)
and since ϕj(t) is zero outside the interval 0 ≤ t ≤ T, we finally get
0
( ) ( ) ( )
T
j j
y T x d
   
  (3.34)
We note that yj(T) = xj, where xj is the jth
correlators output produced by the received signal
x(t) in Fig. 3.8 (a). Thus the detector part of the optimum receiver may also be implemented
as in Fig. 3.9.
Figure 3.9: Detector part of matched filter receiver; the vector receiver part is as shown in Fig. 3.8
A filter whose impulse response is a time-reversed and delayed version of some signal ϕj(t),
as in Eq. (3.31), is said to be matched to ϕj(t). Correspondingly, the optimum receiver based
on the detector of Fig. (3.9) is referred to as the matched filter receiver.
Maximization of Output Signal to noise Ratio:
Figure 3.10: Illustrating the condition for derivation of the matched filter
The output signal to noise ration is used as the optimality criterion for deriving the matched
filter. Consider then a linear filter of impulse response h(t), with an input that consists of a
1
x
( )
x t 2
x
N
x

1( )
T t
 
Observation
vector
x
2 ( )
T t
 
( )
N T t
 
Sample
at =
t T
Matched
filters
Known
Signal ( )
t


+
+
Sample at
=
t T
white Gaussian
noise ( )
w t
Impulse
Response ( )
h t
( )
y t

9. known signal, ϕ(t), and an additive noise conponent, w(t), as shown in Fig. 3.10. We thus
write
x(t) = ϕ(t) + w(t), 0 ≤ t ≤ T (3.35)
where T is the observation instant. In particular, we may choose ϕ(t) to be one of the
orthonormal basis functions. The w(t) is the sample function of a white Gaussian noise
process of zero mean and power spectral density N0/2. Since the filter is linear, the resulting
output y(t), may be expressed as
y(t) = ϕo(t) + n(t) (3.35)
where ϕo(t) and n(t) are produced by the signal and noise components of the input x(t),
respectively. Asimple way of describing the requirement that the output signal component
ϕo(t) be considerably grater than the output noise component n(t) is to have the filter make the
instantaneous powe in the output signal ϕo(t), measured at time t = T, as large as possible
compared with the average power of the output noise n(t). This is equivalent to maximizing
the out signal to noise ratio defined as
2
2
( )
( )
( )
o
o
T
SNR
E n t


 
 
(3.36)
We now show that this maximization occurs when the filter matches with the known signal
ϕ(t) at the input.
Let Φ(f) denote the Fourier transform of the known signal ϕ(t), and H(f) denote the
transfer function of the filter. Then the Fourier transform of the output signal ϕo(t) is equal to
H(f) Φ(f), and ϕo(t) is itself given by the inverse Fourier tranform
( ) ( ) ( )exp( 2 )
o t H f f j ft df
 


 
 (3.37)
Hence, when the filter output is sampled at time t = T, we may write
2
2
( ) ( ) ( )exp( 2 )
o T H f f j fT df
 


 
 (3.38)
Consider next the effect of the noise w(t) alone on the filter output. The power spectral
density SN(f) of the output noise n(t) is equal to the power spectral density of the input noise
w(t) times the squared magnitude of the transfer function H(f). Since w(t) is drawn rom a
process that is white with constant power spectral density N0/2, it follows that
2
0
( ) ( )
2
N
N
S f H f
 (3.39)
The Average power of the output noise n(t) is therefore

10. 2
( ) ( )
N
E n t S f df


  
  
2
0
( )
2
N
H f df


  (3.40)
Thus, substituting Eq. (3.40) and Eq. (3.38) into Eq. (3.36), we may rewrite the expression
for the output signal to noise ratio as
2
2
0
( ) ( )exp( 2 )
( )
( )
2
o
H f f j fT df
SNR
N
H f df









(3.41)
Our problem is to find, while holding the Fourier transform Φ(f) of the input signal fixed, the
form of the transfer function H(f) of the filter that make (SNR)o a maximum. To find the
solution to this optimization problem, we apply a mathematical result known as Schwartz’s
inequality to the numerator of Eq. (3.41).
According to Schwartz’s inequality, it suffices to say that if we have two complex
functions ϕ1(x) and ϕ2(x) in the real variable x, satisfying the conditions
2
1( )
x dx



 

and
2
2 ( )
x dx



 

then we may write
2
2 2
1 2 1 2
( ) ( ) ( ) ( )
x x dx x dx x dx
   
  
  

  
This equality holds if, and only if, we have
*
1 2
( ) ( )
x k x
 

where k is an arbitrary constant, and asterisk denotes complex conjugate.
Returning to the problem, we readily see that by invoking Schwartz’s inequality, and setting
ϕ1(x) = H(f) and ϕ2(x) = Φ(f) exp(j2πfT), the numerator in Eq. (3.41) may be rewritten as
2
2 2
( ) ( )exp( 2 ) ( ) ( )
H f f j fT df H f df f df

  
  
  
   (3.42)
Using this relation in Eq. (3.41), we may simply the output signal to noise ratio as

11. 2
0
2
( ) ( )
o
SNR f df
N


 
 (3.43)
The right side of Eq. (3.43) is uniquely defined by two quantities:
1. The signal energy given by (in accordance with Rayleigh’s energy theorem)
2 2
( ) ( )
t dt f dt

 
 
 
 
2. The noise power spectral density N0/2
As such, the right side of Eq. (3.43) does not depend on the transfer function H(f).
Consequently, the output signal to noise ratio will be a maximum when H(f) is chosen so that
the equality holds; that is
2
,max
0
2
( ) ( )
o
SNR f df
N


 
 (3.44)
For this condition, H(f) assumes its optimum value denoted as Hopt(f). From Schwartz’s
inequality, we also find that, except for a scaling factor, the optimum value of this transfer
function is defined by
*
( ) ( )exp( 2 )
opt
H f f j fT

   (3.45)
where Φ*
(f) is the complex conjugate of the Fourier transform of the input signal ϕ(t). This
relation states that, except for the necessary time delay factor exp(–j2πfT), the transfer
function of the optimum filter is the same as the complex conjugate of the spectrum of the
input signal.
Eq. (3.45) specifies the matched filter in the frequency domain. To characterize it in
time domain, we take the inverse Fourier transform of Hopt(f) in Eq. (3.45) to obtain the
impulse response of the matched filter as
 
*
( ) ( )exp 2 ( )
opt
h t f j f T t df



   

Since for a real valued signal ϕ(t) we have Φ*
(f) = Φ(–f), we may also write
 
( ) ( )exp 2 ( ) ( )
opt
h t f j f T t df T t
 


      
 (3.46)
Eq. (3.46) shows that the impulse response of the optimum filter is a time reversed and
delayed version of the input signal ϕ(t); that is, it is matched to the input signal. Note that the
only assumptions we have made about the input noise w(t) are that it is additive, stationary,
and white with zero mean and power spectral density N0/2.

12. Problem: