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J.1
Passband Data Transmission IReferences
Phase-shift keyingChapter 4.1-4.3, S. Haykin, Communication Systems, Wiley.
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Introduction
In digital passband transmission, the incoming datastream is modulated onto a carrier with fixedfrequency and then transmitted over a band-passchannel.
In baseband pulse transmission, a data streamrepresented in the form of a discrete pulse-amplitudemodulated (PAM) signal is transmitted over a low-pass channel.
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Passband digital transmission allows more efficient use of theallocated RF bandwidth, and flexibility in accommodating differentbaseband signal formats.
ExampleMobile Telephone Systems
GSM: GMSK modulation is used (a variation of FSK)
IS-54: π/4-DQPSK modulation is used (a variation of PSK)
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The modulation process making the transmissionpossible involves switching (keying) the amplitude,frequency, or phase of a sinusoidal carrier inaccordance with the incoming data.
There are three basic signaling schemes:Amplitude-shift keying (ASK)Frequency-shift keying (FSK)Phase-shift keying (PSK)
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ASK
PSK
FSK
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Unlike ASK signals, both PSK and FSK signals have aconstant envelope.
PSK and FSK are preferred to ASK signals forpassband data transmission over nonlinear channel(amplitude nonlinearities) such as micorwave link andsatellite channels.
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Classification of digital modulation techniques
Coherent and Noncoherent
Digital modulation techniques are classified intocoherent and noncoherent techniques, depending onwhether the receiver is equipped with a phase-recovery circuit or not.
The phase-recovery circuit ensures that the localoscillator in the receiver is synchronized to theincoming carrier wave (in both frequency and phase).
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Phase Recovery (Carrier Synchronization)
Two ways in which a local oscillator can be synchronized with anincoming carrier wave
transmit a pilot carrier
use a carrier-recovery circuit such as a phase-locked loop (PPL)
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M-ary signaling
In an M-ary signaling scheme, there are M possible signalsduring each signaling interval of duration T. Usually,
nM 2= and bnTT = where bT is the bit duration.
In passband transmission, we have M-ary ASK, M-aryPSK, and M-ary FSK digital modulation schemes.We can also combine different methods:
M-ary amplitude-phase keying (APK)M-ary quadrature-amplitude modulation (QAM)
In baseband transmission, we have M-ary PAM
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M-ary signaling schemes are preferred over binarysignaling schemes for transmitting digital information overband-pass channels when the requirement is to conservebandwidth at the expense of increased power.
The use of M-ary signaling enables a reduction intransmission bandwidth by the factor Mn 2log= overbinary signaling.
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Coherent PSKThe functional model of passband data transmission systemis
ModulatorSignal
transmission encoder
im
Carrier signal
)(tsi Channel)(tx
Detectoris x Signal transmission
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 .
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The following parameters are considered for a signalingscheme:
Probability of errorA major goal of passband data transmission systems isthe optimum design of the receiver so as to minimizethe average probability of symbol error in the presenceof additive white Gaussian noise (AWGN)
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Power spectraUse to determine the signal bandwidth and co-channelinterference in multiplexed systems.In practice, the signalings are linear operation,therefore, it is sufficient to evaluate the basebandpower spectral density.
Bandwidth Efficiency
Bandwidth efficiency BRb=ρ bits/s/Hz
where bR is the data rate and B is the usedchannel bandwidth.
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In a coherent binary PSK system, the pair of signals )(1 tsand )(2 ts used to represent binary symbols 1 and 0,respectively, is defined by
)2cos(2)(1 tfTEts cb
b π=
)2cos(2)2cos(2)(2 tfTEtf
TEts c
b
bc
b
b πππ −=+=
where bTt ≤≤0 , and bE is the transmitted signal energyper bit.
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For example,
[ ] bb
b
bT
cb
bT
ETTEdttf
TEdttsE
bb=⋅=== ∫∫ 2
2)2(cos2)(0
2
0
21 π
To ensure that each transmitted bit contains an integralnumber of cycles of the carrier wave, the carrier frequency
cf is chosen equal to bTn / for some fixed integer n.
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The transmitted signal can be written as)()(1 tEts bφ= and
)()(2 tEts bφ−=
where bcb
TttfT
t <≤= 0 )2cos(2)( πφ
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Generation of coherent binary PSK signals
To generate a binary PSK signal, we have to represent theinput binary sequence in polar form with symbols 1 and 0represented by constant amplitude levels of bE+ and
bE− , respectively.
ProductModulator
Signal transmission
encoder
10101 is
)2cos(2)( tfT
t cb
πφ =
)(tsi
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• This signal transmission encoder is performed by apolar nonreturn-to-zero (NRZ) encoder.
−+
=0 is symbolinput 1 is symbolinput
b
bi E
Es
• The carrier frequency bc Tnf /= where n is a fixedinteger.
−=−=
===
bicb
b
bicb
b
i
EstfTEts
EstfTEts
ts if)2cos(2)(
if)2cos(2)()(
2
1
π
π
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Detection of coherent binary PSK signals
To detect the original binary sequence of 1s and 0s, weapply 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
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Example: If the transmitted symbol is 1,
)2cos(2)( tfTEtx cb
b π=
and the correlator output is
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
Similarly, If the transmitted symbol is 0, bEx −=1 .
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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−
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There are two possible kinds of erroneous decision:• Signal )(2 ts is transmitted, but the noise is such that
the received signal point inside region with 01 >xand so the receiver decides in favor of signal )(1 ts .
• Signal )(1 ts is transmitted, but the noise is such thatthe received signal point inside region with 01 <xand so the receiver decides in favor of signal )(2 ts .
∫bT
0
)(tφ
X 1x Decisiondevice
0
0 if 00 if 1
1
1
<>
xx
)()( twtsi +
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For the first case, the observable element 1x is related to thereceived signal )(tx by
[ ]
∫∫∫
+−=
+=
=
b
b
b
T
b
T
i
T
dtttwE
dtttwts
dtttxx
0
0
01
)()(
)()()(
)()(
φ
φ
φ
1x is a Gaussian process with mean 1x :
b
T
b
ii
E
dtttwEE
xExb
−=
+−=
=
∫ ])()([
][
0φ
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and variance σ :
2
)(2
)()()(2
)()()]()([
)()()()(
)()(
])[(
0
2
0 0
0 0
0 0
2
0
22
o
To
T To
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
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Therefore, the conditional probability density function of1x , given that symbol 0 was transmitted is
+−=
−−=
o
b
o NEx
N
xxxf
21
2
211
1
)(exp1
2)(exp
21)0|(
π
σσπ
J.26
and 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
∫∞
−=u
dzzu )exp(2)erfc( 2
π
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Similarly, the error of the second kind
==
o
b
NEpp erfc
21
1001 and hence
=
o
be N
Ep erfc21
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Quadriphase-shift keying (QPSK)
QPSK has twice the bandwith efficiency of BPSK,since 2 bits are transmitted in a single modulationsymbol. The data input )(tdk is devided into an in-phase stream )(td I , and a quadrature stream )(tdQ .
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t
1 0 0 1
t
1 0
t
0 1
)(tdk
)(tdI
)(tdQ
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The 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 =
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Each possible value of the phase corresponds to aunique dibit.
For example, 10 for i=1, 00 for i=2, 01 for i=3and 11 for i=4.(only a single bit is change from one dibit to thenext)
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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 ==
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Input dibit Phase of QPSK 1is 2is10 π/4 2/E 2/E−00 3π/4 2/E− 2/E−01 5π/4 2/E− 2/E11 7π/4 2/E 2/E
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The constellation of QPSK is
2/E
)(1 tφ
)(2 tφ
)10(
)11(
)00(
)01(
2/E
2/E−
2/E−
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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 .
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Demulti-plexerPolar NRZ
10101 is
)2cos(2)(1 tfT
t cb
πφ =
)(ts
is1
is2
X
X
+
)2sin(2)(1 tfT
t cb
πφ =
J.38
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
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J.39
Error probability of QPSK
The received signal is)()()( twtstx i +=
and the observation elements are
∫∫
+±=
=
b
b
T
b
T
dtttwE
dtttxx
0 1
0 11
)()(
)()(
φ
φ
∫∫
+±=
=
b
b
T
b
T
dtttwE
dtttxx
0 2
0 22
)()(
)()(
φ
φ
J.40
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
21
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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
J.42
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
22
J.43
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
Ep2
erfct
1 0 0 1
t
1 0
t
0 1
)(tdk
)(tdI
)(tdQ
J.44
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
NE
2erfc
21BER
23
J.45
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.
J.46
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 ofoctaphase-shift keying is
≈
MNEP
oe
πsinerfc 4≥M
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J.47
Power spectra of M-ary PSK signals
The symbol function is
≤≤=otherwise0
02)( Tt
TE
tg
where MTT b 2log= and bT is the bit duration.
As the energy spectral density is the magnitude of thesignal’s Fourier transform, the baseband powerspectral density is
)log(sinclog2)(
)(sin2)(
22
2
2
2
MfTMETf
TfEfS
bb=
=ππ
J.48(Normalized to bfT )
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Bandwidth efficiencyThe bandwidth required to pass M-ary signal (mainlobe) is given by
MR
MT
TB
b
b
2
2
log2log
2
0)2sinc(2
=
=
== Q
Therefore, the bandwidth efficiency is
2log2 MBRb
=
=ρ