- 1 - KyungHee University Digital Communication 1 Chapter 4 Chapter 4 Chapter 4: Bandpass Modulation and Demodulation/Detection
Dec 25, 2015
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Chapter 4: Bandpass Modulation and Demodulation/Detection
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.1 Why Modulate?4.1 Why Modulate?
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.1 Why Modulate?4.1 Why Modulate?
Digital modulation : digital symbol : waveform compatible with the characteristic
of the channel Why use carrier? ⓐ reduce size of antenna (=3108m/fc)
e.g.) fc = 3kHz : antenna span : /4 = 25km
fc = 900 MHz : antenna diameter : /4 = 9cm
ⓑ frequency-division multiplexing ⓒ minimize the effect of interference : spread spectrum ⓓ place a signal in a frequency band where design requirements are met (e.g.)RF->IF
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.1 Why Modulate?4.1 Why Modulate?
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.2 Digital Bandpass Modulation 4.2 Digital Bandpass Modulation TechniqueTechnique
General form of a carrier wave
)](cos[)()(
)()(
)(cos)()(
0
0
tttAts
ttt
ttAts
4.2.1 Phasor Representation of a Sinusoid complex notation of a sinusoidal carrier wave
tjte tj00 sincos0
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.2 Digital Bandpass Modulation 4.2 Digital Bandpass Modulation TechniqueTechnique
Analytical form of transmitted waveform )(),(cos 0 AMt mm
221Re)( 0
tjtjtj
mm eeets
Analytical representation of narrowband FM(NFM)
tjtjtj mm eee
221 Res(t) 0
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.2 Digital Bandpass Modulation 4.2 Digital Bandpass Modulation TechniqueTechnique
4.2.2 Phase Shift Keying
4.2.3 Frequency Shift Keying
4.2.4 Amplitude Shift Keying
MiM
it
Mi
Tttt
T
Ets
i
ii
,...,12
)(
,...,1
0)(cos
2)( 0
Mi
Ttt
T
Ets ii ,...,1
0)cos(
2)(
Mi
Ttt
T
tEts i
i ,...,1
0)cos(
)(2)( 0
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.2 Digital Bandpass Modulation 4.2 Digital Bandpass Modulation TechniqueTechnique
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise
Two-dimensional signal space (M=2) Detector decides which of the signals s1 or s2 was transmitted, after receiving r =>Minimum-error decision rule chooses the signal class s.t. distance is minimized Decision region Decision rule
isrd
1 1
2 2
r Region s sent
r Region s sent
4.3.1 Decision Regions
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise
Step 2 : Choose waveform si(t) that has the largest correlation with r(t) Choose the si(t) whose index corresponds to the max Zi(T)
4.3.2 Correlation Receiver
Received signal Detection process Step 1 : Transform the waveform r(t) into a single random variable(R.V.)
MiTttntstr i ,...,1,0)()()(
),...,1()(..)( ' MiTZVRorTZ i
Matched filter (Correlator) maximizes SNR
T
ii dttstrTZ0
)()()(
Another detection approach (Fig.4.7.(b)) Any signal set can be expressed in terms of some set of basis functions
),...,1()( Mitsi ( ) ( 1,..., )j t j N where N M
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise
Signal NSignal N symbol M symbol M
Signal N< symbol MSignal N< symbol M
Ex) M-ary PSKEx) M-ary PSK
N=2N=2
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise4.3.2.1 Binary Detection Threshold
Decision stage : choose the signal best matched to the coefficients aij (with the set of output Zj(T))
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian Noise4.3 Detection of signals in Gaussian Noise
1 2
1( ) ( | ) ( | )
2p z p z s p z s
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise
Minimum error criterion for equally likely binary signals corrupted by Gaussian noise
For antipodal signals,
1 1 2
2
( ) ( ) ( )
( )
or decide s t if z T z T
s t otherwise
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.4 Coherent Detection4.4 Coherent Detection
4.4.1Coherent Detection of PSK(BPSK) Coherent detector BPSK example
Orthonormal basis function):(
0)cos(2
)cos(2
)(
0)cos(2
)(
0
02
01
symbolperenergysignalE
TttT
E
tT
Ets
TttT
Ets
1 0
1 1
1 11 1 1
2 21 1 1
2( ) cos( ) 0
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
i i
t t t TT
s t a t
s t a t E t
s t a t E t
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.4 Coherent Detection4.4 Coherent Detection
When s1(t) is transmitted, the expected values of product integrator
4.4.1Coherent Detection of PSK(BPSK)
Decision stage Choose the signal with largest value of zi(T)
21 1 1 1
0
22 1 1 1 1 0
0
21 1 0 0
0
22 1 0 0
0
| ( ) ( ) ( )
2| ( ) ( ) ( ) , ( ) cos
2| cos ( ) cos )
2| cos ( ) cos )
T
T
T
T
E z s E E t n t t dt
E z s E E t n t t dt then t tT
E z s E E t n t t dt ET
E z s E E t n t t dt ET
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Example 4.1 Sampled Matched Filter Consider the BPSK waveform set
Illustrate how a sampled matched filter or correlator can be used to detect a received signal, say s1(t), from the BPSK Waveform set, in the absence of noise.
4.4.2 Sampled Matched Filter
ttsandtts cos)(cos)( 21
sec)1sec,25.0,10002..( mTmTge s
Sampled MF (N samples per symbol)
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.4 Coherent Detection4.4 Coherent Detection
Ex) Sampled MF (4 samples per symbol)
sampled s1
sampled s2
1( 3) 2z k
2 ( 3) 2z k
3
0212 ][]3[]3[
n
ncnskz
)(1 tTs
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.4 Coherent Detection4.4 Coherent Detection
Signal space for QPSK(quadri-phase shift keying), M=4 (N=2)
4.4.3 Coherent Detection of Multiple Phase Shift Keying
For typical coherent MPSK system, Orthonormal basis function
MiTtM
it
T
Etsi ,...,1,0)
2cos(
2)( 0
tT
ttT
t 0201 sin2
)(,cos2
)(
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Signal can be written as Received signal
),...,1,0(
)()2
sin()()2
cos(
)()()(
21
2211
MiTt
tM
iEt
M
iE
tatats iii
)arctan(ˆ
)()(:
)()(:
0
2
0
1
XY
dtttrYcorrelatorlower
dtttrXcorrelatorupper
T
T
Demodulator
4.4.3 Coherent Detection of Multiple PSK
)()()( tntstr i
decision i
)(2 t
)(1 t
8M 1i
8i
7i5i
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Demodulator of multiple-PSK
)arctan(ˆ
)()(:
)()(:
0
2
0
1
XY
dtttrYcorrelatorlower
dtttrXcorrelatorupper
T
T
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Typical set of FSK signal waveforms
Orthonormal set
Distance between any two prototype signal vectors is constant
2( ) cos( ) 0 1,...,i i
Es t t t T where i M
T
otherwise
jiforEta
tdtT
tT
Eta
NjtT
Et
ij
j
T
iij
jj
0
)(
cos2
cos2
)(
),...,1(cos)(
0
( , ) 2i j i jd s s s s E for i j
The ith prptotype signal vector is located on the ith coordinated axis a displacement from originE
4.4.4 Coherent Detection of FSK
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.4 Coherent Detection4.4 Coherent Detection
Example:3-ary FSK signal
Mi
i 2
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.5 Non-coherent 4.5 Non-coherent DetectionDetection
Non-coherent detection : actual value of the phase
of the incoming signal is not required
4.5.1 Detection of Differential PSK
• For coherent detection, MF is used
• For non-coherent detection, this is not possible because MF
output is a function of unknown angle α
),...,1,0()(])(cos[2
)(:
)](cos[2
)(:
0
0
MiTttnttT
EtrsignalR
ttT
EtssignalT
ix
iix
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Differential encoding : information is carried by the difference in phase between two successive waveforms. To sent the i-th message (i=0,…,M), the present signal must have its phase advanced by over the previous signal Differential coherent detection : non-coherent because it does not require a reference in phase with received carrier Assuming that αvaries slowly relative to 2T, phase difference is independent of α as
Mi
i 2
)()()(])([])([ 21212 TTTTT ijkjk
4.5.1 Detection of Differential PSK
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.5.1 Detection of Differential PSK
DPSK Vs. PSK DPSK : 3dB worse than PSK PSK compares signal with clean reference DPSK compares two noisy signals, reducing complexity
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.5.2 Binary Differential PSK Example
)()()1()(
)()1()(
hereusedkmkckc
orkmkckc
Sample index k
Original message
Differential message
Correspondng phase
decoder
encoder
1 Arbitrary setting
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.5 .3 Non-coherent Detection of 4.5 .3 Non-coherent Detection of Binary Differential FSK
• Just an energy detector without phase measurement• Twice as many channel branches• Quadrature receiver
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
Three different cases :
)()cos()()3
)(sin)()2
)(cos)()1
1
1
1
tnttr
tnttr
tnttr
Another implementation for non-coherent FSK detection Envelop detector : rectifier and LPF Looks simpler, but (analog) filter require more complexity
4.5 .3 Non-coherent Detection of 4.5 .3 Non-coherent Detection of Binary Differential FSK
- 30 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
In order for the signal set to be orthogonal, any pair of adjacent tones must have a frequency separation of a multiple of 1/T[Hz] cf) Nyquist filter
( ) (cos 2 ) ( )
1 2 2( )0 2
{ ( )} ( )
i i
i i
ts t f t rect TT Tfor t
twhere rect T Tfor t
Fourier transform
F s t Tsinc f f T
Minimum tone separation:1/T[Hz]
4.5.4 Required Tone Spacing for Non-coherent Orthogonal FSK Signaling
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.5 Non-coherent Detection : 4.5 Non-coherent Detection : Example 4.3
⊙ Non-coherent FSK signal :
⊙ Coherent FSK signal :
1 2 1 20
1 2
1cos(2 )cos 2 0
. . 10,000 11,000 ?
1,000 / ,
1,000 / ,
Tf t f tdt for orthogonality f f
Te g two tones f Hz and f Hz orthogonal
if rate symvols s then orthogonal
if rate symvols s then not orthogonal
1 2
10
2f f
T
1 2 1 2cos(2 ) cos 2f t f t where f f
1 20
1 2 1 20 0
1 2 1 2
1 2 1 2
cos(2 )cos 2
cos cos 2 cos 2 sin sin 2 cos 2
sin 2 ( ) cos 2 ( ) 1cos sin
2 ( ) 2 ( )
T
T T
f t f tdt
f t f tdt f t f tdt
f f T f f T
f f f f
1 2 1 2
1 2 1 2
1 2
sin 2 ( ) cos 2 ( )0
2 ( ) 2 ( )
si 0
f f T f f T
f f f f
nce f f
Non-coherent이면 둘 다 0
이어야 함
- 32 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems
4.7.1Probability of Bit Error for Coherently Detected BPSK
Antipodal signals
Basis function Decision rule is
TttT
E
tEtats
tEtatst
T
Ets
tT
Ets
0)cos(2
)()()(
)()()()cos(
2)(
)cos(2
)(
0
11212
1111102
01
TtfortT
t 0cos2
)( 01
otherwisets
Tzifts
)(
0)()(
2
01
- 33 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performnace for Binary 4.7 Error Performnace for Binary SystemsSystems
dzaz
dzszPP
pdfofsymmetrysHPsHPP
sHPsHPP
sPsPsPsHPsPsHPP
u
aau
aaB
B
B
B
2
)(
2
0
2
0
2
)(2
2112
2112
21221112
21
210
2
1exp
2
1
)|(
)|()|(
)|(2
1)|(
2
12
1)()()()|()()|(
x
azudu
uXQ
0
22
,2
exp2
1)(
The same a priori
probability
a1 a2
- 34 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performnace for Binary 4.7 Error Performnace for Binary SystemsSystems
02
2
00
020
21
0
21
2
)(
2
2
2exp
2
1
))(2)(2:)((
2
:,,
22exp
2
1
0
21
N
EQdu
uP
NRNPSDwithnoisewhitetn
NSince
symbolbinaryperenergysignalEEaEa
aaQdu
uP
b
NE
B
n
bbb
u
aau
B
b
- 35 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Another approach (1)
- 36 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Another approach (2)
s1s2
s1
s2
bE bEbE
bE
2
1
bd EE 42)2( bd EE
BPSK BFSK
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KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Probability of bit error for several types of binary systems
TABLE 4.1 Probability of Error for SelectedBinary Modulation Schemes
0
2
N
EQ b
0
exp2
1
N
Eb
0N
EQ b
02exp
2
1
N
Eb
Modulation
PSK(coherent)
DPSK(dfferentially coherent)
Orthogonal FSK(coherent)
PB
Orthogonal FSK(noncoherent)
- 38 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.8 M-ary Signaling and 4.8 M-ary Signaling and PerformancePerformance
(R, Eb/No, BER, BW) : fundamental “trade-off”
M-ary orthogonal
k↑ BER↑ BW↑
4.8.2 M-ary Signaling(M=2k k:bits, M=# of waveforms)
M-ary PSK
k↑ BER↑
same BWShannon
Limit
-1.6dB
k=∞
①
②
- 39 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.8.3 Vectorial View of MPSK Signaling
① (M=2k↑, the same Eb/No)
bandwidth efficiency (R/W) ↑, PB ↑ ② (M=2k↑, the same PB)
bandwidth efficiency (R/W) ↑, Eb/No ↑
- 40 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.8.4 BPSK and QPSK : the same bit error probability
General relationship
R
W
N
S
R
W
N
S
N
Eb 22/
0
QPSK = two orthogonal BPSK channel (I stream, Q stream)
Magnitude
(A)
I stream ( A/root(2) )
Q stream ( A/root(2) )
Power/bit Half Half
Bit rate Half Half
QPSKBPSK
- 41 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.8 M-ary Signaling and Performance4.8 M-ary Signaling and Performance
If original QPSK is given by R[bps], S[watt],
RN
S
R
W
N
S
N
EBPSKeach b 1
2
2:
000
Same BER, BW efficiency : BPSK=1,QPSK=2[bit/s/Hz] Eb/N0 vs. SNR
0
0 0 2 0
2
( ):
1
log
log: , : , 1( )
b
b b b
E N Normalized SNR the most meaningful way of comparing one digital system with another
E E ES W S WT S WT S
N N R N N M N k N N k
M kwhere W Detection BW R data rate WT typical
T TEff
:ect of normalized SNR noise increases as k increases
0bS RE N WN
- 42 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.8 M-ary Signaling and Performance4.8 M-ary Signaling and Performance
Fig. 4.34 : M-ary orthogonal signaling at PE=10-3 in dB(decibel, nonlinear), factor(linear) k=10 (1024-ary symbol), 20SNR(factor)→2SNR per bit(factor); each bit require 2.
10:][10][][
)8(3:][77.4][][
)(2:][3][][
)(1:][][
log10][][
0
0
0
0
0
kdBdBN
SdB
N
E
PSKkdBdBN
SdB
N
E
QPSKkdBdBN
SdB
N
E
BPSKkdBN
SdB
N
E
kdBN
SdB
N
E
b
b
b
b
b
- 43 -
KyungHeeUniversity
Digital Communication 1 Chapter 4Chapter 4
4.9 Symbol Error Performance for M-ary 4.9 Symbol Error Performance for M-ary System(M>2)System(M>2)
4.9.4 Bit Error Probability vs. Symbol Error Probability for Multiple Phase Signaling
Assume that the symbol(011) is transmitted If an error occur, (010) or (100) is likely→3bit errors Gray code : neighboring symbols differ from one another in only one bit position
yprobabiliterroersymbolPPfork
P
M
PP EE
EEB ),1(
log2
BPSK vs. QPSK
QPSKforPP
BPSKforPP
BE
BE
2
1 (1 ) .kE BNote that P P