Post on 13-Mar-2018
Lecture 3
Lecture 3 2
Example of M-ary PAM
-B
B
T
‘01’
3B
T
T
-3B
T
‘00’ ‘10’
‘1’
A.
T
‘0’
T
-A.
Assuming real time transmission and equal energy per
transmission data bit for binary-PAM and 4-ary PAM:
• 4-ary: T=2Tb and Binary: T=T
b
•
4-ary PAM
(rectangular pulse)
Binary PAM
(rectangular pulse)
‘11’
22 10BA
Lecture 3 3
Example of M-aryPAM…
0 Tb 2T
b 3T
b 4T
b 5T
b 6T
b
0 Ts 2T
s
0 T 2T 3T
2.2762 V 1.3657 V
1 1 0 1 0 1
0 T 2T 3T 4T 5T 6T
Rb=1/T
b=3/T
s
R=1/T=1/Tb=3/T
s
Rb=1/T
b=3/T
s
R=1/T=1/2Tb=3/2T
s=1.5/T
s
Lecture 3 4
Today we are going to talk about:
Receiver structure
Demodulation (and sampling)
Detection
First step for designing the receiver
Matched filter receiver
Correlator receiver
Lecture 3 5
Demodulation and detection
Major sources of errors:
Thermal noise (AWGN) disturbs the signal in an additive fashion (Additive)
has flat spectral density for all frequencies of interest (White)
is modeled by Gaussian random process (Gaussian Noise)
Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter, channel and receiver, symbolsare“smeared”.
Format Pulse
modulate
Bandpass
modulate
Format Detect Demod.
& sample
)(tsi)(tgiim
im̂ )(tr)(Tz
channel )(thc
)(tn
transmitted symbol
estimated symbol
Mi ,,1
M-ary modulation
Lecture 3 6
Example: Impact of the channel
Lecture 3 7
Example:Channelimpact…
)75.0(5.0)()( Tttthc
Lecture 3 8
Figure : Three examples of filtering an ideal pulse. (a) Example 1: Good-fidelity output. (b) Example 2: Good-recognition output. (c) Example3: Poor-recognition output.
Lecture 3 9
Receiver tasks
Demodulation and sampling:
Waveform recovery and preparing the received
signal for detection:
Improving the signal power to the noise power (SNR)
using matched filter
Reducing ISI using equalizer
Sampling the recovered waveform
Detection:
Estimate the transmitted symbol based on the
received sample
Lecture 3 10
Receiver structure
Frequency
down-conversion
Receiving
filter
Equalizing
filter
Threshold
comparison
For bandpass signals Compensation for
channel induced ISI
Baseband pulse
(possibly distored) Sample
(test statistic) Baseband pulse
Received waveform
Step 1 – waveform to sample transformation Step 2 – decision making
)(tr)(Tz
im̂
Demodulate & Sample Detect
Lecture 3 11
Baseband and bandpass
Bandpass model of detection process is
equivalent to baseband model because:
The received bandpass waveform is first
transformed to a baseband waveform.
Equivalence theorem:
Performing bandpass linear signal processing followed by
heterodyning the signal to the baseband, yields the same
results as heterodyning the bandpass signal to the
baseband , followed by a baseband linear signal
processing.
12
Detection of Binary Signal in Gaussian Noise
For any binary channel, the transmitted signal over a symbol interval (0,T) is:
The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse response of the channel hc(t), is
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:
1
2
( ) 0 1( )
( ) 0 0i
s t t T for a binarys t
s t t T for a binary
2,1)()(*)()( itnthtstr ci
Ttitntstr i 02,1)()()(
13
Detection of Binary Signal in Gaussian Noise
The recovery of signal at the receiver consist of two parts
Filter
Reduces the effect of noise (as well as Tx induced ISI)
The output of the filter is sampled at t=T. This reduces the received signal to a single variable z(T) called the test statistics
Detector (or decision circuit)
Compares the z(T) to some threshold level 0 , i.e.,
where H1 and H2 are the two possible binary hypothesis
1
2
0( )
H
H
z T
Lecture 3 14
Model the received signal
Simplify the model:
Received signal in AWGN
)(thc)(tsi
)(tn
)(tr
)(tn
)(tr)(tsi
Ideal channels
)()( tthc
AWGN
AWGN
)()()()( tnthtstr ci
)()()( tntstr i
Receiver Functionality
15
Receiver Functionality
The recovery of signal at the receiver consist of two parts:
1. Waveform-to-sample transformation (Blue Block)
Demodulator followed by a sampler
At the end of each symbol duration T, predetection point yields a sample z(T ), called test statistic
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
0( ) ( ) ( ) 1,2iz T a T n T i
2
0
0
0
02
1exp
2
1)(
nnp
16
Then output is another Gaussian random process
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:
Need to achieve maximum (S/N)T
2
11
00
1 1( | ) exp
22
z ap z s
2
22
00
1 1( | ) exp
22
z ap z s
2
0
2
i
T
a
N
S
Receiver Functionality
17
The Matched Filter
Objective: To maximizes (S/N)T
Expressing signal ai(t) at filter output in terms of filter transfer function H(f ) (Inverse Fourier transform of the product H(f)S(f)).
where S(f) is the Fourier transform of input signal s(t)
Output noise power can be expressed as:
Expressing (S/N)T as:
dfefSfHta ftj
i
2)()()(
dffH
N 202
0 |)(|2
dffHN
dfefSfH
N
SfTj
T 20
22
|)(|2
)()(
18
Now according to Schwarz’sInequality:
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:
Substituting to yield:
dffSdffHdfefSfH fTj222
2 )()()()(
dxxfdxxfdxxfxf2
2
2
1
2
21 )()()()(
dffSNN
S
T
2
0
)(2
The Matched Filter
19
Or and energy E of the input signal s(t):
Thus (S/N)T depends on input signal energy
and power spectral density of noise and
NOT on the particular shape of the waveform
Equality for holds for optimum filter transfer function H0(f)
such that:
For real valued s(t):
0
2max
N
E
N
S
T
dffSE2
)(
0
2max
N
E
N
S
T
fTjefkSfHfH 2
0 )(*)()(
fTjefkSth 21 )(*)(
( ) 0( )
0
ks T t t Th t
else where
The Matched Filter
20
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
Defined as:
a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform
( ) 0( )
0
ks T t t Th t
else where
The Matched Filter
21
Correlation realization of Matched filter
A filter that is matched to the waveform s(t), has an impulse response
h(t) is a delayed version of the mirror image of the original signal waveform
( ) 0( )
0
ks T t t Th t
else where
Signal Waveform Mirror image of signal waveform
Impulse response of matched filter
Lecture 3 22
Example of matched filter
T t T t T t 0 2T
)()()( thtsty opti 2A)(tsi )(thopt
T t T t T t 0 2T
)()()( thtsty opti 2A)(tsi )(thopt
T/2 3T/2 T/2 T T/2
2
2A
T
A
T
A
T
A
T
A
T
A
T
A
Lecture 3 23
Properties of the matched filter
The Fourier transform of a matched filter output with the matched signal as input is, except for a time delay factor, proportional to the ESD of the input signal.
The output signal of a matched filter is proportional to a shifted version of the autocorrelation function of the input signal to which the filter is matched.
The output SNR of a matched filter depends only on the ratio of the signal energy to the PSD of the white noise at the filter input.
Two matching conditions in the matched-filtering operation:
spectral phase matching that gives the desired output peak at time T.
spectral amplitude matching that gives optimum SNR to the peak value.
)2exp(|)(|)( 2 fTjfSfZ
sss ERTzTtRtz )0()()()(
2/max
0N
E
N
S s
T
24
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
Substituting h(t) to yield:
When t =T,
dthrthtrtzt
)()()(*)()(0
dtTsr
dtTsrtz
t
t
0
0
)(
)()()(
dsrtzT
)()()(0
Correlation realization of Matched filter
25
The function of the correlator and matched filter are the same
Compare (a) and (b)
From (a)
T
dttstrtz0
)()()(
T
TtdsrTztz
0)()()()(
Correlation realization of Matched filter
26
From (b)
But
At the sampling instant t = T, we have
This is the same result obtained in (a)
Hence
0'( ) ( )* ( ) ( ) ( ) ( ) ( )
t
z t r t h t r h t d r h t d
)()]([)()()( tTstTsthtTsth
t
dtTsrtz0
)()()('
0 0'( ) '( ) ( ) ( ) ( ) ( )
T T
t Tz t z T r s T T d r s d
)(')( TzTz
T
dsrTz0
)()()('
Correlation realization of Matched filter
Lecture 3 27
Implementation of matched filter receiver
Mz
z
1
z)(tr
)(1 Tz)(
*
1 tTs
)(*
tTsM )(TzM
z
Bank of M matched filters
Matched filter output:
Observation
vector
)()( tTstrz ii Mi ,...,1
),...,,())(),...,(),(( 2121 MM zzzTzTzTz z
Lecture 3 28
Implementation of correlator receiver
dttstrz i
T
i )()(0
T
0
)(1 ts
T
0
)(ts M
Mz
z
1
z)(tr
)(1 Tz
)(TzM
z
Bank of M correlators
Correlators output:
Observation
vector
),...,,())(),...,(),(( 2121 MM zzzTzTzTz z
Mi ,...,1
Lecture 3 29
Implementation example of matched filter
receivers
2
1
z
z
z)(tr
)(1 Tz
)(2 Tz
z
Bank of 2 matched filters
T t
)(1 ts
T t
)(2 tsT
T 0
0
T
A
T
A
T
A
T
A
0
0
30
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
1
2
0( )
H
H
z T
31
P[s1], P[s2] a priori probabilities
These probabilities are known before transmission
P[z]
probability of the received sample
p(z|s1), p(z|s2)
conditional pdf of received signal z, conditioned on the class si
P[s1|z], P[s2|z] a posteriori probabilities
After examining the sample, we make a refinement of our previous knowledge
P[s1|s2], P[s2|s1]
wrong decision (error)
P[s1|s1], P[s2|s2]
correct decision
Probabilities Review
32
Maximum Likelihood Ratio test and Maximum a posteriori (MAP) criterion:
If
else
Problem is that a posteriori probabilities are not known.
Solution:UseBay’stheorem:
1 2 1( | ) ( | )p s z p s z H
1 1
2 2
1 1 2 21 1 2 2
)( | ) ( ) ( | ) (( | ) ( ) ( | ) ( )
( ) ( )
H H
H H
p z s P s p z s P sp z s P s p z s P s
P z P z
How to Choose the threshold?
2 1 2( | ) ( | )p s z p s z H
( | ) ( )( | )
( )
p z s P si ip s z
i p z
33
1
2
1 2
2 1
( )( | ) ( )
( )( | ) ( )
H
H
likelihood ratio test LRTp z s P s
L zp z s P s
When the two signals, s1(t) and s2(t), are equally likely, i.e., P(s2) = P(s1) = 0.5, then the decision rule becomes
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
1
2
1
2
max( | )
( ) 1( | )
H
H
likelihood ratio testp z s
L zp z s
MAP criterion
34
Substituting the pdfs
2
11 1
00
1 1: ( | ) exp
22
z aH p z s
2
22 2
00
1 1: ( | ) exp
22
z aH p z s
2
11 1
001
22
2
2 200
1 1exp
22( | )( ) 1 1
( | ) 1 1exp
22
z aH H
p z sL z
p z s z aH H
MAP criterion
35
Taking the log of both sides will give
1
2 2
1 2 1 2
2 2
0 0
2
( ) ( )ln{ ( )} 0
2
H
z a a a aL z
H
1
2 2
1 2 1 2 1 2 1 2
2 2 2
0 0 0
2
( ) ( ) ( )( )
2 2
H
z a a a a a a a a
H
1
2 2
1 2 1 2
2 2
0 0
2
( ) ( )exp 1
2
H
z a a a a
H
Hence:
MAP criterion
36
Hence
where z is the minimum error criterion and 0 is optimum threshold
For antipodal signal, s1(t) = - s2 (t) a1 = - a2
1
2
0 1 2 1 2
2
0 1 2
2
( )( )
2 ( )
H
a a a az
a a
H
1
1 20
2
( )
2
H
a az
H
1
2
0
H
z
H
MAP criterion
37
This means that if received signal was positive, s1 (t) was sent, else s2(t) was sent
MAP criterion
38
Detection of Binary Signal in Gaussian Noise
The output of the filtered sampled at T is a Gaussian random process
39
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 and is given by:
( ) 0( )
0
ks T t t Th t
else where
Matched Filter and Correlation
40
Hence
where z is the minimum error criterion and 0 is optimum threshold
For antipodal signal, s1(t) = - s2 (t) a1 = - a2
1
1 20
2
( )
2
H
a az
H
1
2
0
H
z
H
Bay’s Decision Criterion and Maximum Likelihood
Detector
41
Probability of Error
Error will occur if
s1 is sent s2 is received
s2 is sent s1 is received
The total probability of error is the sum of the errors
0
2 1 1
1 1
( | ) ( | )
( | ) ( | )
P H s P e s
P e s p z s dz
0
1 2 2
2 2
( | ) ( | )
( | ) ( | )
P H s P e s
P e s p z s dz
2
1 1 2 2
1
2 1 1 1 2 2
( , ) ( | ) ( ) ( | ) ( )
( | ) ( ) ( | ) ( )
B i
i
P P e s P e s P s P e s P s
P H s P s P H s P s
42
If signals are equally probable
Numerically, PB is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s2) and is given by:
2 1 1 1 2 2
2 1 1 2
( | ) ( ) ( | ) ( )
1( | ) ( | )
2
BP P H s P s P H s P s
P H s P H s
2 1 1 2 1 2
1( | ) ( | ) ( | )
2
by Symmetry
BP P H s P H s P H s
0 0
0
1 2 2
2
2
00
( | ) ( | )
1 1exp
22
BP P H s dz p z s dz
z adz
Probability of Error
43
The above equation cannot be evaluated in closed form (Q-function)
Hence,
0
1 2
0
2
2
00
2
0
2
( )
2
1 1exp
22
( )
1exp
22
B
a a
z aP dz
z au
udu
1 2
0
.182
B
a aP Q equation B
21( ) exp
22
zQ z
z
Probability of Error
44
Table for computing of Q-Functions
Probability of Error
45
A vector View of Signals and Noise
N-dimensional orthonormal space characterized by N linearly independent basis function {ψj(t)}, where:
From a geometric point of view, each ψj(t) is mutually perpendicular to each of the other {ψj(t)} for j not equal to k.
ji
jidttt
T
ji if 0
if 1 )()(
0
46
Representation of any set of M energy signals { si(t) } as a linear combinations of N orthogonal basis functions where N M.
where:
N
j
jijiMi
Tttats
1 ,...,2,1
0)()(
Nj
Midtttsa j
T
iij,...,2,1
,...,2,1)()(
0
A vector View of Signals and Noise
47
Therefore we can represent set of M energy signals {si(t) } as:
Representing (M=3) signals, with (N=2) orthonormal basis functions
Waveform energy:
N
j
ij
T N
j
jij
T
ii tattadttsE1
2
01
2
0
2 )( )]()([ )(
Miaaas iNiii ,...,2,1).......,,( 21
A vector View of Signals and Noise
48
Question 1: Why use orthormal functions?
In many situations N is much smaller than M. Requiring few matched filters at the receiver.
Easy to calculate Euclidean distances
Compact representation for both baseband and passband systems.
Gram-Schmidt orthogonalization procedure.
Question 2: How to calculate orthormal
functions?
A vector View of Signals and Noise
49
Examples
A vector View of Signals and Noise
50
Examples (continued)
A vector View of Signals and Noise
51
Generalized One Dimensional Signals
One Dimensional Signal Constellation
52
Binary Baseband Orthogonal Signals
Binary Antipodal Signals
Binary orthogonal Signals
Is a method of representing the symbol states of modulated bandpass signals in terms of their amplitude and phase
In other words, it is a geometric representation of signals
There are three types of binary signals:
Antipodal
Two signals are said to be antipodal if one signal is the negative of the other
The signal have equal energy with signal point on the real line
ON-OFF
Are one dimensional signals either ON or OFF with signaling points falling
on the real line 53
Constellation Diagram
)()( 01 tsts
54
With OOK, there are just 2 symbol states to map onto the constellation space
a(t) = 0 (no carrier amplitude, giving a point at the origin)
a(t) = A cos wct (giving a point on the positive horizontal axis at a distance A from the origin)
Orthogonal
Requires a two dimensional geometric representation since there are two linearly independent functions s1(t) and s0(t)
Constellation Diagram
55
Typically, the horizontal axis is taken as a reference for symbols that are Inphase with the carrier cos wct, and the vertical axis represents the Quadrature carrier component, sin wct
Error Probability of Binary Signals
Recall:
Where we have replaced a2 by a0.
18.2 0
01 Bequationaa
QPB
Error Probability of Binary Signal
56
To minimize PB, we need to maximize:
or
We have
Therefore,
02
2
01 )(
aa
0
01
aa
2
1 0
2
0 0 0
( ) 2
/ 2
d da a E E
N N
2
1 0 1 0
2
0 0 0 0
( ) 21 1
2 2 2 2
d da a a a E E
N N
Error Probability of Binary Signal
57
TTT
T
d
tstsdttsdtts
dttstsE
001
2
00
2
01
2
001
)()(2)()(
)()(
02N
EQP d
B
The probability of bit error is given by:
Error Probability of Binary Signal
58
The probability of bit error for antipodal signals:
The probability of bit error for orthogonal signals:
The probability of bit error for unipolar signals:
0
2
N
EQP b
B
02N
EQP b
B
0N
EQP b
B
Error Probability of Binary Signal
59
Bipolar signals require a factor of 2 increase in energy compared to orthogonal signals
Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than orthogonal
Error Probability of Binary Signal
60
Comparing BER Performance
For the same received signal to noise ratio, antipodal provides lower bit error rate than orthogonal
4
,
2
,
0
10x8.7
10x2.9
10/
antipodalB
orthogonalB
b
P
P
dBNEFor
61
Relation Between SNR (S/N) and Eb/N0
In analog communication the figure of merit used is the average signal power to average noise power ration or SNR.
In the previous few slides we have used the term Eb/N0 in the bit error calculations. How are the two related?
Eb can be written as STb and N0 is N/W. So we have:
Thus Eb/N0 can be thought of as normalized SNR.
Makes more sense when we have multi-level signaling.
0 /
b b
b
E ST S W
N N W N R