DIGITAL CARRIER MODULATION SCHEMES
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Transcript of DIGITAL CARRIER MODULATION SCHEMES
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1 Dr. Uri Mahlab
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INTRODUCTION
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 wherein the digital information modifies the amplitude
the phase, or the frequency of the carrier in . discrete steps
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The modulation waveforms for transmitting :binary information over
bandpass channels
ASK
FSK
PSK
DSB
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OPTIMUM RECEIVER FOR BINARY :DIGITAL MODULATION SCHEMS
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 receiver is 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
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Description of binary ASK,PSK, and :FSK schemes
-Bandpass binary data transmission system
ModulatorChannel
)Hc(fDemodulator
)receiver(
{ bk}
Binarydata
Input
{bk}
Transmitcarrier
Clock pulsesNoise
)n(t Clock pulses
Local carrier
Binary data output)Z(t
+
+
)V(t
+ 7ּ
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:Explanation *The input of the system is a binary bit sequence }bk{ with a*
.bit rate r b 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 )1(:
1b if ])1([
0b if ])1([)(
k2
k1
b
b
Tkts
TktstZ
.1
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The waveforms S1(t) and S2(t) have a duration* of Tb and have finite energy,that is,S1(t) and S2(t) =0
],0[ bTtif and
b
b
T
T
dttsE
dttsE
0
222
0
211
)]([
)]([Energy:Term
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: The received signal + noise
dbdb
db
db
tkTttTk
tntTkt
tntTkt
tV
)1(
)(])1([s
or
)(])1([s
)(
2
1
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Choice of signaling waveforms for various types of digital*modulation schemes S1(t),S2(t)=0 for
2
];,0[ ccb fTt
.The frequency of the carrier fc is assumed to be a multiple of rb
Type ofmodulation
ASK
PSK
FSK
bTtTS 0);(1 bTtts 0);(2
)sinor (
cos
twA
twA
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
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: Receiver structure
Thresholddevice or A/D
converter
) V0(t
Filter)H(f output
Sample everyTb seconds
)()()( tntztv
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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
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: The output of the filter at t=kTb can be written as*
)()()( 000 bbb kTnkTskTV
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: The signal component in the output at t=kTb
bkT
bb dkThZkTs )()()(0
termsISI)()()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 )()()(
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Substituting Z(t) from equation 1 and making*change of the variable, the signal component
:will look like that
b
b
T
bb
T
bb
b
kTsdThs
kTsdThs
kTs
0
k012
0
k011
0
1b when )()()(
0b when )()()(
)(
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:The noise component n0(kTb) is given by *
bkT
bb dkThnkTn )()()(0
.The output noise n0(t) is a stationary zero mean Gaussian random process
:The variance of n0(t) is*
dffHfGtnEN n
2200 )()(){(}
:The probability density function of n0(t) is*
n
NNnfn ;
2
n-exp
2
1)(
0
2
00
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The probability that the kth bit is incorrectly decoded*:is given by
{1|)(}2
1
{0|)(}2
1
{)(V and 1
)(V and 0}
00
00
00
00
kb
kb
bk
bke
bTkTVP
bTkTVP
TkTbor
TkTbPP.2
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:The conditional pdf of V0 given bk = 0 is given by*
00
2020
0
01\
00
2010
0
00\
- , 2
)(V-exp
2
1)(
- , 2
)(V-exp
2
1)(
0
0
VN
s
NVf
VN
s
NVf
k
k
bV
bV
:It is similarly when bk is 1*
.3
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Combining equation 2 and 3 , we obtain an*:expression for the probability of error- Pe as
0
0
00
2020
0
00
2010
0
2
)(V-exp
2
1
2
1
2
)(V-exp
2
1
2
1
T
T
e
dVN
S
N
dVN
S
NP
.4
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:Conditional pdf of V0 given bk
:The optimum value of the threshold T0* is*
20201*
0
SST
)( 0
00 v
kv bf )(k
0
01b v
vf
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Substituting the value of T*0 for T0 in equation 4* we can rewrite the expression for the probability
:of error as
00102
0102
2/)(
2
2/)(
00
2010
0
2exp
2
1
2
)(exp
2
1
Nss
ss
e
dZZ
dVN
sV
NP
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The optimum filter is the filter that maximizes*the ratio or the square of the ratio
)maximizing eliminates the requirement S01<S02(
0
0102 )()(
N
TSTS bb
2
2
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:Transfer Function of the Optimum Filter* The probability of error is minimized by an*
appropriate choice of h(t) which maximizes
Where
0
201022 )]()([
N
TsTs bb
bT
bbb dThssTsTs0
120102 )()]()([)()(
And dffHfGN n
2
0 )()(
2
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If we let P(t) =S2(t)-S1(t), then the numerator of the*: quantity to be maximized is
bT
bb
bbb
dThPdThP
TPTSTS
0
00102
)()()()(
)()()(
Since P(t)=0 for t<0 and h( )=0 for <0*:the Fourier transform of P0 is
dffTjfHfPTP
fHfPfP
bb )2exp()()()(
)()()(
0
0
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:Hence can be written as* 22
2
2
)()(
)2exp()()(
dffGfH
dffTjfPfH
n
b (*)
We can maximize by applying Schwarz’s*:inequality which has the form
dffX
dffX
dffXfX2
2
2
2
1
21
)(
)(
)()(
(**)
2
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Applying Schwarz’s inequality to Equation(**) with-
)(
)2exp()()(
)()()(
2
1
fG
fTjfPfX
fGfHfX
n
b
n
and
We see that H(f), which maximizes ,is given by-
)(
)2exp()()(
*
fG
fTjfPKfH
n
b
!!!Where K is an arbitrary constant
(***)
2
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Substituting equation (***) in(*) , we obtain-:the maximum value of as
2
dffG
fP
n )(
)(2
max2
:And the minimum probability of error is given by-
22exp
2
1 max2
2max/
QdZZ
Pe
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:Matched Filter Receiver*
If the channel noise is white, that is, Gn(f)= /2 ,then the transfer- :function of the optimum receiver is given by
)2exp()()( *bfTjfPfH
From Equation (***) with the arbitrary constant K set equal to /2-:The impulse response of the optimum filter is
dfjftjfTfPth b )2exp()]2exp()([)( *
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Recognizing the fact that the inverse Fourier* of P*(f) is P(-t) and that exp(-2 jfTb) represent
: a delay of Tb we obtain h(t) as
)()( 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 called
MATCHED FILTER28 Dr. Uri Mahlab
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:Impulse response of the Matched Filter*
)S2(t
)S1(t2\ Tb
2\ Tb
1
0
0
1-
2
0Tb
t
t
t
t
t
)a(
)b(
)c(
2\ Tb)P(t)=S2(t)-S1(t
)P(-tTb- 0
2)d(
2\ Tb0
Tb
)h(Tb-t)=p(t
2
)e(
)h(t)=p(Tb-t
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:Correlation Receiver*
bT
bb dThVTV )()()(0
The output of the receiver at t=Tb*
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,for 0)( b h
b b
b
T T
T
b
dSVdSV
dSSVTV
0 0
12
0
120
)()()()(
)]()()[()(
(# #)
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Equation(# #) suggested that the optimum receiver can be implemented* as shown in Figure 1 .This form of the receiver is called
A Correlation Receiver
Thresholddevice
)A\D(
integrator
integrator
- +
Sampleevery Tb
seconds
bT
0
bT
0
)(1 tS
)(2 tS
)()(
)()(
)(
2
1
tntS
or
tntS
tV
Figure 1
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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 ) - end of each signaling interval in order to ovoid (I.S.I
!!! Inter symbol interference
:Integrate and dump correlation receiver
Filterto
limitnoisepower
Thresholddevice
)A/D(R)Signal z(t
+
)n(t
+
WhiteGaussian
noise
High gainamplifier
)()( 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
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Example: A band pass data transmission scheme uses a PSK signaling scheme with
sec2.0T ,Tt0 ,cos)(
/10 ,Tt0 ,cos)(
b b1
b2
mtwAtS
TwtwAtS
c
bcc
The carrier amplitude at the receiver input is 1 mvolt andthe psd of the A.W.G.N at input is watt/Hz. Assumethat an ideal correlation receiver is used. Calculate the
.average bit error rate of the receiver
1110
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:Solution
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=Probability of error = Pe*
:Solution Continue
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* Binary ASK signaling schemes:
1b if ])1([
1)T-(k
0b if ])1([
)(
k2
b
k1
b
b
b
Tkts
kTt
Tkts
tz
The binary ASK waveform can be described as
Where andtAtS ccos)(2 0)(1 ts
We can represent :Z(t) as
)cos)(()( tAtDtZ c36 Dr. Uri Mahlab
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Where D(t) is a lowpass pulse waveform consisting of . rectangular pulses
: The model for D(t) is
k
bk Tktgbtd 1or 0b ],)1([)( k
elswhere 0
Tt0 1)( btg
)()( TtdtD
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: The power spectral density is given by
)()([4
)(2
cDcDz ffGffGA
fG
The autocorrelation function and the power spectral density: is given by
b
bD
b
bb
b
DD
Tf
fTffG
T
TT
T
R
22
2sin)(
4
1)(
for 0
for 44
1
)(
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: The psd of Z(t) is given by
)
2
2
22
2
2
(
)(sin
)(
)(sin
)()((16
)(
cb
cB
cb
cb
cz
ffT
ffT
ffT
ffT
ffffA
fG
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If we use a pulse waveform D(t) in which the individual pulsesg(t) have the shape
elsewere 0
Tt0 )2cos(12)( b tra
tg b
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Coherent ASKWe start with The signal components of the receiver output at the
: of a signaling interval are
0)( and cos)( 12 tstAts c
b
b
T
bb
T
b
TA
dttststskT
dttststskTs
0
2
122O2
0
12101
2)]()()[()(S
and
0)]()()[()(
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: The optimum threshold setting in the receiver is
bbb T
AkTskTsT
42
)()( 20201*
0
: The probability of error can be computed as eP
max2
1
22
22max
42exp
2
1
be
b
TAQdz
zp
TA
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: The average signal power at the receiver input is given by
4
2Asav
We can express the probability of error in terms of the: average signal power
bav
e
TSQp
The probability of error is sometimes expressed in* : terms of the average signal energy per bit , as
bavav TsE )(
av
e
EQP
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Noncoherent ASK: The input to the receiver is*
0b when )(
1b when )(cos)(
k
k
tn
tntAtV
i
ic
white.and Gaussian,
mean, zero be toassumed is which
inputreceiver at the noise the)( tni
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Noncoharent ASK Receiver
filter bandpass theof
output at the noise theis n(t)when
0A and 1bbit dtransmitte
kth when theA where
sin)(
cos)(cos
)(cos)(
:have output wefilter At the
kk
k
A
ttn
ttntA
tntAtY
cs
ccck
ck
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:The pdf is
0r ,2
exp)(
0r ,2
exp)(
0
22
00
01|
0
2
00|
N
Ar
N
ArI
N
rrf
N
r
N
rrf
k
k
bR
bR
BT T
BN
N
2
filter. bandpass
theofoutput at thepower noise
0
0
2
0
0 ))cos(exp(2
1)( duuxXI
46 Dr. Uri Mahlab
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pdf’s of the envelope of the noise and the signal* : pulse noise
47 Dr. Uri Mahlab
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2
2exp
)(
ionapproximat theUsing
22
)(exp
2
1
and
8exp
2exp
where2
1
2
1
)1b|error(2
1)0b|error(
2
1
2
2
00
2
0
1
20
2
0
2
00
10
kk
x
x
xQ
N
AQdr
N
Ar
Np
N
Adr
N
r
N
rp
pp
ppp
A
e
Ae
ee
e
: The probability of error is given by
48 Dr. Uri Mahlab
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02
0
2
0
2
20
0
2
20
1
1
A if 8
exp2
1
8exp
2
41
2
1
Hence,
8exp
2
4
to reducecan we x,largefor
NN
A
N
A
A
Np
N
A
A
Np
p
e
e
e
49 Dr. Uri Mahlab
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BINERY PSK SIGNALING SCHEMES
: The waveforms are*
0bfor cos)(
1bfor cos)(
k2
k1
tAts
tAts
c
c
: The binary PSK waveform Z(t) can be described by*
)cos)(()( tAtDtZ c. D(t) - random binary waveform*
50 Dr. Uri Mahlab
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: The power spectral density of PSK signal is
b
bD
cDcDZ
Tf
fTfG
Where
ffGffGA
fG
22
2
2
sin)(
,
)]()([4
)(
51 Dr. Uri Mahlab
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Coherent PSK: The signal components of the receiver output are
b
b
b
b
kT
Tk
bb
kT
Tk
bb
TAdttststskTs
TAdttststskTs
)1(
212202
)1(
212101
)]()()[()(
)]()()[()(
52 Dr. Uri Mahlab
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: The probability of error is given by
bav
av
av
be
T
bc
e
TA
E
A
E
s
TAQp
TAdttA
QP
b
2
and2
s
are scheme
PSK for the bit per energy signal
theend power signal average The
or
4)cos2(
2
where
2
2
2
av
2
0
222
max
max
53 Dr. Uri Mahlab
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av
bave
EQ
Tsp
2
2
:error ofy probabilit theexpresscan we
54 Dr. Uri Mahlab
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DELAY
LOGICNETWORK
LEVELSHIFT
bT
BINERYSEQUENCE
1or o
dk
1kd
1
tA ccos
tA Ccos
Z(t)
DIFFERENTIALLY COHERENT* : PSK
DPSK modulator
55 Dr. Uri Mahlab
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DPSK demodulator
Filter tolimit noise
power
Delay
Lowpassfilter or
integrator
Thresholddevice
(A/D)
Z(t)
)(tn
bT
kb̂
bkTat
sample
56 Dr. Uri Mahlab
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Differential encoding & decoding
InputSeque-nce
1 1 0 1 0 0 0 1 1Encodedsequence 1 1 1 0 0 1 0 1 1 1TransmitPhase 0 0 0 pi pi 0 pi 0 0 0PhaseCompari-sonoutput
+ + - + - - - + +OutputBitsequence 1 1 0 1 0 0 0 1 1
57 Dr. Uri Mahlab
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* BINARY FSK SIGNALING SCHEMES : : The waveforms of FSK signaling
1bfor )cos()(
0bfor )cos()(
k2
k1
ttAtS
ttAtS
dC
dc
: Mathematically it can be represented as
')'(cos)( dttDtAtZ dc
0bfor 1
1bfor 1)(
k
ktD
58 Dr. Uri Mahlab
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Power spectral density of FSK signals
Power spectral density of a binary FSK signal with
bd rf 2
59
2
2
ee
dd
wf
wf
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Coherent FSK: The local carrier signal required is
)cos()cos()()( 12 ttAttAtsts dcdc
The input to the A/D converter at sampling time where)(or )( is 0201 bbb kTskTskTt
b
b
T
b
T
b
dttststskTs
dttststskTs
0
12101
0
12202
)]()()[()(
)]()()[()(
60 Dr. Uri Mahlab
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The probability of error for the correlation receiver is : given by
)cos()(
and )cos()(
when
)]()([2
where
2
1
2
0
212
2max
max
ttAts
ttAts
dttsts
QP
dc
dc
T
e
b
61 Dr. Uri Mahlab
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. Which are usually encountered in practical system
: We now have
bd
bdb
T
TTA
2
2sin1
2 22max
62dbc wTw c w, 1
:When
Dr. Uri Mahlab
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Noncoherent FSK
0r ,2
exp)(
and
0r ,2
exp)(
:isfilter bottom theof )(R envelope theof pdf theinterval,
signalingkth theduring mittedbeen trans has )cos()( that Assuming
20
22
0
22|
10
221
0
10
0
11)(|
1
1
12
11
N
r
N
rrf
n
Ar
N
ArI
N
rrf
kT
tAts
sR
tsR
b
dc
63 Dr. Uri Mahlab
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Noncoharenr demodulator of binary FSK
ENVELOPEDETECTOR
ENVELOPEDETECTOR
THRESHOLDDEVICE
(A/D)
dc ff
filter
Bandpass
dc ff
filter
bandpass
+
-
)(2 bkTR
)(1 bkTR
0*0 T
Z(t)+n(t)
0
2
4exp
2
1
N
APe
64 Dr. Uri Mahlab
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Probability of error for binary digital modulation* :schemes
65 Dr. Uri Mahlab
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M-ARY SIGNALING SCHEMES
: M-ARY coherent PSK
The M possible signals that would be transmitted: during each signaling interval of duration Ts are
sTt0 ,1,...1,0 ,2
cos)(
Mk
M
ktAtS ck
: The digital M-ary PSK waveform can be represented
k
kcs tkTtgAtZ )cos()()( 66 Dr. Uri Mahlab
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k k
skcskc kTtgtAkTtgtAtZ )()(sinsin)()(coscos)(
: In four-phase PSK (QPSK), the waveform are
S
c
c
c
c
Tt
tAtS
tAtS
tAtS
tAtS
0 allfor
sin)(
cos)(
sin)(
cos)(
4
3
2
1
67 Dr. Uri Mahlab
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Phasor diagram for QPSK
)45cos( and )45cos( 00 tAtA cc That are derived from a coherent local carrier
reference tA ccos
68
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If we assume that S 1 was the transmitted signal: during the signaling interval (0,Ts),then we have
0
2
0
01
4cos
2
)4
cos()cos()(
LTA
dttAtATS
s
T
ccs
s
0
2
0
02
4cos
2
A
4cos)cos()(
LT
dttAtATS
s
T
ccs
s
69 Dr. Uri Mahlab
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Z(t)
)(tn
)45cos( tA c
)45cos( tA c
ST
0
ST
0
)(01 SkTV
)(02 SkTV
QPSK receiver scheme
70 Dr. Uri Mahlab
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: The outputs of the correlators at time t=TS are
S
s
T
cs
T
cs
ss
sss
sss
dttAtnTn
dttAtnTn
TnTn
TnTSTV
TnTSTV
0
002
0
001
0201
020202
010101
)45cos()()(
)45cos()()(
by defined variablesrandomGaussian mean zero are )( & )( where
)()()(
)()()(
71 Dr. Uri Mahlab
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Probability of error of QPSK:
2
2
0
0
002
0011
2N
LQ
))((
))((
ecs
s
sec
PTA
Q
LTnP
LTnPP
72 Dr. Uri Mahlab
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sin2
4Mfor
2221
:is system for the
)1)(1(
correctly received is signal ed transmitty that theprobabilit The
22
2
1
21
M
TAQP
TAQPPP
P
PPP
- P
se
secce
e
ececc
c
73 Dr. Uri Mahlab
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Phasor diagram for M-ary PSK ; M=8
74 Dr. Uri Mahlab
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The average power requirement of a binary PSK : scheme are given by
sin
1
)(
)(
Z& small very is If
sin
1
Z)(
)(
2
21
222
21
MS
S
ZP
M
Z
S
S
bav
Mav
e
bav
Mav
75 Dr. Uri Mahlab
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*COMPARISION OF POWER-BANDWIDTH: FOR M-ARY PSK
410eP
Valueof M b
M
Bandwidth
Bandwidth
)(
)(
bav
mav
S
S
)(
)(
48
1632
0.50.333
0.250.2
0.34 dB3.91 dB8.52 dB
13.52 dB
76 Dr. Uri Mahlab
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* M-ary for four-phase Differential PSK:
RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK
Integrateand dump
filter
ST
Delay
ST
Delay
shift
phase
090
Integrateand dump
filter
)(01 tV
)(02 tV
)(tn
Z(t)
77 Dr. Uri Mahlab
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: The probability of error in M-ary differential PSK
M
TAQP S
e 2sin22 2
2
: The differential PSK waveform is
)cos()()( kk
cS tkTtgAtZ
78 Dr. Uri Mahlab
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: Transmitter for differential PSK*
Serial toparallel
converter
Diffphasemod.
Envelopemodulator BPF
(Z(t
3
4
2400br
Data
Binary
Clocksignal
2400 Hz
4
1200
M
rs
Hzfc 1800
600 Hz
79 Dr. Uri Mahlab
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* M-ary Wideband FSK Schemas: Let us consider an FSK scheme witch have the
: following properties
ST
0
2
s
FOR 0
FOR 2)()(
elsewhere 0
Tt0 cos)(
ji
jiTA
tStS
and
tAtS
S
ji
ii
80 Dr. Uri Mahlab
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:Orthogonal Wideband FSK receiver
MAXIMUMSELECTOR
ST
0
ST
0
ST
0
Z(t)
)(tn
noise
gausian
)(1 tS
)(2 tS
)(tSM
.
.
.
.
)(1 tY
)(2 tY
)(tYM
81 Dr. Uri Mahlab
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: The filter outputs are
component noise The-)(
outputfilter th -j theofcomponent signal The-)(
where
)()(
)()()()(
M1,2,....,j ,)]()()[()(
0
0
0 0
1
0
1
S
sj
sj
sjsj
T T
jj
T
jsj
Tn
TS
TnTS
dttntSdttStS
dttStntSTY
S S
s
82 Dr. Uri Mahlab
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: N0 is given by
42
0
sTAN
: The probability of correct decoding as
-
11|andsent
112
1113121
)({|,...,}
{ |,...,,}
11
1
11dyyfyYyYP
sentSYYYYYYpP
SYs
yYM
Mc
: In the preceding step we made use of the identity
dyyfyYyXPYXP Y )()|()(
83 Dr. Uri Mahlab
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The joint pdf of Y2 ,Y3 ,…,YM* : is given by
M
iiYMyYSYY yfyyf
iM2
2:|...2 )(),...,(111
84 Dr. Uri Mahlab
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s
s
SY
Y
SY
My
Y
SY
y y M
iiiYc
ii
iY
TA
S
TA
N
yN
Sy
Nyf
yN
y
Nyf
dyyfdyyf
dyyfdyyfP
yN
y
Nyf
i
i
2
22
and
, 2
)(exp
2
1)(
, 2
exp2
1)(
where
)()(
)()(...
and
, 2
exp2
1)(
2
01
2
0
10
2011
0
1|
0
2
0
-
11|
1
11|2
1
0
2
0
11
11
1
11
1 1
where
85 Dr. Uri Mahlab
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Probability of error for M-ary orthogonal* : signaling scheme
86 Dr. Uri Mahlab
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The probability that the receiver incorrectly* decoded the incoming signal S1(t) is
Pe1 = 1-Pe1
The probability that the receiver makes * an error in decoding is
Pe = Pe1
We assume that , and We can see that increasing values of M lead to smaller power requirements and also to more complex transmitting receiving equipment .
2M )inteegr positive a ( log2 ssb rMrr
87 Dr. Uri Mahlab
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In the limiting case as M the probability of error Pe satisfies
7.0r /S if 0
7.0/S if 1
bav
av
b
e
r
P
The maximum errorless rb at W data can be transmittedusing an M- ary orthogonal FSK signaling scheme
eSS
r avavb 2log
7.0
The bandwidth of the signal set as M 88 Dr. Uri MahlabDr. Uri Mahlab