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Performance of Coherent M-ary Signaling. ENSC 428 – Spring 200 7. Digital Communication System. 1. M-ary PSK. T. sin. cont …. cont …. Integration over IQ plane. cont …. cont …. 2. M-ary Orthogonal Signaling. cont …. 3- ary orthogonal signal space. - PowerPoint PPT Presentation
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Performance of Coherent M-ary
Signaling
ENSC 428 – Spring 2007
Digital Communication System
1. M-ary PSK
T
sin
cont …
cont …
Integration over IQ plane
cont …
2
0 0
2 2 log2 sin 2 sins b
s
E E MP e Q Q
N M N M
cont …
2. M-ary Orthogonal Signaling
cont …
3-ary orthogonal signal space
cont … assume equally likely M-ary symbols a priori
1 2 1 2
{1,2,..., )
( , ,..., ) ( , ,..., )
Optimal decision rule
argmax
M M
m M m
r r r s w w w
r
cont … assume equally likely symbols a priori
1 1
1 11 1
1 2 1 3 1 1 1 1 1
2
2 3 1 100
2
12 0
1 1 (symmetry)
1 , ,..., ,
1, ,..., exp
1exp
M M
s mm m
M r s
s
M
Ms
ii
P e P e s P e s P e sM M
P e s P n r n r n r r x s f x s dx
x EP n x n x n x r x dx
NN
x EP n x r x
N
0
21
00 0
11 exp
/ 2
M
s
dxN
x ExQ dx
NN N
cont … union bound
2
0 0
2
2 2
2
log1 ( 1)
1Also, let us learn exp , 0
2 2
1 1 1 1 exp exp , 0 (Gallager Problem 10.4)
2 22 2
s bs
E E MP e M Q M Q
N N
xQ x x
x xQ x x
x x x
cont …
cont … Performance improves as M increases (??) In the limit (M∞), error probability can be made
arbitrarily small as long as Eb/N0 > ln2 (-1.59 dB). Proof in Gallager Lecture 19 section 4.3 In fact, Information Theory also proves that we cannot
achieve error probability arbitrarily small if Eb/N0 < ln2.
Most practical systems use non-coherent FSK rather than coherent FSK. We will discuss non-coherent FSK soon.
Biorthogonal Signaling
cont …
6-ary biorthogonal signal constellation
Simplex Signaling
The centroid of an orthogonal constellation is located at:
, ,...,
More enegy-efficient signal contellation can be achieved by moving
the centroid to the origin.
, 1, 2
s s s
m m
E E Ec
M M M
s s c m
,...,
Identical probability of error, M
M
P
cont …3-ary simplex signal constellation
cont …
3. M-ary QAM
16-ary QAM
cont …
cont …
Symbol Error Rate (SER) or SEP
Bit Error Rate (BER) or BEP(cf SEP, symbol error probability)
2
,1 1,2
,
#bit errors per symbol( ) #bit errors per bit
#bits per symbol
#bit errors per symbol
log
1ˆ
log
where is the number of bits that differ betwee
b
M M
i i j j ii j j i
i j
EP e E
E
M
P s n P s sM
n
,1 1,2
,,
1 1,2 0
n symbols and .
For equally likely symbols a priori,
1 1ˆ( )
log
1 1 (union bound)
log 2
i j
M M
b i j j ii j j i
M Mi j
i ji j j i
s s
P e n P s sM M
dn Q
M M N
Orthogonal signaling BEP
k bits, M=2k, each bit error pattern corresponds to a unique symbol, which is not the transmitted.
e e e
1
In orthogoanl siganling, 1 kinds of symbol error are equally likely, so
Probability of a particular bit error pattern is .1 2 1
# of bit errors per symbol
1 1
2 1
M Mk
k Mkn
M
P P
ME
BEPk
k Pnnk k
1
1 22
2 1 2 1 2
kk M M M
k k
P P Pk
Example: 8-ary PSK
Gray Coding
Gray-coded MPSK
,1 1,2
12 2
The most probable errorr result in
the erroneous selection of an adjacent phase.
1 1ˆ( )
log
1 1 1 1
log 2 2 log
M M
b i j j ii j j i
MM M M
i
P e n P s sM M
P P P
M M M
cont …
cont …
Gray code (Reflected binary code by Frank Gray) generation
Can be generated recursively by reflecting the bits (i.e. listing them in reverse order and concatenating the reverse list onto the original list), prefixing the original bits with a binary 0 and then prefixing the reflected bits with a binary 1.
Gray code generation: another view
1 2
1
Convert a natural binary string
(If 1, then 1 . Otherwise, .)
0000 0000
0001 0001
0010 0011
0011 0010
0100 0110
0101 0111
0110 0101
0111 0100
1000 1100
1001 1101
1010 1111
1011 1110
n
n n n n n
d d d
d g d g d