Transcript Elements of a Digital Communication System

Elements of a Digital Communication System
• Block diagram of a communication system:
Information
source and
input
transducer
Source
encoder
Channel
encoder
Digital
modulator
Channel
Output
transducer
Source
decoder
Channel
decoder
Digital
demodulator
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Mathematical Models for Communication
Channels
• Additive Noise Channel:
– In presence of attenuation:
r (t )   (t )  s(t )  n(t )
channel
S(t)
+
r(t) = s(t) + n(t)
n(t)
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Mathematical Models for Communication
Channels
• The Linear filter channel:

r (t )  s(t ) * c(t )  n(t )   c( )s(t   )d  n(t )

s(t)
+
Linear filter c(t)
r(t) = s(t)*c(t)+n(t)
channel
n(t)
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Mathematical Models for Communication
Channels
• Linear Time-Variant Filter Channel:
– Are charachterized by a time-variant channel impulse response
c( ; t )
s(t)
r(t)
+
Linear time_variant filter
channel
n(t)
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Representation of Band-Pass Signals and
Systems
• Representation of Band-Pass Signals:
s(t )  x(t ) cos(2    f c  t )  y(t ) sin(2    f c  t )
Energy of the signal:

1
2
   sl (t ) dt
2 
• Representation of Linear Band-Pass Systems:
h(t )  hl (t )e j2  fc t  hl (t )e  j2  fc t
• Response of a Band-Pass System to a Band-Pass Signal:

r(t )  Re rl (t )e j2  fc t

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Orthogonal Expansion of Signals
• We can express M orthonormal signals sn (t ) as a Linear
combination of basis functions  f n (t ) and hence can be defined
N
as
sk (t )   skn f n (t ), k  1,2,.., M
n 1
• Linear digitally modulated signals can be expanded in terms of two
orthonormal basis functions given by:
f1 (t ) 
2
cos 2 f c t
T
and
f 2 (t )  
2
sin 2 f c t
T
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Representation of Digitally Modulated Signals
• Pulse-amplitude-modulated Signals (PAM):

sm (t )  Re Am g (t )e j2  fc t

m=1,2,…M
• Phase-modulated signals (PSK):
s m (t )  g (t ) cos
2
2
(m  1) cos 2 f c t  g (t ) sin
(m  1) sin 2 f c t
M
M
• Quadrature amplitude modulation (QAM):

sm (t )  Re ( Amc  jAms ) g (t )e j 2 fct

m=1,2,..,M,
 Amc g (t ) cos2 f c t  Ams g (t ) sin 2 f c t
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Representation of Digitally Modulated Signals
• Orthogonal multidimensional signals:
Cm  cm1cm2 ...cmN 
• Biorthogonal signals:
• Simplex signals:
sm '  sm  s,
•
m=1, 2,…, M..
• Signal waveforms from binary codes:
Cm  cm1....cmN 
c mj  1  S mj (t ) 
cmj  0  S mj (t )  
2 c
cos 2 f c t
Tc
2 c
cos2 f ct
Tc
0  t  Tc
0  t  Tc
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Optimum Receivers Corrupted by additive White
Gaussian Noise- I
• General Receiver:
channel
Sm(t)
+
r(t) = sm(t) + n(t)
n(t)
r (t )  sm (t )  n(t ), (0  t  T )
Receiver is subdivided into:
– 1. Demodulator.
• (a) Correlation Demodulator.
• (b) Matched Filter Demodulator.
– 2. Detector.
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Optimum Receivers Corrupted by additive White
Gaussian Noise- II
• Correlation Demodulator:
– Decomposes the received signal and noise into a series of
linearly weighted orthonormal basis functions.
• Equations for correlation demodulator:
rk  0 r (t ) f k (t )dt  0 s m (t )  n(t ) f k (t )dt
T
T
s mk 
s
T
0
m
k  1,2,...N
(t ) f k (t )dt,
nkm  0n(t ) f k (t )dt,
T
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Optimum Receivers Corrupted by additive White
Gaussian Noise- III
• Matched Filter Demodulator:
– Equation of a matched filter:
hk (t )  f k (T  t ),
0t T
• Output of the matched filter is given by:
y k (t )  0r (t )hk (t   )d
T
T
  r (t ) f k (T  t   )d
k=1,2, …N
0
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Optimum Receivers Corrupted by additive White
Gaussian Noise- IV
• Optimum Detector:
– The optimum detector should make a decision on the transmitted
signal in each signal interval based on the observed vector.
• Optimum detector is defined by:
N
N
N
n 1
n 1
2
D(r , s m )   r  2 rn s mn   s mn
n 1
2
n
2
 r  2r  s m  s m ,
2
or
m=1,2,… M
D(r, sm )   2r  s m  s m
2
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OFDM
• It is a block modulation scheme where data symbols are transmitted
in parallel by employing a large number of orthogonal sub-carriers.
• Equation of complex envelope of the OFDM signal:

s (t )  A b(t  nT, xn )
n
• where
N 1 

2

(
k

)t 
N 1

2
b(t , xn )  ha (t ) xnk exp j

T
k 0




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General FFT based OFDM system-I
• Block diagram of FFT based OFDM transmitter :
X 
k
X0
X1
...
XN-2
XN-1
IFFT
X 
~
sl (t )
g
In
D/A
X 
n
X0
X1
...
XN-2
XN-1
Insert
Cyclic
prefix
X 
g
n
~
sQ (t )
X 
D/A
g
Qn
• Equations at the transmmitter end:
 j 2kt 
s(t )  A xk exp
u r (t )
NT
k 0
s 

N 1
 j 2kn 
X n  s(nTs )  A xk exp
,
N
k 0
s


N 1
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General FFT based OFDM system-II
•
Block diagram of FFT based
OFDM receiver:
R 
~
rl (t )
k ,n
A/D
~
rQ (t )
A/D
R 
Remove
cyclic
prefix
R0
R1
...
RN-2
RN-1
Qn
FFT
Z 
n
•
Z0
Z1
...
ZN-2
ZN-1
Serial metric
computer
 (vm )
At the demodulator:
1 N 1
Z i   Rn e
N n 0
 j 2ni
N
Z i  i Axi
L
i   g m e
m 0
j
2mi
N
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General FFT based OFDM system-II
• Merits of OFDM:
– 1. the modulation and the demodulation can be achieved in the
frequency-domain by using a DFT.
– 2. the effects of ISI can be eliminated with the introduction of the
guard interval.
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IMPLEMENTATION OF OFDM SYSTEM-I
• Basic implementation of OFDM system:
1 N 1
Z i   Rn e
N n 0
L
i   g m e
 j 2ni
N
j
2mi
N
m 0
Zi  i Axi
n0
X0
BPSK
.
.
.
BPSK
.
.
.
X1
.
.
.
BPSK
X127
n1
1
127
R0
+
.
.
.
.
.
.
Bits
Serial To
Parallel
Converter
0
.
.
.
n127
R1
Detector
+
.
.
.
.
.
.
.
.
.
.
.
.
R127
+
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SIMULATION RESULTS.
• Perfomance charachteristics were obtained for the simulated OFDM
system.
-1
10
Non Fading Channel
Fading Channel
-2
10
-3
10
-4
10
-5
10
-6
10
10
15
20
25
30
35
40
45
50
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Conclusion.
• 1. OFDM communication system exhibits better Pe Vs SNR curves
in case of Non-Fading channel as compared to the Fading channel.
• 2. As the value of the SNR is increased the value of Pe gradually
decreases.
• 3. Perfomance charachteristics of simulated OFDM communication
system are consistent with the performance charachteristics of the
general OFDM communication system.
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