6.8 linear modulation techniques
DESCRIPTION
6.8 Linear Modulation Techniques. a(t) = . (1) Cartesian basis : . s(t) = s I (t)cos(2 π f c t) - s Q (t)sin(2 π f c t). (2) Polar basis : . s(t) = a(t)cos( 2 π f c t + θ (t) ). envelope of s(t) given as . θ (t) =. phase of s(t) given as . 6.8 Linear Modulation Techniques - PowerPoint PPT PresentationTRANSCRIPT
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6.8 Linear Modulation Techniques
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6.8 Linear Modulation Techniques
linear modulation: carrier’s amplitude carrier varies linearly with m(t)• bandwidth efficient • attractive for systems with limited spectrum (e.g. wireless)
s(t) = sI(t)cos(2πfct) - sQ(t)sin(2πfct)(1) Cartesian basis:
s(t) = a(t)cos( 2πfct + θ(t) )(2) Polar basis:
constructing signal constellations of linear modulated signals
a(t) = )()( 22 tsts QI envelope of s(t) given as
θ(t) =
)()(
tan 1
tsts
I
Qphase of s(t) given as
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A = signal amplitude fc = carrier frequency s(t) = transmitted signalm(t) = modulating digital signalmR + jm1(t) = complex envelope representation of m(t)
s(t) = Re[Am(t) exp(j2fct)]
s(t) = A[ mR(t) cos(2fct) - m1(t) sin(2fct) ]6.65
• generally linear modulation doesn’t have constant envelope
• non-linear modulation has either linear or constant carrier envelope
Provides Basis for any transmitted signal
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6.8.1 BPSK
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6.8.1 BPSK
phase of constant amplitude carrier switched between 2 values• phase switches based on 2 possible symbols, m1 and m2
• normally phase separated by 180o
Assume rectangular pulse shape: p(t) =
b
bTTt
rect2/
b
bTE2• Ac =
• carrier given by Accos(2πfct + θc)
• bit energy Eb = ½ Ac2Tb
• c = phase shift in carrier
For a carrier with frequency = fc and amplitude (volts) = Ac
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1. Transmitted BPSK signal is
6.66sBPSK(t)= bccb
b Tt tfTE
0)2cos(2
binary 1
binary 0 sBPSK(t) = bcc
b
b Tt tfTE
0)2cos(2
= bccb
b TttfTE
0)2cos(2
bE bEBPSK constellation
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Basis Signal Set consists of one waveform (symbol), 1(t)
1(t) = bcb
Tt tfT
0)2cos(2 6.60
BPSK signal set expressed in terms of basis
SBPSK = )(),( 11 tEtE bb 6.61
6.67bcb
b Tt tfTE
0)2cos(2 sBPSK(t)= m(t)
generalize m1 and m2 as binary signal m(t)
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BPSK signal is a double sideband amplitude modulated waveform • suppressed carrier
• applied carrier = Accos(2fct)
• data signal = m(t) BPSK signal can be generated using balanced modulator
Spectrum and Bandwidth of BPSK• BPSK signal can be expressed in complex envelope form • use polar baseband data waveform of m(t),
sBPSK = Re[gBPSK(t) exp(j2 fct)] 6.68
gBPSK(t) = cbb jtmTE exp)(2 6.69
where gBPSK(t) is complex envelope of sBPSK(t) given by:
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PSD of baseband complex envelope can be shown to be
PgBPSK(f) =
2sin2
b
bb fT
fTE
6.70
PSD for BPSK at RF passband can be evaluated by translating baseband spectrum according to 6.41.
22
)()(sin
)()(sin
2 bc
bc
bc
bcb
TffTff
TffTffE
6.71PgBPSK(f) =
Ps(f) = ¼[Pg(f-fc) + Pg(-f-fc)]
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1
0
-1
T 2T 3T 4T 5T 6T
1 1 0 0 1 1
Peak PSD
fc-5f0 fc-3f0 fc-f0 fc fc+f0 fc+3f0 fc+5f0
(occurs with 101010…pattern)
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norm
aliz
ed P
SD (d
B)
fc-3Rb fc-2Rb fc-Rb fc fc+Rb fc+2Rb fc+3Rb
0
-10
-20
-30
-40
-50
-60
rectangular pulsesRC shaping with = 0.5
PSD of BPSK signal with rectangular pulse and RC pulse shaping• null-to-null BW = 2Rb (Rb = bit rate)
pulse shape % pulse energy occupied BWrectangular pulse 90% 1.6 Rb
pulse with RC filter, = 0.5 100% 1.5 Rb
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2. Received BPSK Signal
BPSK Receiver can be expressed as
assumes no multipath impairments induced by channel
ch = phase shift from channel time delay
c = phase shift in carrier
fc = carrier frequency
6.72
sBPSK(t)=
= )2cos(2
)( tfTE
tm cb
b
)2cos(2
)( chccb
b tfTE
tm
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BPSK Receiver uses coherent (synchronous) demodulation
• receiver requires information about c & fc
• options to recover fc and c include:
(1) send low level pilot carrier signal & use PLL
(2) synthesize carrier phase & frequency e.g. use Costas loop or - Squaring loop
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BPSK Receiver with squaring loop
sBPSK(t) = m(t)Accos(2fct+)(1) received signal is
(2) sBPSK(t)2 generates dc signal & amplitude varying sinusoid at 4fc
(3) dc signal is filtered using BPF with center frequency = 2fc
(4) frequency divider () used to recreate cos(2fct +)
12 3 4 5 m(t)
bitsynch
integrate & dump
sBPSK(t) square law
frequency f/2
2fc
m2(t)A2ccos2(2fct+)
cos(4fct+2) cos(2fct+)
m(t)Accos2(2fct+)
6
(5) output of mixer
6.73)2(cos2)( 2 tfTEtm cb
b
)24cos(
21
212)( tf
TEtm cb
b
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(6) mixer output applied to integrate & dump • forms LPF segment of detector• if transmit & receive pulse shapes match optimum detection• bit synch facilitates sampling of integrator output at end of Tb
• at end of Tb integrator switch closes & output dumped to decision circuit
(7) decision circuit uses threshold to determine if bit is a 1 or 0• threshold must be tuned to minimize error• if 1 or 0 are equally likely use midpoint of detector voltage output level
Decision Boundary
W
t‘0’
‘1’S
N
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Probability of Bit Error for BPSK• many modulation schemes in AWGN channel – use Q-function of distance between signal points
0
2NEQ bPe,BPSK = 6.74
Bit Error Probability for BPSK from substitution into 6.62
• for BPSK – the distance between points in constellation is given by
bE2d12 =
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6.8.2 DPSK
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6.8.2 DPSK• non-coherent PSK receiver doesn’t need reference signal• simplified receiver - easy & cheap to build, widely used
Let {mk} = input binary sequence
{dk} = differentially encoded output sequence
dk = kth differentially encoded output, generated from compliment of modulo 2 sum of mk and d k-1
dk = mk dk-1
net effect also achieved by following rule: • if mk = 1 dk = dk-1
• if mk = 0 dk = dk-1
no symbol transition with mk = 1 possible synchronization issue
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876543210k
10011011-{dk-1}10101001-{mk}
010011011{dk}
o o = 1 o ō = 0
e.g. for given data stream: {mk}
• less energy efficiency - about 3dB < coherent PSK
• average probability of bit error: PeDPSK =
0exp
21
NEb 6.75
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Transmitter: DPSK obtained by passing dk to product modulator
DPSK signal dkmk
cos(2fct)dk-1
delayTb
DelayTb
DPSK Signal
logic circuit
thresholddevice
integrate & dump
Receiver: • input signal demodulated• original sequence recovered by undoing differential encoding
dk = mk dk-1 mk= dk dk-1
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6.8.3 QPSK
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6.8.3 QPSK• 2 bits transmitted per symbol 2bandwidth efficiency of BPSK • symbol determined from 4 possible phases
• Ts = 2Tb (one symbol time = two bit periods)
• Es = 2Eb bit energy = ½ symbol energy
6.76
2)1(2cos2 itf
TE
cs
ssQPSK(t)= 0 t Ts
QPSK signal sQPSK(t) can be expressed as:
i = 1,2,3,4
s
s
TE2
is the symbol’s amplitude
2)1( i is phase of the symbol (0, 90 , 180 , 270)
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define a Basis for S over interval 0 t Ts: {1(t), 2(t)}
1(t) = )2cos(2 tfT c
s 2(t) = )2sin(2 tf
T cs
6.78)(2
)1(sin)(2
)1(cos 21 tiEtiE ss
si(t)=
define QPSK signal set: S = {s1(t), s2(t), s3(t), s4(t)}
rewrite equation 6.76 over 0 t Ts
6.77)2sin(2
)1(sin2)2cos(2
)1(cos2 tfiTEtfi
TE
cs
sc
s
s
sQPSK(t)=
(cos(α + β) = cosα cosβ - sinα sinβ)
QPSK Basis
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Alternate view of QPSKparallel combination of 2 BPSK modulators operating in quadrature phase to each other
m(t) = 110010100011m1(t) = 101101
m2(t) = 100001
demultiplex binary stream m(t) into m1(t) and m2(t)
1 for 0 ≤ t ≤ 2T 0 otherwise
p(t) =
bk,i = +1 for binary ‘1’
bk,i = -1 for binary ‘0’
m1(t) = k
Ik kTtpb )(, m2(t) = k
Qk kTtpb )(,
p(t) = pulse shape, assume a rectangular pulse:
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0
3/2
/2phase shift key of binary streams • 0 and phase shift mI(t)
• /2 and 3/2 phase shift mQ(t)
Ac = s
s
TE2
sI(t) = AcmI(t)cos(2fct)
sQ(t) = AcmQ(t)sin(2fct)
mI(t) =
2)1(cos i for i = 1, 3
mQ(t) =
2)1(sin i for i = 2,4
sQPSK(t) = sI(t) + sQ(t)
= AcmI(t)cos(2fct) + AcmQ(t)sin(2fct)
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sI(t) = mI(t)cos(2fct)bit = 1 sI(t) = cos(2fct) for 0 ≤ t ≤ 2T
bit = 0 sI(t) = -cos(2fct) for 0 ≤ t ≤ 2T
sQ(t) = mQ(t)sin(2fct)bit = 1 sQ(t) = sin(2fct) for 0 ≤ t ≤ 2T bit = 0 sQ(t) = -sin(2fct) for 0 ≤ t ≤ 2T
sQPSK(t) = cos(2fct) sin(2fct)
normalize Ac =1
I bit Q bit sQPSK(t)1 1 cos(2fct) + sin(2fct)0 1 -cos(2fct) + sin(2fct)1 0 cos(2fct) - sin(2fct)0 0 -cos(2fct) - sin(2fct)
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QPSK Constellation Diagram has four points
Q
I
/4
54 7/4
3/4
47,
45,
43,
4
M2 =
sE
sE2
0
3/2
/2Q
I M1 =
23,,
2,0
• rotate constellation by /4obtain new QPSK signal set
Es = 2Eb
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Signal Space Characterization of QPSK Signal Constellations
ith QPSK signal, based on message points (si1, si2) defined in tables
for i = 1,2 and 0 ≤ t ≤ Ts
si(t) = si1 1(t) + si22(t) (3.36)
π/4003π/4015π/4117π/410
si2 si1 grey coded
QPSK signalbinary symbol
bE bEbEbE
bE
bEbE
bE
00000π/201
0π1103π/210
si2si1 grey coded
QPSK signalbinary symbol
bE2
bE2
bE2
bE2
bE20
3/2
/2/4
54 7/4
3/4
bE
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Thus PeQPSK = PeBPSK
• QPSK has 2 spectral efficiency of BPSK & same energy efficiency
• QPSK can be differentially encoded - allows non-coherent detection
• since Ts = 2Tb Es = 2Eb
• assumes AWGN channel
• distance between adjacent points = bs EE 22
Average probability of bit error: PeQPSK
6.79PeQPSK =
0NEQ s =
0
2NEQ b
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S1 =
23,,
2,0
Baseband QPSK Signal in Time Domain• Ts = 0.1s• Tb = 0.05 Rb = 20bps
t
Ac
1 1 0 1
I1
0
-12T 4T 6T 8T
Q1
0
-1 2T 4T 6T 8T
1 1 0 1
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QPSK data stream• Ts = 0.1s• Tb = 0.05 Rb = 20bps
t
Ac
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QPSK Spectrum & Bandwidth
• PSD of QPSK using rectangular pulses given by PQPSK(f)
• similar to PSD of BPSK, replace Tb with Ts
6.80PQPSK(f) =
22
)()(sin
)()(sin
2 sc
sc
sc
scsTff
TffTff
TffE
22
)(2)(2sin
)(2)(2sin
bc
bc
bc
bcb Tff
TffTff
TffE
PQPSK(f) =
•Wnull-QPSK = Rb
•Wnull-BPSK = 2Rb
Wnull= Null to Null Bandwidth
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norm
aliz
ed P
SD (d
B)
fc-Rb fc-½Rb fc fc+½Rb fc+Rb
0-10-20
-30-40-50-60
• rectangular pulse• RC pulse shaping with = 0.5
PSD of QPSK signal
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QPSK Transmitter - based on modulating 2 modulated BPSK signals
(1) m(t) = bi-polar NRZ input with bit rate = Rb
(2) split m(t) into even and odd stream, mI(t) & mQ(t) each with ½ Rb
(3) modulate each stream with quadrature carriers 1(t), 2(t)
(4) sum two resultant BPSK signals to produce QPSK output
(5) band pass filter confines signal to allocated passband*pulse shaping normally done at baseband, prior to modulation
m(t) at Rb Serial -Parallel
mI(t) at ½ Rb
mQ(t) at ½ Rb
QPSKoutput
LO
90o
2(t)
1(t)
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Coherent QPSK Receiver
(1) front end BPF removes out of band noise
(2) filtered output is split
(3) each part coherently demodulated using I & Q carriers
(4) revover carriers coherently from received signals with squaringloop
(5) demodulated output passed to decision circuit which generates I & Q streams
(6) I & Q streams are multiplexed to recover original binary stream
recovered signal
carrier recovery
90o
receivedsignal symbol timing
recovery MUX
decisioncircuit
decisioncircuit
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6.8.5 Offset QPSK
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6.8.5 Offset QPSK (OQPSK)
QPSK is ideally constant envelope (e.g. amplitude is constant)
Pulse shaped (bandlimited) QPSK signals lose constant envelope
• if phase shift = signal envelope can momentarily pass through 0 (zero crossing)
• hardlimiting or non-linear amplification of zero crossings brings back filtered side lobes
- fidelity of signal at small voltages is lost in transmission- sidelobes result in spectral widening
• Use of linear amplifiers to amplify pulses will avoid this but will result in inefficient power use
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OQPSK (offset QPSK)
• phase (bit) transition instants of mI(t) & mQ(t) are offset by Tb
• phase transitions occur every Tb = ½Ts
• max phase shift = 90o (/2) only one bit stream value changes• ensures smaller phase transitions applied to RF amplifier reduces spectral growth after amplification
QPSK: • phase (bit) transitions of mI(t) & mQ(t) occur at same time instants• phase transitions occur every Ts = 2Tb
• maximum phase transition = 180o () both mI(t) & mQ(t) change• non-linear amplification results in spectral widening
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OQPSK (offset QPSK)
• bit transitions of mI(t) & mQ(t) are offset by Tb in relative alignment
- phase transistions occur every Tb = ½ Ts
- at any time, only one bit stream can change values
maximum phase shift of transmitted signal limited to 90°
m1 m3 m5 m7 m9 m11 m13
-Tb 0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb 11Tb 12Tb13Tb
mI(t)m0 m2 m4 m6 m8 m10 m12
mQ(t)
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0
3/2
/2
0
3/2
/2
OQPSKpossible phase shifts
OQPSKpossible phase shifts
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s2(t) and s2_offset(t)
s1(t) and s2_offset(t)
Offset QPSK
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Ts = symbol periodTb = bit period
mI(t) = even bit streammQ(t-Tb) = odd bit streams, offset by Tb
main differences from QPSK is time alignment of mI(t) & mQ(t)
sI(t) = AcmI(t)cos(2fct)
sQ(t) = AcmQ(t-Tb)sin(2fct)
sOQPSK(t) = sI(t) + sQ(t)
= AcmI(t)cos(2fct) + AcmQ(t-Tb)sin(2fct)
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OQPSK vs QPSK
• OQPSK switching occurs at Tb vs 2Tb for QPSK
• OQPSK eliminates 180° phase transition
• spectrum of OQPSK = spectrum of QPSK – unaffected by offset alignment of bit streams
• OQPSK retains bandlimited nature even with non-linear amplification
- critical for low power operations
• OQPSK appears to perform better than QPSK with phase jitter from noisy reference signals
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6.8.6 /4 QPSK
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6.8.6 /4 QPSK
• compromise between OQPSK & QPSK• either coherent or non-coherent demodulation• maximum phase change limited to 135o
- 180o for QPSK- 90o for OQPSK
• preservation of constant BW property in between 2 variants• performs better than both in multipath spread & fading
/4 DQPSK differential encoded version• facilitate differential detection or coherent modulation
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/4 QPSK modulation • modulated signal selected from 2 QPSK constellations shifted by /4 • for each symbol switch between constellations –total of 8
symbols states 4 used alternately
• phase shift between each symbol = nk = /4 , n = 1,2,3
- ensures minimal phase shift, k = /4 between successive symbols
- enables timing recovery & synchronizationQ
Iall possible signal transitions
= possible states for k for k-1 = n/4= possible states for k for k-1 = n/2
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6.8.7 /4 Transmitter
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6.8.7 /4 QPSK Transmitter
(i) partition input mk into symbol stream mIk, mQk
(ii) produce pulses Ik and Qk by signal mapping during [kT,(k+1)T]
(iii) form I(t) & Q(t) from p(t), Ik, Qk & modulate by quadrature carriers
(iv) pre-modulation pulse shaping
/4 QPSK Transmittercos wct
sin wct
I(t)
Q(t)
/4QPSKoutput
amplifier
mkSerial -Parallel
Signal Mapping
mIk
mQk
Ik
Qk
04/22/23 49
(i) input bit stream m(t) partitioned into symbol streams mIk, mQk by serial-parallel conversion
• each symbol stream with symbol rate, Rs = ½Rb
• for symbol at k+1, phase shift = k is a function of mIk, mQk
-/400-3/4103/401/411
phase shift kinputs
mIk, mQk
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Ik = cosk
= Ik-1cos k - Qk-1sin
k
6.81
Qk = sink
= Ik-1sin k - Qk-1cos k
6.82
k = k-1 + k 6.83
(ii) signal mapping circuit produces Ik & Qk during kT t (k+1)T
• Ik = kth in-phase pulse
• Qk = kth quadrature pulse
• k = phase of kth symbol is a function of k
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(iii) Ik & Qk bit streams are separately modulated by carriers in quadrature to each other (same as QPSK)
6.86
1
0)2/(
N
kssk TkTtpQQ(t) =
1
0)2/(sin
N
kssk TkTtp=
6.85
1
0)2/(
N
kssk TkTtpII(t) =
1
0)2/(cos
N
kssk TkTtp=
s/4(t) = I(t)coswct – Q(t)sinwct 6.84
• transmitted /4 wave form given by:
• p(t) = pulse shape
• Ts = symbol period
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(iv) Ik & Qk often filtered by RC pulse shaping filters, pre-modulation
• reduce bandwidth occupancy• reduces spectral restoration problem
- significant in fully saturated non-linear amplifier systems
Ik , Qk , & peak values of I(t), Q(t) limited to 5 values:
• 0
• 1
• 21
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6.8.8 /4 Detection
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non-coherent differential detection can be used
• even without differential encoding
• bit information is completely contained in k - k = relative phase difference of carrier between 2 adjacent symbols
• easier to implement, compared to coherent detection
• BER performance of non-coherent differential detection ≈ 3dB less than BER performance of QPSK
coherent detection BER performance of = QPSK
6.8.8 /4 QPSK Detection
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types of detection techniques (1) baseband differential detection (2) IF differential detection (3) FM discriminator detection
Differential Detection doesn’t rely on phase synchronization
- offers low error floor for low bit rate, fast fading Rayleigh channels
• in IF & baseband detection - k decision based on determination of cosk & sink
• FM discrimination detects k non-coherently
• all 3 have similar BER performance
• important implementation issues with each
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(1) /4QPSK Base Band Differential Detector
(i) signal quadrature demodulated using local oscillators (LO)
(ii) LPF I & Q arms of demodulator output wk & zk
(iii) pass wk & zk through decoders output xk & yk
(iv) decision circuit used to determine SI from xk & SQ from yk
zk
wk
recovered signal
receivedsignal
MUX-2sin wct
2cos wct
decisioncircuitwkwk-1+zkzk-1
xk
SI
decisioncircuitzkwk-1+wkzk-1
yk
SQ
Ts sample at max output
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(i) incoming /4QPSK signal quadrature demodulated using 2 LOs- LO signal frequency = unmodulated transmitter carrier- LO signals can be out of phase with transmit carrier
wk = cos(k-) 6.87
zk = sin(k-) 6.88
= phase shift due to noise, propagation, & interference
• if Δ << Δk is essentially constantΔ = change in Δk = change in k
• let phase of carrier due to kth bit be: k = tan-1(Qk/Ik) then:(ii) LPF I & Q arms of demodulator output wk & zk
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xk = wkwk-1 + zkzk-1 6.89
yk = zkwk-1 + wkzk-1 6.90
(iii) sequences wk and zk are passed through decoders that perform
• output of differential decoders
= cos(k - k-1)
xk = cos(k-) cos(k-1-) + sin(k-) sin(k-1-) 6.91
= sin(k-k-1)
yk = sin(k-) cos(k-1-) + cos(k-) sin(k-1-) 6.92
• negates relative phase difference with channel effects
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(iv) decision circuit uses table to determine
if xk < 0 SI = ‘0’
if xk > 0 SI = ‘1’6.93
if yk < 0 SQ = ‘0’
if yk > 0 SQ= ‘1’6.94
SQ = detected bits in quadrature arms
SI = detected bits in in-phase arms
• critical that LO frequency = transmit carrier frequency• if LO frequency drifts output phase drifts & BER increases
04/22/23 60
(2) IF Differential Detector• uses delay & 2 phase detectors• doesn’t need LO• received signal first converted to IF then bandpass filtered
BPF designed to match transmitted pulse shape• preserves carrier phase• minimizes noise power
• found that passband of BPF specified at 0.57/Ts minimizes ISI & noise
received IF signal • differentially decoded using delay & 2 mixers• passband at decoder output = 2 based band signal at transmit end
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MUXdemodulatedsignal
decisioncircuit
Ts sample at max output
modulated IF signal
Ts
90 o
decisioncircuit
/4QPSK IF Differential Detector
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FM Discriminator
(1) BPF: input signal put through BPF that is matched to transmitted signal
(2) Limiter: filtered signal is hard-limited to remove envelope fluctuations
- retains phase changes no information lost
(3) FM discriminator determines instantaneous frequency
deviation of received signal
(4) Integrate and Dump: integration of instantaneous frequency deviation over Ts gives phase difference between 2 sampling instants
(5) Threshold Detector: phase difference detected by 4-level threshold comparator obtain original signal
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0BPSK
QPSK
0Acm(t)DSB-Suppressed Carrier0Ac(1+kam(t))AM
SQ(t)SI(t)modulation
k
kc )kTt(pbA
k
,kc )kTt(pbA 1 k
,kc )kTt(pbA 2
Specific Canonical Equations
AM: s(t) = Ac(1+Kam(t))cos(2fct)
s(t) = Acm(t) cost(2fct)DSB-SC:
BPSK: m (t) = k
k )kTt(pb s(t) = Acm(t)cos(2fct)and
s(t) = Acm1(t)cos(2fct) + Acm2(t)sin(2fct)
mi(t) = k
ik kTtpb )(,QPSK: and