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DATA COMMUNICATION 2-dimensional transmission

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DATA COMMUNICATION2-dimensional transmission

A.J. Han Vinck

May 1, 2003

we describe

orthogonal signaling

2-dimensional transmission model

2 signals S1 (t) S2 (t) in time T

Example:

Property:orthogonal

energy E

T

T

1

0

0

S(t)

1

1

0

1/0

+/-

r(t)

1/0

+/-

r(t) = S(t) + n(t)

Note: sin(x)sin(x) = ½ (1 – cos (2x) )

sin(x)cos(x) = ½ sin (2x)

Conclusion: n1 and n2 are Gaussian Random Variables

zero mean

uncorrelated (and thus statistically independent (f(x,y) =f(x)f(y) )

with variance 2.

11

10

00

01

ML receiver:find maximum p(r|s) min p(n) decision regions

From Chapter 1: P(error) =

4-QAM 2 bits

16-QAM

4 bits/s

Channel 2

Channel 1

1

equal density

transmitted

2

noise vector n

received

The noise vector n has length |n| = ( 12+22) ½

n has a spherically symmetric distribution!

Prob (error) = Prob(length noise vector > d/2)

d/2

r

r‘

The error probabiltiy is similar to the 1-dimensional situation:

We have to determine

the minimum d2Euclidean between any two codewords

Example:

C

d2Euclidean =

C‘

The two-code word error probability is then given by:

On-off

FSK

8-PSK

3 bits/s

1 bit/symbol

1 bit/symbol

4-QAM 2 bits

16-QAM

4 bits/s

energy: per information bit must be the same

FSK

d/2

From Chapter 1: P(error) =

FSK

In k symbol transmissions, we transmit k information bits. We use a rate ½ code

In k symbol transmissions, we transmit k bits

ML receiver:

In k symbol transmissions transmit

We can transmit

2k information bits

and k redundant digits

In k symbol transmissions transmit 2k digits

Hence, we can use a code with rate 2/3 with the same energy per info bit!

info

ci

23

encoder

Signal mapper

ci{000,001,010,...111}

transmit

00

00

00

10

10

01

Parity even

Parity odd

11

11

11

or

00

01

00

01

10

11

10

11

Decoder:

1) first detect whether the parity is odd or even

2) do ML decoding given the parity from 1)

Homework: estimate the coding gain

Transmit:s(1):=

s(0):=

Note:

FSK

m

modulator

S(t)

m

r(t)

Select largest

demodulator

m

Transmit:s(1):=

s(0):=

m

m‘

> or < 0?

On-off

BFSK

BPSK

Modulation formats

PERFORMANCE

10-1

10-2

10-3

10-4

10-5

10-6

10-7

Error rate

On-off

BPSKQPSK

5 10 15

Eb/N0 dB