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DATA COMMUNICATION 2-dimensional transmission. A.J. Han Vinck May 1, 2003. we describe orthogonal signaling 2-dimensional transmission model. Content. „orthogonal“ binary signaling. 2 signals S 1 (t) S 2 (t) in time T Example: Property : orthogonal energy E. T. T.

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Data communication 2 dimensional transmission

DATA COMMUNICATION2-dimensional transmission

A.J. Han Vinck

May 1, 2003


Content

we describe

orthogonal signaling

2-dimensional transmission model

Content


Orthogonal binary signaling
„orthogonal“ binary signaling

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

Example:

Property: orthogonal

energy E

T

T



Qam receiver
QAM receiver

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)



About the noise1
about the noise

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.



Geometric presentation 2
Geometric presentation (2)

11

10

00

01

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


Performance
performance

From Chapter 1: P(error) =


Extension
extension

4-QAM  2 bits

16-QAM

 4 bits/s

Channel 2

Channel 1


Geometric presentation 21
Geometric presentation (2)

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!


Geometric presentation 11
Geometric presentation (1)

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

d/2

r

r‘


Error probability for coded transmission
Error probability for coded transmission

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‘


Error probability
Error probability

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


Modulation schemes
modulation schemes

On-off

FSK

8-PSK

 3 bits/s

1 bit/symbol

1 bit/symbol

4-QAM  2 bits

16-QAM

 4 bits/s


Transmitted symbol energy
transmitted symbol energy

energy: per information bit must be the same

FSK


Performance1
performance

d/2

From Chapter 1: P(error) =

FSK


Coding with same symbol speed
Coding with same symbol speed

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:


Famous ungerb ck coding
Famous Ungerböck coding

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!


Modulator
modulator

info

ci

23

encoder

Signal mapper

ci{000,001,010,...111}


Example
example

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


Example frequency shift keying fsk
Example: Frequency Shift Keying-FSK

Transmit: s(1):=

s(0):=

Note:

FSK


Modulator demodulator
Modulator/demodulator

m

modulator

S(t)

m

r(t)

Select largest

demodulator

m


Ex binary phase shift keying bpsk
Ex: Binary Phase Shift Keying-BPSK

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


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