Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK)...
Transcript of Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK)...
PB.19
Coherent PSKThe functional model of passband data transmission systemis
ModulatorSignal
transmissionencoder
im
Carrier signal
)(tsi Channel)(tx
Detectoris x Signaltransmission
decoder m̂
• im is a sequence of symbol emitted from a messagesource.
• The channel is linear, with a bandwidth that is wideenough to transmit the modulated signal and thechannel noise is Gaussian distributed with zeromean and power spectral density 2/oN .
PB.20
Coherent PSK
The following parameters are considered for a signaling scheme:
Probability of error
A major goal of passband data transmission systems is the optimum design of the receiver so as to minimize the average probability of symbol error in the presence of additive white Gaussian noise (AWGN)
eP
PB.21
Coherent PSK
Power spectra
Use to determine the signal bandwidth and co-channel interference in multiplexed systems.
In practice, the signalings are linear operation, therefore, it is sufficient to evaluate the baseband power spectral density.
B 2B
Multiplexer
interference
PB.22
Coherent PSK
Example:– Raised cosine spectrum (α=0.5)– Binary Phase shift keying (BPSK)
=(1+α)Rb=1.5Rb
2Rb
PB.23
Coherent PSK
Bandwidth Efficiency
– Bandwidth efficiency bits/s/Hz
where is the data rate and B is the used channel bandwidth.
Example: Nyquist channel for baseband data transmission
Bandwidth B = W = 1/2Tb.
BRb=ρ
bR
W
bits/s/Hz 22/1/1
===∴b
bb
TT
BR
ρ
PB.24
Coherent PSK
In a coherent binary PSK system, the pair of signals and
used to represent binary symbols 1 and 0, respectively, is defined by
where , and is the transmitted signal energy per bit
)(1 ts
)(2 ts
)2cos(2
)(1 tfTE
ts cb
b π=
)2cos(2
)2cos(2
)(2 tfTE
tfTE
ts cb
bc
b
b πππ −=+=
bT
bTt ≤≤0 bE
PB.25
Coherent PSK
Example:
To ensure that each transmitted bit contains an integral number of cycles of the carrier wave, the carrier frequency
is chosen equal to for some fixed integer n.
[ ] bb
b
bT
cb
bTE
TTE
dttfTE
dttsE bb =⋅=== ∫∫ 22
)2(cos2
)(0
2
0
21 π
cf bTn /
cb fT /2=
PB.26
Coherent PSK
The transmitted signal can be written as
and
where
)()(1 tEts bφ=
)()(2 tEts bφ−=
bcb
TttfT
t <≤= 0 )2cos(2)( πφ
1 )2cos(2)( :Note0
2
2 =
= ∫
bT
cb
dttfT
t πφ
PB.27
Generation of coherent binary PSK signalsTo generate a binary PSK signal, the first step is representing the input binary sequence in polar form with symbols 1 and 0 represented by constant amplitude levels of and , respectively.
This signal transmission encoder is performed by a polar nonreturn-to-zero (NRZ) encoder.
Signaltransmission
encoder
101011 is 101011
−+
=0 is symbolinput 1 is symbolinput
b
bi E
Es
PB.28
Generation of coherent binary PSK signals
The second step is multiplying the carrier encoder output with the carrier
ProductModulator
is
)2cos(2)( tfT
t cb
πφ =
)(tsi
−=−=
===
bicb
b
bicb
b
i
EstfTE
ts
EstfTE
tsts
if)2cos(2
)(
if)2cos(2
)()(
2
1
π
π
bc Tnf /=
PB.29
Detection of coherent binary PSK signals
To detect the original binary sequence of 1s and 0s, we apply the noisy PSK signal to a correlator. The correlatoroutput is compared with a threshold of zero volts.
∫bT
0)(tx
)(tφ
Correlator
X1x Decision
device
0
0 if 00 if 1
1
1
<>
xx
PB.30
Detection of coherent binary PSK signals
Example
If the transmitted symbol is 1,
and the correlator output is
Similarly, if the transmitted symbol is 0, .
)2cos(2
)( tfTE
tx cb
b π=
b
T
cb
b
T
cb
cb
b
T
E
dttfT
E
dttfT
tfTE
dtttxx
b
b
b
=
⋅=
⋅=
=
∫
∫
∫
0
2
0
01
)2(cos2
)2cos(2)2cos(2
)()(
π
ππ
φ
bEx −=1
PB.31
Error probability of binary PSK
We can represent a coherent binary system with a signalconstellation consisting of two message points.
• The coordinates of the message points are all thepossible correlator output under a noiselesscondition.
• The coordinates for BPSK are bb EE − and .
)(tφ
Decisionboundary
bEbE−
PB.32
Error probability of binary PSK
There are two possible kinds of erroneous decision:
– Signal is transmitted, but the noise is such that the received signal point inside region with and so thereceiver decides in favor of signal .
– Signal is transmitted, but the noise is such that the received signal point inside region with and so the receiver decides in favor of signal .
)(2 ts01 >x
)(1 ts
01 <x)(1 ts
)(2 ts
∫ bT
0
)(tφ
X 1x Decisiondevice
0
0if00if1
1
1
<>
xx
)()( twtsi +
PB.33
Error probability of binary PSK
For the first case, the observable element is related to the received signal by
is a Gaussian process with mean :
1x)(tx
[ ]
∫∫∫
+−=
+=
=
b
b
b
T
b
T
i
T
dtttwE
dtttwts
dtttxx
0
0
01
)()(
)()()(
)()(
φ
φ
φ
1x
b
T
b
ii
E
dtttwEE
xExb
−=
+−=
=
∫ ])()([
][
0φ
PB.34
Error probability of binary PSK
Variance is
2
)(2
)()()(2
)()()]()([
)()()()(
)()(
])[(
0
2
0 0
0 0
0 0
2
0
22
o
To
T T o
T T
T T
T
ii
N
dttN
dtduututN
dtduutuwtwE
dtduutuwtwE
dtttwE
xxE
b
b b
b b
b b
b
=
=
−=
=
=
=
−=
∫
∫ ∫
∫ ∫
∫ ∫
∫
φ
φφδ
φφ
φφ
φ
σ
PB.35
Error probability of binary PSK
Therefore, the conditional probability density function of
, given that symbol 0 was transmitted is1x
+−=
−−=
o
b
o NEx
N
xxxf
21
2
211
1
)(exp1
2)(exp
21)0|(
π
σσπ
PB.36
Error probability of binary PSKand the probability of error is
∫
∫∞
∞
+−=
=
0 1
21
0 1110
)(exp1
)0|(
dxN
ExN
dxxfp
o
b
oπ
Putting )(1b
o
ExN
z += , we have
[ ]
=
−= ∫∞
o
b
NE
NE
dzzpob
erfc21
exp10/
210 π
PB.37
Error probability of binary PSK
Similarly, the error of the second kind
==
o
b
NEpp erfc
21
1001 and hence
=
o
be N
Ep erfc21
PB.38
Error probability
The probability of bit error rate is proportional to the distance between the closest points in the constellation.
BPSK Binary FSK
=
=
oo
be N
derfcNEerfcP
221
21
bEbE−
=
=
oo
be N
derfcNEerfcP
221
221
bE
bEd d
PB.39
Transmission Bandwidth
The power spectral density (PSD) of the BPSK for both rectangular and raised cosine rolloff pulse shapes are plotted.
null-to-null bandwidth = 2Rb
=(1+α)Rb=1.5Rb
bps/Hz 5.02
===b
bb
RR
BRρ
PB.40
Quadriphase-shift keying (QPSK)
QPSK has twice the bandwith efficiency of BPSK, since 2 bits are transmitted in a single modulation symbol. The data input is divided into an in-phase stream , and a quadrature stream .
)(tdk )(td I)(tdQ
)(tdk
)(td I
)(tdQ
:1001
:10
:01
PB.41
QPSK
t
1 0 0 1
t
1 0
t
0 1
)(tdk
)(tdI
)(tdQ
bT
bTT 2=
PB.42
QPSKThe phase of the carrier takes on one of four equallyspaced values, such as π/4, 3π/4, 5π/4, and 7π/4.
≤≤−+=elsewhere0
0]4/)12(2cos[2)( Ttitf
TE
ts ci
ππ
where .4,3,2,1=iE is the transmitted signal energy per symbol;T is the symbol duration;
Tnfc /= ;
)2 :(Note bTT =
PB.43
QPSK
The transmitted signal can be written as
)()(
]4/)12sin[(]2sin[2
]4/)12cos[(]2cos[2
]4/)12(2cos[2)(
2211 tsts
itfTE
itfTE
itfTEts
ii
c
c
ci
φφ
ππ
ππ
ππ
+=
−−
−=
−+=
where
]2sin[2)(;]2cos[2)( 21 tfT
ttfT
t cc πφπφ ==
PB.44
QPSK
1is = 2/E or 2/E−
2is = 2/E or 2/E−
1000
01 11
2/E2/E
)(1 tφ
)(2 tφ
PB.45
QPSKEach possible value of the phase corresponds to a unique dibit. For example: Gray code
• only a single bit is change from one dibit to the next
1000
01 11
PB.46
QPSKDifferent QPSK sets can be derived by simply rotating the constellation.
1000
01
11
PB.47
PB.48
Generation of coherent QPSK signals
The incoming binary data sequence is first transformed intopolar form by a nonreturn-to-zero level encoder. The binarywave is next divided by means of a demultiplexer into twoseparate binary sequences.
The result can be regarded as a pair of binary PSKsignals, which may be detected independently due tothe orthogonality of )(1 tφ and )(2 tφ .
PB.49
Demulti-plexerPolar NRZ
10101 is
)2cos(2)(1 tfT
t cb
πφ =
)(ts
is1
is2
X
X
+
)2sin(2)(1 tfT
t cb
πφ =
PB.50
Detection of coherent QPSK signals
∫T
0)(tx
)(1 tφIn-phase channel
X1x Decision
device
0
multiplexer
∫T
0
)(2 tφ
X2x Decision
device
0
0 if 00 if 1
1
1
<>
xx
0 if 00 if 1
2
2
<>
xx
Quadrature channel
PB.51
Error probability of QPSK
The received signal is)()()( twtstx i +=
and the observation elements are
∫∫
+±=
=T
T
dtttwE
dtttxx
0 1
0 11
)()(2/
)()(
φ
φ
∫∫
+±=
=T
T
dtttwE
dtttxx
0 2
0 22
)()(2/
)()(
φ
φ
PB.52
As a coherent QPSK is equivalent to two coherent binaryPSK systems working in parallel and using two carriers thatare in phase quadrature.
Hence, the average probability of bit error in each channelof the coherent QPSK system is
=
=
oo NE
NEp
2erfc
212/erfc
21
PB.53
Error probability of QPSK
As the bit error in the in-phase and quadrature channels ofthe coherent QPSK system are statistically independent, theaverage probability of a correct decision resulting from thecombined action of the two channels is
+
−=
−=
−=
oo
o
c
NE
NE
NE
pp
2erfc
41
2erfc1
2erfc
211
)1(
2
2
2
PB.54
The average probability of symbol error for coherent QPSKis therefore
12/if2
erfc
2erfc
41
2erfc
1
2
>>
≈
−
=
−=
oo
oo
ce
NENE
NE
NE
pp
PB.55
In a QPSK system, since there are two bits per symbol, thetransmitted signal energy per symbol is twice the signalenergy per bit,
bEE 2=
and then
≈
o
be N
Ep erfc t
1 0 0 1
t
1 0
t
0 1
)(tdk
)(tdI
)(tdQ
PB.56
With Gray encoding, the bit error rate of QPSK is
Therefore, a coherent QPSK system achieves the sameaverage probability of bit error as a coherent binary PSKsystem for the same bit rate and the same ob NE / but usesonly half the channel bandwidth.
=
o
b
NEerfc
21BER
PB.57
Note: The probability of bit error rate is also proportional to the distance between the closest points in the constellation.
=
=
oo
b
Nd
NE
2erfc
21erfc
21BER
b
b
E
E
Ed
2
2/22
2/2
=
=
=
PB.58
Transmission Bandwidth
Power spectral density (PSD) of the QPSK for both rectangular and raised cosine rolloff pulse shapes:
null-to-null bandwidth = Rb
=(1+α)Rb/2=0.75Rb
bps/Hz 1===b
bb
RR
BRρ
PB.59
M-ary PSK
During each signaling interval of duration T, one ofthe M possible signals
iiM
tfTEts ci ,...,2,1)1(22cos2)( =
−+=
ππ
is sent.
MTTMEE
b
b
2
2
loglog
==
PB.60
M-ary PSK
( ) ( ) )(2
1cos)(2
1cos)( 21 tiEtiEtsi φπφπ
−−
−=
tfT
t cπφ 2cos2)(1 =
tfT
t cπφ 2sin2)(2 =
PB.61
M-ary PSK
The signal constellation of M-ary PSK consists of Mmessage points which are equally spaced on a circle ofradius E . For example, the constellation of8-ary phase-shift keying is
≈
MNEP
oe
πsinerfc 4≥ME
The average probability of symbol error is
=
oNderfc
2d( )ME /sin π
M/π
PB.62
Transmission Bandwidth
Power spectral density (PSD) of the M-ary PSK for both rectangular and raised cosine rolloff pulse shapes:
PB.63
Transmission Bandwidth
Null-to-null bandwidth efficiency of a M-ary PSK signal:
28.523.418.51410.510.5
32.521.510.5
643216842M
bps/Hz log21
log/2 22
MMR
RBR
b
bb ===ρ
ρ
)10BER(/ 6−=ob NE