performance analysis of optimum receiver

8/13/2019 Performance Analysis of Optimum Receiver

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Average Probability of Error


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cont


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cont


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cont


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cont


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Binary Coherent FSK


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cont


11/62

cont


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cont

FSK is approximately 3dB worse than BPSK


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Binary Coherent ASK

P4


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cont


15/62

cont


16/62

QPSK


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cont

Decision regions


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cont

45 degree rotation


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cont


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cont


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cont


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Union Bound


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cont


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cont


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Illustrating the Union Bound


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Pairwise Error Probability


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cont


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Union Bound


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For example: QPSK


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cont


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M-ary PSK

T

sin


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cont


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cont


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cont


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M-ary Orthogonal Signaling


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cont

3-ary orthogonal signal space

ll lik l M


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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


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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


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cont


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cont

Performance improves as M increases (??)

In the limit (M), error probability can bemade arbitrarily small as long as Eb/N0> ln2

(-1.59 dB). Information Theory also proves that we cannot

achieve error probability arbitrarily small if Eb/N0< ln2.

Most practical systems use non-coherent FSKrather than coherent FSK.


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Biorthogonal Signaling


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cont

6-ary biorthogonal signal constellation


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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


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cont

3-ary simplex signal constellation


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cont


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M-ary QAM

16-ary QAM


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cont


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cont


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Symbol Error Rate (SER)

Bit Error Rate (BER) or BEP


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Bit Error Rate (BER) or BEP

(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 log

where is the number of bits that differ betwee

b

M M

i i j j i

i 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 1( )

log

1 1 (union bound)

log 2

i j

M M

b i j j i

i j j i

M Mi j

i j

i j j i

s s

P e n P s sM M

dn Q

M M N

O h l i li B P


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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 M

k

k M

kn

M

P P

M

EBEPk

k Pn

nk k

11 2

22 1 2 1 2

kk M M M

k k

P P Pk


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8-ary PSK


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Gray Coding


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Gray-coded MPSK

,

1 1,2

12 2

The most probable errorr result in

the erroneous selection of an adjacent phase.

1 1( )

log1 1

1 1log 2 2 log

M M

b i j j i

i j j i

MM M M

i

P e n P s s

M MP P P

M M M


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cont


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cont

Gray code (Reflected binary code)


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Gray code (Reflected binary code)

generation

Can be generated recursively by reflecting the bits (i.e.listing them in reverse order and concatenating thereverse list onto the original list), prefixing the original

bits with a binary 0 and then prefixing the reflected bitswith a binary 1.
http://upload.wikimedia.org/wikipedia/commons/a/af/Gray_code_reflect.png

performance analysis of optimum receiver

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