# Digital Communication 1 Chapter 4

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Chapter 4:
Bandpass Modulation and
Demodulation/Detection
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Chapter 4
4.1 Why Modulate?
Digital Communication 1
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Chapter 4
4.1 Why Modulate?
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 Digital modulation :
digital symbol : waveform compatible with the characteristic
of the channel
 Why use carrier?
ⓐ reduce size of antenna (=3108m/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
Digital Communication 1
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Chapter 4
4.1 Why Modulate?
Digital Communication 1
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Chapter 4
4.2 Digital Bandpass Modulation Technique
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General form of a carrier wave
s(t )  A(t ) cos (t )
 (t )  0t   (t )
s(t )  A(t ) cos[0t   (t )]
4.2.1 Phasor Representation of a Sinusoid
complex notation of a sinusoidal carrier wave
e j0t  cos0t  j sin 0t
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Chapter 4
4.2 Digital Bandpass Modulation Technique
 Analytical form of transmitted waveform
 j0t 
e jmt e  jmt
s (t )  Re e 1 

2
2





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(cosmt , m  0 ) ( AM )
 Analytical representation of narrowband FM(NFM)





s(t)  Re e j0t 1  e  jmt  e jmt 
2
2



Digital Communication 1
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Chapter 4
4.2 Digital Bandpass Modulation Technique
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4.2.2 Phase Shift Keying
si (t ) 
i (t ) 
2E
cos0t  i (t )
T
2i
M
0  t  T 


i

1
,...,
M


i  1,...,M
4.2.3 Frequency Shift Keying
si (t ) 
2E
cos(i t   )
T
0  t  T 


i

1
,...,
M


4.2.4 Amplitude Shift Keying
si (t ) 
Digital Communication 1
2Ei (t )
cos(0t   )
T
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0  t  T 


i  1,...,M 
Chapter 4
4.2 Digital Bandpass Modulation Technique
Digital Communication 1
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Chapter 4
4.3 Detection of signals in Gaussian Noise
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4.3.1 Decision Regions
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.
d  r  siis minimized
distance
 Decision region
 Decision rule
r  Region1  s1 sent
r  Region2  s2 sent
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Chapter 4
4.3 Detection of signals in Gaussian Noise
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 Detection process
r (t )  si (t )  n(t )
0  t  T,
i  1,...,M
Step 1 : Transform the waveform r(t) into a single random variable(R.V.)
Z (T ) or R.V .' Zi (T ) (i  1,...,M )
Matched filter (Correlator) maximizes SNR
T
Z i (T ) 
 r (t ) s
i
(t ) dt
0
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)
 Another detection approach (Fig.4.7.(b)) Any signal set
expressed in terms of some set of basis functions
can be
si (t ) (i  1,...,M )  j (t ) ( j  1,..., N ) where N  M
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Chapter 4
4.3 Detection of signals in Gaussian Noise
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Signal N symbol M
Signal N< symbol M
Ex) M-ary PSK
N=2
Digital Communication 1
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Chapter 4
4.3 Detection of signals in Gaussian Noise
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4.3.2.1 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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Chapter 4
4.3 Detection of signals in Gaussian Noise
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 Two conditional pdfs : likelihood of s1(s2)
p( z ) 
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1
 p( z | s1 )  p( z | s2 ) 
2
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Chapter 4
4.3 Detection of signals in Gaussian Noise
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 Minimum error criterion for equally likely binary signals
corrupted by Gaussian noise
 For antipodal signals,
or decide s1 (t ) if z1 (T )  z2 (T )
s2 (t ) otherwise
Digital Communication 1
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Chapter 4
4.4 Coherent Detection
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4.4.1Coherent Detection of PSK(BPSK)
 Coherent detector
 BPSK example
s1 (t ) 
2E
cos(0 t   ) 0  t  T
T
s2 (t ) 
2E
cos(0 t     )
T
2E
cos(0 t   ) 0  t  T
T
( E : signalenergy per sym bol)

 Orthonormal basis function
2
cos(0t ) 0  t  T
T
si (t )  ai1 1 (t )
 1 (t ) 

s1 (t )  a11 1 (t ) 
E 1 (t )
s2 (t )  a21 1 (t )   E 1 (t )
Digital Communication 1
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Chapter 4
4.4 Coherent Detection
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4.4.1Coherent Detection of PSK(BPSK)
 When s1(t) is transmitted, the expected values of product integrator
T

E  z1 | s1  E   E 12 (t )  n(t ) 1 (t ) dt 
0

T

2
E  z2 | s1  E    E 12 (t )  n(t ) 1 (t ) dt 
then,  1 (t )  cos 0t
T
0

T 2

E  z1 | s1  E  
E cos 2 0t  n(t ) cos 0t ) dt  E 
0 T

T 2

E  z2 | s1  E   
E cos 2 0t  n(t ) cos 0t )dt   E 
0 T

 Decision stage
Choose the signal with largest value of zi(T)
Digital Communication 1
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Chapter 4
4.4.2 Sampled Matched Filter
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 Example 4.1 Sampled Matched Filter Consider the BPSK waveform
set
s1 (t )  cost and s2 (t )   cost
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.
(e.g.   21000, Ts  0.25m sec, T  1m sec)
Sampled MF (N samples per symbol)
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Chapter 4
4.4 Coherent Detection
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Ex) Sampled MF (4 samples per symbol)
sampled s1
z1 (k  3)  2
s1 (T  t )
sampled s2
3
z2 [k  3]   s1[3  n]c2 [n]
z2 (k  3)  2
n 0
Digital Communication 1
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Chapter 4
4.4 Coherent Detection
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4.4.3 Coherent Detection of Multiple Phase Shift Keying
 Signal space for QPSK(quadri-phase shift keying), M=4 (N=2)
 For typical coherent MPSK system,
si (t ) 
 Orthonormal basis function
2E
2i
cos(0t 
) 0  t  T , i  1,...,M
T
M
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 1 (t ) 
2
cos0t ,  2 (t ) 
T
2
sin 0t
T
Chapter 4
4.4.3 Coherent Detection of Multiple PSK
 Signal can be written as
si (t )  ai1 1 (t )  ai 2 2 (t )
2i
2i
) 1 (t )  E sin(
) 2 (t )
M
M
(0  t  T , i  1,...,M )
 E cos(
 Demodulator
i 1
T
 1 (t )
i7
Digital Communication 1
T
0
i 8
i5
r (t )  si (t )  n(t )
upper correlator : X   r (t ) 1 (t )dt
 2 (t )
M 8
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lower correlator : Y   r (t ) 2 (t )dt
0
ˆ  arct an(Y X ) decision
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i
Chapter 4
Demodulator of multiple-PSK
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T
upper correlator : X   r (t ) 1 (t )dt
0
T
lower correlator : Y   r (t ) 2 (t )dt
0
ˆ  arct an(Y X )
Digital Communication 1
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Chapter 4
4.4.4 Coherent Detection of FSK
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 Typical set of FSK signal waveforms
si (t ) 
2E
cos(i t   ) 0  t  T
T
where
i  1,..., M
 Orthonormal set
 j (t ) 
E
cos j t ( j  1,...,N )
T
T
 aij (t )  
0
2E
2
cosi t
cos j tdt
T
T
 aij (t ) 
E for i  j
 Distance between any two prototype
signal vectors is constant
d ( si , s j )  si  s j 
2 E for i  j
0 otherwise
 The ith prptotype signal vector is located on the ith coordinated axis a displacement
from origin
E
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Chapter 4
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4.4 Coherent Detection
 Example:3-ary FSK signal
i  2i M
Digital Communication 1
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Chapter 4
4.5 Non-coherent Detection
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4.5.1 Detection of Differential PSK
 Non-coherent detection : actual value of the phase
of the incoming signal is not required
Tx signal: si (t ) 
2E
cos[0t  i (t )]
T
Rx signal: r (t ) 
2E
cos[0t  i (t )   ]  n(t ) (0  t  T , i  1,...,M )
T
• For coherent detection, MF is used
• For non-coherent detection, this is not possible because MF
output is a function of unknown angle α
Digital Communication 1
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Chapter 4
4.5.1 Detection of Differential PSK
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 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
i  2i
M
 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
[k (T2 )   ]  [ j (T1 )   ]  k (T2 )  j (T1 )  i (T2 )
Digital Communication 1
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Chapter 4
4.5.1 Detection of Differential PSK
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 DPSK Vs. PSK
 DPSK : 3dB worse than PSK
 PSK compares signal with clean reference
 DPSK compares two noisy signals,
reducing complexity
Digital Communication 1
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Chapter 4
4.5.2 Binary Differential PSK Example
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c(k )  c(k  1)  m(k ) or
c(k )  c(k  1)  m(k ) (used here)
Sample index k
Original message
encoder
Differential message
Correspondng phase
1 Arbitrary setting
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decoder
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Chapter 4
4.5 .3 Non-coherent Detection of Binary Differential FSK
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• Just an energy detector without phase measurement
• Twice as many channel branches
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Chapter 4
4.5 .3 Non-coherent Detection of Binary Differential FSK
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 Three different cases :
1) r (t )  cos1t  n(t )
2) r (t )  sin 1t  n(t )
3) r (t )  cos(1t   )  n(t )
 Another implementation for
non-coherent FSK detection
 Envelop detector :
rectifier and LPF
Looks simpler, but (analog)
filter require more complexity
Digital Communication 1
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Chapter 4
4.5.4 Required Tone Spacing for Non-coherent
Orthogonal FSK Signaling
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 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
si (t )  (cos 2 f i t )rect ( t )
T
 1 for  T  t  T

2
2
where rect ( t )  
T
0 for t  T


2
Fourier transform
F{si (t )}  Tsinc( f  f i )T
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Minimum tone separation:1/T[Hz]
Chapter 4
4.5 Non-coherent Detection : Example 4.3
cos(2 f1t   ) cos 2 f 2t
⊙ Non-coherent FSK signal :

T
0
cos(2 f1t   ) cos 2 f 2tdt  0 for orthogonality 
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where f1  f 2
f1  f 2 
1
T
e.g. two tones f1  10, 000 Hz and f 2  11, 000 Hz orthogonal ?
if rate  1, 000 symvols / s, then orthogonal
if rate  1, 000 symvols / s, then not orthogonal
⊙ Coherent FSK signal :

T
0
 0

f1  f 2 
1
2T
cos(2 f1t   ) cos 2 f 2tdt
T
T
0
0
 cos   cos 2 f1t cos 2 f 2tdt  sin   sin 2 f1t cos 2 f 2tdt
 cos 
sin 2 ( f1  f 2 )T
cos 2 ( f1  f 2 )T  1
 sin 
2 ( f1  f 2 )
2 ( f1  f 2 )
sin 2 ( f1  f 2 )T cos 2 ( f1  f 2 )T

0
2 ( f1  f 2 )
2 ( f1  f 2 )
since
Digital Communication 1
Non-coherent이면
둘 다 0이어야 함
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f1  f 2  0
Chapter 4
4.7 Error Performance for Binary Systems
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4.7.1Probability of Bit Error for Coherently Detected BPSK
 Antipodal signals
s1 (t ) 
2E
cos(0t   )
T
s2 (t ) 
2E
cos(0t     )
T


s (t )  a11 1 (t )  E 1 (t )
 1

s2 (t )  a21 1 (t )   E 1 (t )
2E
cos(0t   ) 0  t  T
T
 Basis function
 1 (t ) 
 Decision rule is
2
cos0t for 0  t  T
T
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s1 (t ) if z (T )   0  0

s2 (t ) otherwise
Chapter 4
4.7 Error Performnace for Binary Systems
1
PB  P( H 2 | s1 ) P( s1 )  P( H1 | s2 ) P( s2 )  P( s1 )  P ( s2 ) 
2
1
1
PB  P( H 2 | s1 )  P( H1 | s2 )
2
2
The same
PB  P( H 2 | s1 )  P( H1 | s2 )  sym m etry of pdf
priori

0
a
probability
 P( z | s2 )dz
PB 
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( a1  a2 )
2
2

1
1  z  a2  
 dz

exp 

2  0  

( a1  a2 )  0 2


u
u 
a1
a2
2

1
  Q( X ) 
2

Digital Communication 1
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 u2 
z  a2 
x exp  2 du, u   0 


Chapter 4
4.7 Error Performnace for Binary Systems
u 
PB 
u

 a1 a2 
 u2 
1

exp  du  Q
2
 2 
 2 0 
a1 
Eb , a2   Eb , Eb : signal energy per binary sym bol
( a1  a2 )
2
Since  02 
N0
2
(n(t ) : white noise with PSD

PB 
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
2 Eb N 0
N0
2
 Rn ( ) 
 2 Eb
 u2 
1
exp  du  Q
2
 2 
 N0
Digital Communication 1
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N0
2
 ( ))




Chapter 4
4.7 Error Performance for Binary Systems
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 Another approach (1)
Digital Communication 1
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Chapter 4
4.7 Error Performance for Binary Systems
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 Another approach (2)
2
BPSK
s2
Ed  4Eb
Eb
BFSK
s2
s1
Eb
Eb
Digital Communication 1
Ed  ( 2Eb ) 2
s1
Eb
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1
Chapter 4
4.7 Error Performance for Binary Systems
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 Probability of bit error for several types of binary systems
TABLE 4.1 Probability of Error for Selected
Binary Modulation Schemes
Modulation
PSK(coherent)
DPSK
(dfferentially coherent)
Orthogonal FSK
(coherent)
Orthogonal FSK
(noncoherent)
Digital Communication 1
PB
 2 Eb
Q 
 N0




 E 
1
exp   b 
2
 N0 

Q

Eb
N0




 E 
1
exp  b 
2
 2 N0 
- 36 -
Chapter 4
4.8 M-ary Signaling and Performance
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4.8.2 M-ary Signaling(M=2k k:bits, M=# of waveforms)
①
M-ary orthogonal
k↑ BER↑ BW↑
M-ary PSK
k↑ BER↑
Shannon
Limit
-1.6dB
k=∞
same BW
②
(R, Eb/No, BER, BW) : fundamental “trade-off”
Digital Communication 1
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Chapter 4
4.8.3 Vectorial View of MPSK Signaling
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① (M=2k↑, the same Eb/No)
 bandwidth efficiency (R/W) ↑, PB ↑
② (M=2k↑, the same PB)
 bandwidth efficiency (R/W) ↑, Eb/No ↑
Digital Communication 1
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Chapter 4
4.8.4 BPSK and QPSK : the same bit error probability
 General relationship
BPSK
Eb S / 2  2W


N0
N  R
QPSK
 S W 
  
 NR
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 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
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Chapter 4
S  REb
4.8 M-ary Signaling and Performance
 If original QPSK is given by R[bps], S[watt],
each BPSK:
N  WN0
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Eb S 2  W  S  1 



 
N0
N 0  R 2  N 0  R 
Same BER, BW efficiency : BPSK=1,QPSK=2[bit/s/Hz]
 Eb/N0 vs. SNR
 Eb N 0 ( Normalized SNR ):the most meaningful way of comparing one digital system with another
Eb
E
Eb
S W 
S  WT  S  WT 
S 1
   b  
  
 

N0 N  R 
N 0 N  log 2 M  N  k 
N0 N  k 
log 2 M k
where W : Detection BW , R 
 : data rate, WT  1(typical )
T
T
 Effect of normalized SNR : noise increases as k increases
Digital Communication 1
- 40 -
Chapter 4
4.8 M-ary Signaling and Performance
KyungHee
University
 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.

Eb
S
[dB]  [dB]  10 log k
N0
N
 Eb
 N [dB] 
 0
 Eb
 N [dB] 
 0

 Eb [dB] 
 N0

 Eb [dB] 

 N0

: k  1( BPSK) 

S

[dB]  3[dB]
: k  2(QPSK)
N


S
[dB]  4.77[dB] : k  3(8 PSK) 

N

S

[dB]  10[dB] : k  10

N

S
[dB]
N
Digital Communication 1
- 41 -
Chapter 4
4.9 Symbol Error Performance for M-ary System(M>2)
KyungHee
University
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
PE
P
 E ( for PE  1), PE  sym bol erroer probability
log2 M
k
Note that PE  1  (1  PB )k .
 BPSK vs. QPSK
PB 
PE  PB
for BPSK
PE  2 PB for QPSK
Digital Communication 1
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Chapter 4
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