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 bs

E E MP e Q QN M N M

π π ≈ =

cont …

2. M-ary Orthogonal Signaling

cont …

3-ary orthogonal signal space

cont … assume equally likely M-arysymbols a priori

1 2 1 2

{1,2,..., )

( , ,..., ) ( , ,..., )Optimal decision ruleargmax

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

1 exp

M Ms mm m

M r s

s

M

M s

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

π

π

= =

∞

−∞

∞

−∞

∞

=−∞

= = =

− = < < < =

− = < < < = −

−= < = −

∑ ∑

∫

∫

∏∫

( )

0

21

00 0

11 exp/ 2

Ms

dxN

x ExQ dxNN Nπ

−∞

−∞

− = − −

∫

cont … union bound

( ) ( )

( )

( )

2

0 0

2

2 2

2

log1 ( 1)

1Also, let us learn exp , 02 2

1 1 1 1 exp exp , 0 (Gallager Problem 10.4)2 22 2

s bs

E E MP e M Q M QN N

xQ x x

x xQ x xx x xπ π

≤ − = −

≤ − ∀ >

− − ≤ ≤ − ∀ >

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 4.3In 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 movingthe 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

1 ˆ 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

EM

P s n P s sM

n

= = ≠

= =

=

= ∑ ∑

,1 1,2

,,

1 1,2 0

n symbols and .

For equally likely symbols a priori,1 1 ˆ( )

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

MP PM

EBEP

kk Pnnk k=

−

=− −

=

= = − ∑ ( )

11 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 1 ˆ( )log

1 1 1 1log 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 PM 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 00000001 00010010 00110011 00100100 01100101 01110110 01010111 01001000 11001001 11011010 11111011 1110

n

n n n n n

d d dd g d g d− = = − =

→→→→→→→→→→→→