dr. uri mahlabn. binary signal transmission binary data consisting of a sequence of 1 ’ s and 0...
Post on 14-Dec-2015
216 Views
Preview:
TRANSCRIPT
Dr. Uri Mahlabn
Dr. Uri Mahlabn
Dr. Uri Mahlabn
Binary Signal Transmission
Binary data consisting of a sequence of 1’s and 0’s.
• Tb - Bit time interval
Dr. Uri Mahlabn
+
Noise PSD
AWGN
AWGN - Channel
Dr. Uri Mahlabn
+ Receiver
•The receiver task is to decide whether a O or 1 was transmitter•The receiver is designed to minimize the error probability.•Such receiver is called the Optimum receiver.
Dr. Uri Mahlabn
Optimum Receiver for the AWGN Channel
s t0 ( )
s t1( )
( )dr0
t
( )dr0
t
r1
r0
r t( )
Dr. Uri Mahlabn
Signal Correlator
Output
data
r1
r0
Sampling @ t=Tb
Dr. Uri Mahlabn
detector Output data
s t0 ( )
s t1( )
( )dr0
t
( )dr0
t
r1
r0
r t( )
Dr. Uri Mahlabn
Example 5.1: suppose the signal waveforms s0(t) and s1(t) are the ones shown in figure 5.2, and let s0(t) be the transmitted signal. Then, the received signal is
Answerip_05_01MATLAB.lnk
0
A s t0 ( )
tbt
0
A s t1( )
tb t
A-
Figure 5.2: Signal waveforms s0(t) and s1(t) for a binary communication system
r t s t n t( ) ( ) ( ), 0 0 t Tb
Determine the correlator outputs at the sampling instants.
Dr. Uri Mahlabn
Figure 5.3 illustrates the two noise-free correlator outputs in the interval
for each of the two cases-I.e., when s0(t) is transmitted and when s1(t) is transmitted.
0 t Tb
0 tb
E
t
Output of correlator 0
0 tb t
Output of correlator 1
E
2tb2
0 tb t
Output of correlator 1
E
2tb2
0 tb
E
t
Output of correlator 0
)a( )b(
Figure 5.3:Noise-free correlator outputs.(a) s0(t) was transmitted.(b) s1(t) was transmitted.
Dr. Uri Mahlabn
0 rProbability density function p(r0|0) and p(r1|0)
when s0(t) is transmitted
Dr. Uri Mahlabn
Matched Filter•Provides an alternative to the signal correlator for demodulating the received signal r(t).•A filter that is matched to the signal waveform s(t) has an impulse response;
Dr. Uri Mahlabn
The matched filter output at the sampling instant t=Tb is identical to the output of the signal correlator.
Dr. Uri Mahlabn
Answerip_05_02MATLAB.lnk
Example 5.2: Consider the use of matched filters for the demodulation of the two signal waveforms shown in the figure and determine the outputs
0
A s t0 ( )
tbt
0
A s t1( )
tb t
A-
Dr. Uri Mahlabn
0
A h t s T tb0 0( ) ( )
tbt 0
A
A-
h t s T tb1 1( ) ( )
Tbt
Figure 5.5:Impulse responses of matched filters for signals s0(t) and s1(t).
0
y t0 ( )
t
A Tb2
2TbTb 0
y t0 ( )
t2TbTb
)a( )b(
Figure 5.6:Signal outputs of matched filters when s0(t) is transmitted
Dr. Uri Mahlabn
The DetectorThe detector observes the correlator or the matched filter outputr0 and r1 and decided on whether the transmitted signal waveformis s1(t) or s0(t), which corresponding to “1” or “0”, respectively.
The optimum detector is defined the detector that minimizes the probability of error.
Dr. Uri Mahlabn
Example 5.3: Let us consider the detector for the signals shown in Figure 5.2 which are equally probable and have equal energies. The optimum detector for these signals compares r0 and r1 and decides that a 0 was transmitted when r0>r1 and that a 1 was transmitted when r0>r1 . Determine the probability of error.
Answerip_05_03MATLAB.lnk
0
A s t0 ( )
tbt 0
As t1( )
tb t
A-
Dr. Uri Mahlabn
Monte Carlo Simulation Communication System
Monte Carlo computer simulations are usually performed in practice to estimate the probability of error of a digital communication system, especially in cases where the analysis of the detector performance is difficult to perform.
Dr. Uri Mahlabn
Example 5.4: use Monte Carlo simulation to estimate an plot Pe versus SNR for a binary communication system that employs correlators or matched filters. The model of the system is illustrated in figure 5.8.
Answerip_05_04MATLAB.lnk
Uniform random number generator
Binary data source
0 / E
1/ E
r0r1
detector
Output data
Compare
Error counter
Gaussian random number generator
n0
Gaussian random number generator
n1
Figure 5.8: Simulation model for Illustrative
Dr. Uri Mahlabn
Dr. Uri Mahlabn
Other Binary Signal Transmission Methods
Dr. Uri Mahlabn
Antipodal Signal for Binary Signal Transmission
Antipodal signal If one signal waveform is negative of the other.
Dr. Uri Mahlabn
0
A s t0 ( )
Tbt 0
A-
s t1( )
Tb t
(a) A pair of antipodal signal
0
As t0 ( )
Tb t
A-
0
As t0 ( )
Tbt
A-(b) Another pair of antipodal signal
Dr. Uri Mahlabn
Matched filter demodulator
Correlator demodulator
The received signal is
Dr. Uri Mahlabn
p r( )1 1
0
p r( )0 0
r
probability density function for the input to the detector
Dr. Uri Mahlabn
The DetectorThe detector observes the correlator or the matched filter outputr0 and r1 and decided on whether the transmitted signal waveformis s1(t) or s0(t), which corresponding to “1” or “0”, respectively.
The optimum detector is defined the detector that minimizes the probability of error.
For antipodal signal we have :
Dr. Uri Mahlabn
Answerip_05_05MATLAB.lnk
Example 5.5: use Monte Carlo simulation to estime and plot the error probability performance of binary communication system. The model of the system is illustrated in Figure 5.13.
Uniform random number generator
Binary data source
Compare
Error counter
detectorE
rn
Gaussian random number generator
Output
data
Figure 5.13: Model of binary communication system employing antipodal signal
Dr. Uri Mahlabn
On-Off Signal for Binary Signal Transmission
The received signal is:
Binary information sequence may also be transmitted by use of ON-OFF signals
Dr. Uri Mahlabn
p r( )0
0
p r( )1
r
Figure 5.15: The probability density function for the received signal at the output of te correlator for on-off signal.
2
Dr. Uri Mahlabn
0
p r( )0
r
Probability density function for ON-OFF signals
p r( )1
/ 2
Dr. Uri Mahlabn
The DetectorFor antipodal signal we have :
For On-OFF signal we have :
Dr. Uri Mahlabn
Answerip_05_06MATLAB.lnk
Example 5.6:use Monte Carlo simulation to estimate and plot the performance of a binary communication system employing on-off signaling
Uniform random number generator
Binary data source
Compare
Error counter
detectorE
rn
Gaussian random number generator
Output
data
Dr. Uri Mahlabn
E E0
(a)E0
(b)
E0
(b)
E
Figure 5.17: signal point constellation for binary signal.(a) Antipodal signal.(b) On-off signals.(c) Orthogonal signals.
Signal Constellation diagramsfor Binary Signals
Dr. Uri Mahlabn
Answerip_05_07MATLAB.lnk
Example 5.7: The effect of noise on the performance of a binary communication system can be observed from the received signal plus noise at the input to the detector. For example, let us consider binary orthogonal signals, for which the input to the detector consists of the pair of random variables (r0,r1), where either.
The noise random variables n0 and n1 re zero-mean, independent Gaussian random variables with variance .as in Illustrative Problam 5.4 use Monte Carlo simulation to generate 100 samples of (r0,r1) for each value of =0.1, =0.3, and =0.5, and plot these 100 samples for each on different two-dimensional plots. The energy E of the signal may by normalized to unity.
( , ) ( , )
( , ) ( , )
r r E n n
r r n E n
0 1 0 1
0 1 0 1
2
Dr. Uri Mahlabn
Receiver signal points at input to the selector for orthogonalsignals
Dr. Uri Mahlabn
Multiamplitude Signal transmission
Transmitting multiple bits per signal waveform
Symbol = several bits in a single waveform
Dr. Uri Mahlabn
t0T
s t0 ( )
3d
VT
0 tT
s t1( )
dVT
0 tT
s t2 ( )
d
VTt
0 T
s t3( )
d
VT
Figure 5.19: Multi amplitude signal waveforms.
-3d -d 0 d 3d
00 01 11 10
Signal Waveforms with Four Amplitude Levels
Dr. Uri Mahlabn
Optimum receiver for AWGN Channel
Signal correlator
Dr. Uri Mahlabn
The detector
Observes the correlator output r and decides whichof the four PAM signals was transmitted in the signal interval.The optimum amplitude detector computes the distances
The detector selects the amplitude correspondingto the smallest distance.
Dr. Uri Mahlabn
Example 5.8:Perform a Monte Carlo simulation of four - level PAM communicationsystem that employs a signal correlator, followed by an amplitude detector. The model for the system to be simulated is shown in Fig 5.2.
Answerip_05_08MATLAB.lnk
UniformRG
Gaussian randomNumber Generator
compare
Error counter
detectorMapping to Amplitude levels +
X Am r
( , )0 2 Am^
Figure 5.22: Block diagram of four level PAM for Monte Carlo Simulation
Example 5.8:
Dr. Uri Mahlabn
Signal Waveforms with Multiple Amplitude Levels
Dr. Uri Mahlabn
Answerip_05_09MATLAB.lnk
Example 5.9: perform a Monte Carlo simulation of a 16-level PAM digital communication system and measure its error-rate performance.
Dr. Uri Mahlabn
Multidimensional signals
Signal waveform having M=2k amplitude levels
We able to transmit k=log2(M) bits of informationper signal waveform.
Multidimensional Orthogonal signals
Dr. Uri Mahlabn
A
T
s t3( )
3
4
Tt
A
T
s t2 ( )
3
4
TT
2
t
T
4T
s t0 ( )
A
t
A
T
s t1( )
T
4
T
2
t
Dr. Uri Mahlabn
M=2
E s0
s1
E
M=3
E s0
s1
E
E
s3
Figure 5.27: Signal constellation for M=2 and M=3 orthogonal signals.
( , , , , , , , )
( , , , , , , , )
( , , , , , , )
( , , , , , , , )
E
E
E
E
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
Dr. Uri Mahlabn
detector
Optimum receiver for multidimensional orthogonal signals.
Dr. Uri Mahlabn
Detector algorithm:
Dr. Uri Mahlabn
Answerip_05_10MATLAB.lnk
Example 5.10: perform a Monte Carlo simulation of a digital communication system that employs M=4 orthogonal signals. The model of the system to be simulated is illustrated in Figure 5.30.
Gaussian RNG
Gaussian RNG
Gaussian RNG
Gaussian RNG
E n0 r0
n1 r1
n2 r2
0
0
0r3
n3
Compare si with
^si
Error counter
Mapping to signal points
Uniform RNG
detector
Output decision
si^
Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation
Dr. Uri Mahlabn
T
2
s t0 ( )
A
t
T
2
s t2 ( )
A-
tT
s t3( )
A
t
T
2
T
s t1( )
A
tT
2
Dr. Uri MahlabnAnswer
ip_05_11MATLAB.lnk
Example 5.11: perform a Monte Carlo simulation of a digital communication system that employs M=4 orthogonal signals. The model of the system to be simulated is illustrated in Figure 5.30.
Gaussian RNG
Gaussian RNG
E n0 r0
n1 r1
0
Compare si with
^si
Error conter
Mapping to signal points
Uniform RNG
detectorOutput
decision
si^
Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation
top related