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