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DIGITAL CARRIER MODULATION SCHEMES. Dr.Uri Mahlab. 1. Dr. Uri Mahlab. 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 * - PowerPoint PPT Presentation
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1 Dr. Uri Mahlab
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
2 Dr. Uri Mahlab
The modulation waveforms for transmitting :binary information over
bandpass channels
ASK
FSK
PSK
DSB
3 Dr. Uri Mahlab
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
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
5 Dr. Uri Mahlab
: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
6 Dr. Uri Mahlab
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
7 Dr. Uri Mahlab
: The received signal + noise
dbdb
db
db
tkTttTk
tntTkt
tntTkt
tV
)1(
)(])1([s
or
)(])1([s
)(
2
1
8 Dr. Uri Mahlab
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
9 Dr. Uri Mahlab
: Receiver structure
Thresholddevice or A/D
converter
) V0(t
Filter)H(f output
Sample everyTb seconds
)()()( tntztv
10 Dr. Uri Mahlab
:}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*
)()()( 000 bbb kTnkTskTV
12 Dr. Uri Mahlab
: 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 )()()(
13 Dr. Uri Mahlab
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 )()()(
)(
14 Dr. Uri Mahlab
: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
15
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
16 Dr. Uri Mahlab
: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
17 Dr. Uri Mahlab
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
18 Dr. Uri Mahlab
: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
19 Dr. Uri Mahlab
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
20 Dr. Uri Mahlab
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
21 Dr. Uri Mahlab
: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
22 Dr. Uri Mahlab
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
23 Dr. Uri Mahlab
: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
24 Dr. Uri Mahlab
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
25 Dr. Uri Mahlab
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
26 Dr. Uri Mahlab
: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()([)( *
27 Dr. Uri Mahlab
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
: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
29 Dr. Uri Mahlab
: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
)()()()(
)]()()[()(
(# #)
30 Dr. Uri Mahlab
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
31 Dr. 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 ) - 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
32
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
33 Dr. Uri Mahlab
:Solution
34 Dr. Uri Mahlab
=Probability of error = Pe*
:Solution Continue
35 Dr. Uri Mahlab
* 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
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
37 Dr. Uri Mahlab
: 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
)(
38 Dr. Uri Mahlab
: 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
39 Dr. Uri Mahlab
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
40 Dr. Uri Mahlab
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)]()()[()(
41 Dr. Uri Mahlab
: 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
42 Dr. Uri Mahlab
: 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
43 Dr. Uri Mahlab
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
44 Dr. Uri Mahlab
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
45
: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
pdf’s of the envelope of the noise and the signal* : pulse noise
47 Dr. Uri Mahlab
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
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
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
: 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
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
: 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
av
bave
EQ
Tsp
2
2
:error ofy probabilit theexpresscan we
54 Dr. Uri Mahlab
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
DPSK demodulator
Filter tolimit noise
power
Delay
Lowpassfilter or
integrator
Thresholddevice
(A/D)
Z(t)
)(tn
bT
kb̂
bkTat
sample
56 Dr. Uri Mahlab
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
* 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
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
Dr. Uri Mahlab
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
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
. 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
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
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
Probability of error for binary digital modulation* :schemes
65 Dr. Uri Mahlab
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
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
Phasor diagram for QPSK
)45cos( and )45cos( 00 tAtA cc That are derived from a coherent local carrier
reference tA ccos
68
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
Z(t)
)(tn
)45cos( tA c
)45cos( tA c
ST
0
ST
0
)(01 SkTV
)(02 SkTV
QPSK receiver scheme
70 Dr. Uri Mahlab
: 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
Probability of error of QPSK:
2
2
0
0
002
0011
2N
LQ
))((
))((
ecs
s
sec
PTA
Q
LTnP
LTnPP
72 Dr. Uri Mahlab
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
Phasor diagram for M-ary PSK ; M=8
74 Dr. Uri Mahlab
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
*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
* 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
: 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
: 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
* 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
: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
: 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
: 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
The joint pdf of Y2 ,Y3 ,…,YM* : is given by
M
iiYMyYSYY yfyyf
iM2
2:|...2 )(),...,(111
84 Dr. Uri Mahlab
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
Probability of error for M-ary orthogonal* : signaling scheme
86 Dr. Uri Mahlab
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
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
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