dc digital communication part7
TRANSCRIPT
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Modulation, Demodulation and
Coding Course
Period 3 - 2005
Sorour Falahati
Lecture 3
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2005-01-26 Lecture 3 2
Last time we talked about:
Transforming the information source to aform compatible with a digital system
Sampling
Aliasing
Quantization
Uniform and non-uniform
Baseband modulation
Binary pulse modulation M-ary pulse modulation
M-PAM (M-ay Pulse amplitude modulation)
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2005-01-26 Lecture 3 3
Formatting and transmission of basebandsignal
Information (data) rate:
Symbol rate : For real time transmission:
Sampling at rate
(sampling time=Ts)
Quantizing each sampled
value to one of theL levels in quantizer.
Encoding each q. value tobits
(Data bit duration Tb=Ts/l)
Encode
PulsemodulateSample Quantize
Pulse waveforms(baseband signals)
Bit stream(Data bits)
Format
Digital info.
Textualinfo.
Analoginfo.
source
Mapping every data bits to a
symbol out of M symbols and transmittinga baseband waveform with duration T
ss Tf /1! Ll 2log!
Mm 2log!
[bits/sec]/1 bbR !ec][symbols/s/1 TR !
mRRb
!
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2005-01-26 Lecture 3 4
Qunatization example
t
Ts: samplingtime
x(
nTs): sampledvaluesxq(nTs): quantizedvalues
boundaries
Quant. levels
111 3.1867
110 2.2762
101 1.3657
100 0.4552
011 -0.4552
010 -1.3657
001 -2.2762
000 -3.1867
PCM
codeword 110 110 111 110 100 010 011 100 100 011 PCM sequence
amplitudex(t)
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2005-01-26 Lecture 3 5
Example of M-ary PAM
0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb
0 Ts 2Ts
0T 2T 3T
2.2762 V 1.3657 V
1 1 0 1 0 1-B
B
T
01
3B
T
T
-3B
T
0010
1
A.
T
0
T
-A.
Assuming real time tr. and equal energy per tr. data bit forbinary-PAM and 4-ary PAM:
4-ary: T=2Tb and Binay: T=Tb
4-ary PAM(rectangular pulse)
Binary PAM(rectangular pulse)
11
0 T 2T 3T 4T 5T 6T
22 10BA !
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2005-01-26 Lecture 3 6
Today we are going to talk about:
Receiver structure Demodulation (and sampling)
Detection
First step for designing the receiver Matched filter receiver
Correlator receiver
Vector representation of signals (signal
space), an important tool to facilitate Signals presentations, receiver structures
Detection operations
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2005-01-26 Lecture 3 7
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,
symbols are smeared.
Format Pulsemodulate
Bandpassmodulate
Format DetectDemod.
& sample
)(tsi)(tgiim
im )(tr)(Tz
channel)(th
c
)(tn
transmitted symbol
estimated symbol
Mi ,,1 -!M-ary modulation
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2005-01-26 Lecture 3 8
Example: Impact of the channel
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2005-01-26 Lecture 3 9
Example: Channel impact
)75.0(5.0)()( Tttthc
! HH
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2005-01-26 Lecture 3 10
Receiver job
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
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2005-01-26 Lecture 3 11
Receiver structure
Frequencydown-conversion
Receivingfilter
Equalizingfilter
Thresholdcomparison
For bandpass signals Compensation forchannel induced ISI
Baseband pulse(possibly distored)
Sample(test statistic)
Baseband pulseReceived waveform
Step 1 waveform to sample transformation Step 2 decision making
)(tr )(Tz im
Demodulate & Sample Detect
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2005-01-26 Lecture 3 12
Baseband and bandpass
Bandpass model of detection process isequivalent to baseband model because:
The received bandpass waveform is firsttransformed to a baseband waveform.
Equivalence theorem:
Performing bandpass linear signal processingfollowed by heterodying the signal to thebaseband, yields the same results as
heterodying the bandpass signal to thebaseband , followed by a baseband linear signalprocessing.
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2005-01-26 Lecture 3 13
Steps in designing the receiver
Find optimum solution for receiver designwith the following goals:1. Maximize SNR
2. Minimize ISI
Steps in design: Model the received signal
Find separate solutions for each of the goals.
First, we focus on designing a receiver
which maximizes the SNR.
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2005-01-26 Lecture 3 14
Design the receiver filter to maximizethe SNR
Model the received signal
Simplify the model:
Received signal in AWGN
)(thc
)(tsi
)(tn
)(tr
)(tn
)(tr)(tsiIdeal channels)()( tthc
H!
AWGN
AWGN
)()()()( tnthtstrci !
)()()( tntstr i !
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2005-01-26 Lecture 3 15
Matched filter receiver
Problem: Design the receiver filter such that the SNR is
maximized at the sampling time when
is transmitted.
Solution: The optimum filter, is the Matched filter, given by
which is the time-reversed and delayed version of theconjugate of the transmitted signal
)(th
)()()(*
tTsthth iopt !!)2exp()()()(
*fTjfSfHfH iopt T!!
Mitsi ,...,1),( !
T0 t
)(tsi
T0 t
)()( thth opt!
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2005-01-26 Lecture 3 16
Example of matched filter
T t T t T t0 2T
)()()( thtsty opti !2A)(tsi )(thopt
T t T t T t0 2T
)()()( thtsty opti !2A)(tsi )(thopt
T/2 3T/2T/2 T T/2
2
2TA
T
A
T
A
T
A
T
A
T
A
T
A
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2005-01-26 Lecture 3 17
Properties of the matched filter1. The Fourier transform of a matched filter output with the matched
signal as input is, except for a time delay factor, proportional to theESD of the input signal.
2. The output signal of a matched filter is proportional to a shiftedversion of the autocorrelation function of the input signal to whichthe filter is matched.
3. The output SNR of a matched filter depends only on the ratio of thesignal energy to the PSD of the white noise at the filter input.
4. 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 T!
sss ERTzTtRtz !!! )0()()()(
2/max
0N
E
N
S s
T
!
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2005-01-26 Lecture 3 18
Correlator receiver
The matched filter output at thesampling time, can be realized as thecorrelator output.
"!!
!
)(),()()(
)()()(
*
0
tstrdsr
TrThTz
i
T
opt
XXX
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2005-01-26 Lecture 3 19
Implementation of matched filterreceiver
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
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2005-01-26 Lecture 3 20
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!
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2005-01-26 Lecture 3 21
Example of implementation ofmatched filter receivers
2
1
z
zz!
)(tr
)(1 Tz
)(2 Tz
z
Bank of 2 matched filters
T t
)(1 ts
T t
)(2 tsT
T0
0
T
A
T
AT
A
T
A
0
0
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2005-01-26 Lecture 3 22
Signal space
What is a signal space? Vector representations of signals in an N-
dimensional orthogonal space
Why do we need a signal space? It is a means to convert signals to vectors and vice
versa.
It is a means to calculate signals energy andEuclidean distances between signals.
Why are we interested in Euclidean distances
between signals? For detection purposes: The received signal is
transformed to a received vectors. The signal whichhas the minimum distance to the received signal isestimated as the transmitted signal.
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2005-01-26 Lecture 3 23
Schematic example of a signal space
),()()()(
),()()()(
),()()()(
),()()()(
212211
323132321313
222122221212
121112121111
zztztztz
aatatats
aatatats
aatatats
!!
!!
!!
!!
z
s
s
s
]]
]]
]]
]]
)(1 t]
)(2 t]
),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
),( 21 zz!z
Transmitted signalalternatives
Received signal atmatched filter output
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2005-01-26 Lecture 3 24
Signal space
To form a signal space, first we need toknow the inner product between twosignals (functions):
Inner (scalar) product:
Properties of inner product:
g
g
"! dttytxtytx )()()(),( *
= cross-correlation between x(t) and y(t)
""! )(),()(),( tytxatytax
""! )(),()(),( * tytxataytx
"""! )(),()(),()(),()( tztytztxtztytx
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2005-01-26 Lecture 3 25
Signal space contd
The distance in signal space is measure bycalculating the norm.
What is norm? Norm of a signal:
Norm between two signals:
We refer to the norm between two signals asthe Euclidean distance between two signals.
xEdttxtxtxtx !!"! g
g2)()(),()(
)()( txatax !
)()(, tytxd yx !
= length of x(t)
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2005-01-26 Lecture 3 26
Example of distances in signal space
)(1
t]
)(2 t]),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
),( 21 zz!z
zsd ,1
zsd ,2zsd
,3
The Euclidean distance between signalsz(t) ands(t):
3,2,1
)()()()( 2222
11,
!
!!
i
zazatztsd iiizsi
1E
3E
2E
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2005-01-26 Lecture 3 27
Signal space - contd
N-dimensional orthogonal signal space ischaracterized by N linearly independent functions
called basis functions. The basis functionsmust satisfy the orthogonality condition
where
If all , the signal space is orthonormal.
_ aNjj
t1
)(!
]
jiij
T
iji Kdttttt H]]]] !"! )()()(),( *0
Tt ee0
Nij ,...,1, !
{p
!p
! ji
ji
ij 0
1
H
1!iK
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2005-01-26 Lecture 3 28
Example of an orthonormal basisfunctions
Example: 2-dimensional orthonormal signal space
Example: 1-dimensional orthonornal signal space
1)()(
0)()()(),(
0)/2sin(2
)(
0)/2cos(2
)(
21
2
0
121
2
1
!!
!"!
e!
e!
tt
dttttt
TtTtT
t
TtTtT
t
T
]]
]]]]
T]
T]
T t
)(1 t]
T
1
0
)(1 t]
)(2 t]
0
1)(1 !t] )(1 t]0
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2005-01-26 Lecture 3 29
Signal space contd
Any arbitrary finite set of waveformswhere each member of the set is of duration T,can be expressed as a linear combination of N
orthonogal waveforms where .
where
_ aM
ii ts 1)( !
_ aNjj
t1
)(!
] MN e
!
!N
j
jiji tats1
)()( ] Mi ,...,1!MN e
dtttsttsa
T
ji
j
ji
j
ij )()(1)(),(10
*
"!! ]] Tt ee0Mi ,...,1! Nj ,...,1!
),...,,( 21 iNiii aaa!s2
1
ij
N
j
ji aE !
!Vector representation of waveform Waveform energy
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Signal space - contd
!
!N
j
jiji tats1
)()( ] ),...,,( 21 iNiii aaa!s
iN
i
a
a
/
1
)(1 t]
)(tN
]
1ia
iNa
)(tsi
T
0
)(1 t]
T
0
)(tN
]
iN
i
a
a
/
1
ms!)(tsi
1ia
iNa
ms
Waveform to vector conversion Vector to waveform conversion
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2005-01-26 Lecture 3 31
Example of projecting signals to anorthonormal signal space
),()()()(
),()()()(
),()()()(
323132321313
222122221212
121112121111
aatatats
aatatats
aatatats
!!
!!
!!
s
s
s
]]
]]
]]
)(1 t]
)(2 t]
),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
Transmitted signalalternatives
dtttsa
T
jiij
)()(0
! ]Ttee0
M
i,...,1!Nj ,...,1!
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2005-01-26 Lecture 3 32
Signal space contd
To find an orthonormal basis functions for a givenset of signals, Gram-Schmidt procedure can beused.
Gram-Schmidt procedure: Given a signal set , compute an orthonormal
basis1. Define
2. For compute
If let
If , do not assign any basis function.
3. Renumber the basis functions such that basis is
This is only necessary if for any i in step 2.
Note that
_ aMii
ts1
)(!_ aN
jjt
1)(
!]
)(/)(/)()( 11111 tstsEtst !!]
Mi ,...,2!
!
"!1
1
)()(),()()(i
j
jjiii tttststd ]]
0)( {tdi )(/)()( tdtdt iii !]
0)( !tdi
_ a)(),...,(),( 21 ttt N]]]
0)( !tdiMN e
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2005-01-26 Lecture 3 33
Example of Gram-Schmidt procedure
Find the basis functions and plot the signal space forthe following transmitted signals:
Using Gram-Schmidt procedure:
T t
)(1 ts
T t
)(2 ts
)()(
)()(
)()(
21
12
11
AA
tAts
tAts
!!
!
!
ss
]
]
)(1 t]-A A0
1s2s
T
A
T
A
0
0
T t
)(1 t]
T
1
0
0)()()()(
)()()(),(
/)(/)()(
)(
122
01212
1111
0
22
11
!!
!"!
!!
!!
tAtstd
Adtttstts
AtsEtst
AdttsE
T
T
]
]]
]
1
2
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2005-01-26 Lecture 3 34
Implementation of matched filter receiver
)(tr
1z)(1 tT ]
)( tTN ]N
z
Bank of N matched filters
Observationvector
)()( tTtrz jj ! ] Nj ,...,1!
),...,,( 21 Nzzz!z
!!
N
j
jiji tats1
)()( ]
MN e
Mi ,...,1!
Nz
z1
z!
z
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2005-01-26 Lecture 3 35
Implementation of correlator receiver
),...,,( 21 Nzzz!z
Nj ,...,1!dtttrz jT
j )()(0
]!
T
0
)(1 t]
T
0
)(tN]
Nr
r
/
1
z!
)(tr
1z
Nz
z
Bank of N correlators
Observation
vector
!!
N
jjiji tats
1)()( ] Mi ,...,1!
MN e
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2005-01-26 Lecture 3 36
Example of matched filter receivers usingbasic functions
Number of matched filters (or correlators) is reduced by 1 compared tousing matched filters (correlators) to the transmitted signal.
T t
)(1 ts
T t
)(2 ts
T t
)(1 t]
T
1
0
? A1z z!)(tr z
1 matched filter
T t
)(1 t]
T
1
0
1z
T
A
T
A
0
0
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2005-01-26 Lecture 3 37
White noise in orthonormal signal space
AWGN n(t) can be expressed as)(~)()( tntntn !
Noise projected on the signal space
which impacts the detection process.
Noise outside on the signal space
"! )(),( ttnn jj ]
0)(),(~ "! ttn j]
)()(1
tntnN
j
jj!
! ]
Nj ,...,1!
Nj ,...,1!
Vector representation of
),...,,( 21 Nnnn!n
)( tn
independent zero-meanGaussain random variables withvariance
_ aN
jjn 1!
2/)var( 0Nnj !
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Digital Communications I: Modulation
and Coding Course
Period 3 - 2007
Catharina LogothetisLecture 4
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Lecture 4 39
Last time we talked about:
Receiver structure Impact of AWGN and ISI on the transmitted
signal
Optimum filter to maximize SNR
Matched filter receiver and Correlator receiver
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Lecture 4 40
Receiver job
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
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Lecture 4 41
Receiver structure
Frequency
down-conversion
Receiving
filter
Equalizing
filter
Thresholdcomparison
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
im
Demodulate & Sample Detect
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Lecture 4 42
Implementation of matched filter receiver
Mz
z
/
1
z!)(tr
)(Tz)(
*tTs
)(*
tTsM )(TzM
z
Bank of M matched filters
Matched filter output:
Observation
vector
)()( tTstrz ii
! i ,...,1!
),...,,())(),...,(),(( 2121 MM zzzTzTzTz !!z
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Lecture 4 43
Implementation of correlator receiver
dttstrzi
T
i)()(
0
!
T
0
)(1 ts
T
0
ts M
Mz
z
/
1
z!
)(t
)(Tz
)(TzM
z
Bank of M correlators
Correlators output:
Observationvector
),...,,())(),...,(),(( 22 MM zzzTzTzTz !!z
Mi ,...,1!
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Lecture 4 44
Today, we are going to talk about:
Detection:
Estimate the transmitted symbol based on the
received sample
Signal space used for detection Orthogonal N-dimensional space
Signal to waveform transformation and vice versa
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Lecture 4 45
Signal space
What is a signal space? Vector representations of signals in an N-dimensional
orthogonal space
Why do we need a signal space?
It is a means to convert signals to vectors and vice versa.
It is a means to calculate signals energy and Euclideandistances between signals.
Why are we interested in Euclidean distances between
signals?
For detection purposes:T
he received signal is transformed toa received vectors. The signal which has the minimum
distance to the received signal is estimated as the transmitted
signal.
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Lecture 4 46
Schematic example of a signal space
),()()()(
),()()()(
),()()()(
),()()()(
212211
323132321313
222122221212
121112121111
zztztztz
aatatats
aatatats
aatatats
!!
!!
!!!!
z
s
s
s
]]
]]
]]
]]
)(1 t]
)(2 t]
),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
),( 21 zz!z
Transmitted signal
alternatives
Received signal at
matched filter output
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Lecture 4 47
Signal space
To form a signal space, first we need to knowthe inner product between two signals
(functions):
Inner (scalar) product:
Properties of inner product:
"! dtttxttx )()()(),(
= cross-correlation between x(t) and y(t)
"" )(),()(),( tytxatytax
"" )(),()(),( * tytxataytx
"""! )(),()(),()(),()( tztytztxtztytx
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Lecture 4 48
Signal space
The distance in signal space is measure by calculatingthe norm.
What is norm?
Norm of a signal:
Norm between two signals:
We refer to the norm between two signals as the
Euclidean distance between two signals.
xEttxtxtxtx !!"! g
g
2
)()(),()(
)()( tata !
)()(, ttd
= length of x(t)
-
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Lecture 4 49
Example of distances in signal space
)(1 t]
)(t] ),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
),( 21 zz!z
zsd ,1
zsd ,2zsd ,3
The Euclidean distance between signalsz(t) ands(t):
3,2,1
)()()()( 2222
11,
!
!!
i
zazatztd iiizsi
1E
3E
2E
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Lecture 4 50
Orthogonal signal space
N-dimensional orthogonal signal space is characterized byN linearly independent functions called basisfunctions. The basis functions must satisfy the orthogonalitycondition
where
If all , the signal space is orthonormal.
_ aNjj
t1
)(!
]
jiij
T
iji Kdttttt H" )()()(),(*
0
Ttee0
Nij ,...,1, !
p
!p!
ji
jiij
0
1H
1iK
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Lecture 4 51
Example of an orthonormal bases
Example: 2-dimensional orthonormal signal space
Example: 1-dimensional orthonornal signal space
1)()(
0)()()(),(
0)/2sin(2
)(
0)/2cos(2)(
21
2
0
121
2
1
!!
!"!
!
!
tt
ttttt
TtTtT
t
TtTtT
t
T
]]
]]]]
T]
T]
T t
)(1 t]
T
1
0
)(1 t]
)(t]
0
1)(1 !t] )(1 t]
0
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Lecture 4 52
Signal space
Any arbitrary finite set of waveformswhere each member of the set is of duration T, can be
expressed as a linear combination of N orthonogal
waveforms where .
where
_ aMi
it
1
)(!
_ aNjj
t1
)(!
MN e
!
!N
j
jiji tats1
)()( ] Mi ,...,1!
MN e
dtttK
ttK
a
T
ji
j
ji
j
ij )()(1)(),(1 "!! ]] Ttee0Mi ,...,1! Nj ,...,1!
),...,,( 21 iNiii aaa!s2
1
ij
N
j
ji aKE !
!
Vector representation of waveform Waveform energy
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Lecture 4 53
Signal space
N
j
jiji tats1
)()( ] ),...,,( 21 iNiii aaa!s
i
i
a
a
/
1
)(1 t]
)(tN]
1ia
ia
)(tsi
T
0
)(1 t]
T
0
)(tN]
iN
i
a
a
/
1
ms!)(ts
i
1i
iN
ms
Waveform to vector conversion Vector to waveform conversion
E l f j ti i l t
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Lecture 4 54
Example of projecting signals to an
orthonormal signal space
),()()()(
),()()()(
),()()()(
323132321313
222122221212
121112121111
aatatats
aatatats
aatatats
!!
!!
!!
s
s
s
]]
]]
]]
)(1 t]
)(2 t]
),( 12111 aa!s
),( 22212 aa!s
),( 32313 aa!s
Transmitted signal
alternatives
dttta
T
jiij)()(
0! ] Ttee0M,...,1!Nj ,...,1!
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Lecture 4 55
Signal space contd
To find an orthonormal basis functions for a given
set of signals, Gram-Schmidt procedure can beused.
Gram-Schmidt procedure: Given a signal set , compute an orthonormal basis
1. Define
2. For compute
If let
If , do not assign any basis function.
1. Renumber the basis functions such that basis is
This is only necessary if for any i in step 2.
Note that
_ aMii
ts1
)( ! _ aN
jjt
1)(
!]
)(/)(/)()( tstsEtst !!]
Mi ,...,2!
!"!
1
1
)()(),()()(i
j
jjiii tttststd ]]
0)( {ti )(/)()( ttt iii !
0)( !ti
_ a)(),...,(),( 21 ttt N]]]
0)( !tiMN e
E l f G S h id d
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Lecture 4 56
Example of Gram-Schmidt procedure
Find the basis functions and plot the signal space for the following
transmitted signals:
Using Gram-Schmidt procedure:
T t
)(1 t
T t
)(2 t
)()(
)()(
)()(
21
12
11
AA
tAt
tAt
!!
!
!
ss
]
]
)(t]-A A0
1s2s
T
A
T
A
0
0
T t
)(t]
T
1
0
0)()()()(
)()()(),(
/)(/)()(
)(
122
01212
1111
0
22
11
!!
!"!
!!
!!
tAtst
Atttstts
AtsEtst
AttsE
T
T
]
]]
]
1
2
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Lecture 4 57
Implementation of matched filter receiver
)(tr
1)(1 tT
)( tN ]Nz
Bank of N matched filters
Observation
vector
)()( tTtrzjj
! ] Nj ,...,!
),...,,( 21 Nzzz!z!!
N
j
jiji tats1
)()( ]
MN e
Mi ,...,1!
Nz
z1
z!z
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Lecture 4 58
Implementation of correlator receiver
),...,,( 21 Nzzz!z
Nj ,...,1!dtttrz jT
j )()(0
]!
T
0
)(t]
T
0
)(tN]
Nr
r
/
1
z!)(tr
1
Nz
z
f c rrel t rs
Observ ti
vect r
!!N
jjiji tat
s
1 )()( ]Mi
,...,1!
MN e
E l f t h d filt i i
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Lecture 4 59
Example of matched filter receivers using
basic functions
Number of matched filters (or correlators) is reduced by 1 compared to usingmatched filters (correlators) to the transmitted signal.
T t
)(1
t
T t
)(2
t
T t
)(1
t]
T
1
0
? A1z z!)(tr z
1 matched filter
T t
)(1 t]
T
1
0
1z
T
A
T
A
0
0
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Lecture 4 60
White noise in orthonormal signal space
AWGN n(t) can be expressed as)(~)()( tntntn !
Noise projected on the signal space
which impacts the detection process.
Noise outside on the signal space
"! )(),( ttnn jj ]
0)(),(~ "! tt j]
)()(1
tntnN
j
jj!
! ]
Nj ,...,1!
Nj ,...,1!
Vect r represent ti n f
),...,,( 21 Nnnn!n
)( tn
independent zero-meanGaussain random variables with
variance
_ aN
jjn 1!
2/)var( 0Nnj !
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S-72.227 Digital Communication Systems
Spreadspectrum and
Code Division Multiple Access (CDMA)communications
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Spread Spectrum Communications - Agenda Today
Basic principles and block diagrams of spread spectrum communication
systems
Characterizing concepts
Types of SS modulation: principles and circuits
direct sequence (DS)
frequency hopping (FH)
Error rates
Spreading code sequences; generation and properties
Maximal Length (a linear, cyclic code)
Gold Walsh
Asynchronous CDMA systems
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How Tele-operators* Market CDMA
Coverage
ForCoverage, CDMA saveswireless carriers from deployingthe 400% more cell site thatare required by GSM
CDMAs capacity supports atleast 400% more revenue-producingsubscribers in the same spectrumwhen compared to GSM
Capacity Cost
$$A carrier who deploys CDMAinstead of GSM will havea lower capital cost
Clarity
CDMA with PureVoiceprovides wireline clarity
Choice
CDMA offers the choice of simultaneousvoice, async and packet data, FAX, andSMS.
Customer satisfaction
The Most solid foundation forattracting and retaining subscriberis based on CDMA
*From Samsumgs narrowbandCDMA (CDMAOne) marketing(2001)
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Direct Sequence Spread Spectrum (DS-SS)
This figure shows BPSK-DS transmitter and receiver
(multiplication can be realized by RF-mixers)
DS-CDMA is used in WCDMA, cdma2000 and IS-95 systems
2
22
av av
A P A P ! !
spreading
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Characteristics of Spread Spectrum Bandwidth of the transmitted signal Wis much greater than the original
message bandwidth (or the signaling rate R)
Transmission bandwidth is independent of the message. Applied code isknown both to the transmitter and receiver
Interference and noise immunity of SS system is larger, the larger the
processing gain
Multiple SS systems can co-exist in the same band (=CDMA). Increased
user independence (decreased interference) for(1) higher processing
gain and higher(2) code orthogonality
Spreading sequence can be very long -> enables low transmitted PSD->
low probability of interception (especially in military communications)
Narrow band signal(data)
Wideband signal(transmitted SS signal)
/ /c b c
L W R T T! !
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Characteristics of Spread Spectrum (cont.)
Processing gain, in general
LargeLc
improves noise immunity, but requires a larger
transmission bandwidth
Note that DS-spread spectrum is a repetition FEC-coded systems
Jamming margin
Tells the magnitude of additional interference and noise that can be
injected to the channel without hazarding system operation.
Example:
, 10/ (1/ ) /(1/ ) / , 10 log ( )
c c b b c c dB cL W R T T T T L L! ! ! !
[ ( ) ]J c sys desp
M L L SNR!
30 dB,available processing gain2 dB,margin or system losses
10dB,required a ter despreading (at the )
18dB,additional inter erence and noise can deteriorate
receive
c
sys
desp
j
L
L
SNR
M
!!
!
!
d by this amount
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Timo O.
orhonen,
UTCommunication Laboratory 67
Characteristics of Spread Spectrum (cont.)
Spectral efficiencyEeff: Describes how compactly TX signal fits into the
transmission band. For instance for BPSKwith some pre-filtering:
Energy efficiency (reception sensitivity): The value of
to obtain a specified error rate (often 10-9). For BPSK the error rate is
QPSK-modulation can fit twice the data rate of BPSK in the same
bandwidth. Therefore it is more energy efficient than BPSK.
/ /b T b Fff Re
R B RE B! !
,
2 2log log
1/RF fil t c cRF
b
B T LB
k M T M} } !
/ / 1/c b c c b c
L T T L T T ! !
1beff
RF b
RE
B T ! } b
T 2 2log log
c c
M M
L L!
0/
b bE NK !
2/1( 2 ), ( ) exp( 2)
2e b
k
p Q Qk dK P PT
g
! !
22 logkM k M! !
,: bandwidth for polar mod.
: number of levels
: number of bits
RF filtB
M
k
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UTCommunication Laboratory 68
A QPSK-DS Modulator
After serial-parallel conversion (S/P) data modulates the orthogonal
carriers
Modulation on orthogonal carriers spreaded by codes c1
and c2
Spreading codes c1
and c2 may or may not be orthogonal (System
performance is independent of their orthogonality, why?)
What kind of circuit can make the demodulation (despreading)?
2 coso
P t[
2 sino
P t[
1( )c t
2( )c t/S P
( )s t( )d t
i
q
Constellationdiagram
QPSK-modulator
2 cos( ) and 2 sin( )o o
P t P t[ [
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DS-CDMA (BPSK) Spectra (Tone Jamming)
Assume DS - BPSK transmission, with a single tone jamming (jamming
powerJ[W] ). The received signal is
The respective PSD of the received chip-rate signal is
At the receiverr(t) is multiplied with the local code c(t) (=despreading)
The received signal and the local code are phase-aligned:
1 0 0( ) 2 ( )cos 2 cos( ) 'd dr t Pc t T t tt J [[ U N !
_ a
2 2
0 0
0 0
1 1( ) sinc si
1 ( ) ( )2
nc2 2
c cr c cT TS f PT f f PT f f
J f f f f H H
!
1 0
0
( ) 2 ( ) cos ( )
2 cos
( )
( )
d dd
d
d t Pc t T tc t T
c t T
t
J t
[ U
[ N
!
! 1
( ) ( ) 1d d
c t T c t T
_ a
2 2
0 0
0
2 2
0 0
2 ( )cos
1 1( ) sinc s
1 1sinc sinc
inc2
2 2
2d b b b b
d
c c c c
Jc t T t
S f PT f f T PT f
JT f f T
f
J
T
T f f T
[ N
!
1 4 4 4 4 4 4 4 44 2 4 4 4 4 4 4 4 4 43
F
data
Data spectraafter phase modulator
Spreading of jammer power
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Tone Jamming (cont.)
Despreading spreads the jammer power and despreads the signal power:
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orhonen,!
UTCommunication Laboratory 71
Tone Jamming (cont.)
Filtering (at the BW of the phase modulator) after despreading
suppresses the jammer power:
E R t f BPSK DS S t *
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orhonen,#
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Error Rate of BPSK-DS System* DS system is a form of coding, therefore number chips, eg code weight
determines, from its own part, error rate (code gain)
Assuming that the chips are uncorrelated, prob. of code word error for abinary-block coded BPSK-DS system with code weight w is therefore
This can be expressed in terms of processing gainLcby denoting the
average signal and noise power by , respectively, yielding
Note that the symbol error rate is upper bounded due to repetition code
nature of the DS by
where tdenotes the number of erroneous bits that can be corrected in
the coded word
0
2, / ( code rate)be c m c
EP Q R w R k n
N
! ! !
0,b av b av c E P T N N T! ! ,av avP N
2 2av b av
e c m c c m
av c av
P T PP Q R w Q L R w
N T N
! !
min1
1(1 ) , ( 1)2
nm n m
esm t
n P p p t d
m
!
e !
*For further background, see J.G.Proakis:
Digital Communications (IV Ed), Section 13.2
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orhonen,%
UTCommunication Laboratory 73
Example: Error Rate of Uncoded Binary BPSK-DS
For uncoded DS w=n, thus and
We note that and yielding
Therefore, we note that increasing system processing gain W/R, error
rate can be improved
(1/ ) 1cR n n! !
0 0
2 2b b
e c m
E EP Q R w Q
N N
! !
/b av b av b E P T P R! ! 0 /av J J W !
0
/ /
/ /
b av
av av av
E P R W R
J J W J P! !
2 /
/e
av av
W RP Q
J P
!
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orhonen,'
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Code Generation in DS-SS
DS modulator Spreading sequence period
chip interval maximal length (ML)spreading code
ML code generator
delay elements (D-flip-flops) -> XOR - circuit
- code determined by feedback taps- code rate determined by clock rate
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Timo O.(
orhonen,)
UTCommunication Laboratory 75
Some Cyclic BlockCodes
(n,1) Repetition codes. High coding gain, but low rate
(n,k) Hamming codes. Minimum distance always 3. Thus can detect 2errors and correct one error. n=2m-1, k = n - m,
Maximum-length codes. For every integer there exists a
maximum length code (n,k) with n = 2k- 1,dmin = 2k-1.Hamming codes
are dual1 of of maximal codes.
BCH-codes. For every integer there exist a code with n = 2m-1,and where tis the error correction capability
(n,k) Reed-Solomon (RS) codes. Works with ksymbols that consist of
mbits that are encoded to yield code words ofn symbols. For these
codes and
Nowadays BCH and RS are very popular due to large dmin, large numberof codes, and easy generation
For further code references have a look on self-study material!
3ku
3umu k n mt
min2 1u d t
2 1,number of check symbols 2! !mn n k tmin
2 1! d t
1: Task: findoutfrom netwhatis meantbydualcodes!
3m u
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orhonen,1
UTCommunication Laboratory 76
Maximal Length Codes
autocorrelation
power spectral density
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orhonen,3
UTCommunication Laboratory 77
Maximal Length Codes (cont.)
Have very good autocorrelationbut cross correlation not granted
Are linear,cyclic block codes - generated by feedbacked shift registers
Number of available codes* depends on the number of shift register
stages:
Code generator design based on tables showing tap feedbacks:
5 stages->6 codes, 10 stages ->60 codes, 25 stages ->1.3x106 codes
*For the formula see: Peterson, Ziemer: Introduction to Spread Spectrum Communication, p. 121
Design of Maximal Length Generators
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orhonen,5
UTCommunication Laboratory 78
Design of Maximal Length Generators
by a Table Entry
Feedback connections can be written directly from the table:
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orhonen,7
UTCommunication Laboratory 79
Other Spreading Codes
Walsh codes: Orthogonal, used insynchronous systems, also in
WCDMA downlink Generation recursively:
All rows and columns of the matrix are orthogonal:
Gold codes: Generated by summing preferredpairs of maximal length
codes. Have a guarantee 3-level crosscorrelation:
ForN-length code there exists N+ 2 codes in a code family and
Walsh and Gold codes are used especially in multiple access systems
Gold codes are used in asynchronous communicationsbecause their
crosscorrelation is quite good as formulated above
!0 [0]H
!
1 1
11
n n
nnn
H HH
H H !
2
0 0 0 0
0 1 0 1
0 0 1 1
0 1 1 0
H
!( 1)( 1) ( 1)1 1( 1) 1 1 0
_ a ( ) / ,1/ ,( ( ) 2) /t n N N t n N
!
( 1) / 2
( 2) / 21 2 , or odd( )1 2 , or even
n
nnt nn
! 2 1 andnN
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Frequency Hopping Transmitter and Receiver
In FH-SS hopping frequencies are determined by the code and the
message (bits) are usually non-coherently FSK-modulated
This method is applied in BlueTooth
! dBW W
!dBW W
!sBW W
!sBW W
Frequency Hopping Spread Spectrum (FH-SS)
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Frequency Hopping Spread Spectrum (FH-SS)(example: transmission of two symbols/chip)
2 levelsL
2 slotsk
:chip duration
: bit duration
: symbol duration
c
b
s
T
T
T
2 ( data modulatorBW)
2 ( total FH spectral width)
L
d d
k
s d
W f
W W
! }
! }bT
2L
n p
!
4-level FSK modulation
Hopped frequencyslot determined by
hopping code
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orhonen,HUTCommunication Laboratory 82
Error Rate in Frequency Hopping
If there are multiple hops/symbol we have a fast-hopping system. If
there is a single hop/symbol (or below), we have a slow-hoppingsystem.
For slow-hopping non-coherent FSK-system, binary error rate is
and the respective symbol error rate is (hard-decisions)
A fast-hopping FSK system is a diversity-gain system. Assuming non-
coherent, square-law combining of respective output signals from
matched filters yields the binary error rate (withL hops/symbol)
(For further details, see J.G.Proakis: Digital Communications (IV Ed), Section 13.3)
01 exp / 2 , /2e b b bP E NK K! !
12 1 0
1
0
1exp / 2 / 2 ,
22 11
!
L i
e b i b b cL i
L i
i r
P K L
LK
ri
K K K K
!
!
! !
!
1 exp / 2 , / 12es b c cP R R k nK! !
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DS and FH compared
FH is applicable in environments where there exist tone jammers that
can be overcame by avoiding hopping on those frequencies
DS is applicable formultiple accessbecause it allowsstatistical
multiplexing(resource reallocation) to other users (power control)
FH applies usually non-coherent modulation due to carrier
synchronization difficulties -> modulation method degrades
performance
Both methods were first used in militarycommunications,
FH can be advantageous because the hopping span can be very
large (makes eavesdroppingdifficult)
DS can be advantageous because spectral density can be much
smaller than background noise density (transmission is unnoticed)
FH is an avoidance system: does not suffer on near-fareffect!
By using hybrid systems some benefits can be combined: The system
can have a low probability of interception and negligible near-far effect
at the same time. (Differentiallycoherentmodulation is applicable)
2 710 ...10c
L p
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Multiple access: FDMA, TDMA and CDMA
FDMA, TDMA and CDMA yieldconceptually the same capacity
However, in wireless communicationsCDMA has improved capacity due to
statistical multiplexing graceful degradationPerformance can still be improved by
adaptive antennas, multiuser detection,FEC, and multi-rate encoding
Example of DS multiple access waveforms
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Example of DS multiple access waveforms
channel->
detecting A ... ->
polar sig.->
FDMA TDMA d CDMA d ( )
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FDMA, TDMA and CDMA compared (cont.) TDMA and FDMA principle:
TDMA allocates a time instant for a user
FDMA allocates a frequency band for a user
CDMA allocates a code for user
CDMA-system can besynchronous orasynchronous:
Synchronous CDMA can not be used in multipath channels that
destroy code orthogonality Therefore, in wireless CDMA-systems as in IS-95,cdma2000,
WCDMA and IEEE 802.11 user are asynchronous
Code classification:
Orthogonal, as Walsh-codes for orthogonal or
near-orthogonal systems Near-orthogonal and non-orthogonal codes:
Gold-codes, for asynchronous systems
Maximal length codes for asynchronous systems
C i f ll l CDMA
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Capacity of a cellularCDMA system
Consider uplink (MS->BS)
Each user transmitsGaussian noise (SS-signal) whose
deterministic characteristics
are stored in RX and TX
Reception and transmission
are simple multiplications Perfect power control: each
users power at the BS the same
Each user receives multiple copies of powerPrthat is other users
interference power, therefore each user receives the interference power
where Uis the number of equal power users
( 1)k rI U P! (1)
C i f ll l CDMA ( )
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Each user applies a demodulator/decoder characterized by a certain
reception sensitivityEb/Io (3 - 9 dB depending on channel coding,channel, modulation method etc.)
Each user is exposed to the interference power density (assumed to be
produced by other users only)
where BTis the spreading (and RX) bandwidth
Received signal energy / bit at the signaling rate R is
Combining (1)-(3) yields the number of users
This can still be increased by using voice activity coefficient Gv = 2.67
(only about 37% of speech time effectively used), directional antennas,
for instance for a 3-way antenna GA = 2.5.
0/ [W/Hz]k TI I B! (2)
/ [ ] [ ][ ]b r
E P R J W s! ! (3)
0 0
1/ /1
1/ /
Tk o T
r b b b
R BI I B W RU
P ER E I E I ! ! ! ! (4)
Capacity of a cellularCDMA system (cont.)
( 1)k r
I P!
C it f ll l CDMA t ( t )
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In cellular system neighboring cells introduce interference that decreases
capacity. It has been found out experimentally that this reduces thenumber of users by the factor
Hence asynchronous CDMA system capacity can be approximated by
yielding with the given values Gv=2.67, GA=2.4, 1+f= 1.6,
Assuming efficient error correction algorithms, dual diversity antennas,and RAKE receiver, it is possible to obtainEb/Io=6 dB = 4, and then
1 1.6f }
/
/ 1v A
b o
G GW R
E I f!
4 /
/b o
W R
E I!
WU
R} This is of order of magnitude larger value than
with the conventional (GSM;TDMA) systems!
Capacity of a cellularCDMA system (cont.)
L L d
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Timo O.Korhonen,HUTCommunication Laboratory 90
Lessons Learned
You understand what is meant by code gain, jamming margin, and
spectral efficiency and what is their meaning in SS systems You understand how spreading and despreading works
You understand the basic principles of DS and FH systems and know
their error rates by using BPSK and FSK modulations
You know the bases of code selection for SS system. (What kind of
codes can be applied in SS systems and when they should be applied.) You understand how the capacity of asynchronous CDMA system can
be determined
March. 2007 doc.: 15-07-0624-00-004c
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a c . 007 doc.: 5 07 06 00 00 c
Slide 91Submission Liang Li
Outline
One formal PHY tech proposal from Chinese
companies, universities and partners.
Major tech points are upgraded from the OQPSKModulation on IEEE 802.15.4-2006
Spreading sequence
SFD design
Pulse shaping filter
Synchronization and demodulation performance
March. 2007 doc.: 15-07-0624-00-004c
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Slide 92Submission Liang Li
Proposed New Operation Frequency Bands
Fc= 314.3, 314.8, 315.3, 315.8 (MHz); BW=400kHz
Fc= 430.3, 430.8, 431.3, 431.8 (MHz); BW=400kHz
Fc= 433.3, 433.8, 434.3, 434.8 (MHz); BW=400kHz
Fc=780, 782,786, 788 (MHz) BW=2MHz
March. 2007 doc.: 15-07-0624-00-004c
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Slide 93Submission Liang Li
Communication Modulation and Data Rate
PHY
(MHz)
Frequency
Bands
(MHz)
Spreading Parameters Data Parameters
Chip Rate
(Mchip/s)Modulation
Bit Rate
(kb/s)
Symbol Rate
(ksymbol/s)
315 0.4 0.2 Chirp Sequence+ MPSK 50 12.5
430 0.4 0.2Chirp Sequence
+ MPSK50 12.5
780 2 1Chirp Sequence
+ MPSK250 62.5
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Slide 94Submission Liang Li
Upgrade from PHY Layer Operation of IEEE802.15.4-2006
The new PHY tech proposal is similar to the OQPSKones used in IEEE802.15.4-2006 at sub 1GHZ on:
The principal structures of the transmitter and
receiver Operation procedure of the transceiver
The important parameters forRF parts
Some different techniques are applied in this new
PHY proposal.
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Slide 95Submission Liang Li
Reference Design of the Wireless Transceiver based on
New PHY tech Proposal
VCO
LP
witc
h
ADC
ADC
Tra eiver
LP
LNA
RF
PHY
Q
I
MAC
Bit Stream
90o
Pul e
ilter
PAI
Q
BP
Pre-
proce i gMappi g
Acqui itio
Sy c
Demodulatio
VGA
Pul e
ilter
Digital Sig al
Proce i g
Bit Stream
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Slide 96Submission Liang Li
Difference 1: Spreading Sequence and Mapping
The Direct Sequence Spread Spectrum (DSSS) tech is applied
16 orthogonal spreading sequences are designed to map 4
information bits. The base sequence is a 16 length chirp
sequence and the other 15 sequences are its cyclic shifts.
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Slide 97Submission Liang Li
Spreading Sequence
New Proposal:
Chirp code is orthogonal among its cyclic shifts
Perfect auto-correlation property of the Preamble sequence,
Perfect orthogonal property of the 16 spreading sequences,
Reduce inter-chip interference in multipath environments.
Chirp code is robust to frequency offset
Low cost implementation of transmitter and receiver.
OQPSK in 15.4-2006 in sub-1GHz :
16 sequences are quasi-orthogonal. The auto-correlation property of the Preamble sequence is notvery well.
The code is susceptible to frequency offset.
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Slide 98Submission Liang Li
Sliding Correlation Values of the Preamble Sequence
-15 -10 -5 0 5 10 15 0
2
4
6
8
10
12
14
16
Sliding Chips
Auto-correlation
Values
-15 -10 -5 0 5 10 15 0
2
4
6
8
10
12
14
16
Sliding Chips
Auto-correlation
Values
With Sequence in New Proposal With Sequence in 15.4-2006 Std
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Slide 99Submission Liang Li
Difference 2: Pre-processing on Transmission
Conjugate the first sequence to obtain SFD The SFD sequence is the conjugate of the Preamble sequence,
that means the phases of the SFD chips are adverse to thephases of the Preamble chips.
DC component removal All chips of each symbol in the head and load of PPDU should
be multiplied by 1 or 1 based on the follow PN code serials.
r6 r5 r4 r3 r2 r1 r0 PNcode
1)( 37 ! xxxG
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Slide100
Submission Liang Li
SFD Design
New Proposal : The 16 spreading sequences are cyclic shift of each
other.
The SFD sequence is the complex conjugate of the
first spreading sequence.
OQPSK in 15.4-2006 in sub-1GHz : The first 8 spreading sequences are cyclic shift of
each other, and the last 8 spreading sequences are
the complex conjugate of the first 8 spreadingsequences.
The SFD sequences are chosen from the spreadingsequences.
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Slide101
Submission Liang Li
The Pulse Shaping Filter on I and Q path is designed as a raised cosine
filter with roll-off factor 0.5:
222 /41
)/cos(
/
)/sin()(
c
c
c
c
Ttr
Ttr
Tt
Tttp
!
T
T
T
The Transmit waveform and Spectrum are :
Difference 3: Pulse Shaping on Transmission andSpectrum ofTransmit Waveform
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Slide102
Submission Liang Li
Pulse Shaping Filter
New Proposal: Raised cosine filterThe chip duration is 1us.The zero-to-zero bandwidth is 1.5MHz.
15.4-2
006: Half-sine filterTo O-QPSKPHY, the zero-to-zero bandwidthis 1.5MHz.To New PHY proposal, the zero-to-zerobandwidth is 3MHz.
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Slide103
Submission Liang Li
Synchronization Performance of
3 4 5 6 7 8 9 10 10-6
10-5
10-4
10-3
10-2
10-1
100
Eb/N
0
A
ync
Error
Rate
B -W PC
N
15.4-2006 Simulation Conditions:1) AWGN channel
environment
2) Chip rate sampling
3) Basic sliding correlation
receiver4) Synchronized on other chips
means an error
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Slide104
Submission Liang Li
System Packet Error Performance
3 4 5 6 7 8 9 10 10-4
10-3
10-2
10-1
100
Eb/n
0
PER
C-W PA N
15.4-2006 Simulation Conditions:1) AWGN channel
environment
2) Chip rate sampling
3) Basic sliding correlation
receiver4) Ideal synchronization5) 32 data octets in each
packet
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Slide105
Submission Liang Li
The Independent Simulation Results from I2R
0 2D
E F G 0 1210
-4
10-3
10-2
10-1
100
AWGNH hannel I erformance P I Q RR
Eb
/NoP d S )
T
ER
COBI16 P Ideal)
U SSS P Ideal)
COBI16 P Sync)
U SSS P Sync)
V W X Y ` a V a W
a V
b
c
a V
b 3
a V
b
d
a V
b
e
a V
f
Egh
i
p
PE
q
a YF S K ( i o r s ohere r t)
D S S S ( i o r s ohere r t)
s OB Ia Y
(i o r s ohere r t)
D S S S ( s ohere r t)
sOB I
a Y
(s
oherer
t)
I2R implements the independent Simulation based on New PHY proposal. The left figure showsits results.
BUAA obtained the Same results in the right picture. And compare with the ones of I2R on samepage. Simulation condition are
Packet Length = 20bytes; AWGN channel, Ideal Sync.
Coherent detection: Decision is based on the real parts of the correlation values
Noncoherent detection: Decision is based on the norms of the correlation values
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Slide106
Submission Liang Li
System Performance ofReceiver based on New Proposal
in Multipath Channel
Tapped-Delay-Line Channel Model
IEEE P802.15 Working Group for WPANs, Multipath Simulation
Models for Sub-GHz PHY Evaluation, 15-04-0585-00-004b, Oct.
2004.
Power delay profile is exponentially declined. Each path is independently Rayleigh fading.
The average power of the channel response over many packets
is 1, but in each packet the power is varied.
Short Delay Environments
Without rake receiver
Long Delay Environments
With 3-tap rake receiver
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Slide107
Submission Liang Li
System Performance ofRake Receiver based on New
Proposal in Multipath Channel
Test Conditions
RMS delay spread 0~600ns
Tx nonlinear amplifier, Rapps model, p=3, backoff=1.5dB
Tx and Rx frequency offset80ppm, phase noise -110dBc/Hz
@1MHz Tx and Rx IQ imbalance 2dB, 10o
3bit AD sampling, 8bit baseband processing
Rx will implement time and frequency synchronization and data
detection
5000 packets are tested for each SNR, each packet comprises20 octets
The packet error rate is counted for 90% coverage
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Slide108
Submission Liang Li
PER in Short Delay Environments without Rake Receiver
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Slide109
Submission Liang Li
PER in Long Delay Environments with 3-tap Rake Receiver
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Slide110
Submission Liang Li
Thank you!
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1 Dr. ri M hl
INTRODUCTION
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In order to transmit digital information over *bandpass channels, we have to transfer
the information to a carrier wave of.appropriate frequency
We will study some of the most commonly *
used digital modulation techniques whereinthe digital information modifies the amplitudethe phase, or the frequency of the carrier in.discrete steps
2 Dr. Uri Mahlab
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OPTIMUM RECEIVERFORBINARY:DIGITAL MODULATION SCHEMS
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The function of a receiver in a binary communication *system is to distinguish between two transmitted signals.S1(t) and S2(t) in the presence of noise
The performance of the receiver is usually measured *in terms of the probability of error and the receiveris said to be optimum if it yields the minimum
.probability of error
In this section, we will derive the structure of an optimum *receiver that can be used for demodulating binary
.ASK,PSK,and FSK signals
4 Dr. Uri Mahlab
Description of binary ASK,PSK, and
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p y , ,: FSK schemes
-Bandpass binary data transmission system
ModulatorChannel(Hc(f
Demodulator(receiver)
{bk}
Binarydata
Input
{bk}
Transmitcarrier
Clock pulses
Noise
(n(t Clock pulses
Local carrier
Binary data output(Z(t
+
+
(V(t
+
5 Dr. Uri Mahlab
:Explanation *The input of the system is a binary bit sequence {bk} with a *
bi d bi d i T
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.bit rate rb and bit duration Tb
The output of the modulator during the Kth bit interval *.depends on the Kth input bit bk
The modulator output Z(t) during the Kth bit interval is *a shifted version of one of two basic waveforms S1(t) or S2(t) and
:Z(t) is a random process defined by
bb kTtTkfor ee )1(:
!
!
! 1bif])1([
0bif])1([
)( k2
k1
b
b
Tkts
Tkts
tZ
.1
6 Dr. Uri Mahlab
The waveforms S1(t) and S2(t) have a duration *f T d h fi i h i S1( ) d S2( ) 0
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of Tb and have finite energy,that is,S1(t) and S2(t) =0
],0[ bTtif and
g!
g!
b
b
T
T
dttsE
dttsE
0
2
22
0
2
11
)]([
)]([Energy:Term
7 Dr. Uri Mahlab
:The received signal + noise
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dbdb
db
db
tkTttTk
tntTkt
tntTkt
tV ee
! )1(
)(])1([s
or
)(])1([s
)(
2
1
8 Dr. Uri Mahlab
Choice of signaling waveforms for various types of digital*modulation schemesS (t) S (t)=0 for
[]0[ cfT
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S1(t),S2(t)=0 for
T2];,0[ c
cb fTt !
.The frequency of the carrier fc is assumed to be a multiple of rb
Type ofmodulation
ASK
PSK
FSK
bTtTS ee0);(1 bTtts ee0);(2
)sinor(cos
twAtwA
c
c
)sin(
cos
twAor
twA
c
c
0
)sin(
cos
twA
twA
c
c
}])sin{([(
})cos{(
twwAor
twwA
dc
dc
}])sin{(or[
})cos{(
twwA
twwA
dc
dc
9 Dr. Uri Mahlab
:Receiver structure
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Thresholddevice or A/D
converter
(V0(t
Filter(H(f
output
Sample everyTb seconds
)()()( tntztv !
10 Dr. Uri Mahlab
:{Probability of Error-{Pe*
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:{Probability of Error {Pe
The measure of performance used for comparing *!!!digital modulation schemes is the probability of error
The receiver makes errors in the decoding process *
!!! due to the noise present at its input
The receiver parameters as H(f) and threshold setting are *!!!chosen to minimize the probability of error
11 Dr. Uri Mahlab
:The output of the filter at t=kTb can be written as *
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)()()( 000 bbb kTnkTskTV !
12 Dr. Uri Mahlab
:The signal component in the output at t=kTb
bkT
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g
! bb dkThZkTs ]]] )()()(0
termsI I)()()1
!
]]] dkThZ b
kT
Tk
b
b
h( ) is the impulse response of the receiver filter*ISI=0*
]
!b
b
kT
Tk
bb dkThZkTs)1(
0 )()()( ]]]
13 Dr. Uri Mahlab
Substituting Z(t) from equation 1 and making*change of the variable the signal component
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change of the variable, the signal component:will look like that
!!
!!!
b
b
T
bb
T
bb
b
kTsdThs
kTsdThskTs
0
k012
0k011
0
1bwhen)()()(
0bwhen)()()()(
]]]
]]]
14 Dr. Uri Mahlab
:The noise component n0(kTb) is given by *
kT
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g!
bkT
bb
dkThnkTn ]]] )()()(0
he output noise n0(t) is a stationary zero mean Gaussian random process
:The variance of n0(t) is*
g
g
!! dffHfGtnEN n22
00 )()()}({
:The probability density function of n0(t) is*
gg
! nNN
nfn ;2
n-exp
2
1)(
0
2
0
0 T
15
The probability that the kth bit is incorrectly decoded*:is given by
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:is given by
}1|)({21
}0|)({2
1
})(and1
)(and0{
00
00
00
00
!
!u!
!
e!!
kb
kb
bk
bke
bTkTVP
bTkTVP
TkTbor
TkTbPP.2
16 Dr. Uri Mahlab
:The conditional pdf of V0 given bk= 0 is given by*
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gg
!
gg
!
!
!
0
0
2
020
0
01\
0
0
2010
0
00\
-,2
)(-exp
2
1)(
-,2
)(-exp
2
1)(
0
0
VN
s
N
Vf
VN
s
N
Vf
k
k
bV
bV
T
T
:It is similarly when bk is 1*
.3
17 Dr. Uri Mahlab
Combining equation 2 and 3 , we obtain an*i f th b bilit f P
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:expression for the probability of error- Pe as
g
g
!
0
0
0
0
2
020
0
0
0
2
010
0
2
)(-exp
2
1
2
1
2
)(-exp
2
1
2
1
T
T
e
dVN
S
N
dVN
S
N
P
T
T
.4
18 Dr. Uri Mahlab
:Conditional pdf of V0 given bk
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:The optimum value of the threshold T0* is*
2
0201*0
SST
!
)( 0
00 vkv bf !
)(
k0
01b
v
vf !
19Dr. Uri
Mahlab
Substituting the value of T*0 for T0 in equation 4*we can rewrite the expression for the probability
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:of error as
g
g
!
!
00102
0102
2/)(
2
2/)(
0
0
2
010
0
2exp2
1
2
)(exp
2
1
Nss
ss
e
dZ
Z
dVN
sV
NP
T
T
20Dr. Uri
Mahlab
he optimum filter is the filter that maximizes*he ratio or the square of the ratio
\
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qmaximizing eliminates the requirement S01
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The probability of error is minimized by an *appropriate choice of h(t) which maximizes
Where
0
2
01022 )]()([
N
TsTs bb !\
!bT
bbb dThssTsTs0
120102 )()]()([)()( \\\\
And dffHfGN n
2
0 )()(g
g
!22
Dr. UriM
ahlab
If we let P(t) =S2(t)-S1(t), then the numerator of the*:quantity to be maximized is
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!!!
g
g
bT
bb
bbb
dThPdThP
TPTSTS
0
00102
)()()()(
)()()(
\\\K\\
Since P(t)=0 for t
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2
2
)()(
)2exp()()(
g
g
g
g!dffGfH
dffTjfPfH
n
bT
K (*)
We can maximize by applying Schwarzs*:inequality which has the form
g
g
g
g
g
g
e dffX
dffX
dffXfX2
2
2
2
1
21
)(
)(
)()((**)
2K
24Dr. Uri
Mahlab
Applying Schwarzs inequality to Equation(**) with-
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)(
)2exp()()(
)()()(
2
1
fG
fTjfPfX
fGfHfX
n
b
n
T!!and
We see that H(f), which maximizes ,is given by-
)(
)2exp()(
)(
*
fG
fTjfPKfH
n
bT!
!!! Where Kis an arbitrary constant
(***)
25Dr. Uri
Mahlab
Substituting equation (***) in(*) , we obtain-:the maximum value of as 2
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2K
g
g
! dffG
fP
n )(
)(2
max2K
:And the minimum probability of error is given by-
!
!
g
22exp
21 max
2
2max/
KTK
QdZZPe
26Dr. Uri
Mahlab
:Matched Filter Receiver*
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If the channel noise is white, that is, Gn(f)= /2 ,then the transfer -
:function of the optimum receiver is given by
)2exp()()( * bfTjfPfH T!
From Equation (***) with the arbitrary constant K set equal to /2-:The impulse response of the optimum filter is
L
L
g
g! dfjftjfTf
Pth b )2exp()]2exp()([)(
* TT
27Dr. Uri
Mahlab
Recognizing the fact that the inverse Fourier *of P*(f) is P(-t) and that exp(-2 jfTb) representT
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:a delay of Tb we obtain h(t) as T
)()( tTpth b !:Since p(t)=S1(t)-S2(t) , we have*
)()()( 12 tTStTSth bb !The impulse response h(t) is matched to the signal *:S1(t) and S2(t) and for this reason the filter is calledMATCHED FILTER
28Dr. Uri
Mahlab
:Impulse response of the Matched Filter *
(S2(t 1
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(S2(t
(S1(t2 \Tb
2 \Tb
0
0
1-
2
0
Tb
t
t
t
t
t
(a)
(b)
(c)
2 \Tb(P(t)=S2(t)-S1(t
(P(-t
Tb- 02
(d)
2 \Tb0
Tb
(h(Tb-t)=p(t
2
(e)
(h(t)=p(Tb-t
29Dr. Uri
Mahlab
:Correlation Receiver*
T
The output of the receiver at t=Tb*
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g !bT
bb dThVTV \\\ )()()(0
Where V( ) is the noisy input to the receiver
Substituting and noting *: that we can rewrite the preceding expression as)()()(12 \\\ ! bb TSTSh
)T(0,or0)( b! \\h
\
!
!
b b
b
T T
T
b
dSVdSV
dSSVTV
0 0
12
0
120
)()()()(
)]()()[()(
\\\\\\
\\\\(# #)
30Dr. Uri
Mahlab
Equation(# #) suggested that the optimum receiver can be implemented *as shown in Figure 1 .This form of the receiver is called
A C l ti R i
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A Correlation Receiver
Thresholddevice(A\D)
integrator
integrator
-+
Sample
every Tbseconds
bT
0
bT
0
)(1 tS
)(2 tS
!
)()(
)()()(
2
1
tntS
ortntS
tV
Figure 1
31Dr. Uri
Mahlab
In actual practice, the receiver shown in Figure 1 is actually *.implemented as shown in Figure 2
In this implementation, the integrator has to be reset at the
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- (end of each signaling interval in order to ovoid (I.S.I!!!Inter symbol interference
:Integrate and dump correlation receiver
Filter
tolimitnoisepower
Thresholddevice(A/D)
R(Signal z(t
+
(n(t
+
WhiteGaussian
noise
High gain
amplifier)()( 21 tStS
Closed every Tb seconds
c
Figure 2
The bandwidth of the filter preceding the integrator is assumed *!!! to be wide enough to pass z(t) without distortion
32
Example: A band pass data transmission schemeS i li h i h
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uses a PSK signaling scheme with
sec2.0T,Tt0,cos)(
/10,Tt0,cos)(
bb1
b2
mtwAtS
TwtwAtS
c
bcc
!ee!
!ee! T
The carrier amplitude at the receiver input is 1 mvolt andthe psd of the A.W.G.N at input is watt/Hz. Assume
that an ideal correlation receiver is used. Calculate the.average bit error rate of the receiver
1110
33Dr. Uri
Mahlab
:Solution
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34Dr. Uri
Mahlab
=Probability of error = Pe *
:Solution Continue
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=Probability of error = Pe
35Dr. Uri
Mahlab
* Binary ASK signaling
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Binary ASK signaling
schemes:
!
ee
!
!
1bif])1([
1)T-(k
0bif])1([
)(
k2
b
k1
b
b
b
Tkts
kTt
Tkts
tz
The binary ASK waveform can be described as
Where andtAtS
c[cos)(2 ! 0)(1 !ts
We can represent:Z(t) as
)cos)(()( tAtDtZc
[!36
Dr. UriM
ahlab
Where D(t) is a lowpass pulse waveform consisting of.rectangular pulses
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.rectangular pulses
:The model for D(t) is
g
g!
!!k
bk Tktgbtd 1or0b],)1([)( k
ee
!
elswhere0
Tt01)(
btg
)()( TtdtD !
37Dr. Uri
M
ahlab
:The power spectral density is given by
2A
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)()([4
)(2
c
Dc
Dz
ffGffGA
fG !
The autocorrelation function and the power spectral density:is given by
!
u
e
!
b
bD
b
bb
b
DD
Tf
fTffG
T
TT
T
R
22
2sin)(
4
1)(
or0
for44
1
)(
T
TH
\
\\
\
38Dr. Uri
M
ahlab
:The psd of Z(t) is given by
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)22
22
2
2
(
)(sin
)()(sin
)()((16
)(
cb
cB
cb
cb
cz
ffT
ffT
ffTffT
ffffA
fG
!
T
T
TT
HH
39
Dr. UriM
ahlab
If we use a pulse waveform D(t) in which the individual pulsesg(t) have the shape
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? A
ee
!
elsewere0
Tt0)2cos(12)(
bTT tratg
b
40Dr. Uri
M
ahlab
Coherent ASK
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We start with
The signal components of the receiver output at the:of a signaling interval are
0)(andcos)( 12 !! tstAts c[
!!
!!
b
b
T
bb
T
b
TA
dttststskT
dttststskTs
0
2
122O2
0
12101
2)]()()[()(S
and
0)]()()[()(
41Dr. Uri
M
ahlab
:The optimum threshold setting in the receiver is
AkTskTs )()( 2*
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b
bb TAkTskTs
T42
)()( 0201*
0
!
!
:The probability of error can be computed aseP
g
!
!
!
max2
1
22
22
max
42exp
2
1
KLT
LK
be
b
TAQdz
zp
TA
42Dr. Uri
M
ahlab
:The average signal power at the receiver input is given by
2A
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4
2A
sav !We can express the probability of error in terms of the:average signal power
! L
bave TSQp
The probability of error is sometimes expressed in *: terms of the average signal energy per bit , as
bavav TsE )(!
! Lav
e
E
QP
43 Dr. Uri Mahlab
Noncoherent ASK
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Noncoherent ASK:The input to the receiver is *
!
!!
0bhen)(
1bhen)(cos)(
k
k
tn
tntAtV
i
ic[
hite.andaussian,
mean,zerobetoassumedishichinputreceiverat thenoisethe)( tni
44 Dr. Uri Mahlab
Noncoharent ASKReceiver
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filterbandpasstheof
outputat thenoisetheisn(t)when
0Aand1bbitdtransmitte
kthwhen theAwhere
sin)(
cos)(cos
)(cos)(
:haveoutput wefilterAt the
kk
k
!!!
!
!!
A
ttn
ttntA
tntAtY
cs
ccck
ck
[
[[
[
45
:The pdf is0r,
2exp)(
0
2
0
0| "
!!
N
r
N
rrf
kbR
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0r,2
exp)(
2
0
22
0
0
0
1|
00
"
!
!N
Ar
N
ArI
N
rrf
NN
kbR
B
T
TBN
N
LL
2
filter.bandpasstheofoutputat thepo er noise
0
0
}!
!T
T
2
0
0 ))cos(exp(2
1)( duuxXI
46 Dr. Uri Mahlab
pdfs of the envelope of the noise and the signal *:pulse noise
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47 Dr. Uri Mahlab
)1b|error(2
1)0b|error(
2
1kk pppe !!!
:The probability of error is given by
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T
T
2
2exp
)(
ionapproximattheUsing
22
)(exp
2
1
and
8exp
2exp
where
2
1
2
1
22
2
2
00
2
0
1
2
0
2
0
2
0
0
10
x
x
xQ
N
AQdr
N
Ar
Np
N
Adr
N
r
N
rp
pp
A
e
A
e
ee
!
!
!
!
!
!
g
g
48 Dr. Uri Mahlab
1 toreducecanex,largefor p e
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0
2
0
2
0
2
2
0
0
2
2
01
Aif8
exp2
1
8exp
2
41
2
1
ence,
8exp
24
NN
A
N
A
A
Np
N
AA
Np
e
e
""
}
}
}
T
T
49 Dr. Uri Mahlab
BINERY PSK SIGNALING
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SCHEMES:The waveforms are *
0bforcos)(
1bforcos)(
k2
k1
!!
!!
tAts
tAts
c
c
[
[
:The binary PSK waveform Z(t) can be described by *
)cos)(()( tAtDtZc
[!.D(t) - random binary waveform *
50 Dr. Uri Mahlab
:The power spectral density of PSK signal is
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b
bD
cDcDZ
Tf
fTfG
Where
ffGffGA
fG
22
2
2
sin)(
,
)]()([4
)(
T
T!
!
51 Dr. Uri Mahlab
Coherent PSK
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:The signal components of the receiver output are
!!
!!
b
b
b
b
kT
Tk
bb
kT
Tk
bb
TAdttststskTs
TAdttststskTs
)1(
2
12202
)1(
2
12101
)]()()[()(
)]()()[()(
52 Dr. Uri Mahlab
:The probability of error is given by
e QP
! max
K
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bav
av
av
b
e
T
bc
e
TA
E
A
E
s
TA
Qp
TAdttA
Q
b
!
!
!
!!
2
and
2s
arescheme
PSKfor thebitperenergysignal
theendpowersignalaverageThe
or
4)cos2(
2where
2
2
2
av
2
0
222
max
L
L[
LK
53 Dr. Uri Mahlab
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!
!
!
L
L
av
bave
EQ
Tsp
2
2
:errorofyprobabilittheexpresscanwe
54 Dr. Uri Mahlab
DIFFERENTIALLY COHERENT *:PSK
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DELAY
LOGICNETWORK
LEVELSHIFT
bT
BINERYSEQUENCE
_ a
1oro
dk
_ a1kd
1s
tAc
[cos
tAC
[coss
Z(t)
DPSK modulator
55Dr. Uri Mahlab
DPSK demodulator
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Filter tolimit noisepower
Delay
Lowpassfilter orintegrator
Thresholddevice(A/D)
Z(t)
)(tn
bT
_ akb
bkTat
sample
56 Dr. Uri Mahlab
Differential encoding & decoding
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In t
e e-nce
1 1 0 1 0 0 0 1 1
nco ese ence 1 1 1 0 0 1 0 1 1 1
ransmit
ase 0 0 0 i i 0 i 0 0 0
ase
om ari-son
o t t- - - -
t tit
se ence 1 1 0 1 0 0 0 1 1
57 Dr. Uri Mahlab
* BINARYFSK SIGNALING
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SCHEMES ::The waveforms ofFSK signaling
1bfor)cos()(
0bfor)cos()(
k2
k1
!!
!!
ttAtS
ttAtS
dC
dc
[[
[[
:Mathematically it can be represented as
! U[[ ')'(cos)( dttDtAtZ dc
!
!!
0bfor1
1bfor1)(
k
ktD
58 Dr. Uri Mahlab
Power spectral density ofFSK