ch. 4 bandpass modulation and demodulation...6 bandpass modulation • bandpass modulation – the...
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Department of Electrical EngineeringUniversity of Arkansas
ELEG5663 Communication Theory
Ch. 4 Bandpass Modulation and Demodulation
Dr. Jingxian Wu
2
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
3
INTRODUCTION
• Bandpass modulation
– Recall: baseband modulation
• Mapping digital symbols onto baseband waveforms (pulse shapes)
that are compatible with channel.
– Bandpass modulation
• The baseband pulse shapes are translated to a higher frequency by
using a carrier wave (a high frequency sinusoid)
• Why bandpass modulate?
– To transmit a signal, antennas are usually ¼ of wavelength
• : higher frequency smaller wavelength smaller
antenna.
– E.g. 3KHz baseband signal,
– 1GHz bandpass signal,
– Translate the signal to a pre-allocated channel
• E.g. frequency division multiple access (FDMA)
fc /
4
INTRODUCTION
• Bandpass modulation and demodulation
5
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
6
BANDPASS MODULATION
• Bandpass modulation
– The amplitude, frequency, or phase of a radio frequency (RF) carrier, or a
combination of them, is varied in accordance with the information to be
transmitted.)](2cos[)()( ttftAts c
– Through bandpass modulation, a baseband signal is shifted to a higher
frequency.
• Example, amplitude shift keying.
• Time domain:
• Frequency domain:
tftxtx cc 2cos)()(
)()(2
1)( ccc ffXffXfX
7
BANDPASS MODULATION
• Types of bandpass modulation
– Coherent (the receivers exploits knowledge of the carrier’s phase for
detection)
• Phase shift keying (PSK)
• Frequency shift keying (FSK)
• Amplitude shift keying (ASK)
• Continuous phase modulation (CPM)
• Hybrid: Quadrature amplitude modulation (QAM)
– Noncoherent (the receivers operate without knowledge of the absolute
value of the signal’s phase)
• Differential phase shift keying (DPSK)
• Frequency shift keying (FSK)
• Amplitude shift keying (ASK)
• Continuous phase modulation (CPM)
• Hybrid: Quadrature amplitude modulation (QAM)
8
BANDPASS MODULATION: VECTOR REPRESENTATION
• Most bandpass modulated signals can be represented as
tfT
stfT
sts QI 00 2sin2
2cos2
)(
– Orthogonal representation of signal
tfT
t 01 2cos2
)( tfT
t 02 2sin2
)(
• and are orthonormal
– Proof:
)(1 t )(2 t
– The bandpass modulated signal can be represented as
• : there is a one-to-one relationship between and
the two dimensional vector
)()()( 21 tststs QI
],[)( QI ssts )(ts
QI ss ,
)(1 t
)(2 t
Is
Qs
Tt 0
Tt 0
9
BANDPASS MODULATION: VECTOR REPRESENTATION
• Vector representation of bandpass modulated signal
– Inphase component:
– Quadrature component:
– The inphase carrier, , and quadrature carrier,
are orthonomral.
– The bandpass modulated signal can be equivalently represented as vector
Is
Qs
tT
t 01 cos2
)(
],[)( QI ssts s
• Complex number representation of bandpass modulated signal
– There is a one-to-one relationship between a 2D vector and complex
number
)(1 t
)(2 t
Is
Qs
QIQI jssssts ],[)( s
– The bandpass modulated signal can be equivalently represented as a
complex number
QI jssts )(
– The complex number representation is only used
for mathematical convenience.
tT
t 01 sin2
)(
10
BANDPASS MODULATION: PSK
• Phase shift keying (PSK)
)](cos[2
)( 0 ttT
Ets ii Mi ,,1
Tt 0
– : carrier frequency
– Each digital symbol is mapped to a different phase
– Why ?
00 2 f
MiM
iti ,,1,
2)(
T
E2
– An example of BPSK
11
BANDPASS MODULATION: FSK
• Frequency shift keying (FSK)
]cos[2
)( tT
Ets ii
–
– Each digital symbol is mapped to a different frequency.
– The set of signals, , could be orthogonal or non-orthogonal.
– An example of orthogonal FSK.
Mi ,,1
Tt 0
ii f 2
M
ii ts 1)}({
ij
T
jiT
Edttsts 0 )()(
12
BANDPASS MODULATION: ASK
• Amplitude shift keying (ASK)
]cos[)(2
)( 0 tT
tEts i
i Mi ,,1
Tt 0
– Each digital symbol is mapped to a different amplitude
– An example of BASK (on-off keying)
13
BANDPASS MODULATION: ASK
• Amplitude phase keying (APK)
]cos[)(2
)( 0 tT
tEts i
i
– Both amplitude and phase are altered by the digital symbol.
– An example of a 8-ary APK
Mi ,,1
Tt 0
14
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
15
COHERENT DETECTION: BPSK
• Recall: correlation receiver of binary baseband signal
– the output of correlation receiver is the same as the output of matched
filter and sampler
2
210
aa
T
dttststsa0
2111 )()()( T
dttststsa0
2122 )()()(
– Bit error probability
02)(
N
EQEP d
T
d dttstsE0
2
21 )]()([
16
COHERENT DETECTION: BPSK
• Recall: correlation receiver of binary baseband signal (Cont’d)
– Bit error probability with vector signal representation
• is the squared distance between the two modulation points.dE
)()()( 2121111 tatats )()()( 2221212 tatats
T
d dttstsE0
2
21 )]()([
– Bipolar signal
0
2)(
N
EQEP b
– Orthogonal signal
0
)(N
EQEP b
17
COHERENT DETECTION: BPSK
• Binary Phase Shift Keying (BPSK)
]cos[2
)( 01 tT
Ets ]cos[
2]cos[
2)( 002 t
T
Et
T
Ets
– Orthogonal representation
tT
t 01 cos2
)( tT
t 02 sin2
)( Tt 0
)()( 11 tEts )()( 12 tEts
– Correlation receiver
tT
t 01 cos2
)(
2
210
aa
T
dtttsa0
111 )()(
T
dtttsa0
122 )()(
18
COHERENT DETECTION: BPSK
• Bit error probability of BPSK
02)(
N
EQEP d
T
d dttstsE0
2
21 )]()([
T
d dttstsE0
2
21 )]()([
0
2)(
N
EQEP b
19
COHERENT DETECTION: BPSK
• BPSK: revisit maximum likelihood detector
)()()( tntstr i
nadtttntsdtttrTz i
T
i
T
01
01 )()()()()()(
2
2
2
)(exp
2
1)|(
ii
azszp
bEa 1 bEa 2
– Likelihood function
– Maximum likelihood detection rule:
• If , detect
• If , detect
• Equivalently:
– In the orthogonal representation of signal, choose the signal that has the smallest Euclidean distance with the received sample
21 azaz )(1 ts
21 azaz )(2 ts
20
COHERENT DETECTION: BPSK
• Example
– Find the expected number of bit errors made in one day by the following
continuously operating coherent BPSK receiver. The data rate is 1 kbps.
The input waveforms are and where A =
2mV and the double-sided noise PSD is .
tAts 01 cos)( tAts 02 cos)(
HzW /10 10
COHERENT DETECTION: MAXIMUM LIKELIHOOD
• Maximum Likelihood Detection
– Signal:
– Rx Signal:
– Correlation receiver:
– Likelihood function
21
n
j
jiji tats1
)()(
)()()( tntstr i
…
)(1 t
)(tn
Max
imu
m L
ikel
ihoo
d
1z
nz
is
COHERENT DETECTION: MAXIMUM LIKELIHOOD
• Maximum Likelihood Detection
– Choose that minimizes
22
)(tsi
n
k
ikk az1
2)(
• Graphical Interpretation
– M-ary modulation, there are M constellation points:
– After correlation detector, there is an n-dimension point:
– Detection: choose such that the Euclidean distance is minimized
),,( 1 mnmm aa s
),,( 1 nzz z
is
zs i
23
COHERENT DETECTION
• Maximum Likelihood detection: 2-Dimension
– Choose such that the Euclidean distance between the vector and is
22 )()(|||| iQQiIIi srsr sr
ris
are minimized.
– Decision region example
• Whenever the received signal is located in region 1, choose signal
• Whenever the received signal is located in region 2, choose signal
r 1s
r 2s
24
COHERENT DETECTION: MPSK
• Multiple Phase Shift Keying (MPSK)
M
it
T
Etsi
2cos
2)( 0
Tt 0
Mi ,,1
– Orthogonal representation of MPSK signals
tTM
iEt
TM
iEtsi 00 sin
22cossincos
22cos)(
)(2
cossin)(2
cos)( 21 tM
iEt
M
iEtsi
– Inphase component
M
iEsI
2cos
– Quadrature component
M
iEsQ
2sin
25
COHERENT DETECTION: MPSK
• MPSK coherent detection
– Structure of receiver
– Likelihood functions
IiI
T
i nsdtttntsX 0 1 )())()(( QiQ
T
i nsdtttntsY 0 2 )())()((
2
2
2
)(exp
2
1)|(
iIi
sXsXp
2
2
2
)(exp
2
1)|(
QI
i
sYsYp
2
22
2 2
)()(exp
2
1)|,(
iQiI
i
sYsXsYXp
– Maximum likelihood detection
• Choose that minimizes )(tsi22 )()( iQiI sYsX
26
COHERENT DETECTION: MPSK
• MPSK coherent detection (Cont’d)
– The distance between and
– Maximum likelihood decision rule
T
dtttrX0
1 )()( T
dtttrY0
2 )()(
• Choose that minimizes the distance between and
• Equivalently, choose with phase that is closest to the phase
of the signal at the output of the correlator:
– Find that minimize
)(tsi
)(tsi i
X
Yarctanˆ )(tsi
|ˆ|
),( iQiIi sss
22 )()( iQiIi sYsXd
),( YXr
),( YXr
),( iQiIi sss
27
COHERENT DETECTION: MPSK
• Example:
– For a system with 8PSK, if the received symbols are:
28
COHERENT DETECTION: MFSK
• Multiple frequency shift keying (MFSK)
tT
Ets ii cos
2)(
– The value of can be chosen such that are mutually
orthogoanal orthogonal MFSK
• Orthogonal MFSK is a special case of MFSK
• We are only going to examine orthogonal MFSK
– Orthogonal MFSK
Tt 0
Mi ,,1
i M
iit 1cos
tT
t ii cos2
)( ij
T
ji dttt 0 )()(
– Example: Show the following system is orthogonal BFSK if
Tff c
2
11
Tff c
2
12
Tfc
1
29
COHERENT DETECTION: MFSK
• Coherent receiver structure of MFSK
– Structure of a receiver
– Output of the correlation detector for MFSK
mim
T
miim nsdtttntsr 0 )())()((
)()()()( 2211 tstststs MiMiii
• Output of the correlation detector of MFSK is coordinate in
orthogonal signal representation
),,,( 21 iMii rrr r),,,( 21 iMiii sss s
30
COHERENT DETECTION: MFSK
• Maximum Likelihood detection
– Choose that minimizes the Euclidean distance between)(tsi
),,,( 21 iMii rrr r),,,( 21 iMiii sss s
• Bit error probability of BFSK
31
COHERENT DETECTION
• Vector representation of bandpass communication system
– Recall vector representation of bandpass modulated signal
)()()( 21 tststs iQiIi
tfT
t 01 2cos2
)( tfT
t 02 2sin2
)( Tt 0
)()()( 21 tntntn QI
)()()()()()( tnstnstntstr QQQIIIi )()()( trtrtr QQII
IiII nsr
QiQQ nsr
– The bandpass communication system is equivalently represented as the
summation of signal vector and noise vector
nsr i
],[ QI rrr ],[ iQiIi sss ],[ QI nnn
32
COHERENT DETECTION
• Maximum likelihood detection
– For AWGN with two-sided PSD
• The noise variance per dimension is
– Likelihood functions
2
2
2 2
)(exp
2
1)|(
iIIiII
Srsrp
2
0N
2
02 N
2
2
2 2
)(exp
2
1)|(
iQQ
iQQ
Srsrp
– and are independent In Qn
2
22
4 2
)()(exp
2
1),|,(
iQQiII
iQiIQI
srsrssrrp
– Maximize Minimize ),|,( iQiIQI ssrrp ),|,( iQiIQI ssrrp 22 )()( iQQiII srsr
22 )()(|||| iQQiIIi srsr sr
– Minimize the Euclidean distance between the vector and ris
33
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
34
NONCOHERENT DETECTION
• Why noncoherent detection?
– Coherent detection requires the exact knowledge of the phase of the received signal
– Example: BPSK
• The distance between Tx and Rx is d
• At Tx, the signal is tfT
Ets b
02cos2
)(
• At Rx, the signal is
• : the amount of time the signal travels from Tx to Rx
• : the phase of the signal at the receiver
• In order to perform coherent detection, the Rx needs to know
– can be estimated through a circuitry called phase locked loop (PLL)
)(2cos2
)(2cos2
)( 00 ttfT
ETtf
T
Etr b
db
cdTd /
dTft 02)(
)(t
)(t
35
NONCOHERENT DETECTION
• Why noncoherent detection? (Cont’d)
– What if the Rx doesn’t have the knowledge of ?
– Example:
• Assume a system operates a 1GHz. If the distance between Tx and Rx
is 24.075m, find out the phase of the Rx signal.
)(t
• Noncoherent detection
– The Rx doesn’t require the knowledge of the absolute phase of the Rx
signal.
36
NONCOHERENT DETECTION: DPSK
• Differential PSK
– The information is carried by the phase difference between the current
symbol and the previous symbol
• Recall: for coherent PSK, the information is carried by the absolute
phase of one symbol.
– Example:
• The kth Rx symbol is
• The (k+1)th Rx symbol is
• The phase difference between the two consecutive symbols
ktfT
Etr 02cos
2)(
102cos2
)( ktfT
Etr
ikk 1
• The information is carried by the phase difference
– The Rx doesn’t need the knowledge of the absolute phase . The
information is carried by the phase difference.
M
ii
2
37
NONCOHERENT DETECTION: DPSK
• Binary DPSK
– The essence of differential detection is that the data is carried in the phase
difference between two consecutive symbols
– Tx: differential encoding; Rx: differential decoding.
– Binary differential encoding
)()1()( kmkckc
• m(k): information
• c(k): differentially encoded bit
• : modulo-2 addition
• – : complement
• The information, m(k), is carried by the difference between c(k) and
c(k-1)
38
NONCOHERENT DETECTION: DPSK
• Binary DPSK
– Binary differential decoding
39
NONCOHERENT DETECTION: DPSK
• DPSK: pros and cons
– Pro:
• Doesn’t require the absolute value of the signal phase simpler
receiver
– Con:
• Two noisy signal are compared to detect the signal there are twice
as much noise as in coherent detection the performance is worse
compared to coherent detection
– Trade-off between complexity and performance
40
NONCOHERENT DETECTION: FSK
• Non-coherent detection of FSK
NONCOHERENT DETECTION: FSK
• Non-coherent detection of FSK
– If has been transmitted
• What are the values at the output of the non-coherent detector of FSK?
41
NONCOHERENT DETECTION: FSK
• Non-coherent detection of FSK
– The minimum tone space for non-coherent orthogonal FSK
42
NONCOHERENT DETECTION: FSK
• Minimum Tone Spacing for Coherent Orthogonal FSK
43
NONCOHERENT DETECTION
• Coherent detection v.s. non-coherent detection
44
45
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
46
COMPLEX ENVELOPE
• Complex representation of bandpass modulated signal
tfT
tstfT
tsts QI 00 2sin2
)(2cos2
)()( tfj
QI eT
tjststs 022)]()([)(~
)(~Re)( tsts
– There is a one to one relationship between and )(ts )(~ ts
• Complex envelope
– The complex baseband signal is called complex envelope
• The envelope of the bandpass signal.
– The complex envelope is the same as the vector representation of the
signal up to a scaling factor
)()()( tjststg QI
)(tg
)()()(),()( tjststststs QIQI
47
COMPLEX ENVELOPE
• Complex envelope
)(|)(|)( tjetgtg – Polar representation
)()()( tjytxtg
• Amplitude )()(|)(| 22 tytxtg
• Phase)(
)(tan)( 1
tx
tyt
48
COMPLEX ENVELOPE
• Bandpass modulation can be divided into two steps (Constellation)
– 1. Baseband modulation:
• Transfer information (‘1’s and ‘0’s) into complex envelope
• Example: 8PSK
1 0 0 0 1 0 1 1 1 0 0 1
– 2. Frequency upconversion
• Multiply the complex envelope with tfje 02
})(Re{)( 02 tfjetgts
49
COMPLEX ENVELOPE
• Quadrature implementation of a modulator
– Baseband modulation:
• Mapping ‘0’s and ‘1’s to the values of x(t) and y(t).
– Frequency upconversion:
• Upconverting the frequency of the baseband signal through
quadrature modulation.
50
COMPLEX ENVELOPE
• Quadrature implementation of a demodulator
– Frequency downconversion:
• Downconverting the frequency of the bandpass signal.
– Baseband demodulation:
• Mapping the baseband signal to ‘0’s and ‘1’s
51
COMPLEX ENVELOPE: D8PSK
• D8PSK (Differential 8PSK)
– Baseband modulation
– Frequency upconversion
52
COMPLEX ENVELOPE: D8PSK
• D8PSK Demodulation
– Frequency downconversion
– Baseband demodulation
53
OUTLINE
• Introduction
• Bandpass Modulation
• Coherent Detection
• Noncoherent Detection
• Complex Envelope
• Error Probability
54
ERROR PROBABILITY: BINARY MODULATION
• Comparison of error performance of binary system
– BER of BPSK
0
2
N
EQP b
B
– BER of coherently detected, differentially encoded binary PSK
00
21
22
N
EQ
N
EQP bb
B
– BER of differentially detected, differentially encoded binary PSK (DPSK)
0
exp2
1
N
EP b
B
55
ERROR PROBABILITY: BINARY MODULATION
• Comparison of error performance of binary system
– BER for coherently detected binary orthogonal FSK
0N
EQP b
B
– BER for non-coherently detected binary orthogonal FSK
02exp
2
1
N
EP b
B
56
ERROR PROBABILITY: BINARY MODULATION
Coherent detection PSK > coherent detection of differentially encoded PSK > Differential detection of differentially encoded PSK (DPSK) > Coherent detection of orthogonal FSK > Noncoherent detection of orthogonal FSK.
57
ERROR PROBABILITY: MPSK
• Symbol error rate for MPSK (Fig. 4.35)
– Symbol error rate (SER): # of error symbols/# of symbols transmitted
MN
EQMP s
E
sin
22)(
0
– : symbol energy MEE bs 2log
58
ERROR PROBABILITY: MFSK
• SER for MFSK
0
)1()(N
EQMMP s
E
59
ERROR PROBABILITY: BER V.S. SER
• Relationship between BER and SER for MPSK (Fig. 4.39)
– Gray encoding: two adjacent symbols differ in 1 bit
• At high SNR, Most of the errors are the confusion between adjacent symbols
• At high SNR, 1 symbol error approximately corresponds to 1 bit error
EB PM
P2log
1
60
ERROR PROBABILITY: BER V.S. SER
• Relationship between BER and SER for orthogonal signals
EB PM
MP
1
2/
– The relationship is exact