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# Digital Communication Vector Space concept - PowerPoint PPT Presentation

Digital Communication Vector Space concept. Signal space. Signal Space Inner Product Norm Orthogonality Equal Energy Signals Distance Orthonormal Basis Vector Representation Signal Space Summary. Signal Space. S(t). S=(s1,s2,…). Inner Product (Correlation) Norm (Energy)

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Digital Communication Vector Space concept

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#### Presentation Transcript

Digital

Communication

Vector Space

concept

1

Signal space

• Signal Space

• Inner Product

• Norm

• Orthogonality

• Equal Energy Signals

• Distance

• Orthonormal Basis

• Vector Representation

• Signal Space Summary

2

### Signal Space

S(t)

S=(s1,s2,…)

• Inner Product (Correlation)

• Norm (Energy)

• Orthogonality

• Distance (Euclidean Distance)

• Orthogonal Basis

3

ONLY CONSIDER SIGNALS, s(t)

T

t

Energy

4

### Inner Product - (x(t), y(t))

Similar to Vector Dot Product

5

Example

A

T

t

-A

2A

A/2

t

T

6

### Norm - ||x(t)||

Similar to norm of vector

A

T

-A

7

### Orthogonality

A

T

-A

Y(t)

B

Similar to orthogonal vectors

T

8

X(t)

• ORTHONORMAL FUNCTIONS

{

T

Y(t)

T

9

Correlation Coefficient

1    -1

=1 when x(t)=ky(t) (k>0)

• In vector presentation

10

Example

Y(t)

X(t)

10A

A

t

t

-A

T

T/2

7T/8

Now,

shows the “real” correlation

11

### Distance, d

• For equal energy signals

• =-1 (antipodal)

• =0 (orthogonal)

• 3dB “better” then orthogonal signals

12

### Equal Energy Signals

• To maximize d

(antipodal signals)

• PSK(phase Shift Keying)

13

• EQUAL ENERGY SIGNALS

• ORTHOGONAL SIGNALS (=0)

PSK (Orthogonal Phase Shift Keying)

(Orthogonal if

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### Signal Space summary

• Inner Product

• Norm ||x(t)||

• Orthogonality

15

• Corrolation Coefficient, 

• Distance, d

16

### Modulation

QAM

BPSK

QPSK

BFSK

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• Modulation

• BPSK

• QPSK

• MPSK

• QAM

• Orthogonal FSK

• Orthogonal MFSK

• Noise

• Probability of Error

18

-

19

### Binary antipodal signals vector presentation

• Consider the two signals:

The equivalent low pass waveforms are:

20

The vector representation is –

Signal constellation.

21

The cross-correlation coefficient is:

The Euclidean distance is:

Two signals with cross-correlation coefficient

of -1 are called antipodal

22

### Multiphase signals

• Consider the M-ary PSK signals:

The equivalent low pass waveforms are:

23

The vector representation is:

Or in complex-valued form as:

24

Their complex-valued correlation coefficients are :

and the real-valued cross-correlation coefficients are:

The Euclidean distance between pairs of signals is:

25

The minimum distance dmin corresponds to the case which

| m-k |=1

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(00)

(10)

(11)

(01)

*

27

X(t)

28

(00)

(10)

(11)

(01)

29

Exrecise

30

31

32

### Multi-amplitude Signal

Consider the M-ary PAM signals

m=1,2,….,M

Where this signal amplitude takes the discrete values (levels)

m=1,2,….,M

The signal pulse u(t) , as defined is rectangular

U(t)=

But other pulse shapes may be used to obtain a narrower signal spectrum .

33

Clearly , this signals are one dimensional (N=1) and , hence, are represented by the scalar components

M=1,2,….,M

The distance between any pair of signal is

M=2

0

M=4

0

Signal-space diagram for M-ary PAM signals .

34

The minimum distance between a pair signals

35

### Multi-Amplitude MultiPhase signalsQAM Signals

Where and are the information bearing signal amplitudes of the quadrature carriers and u(t)= .

36

QAM signals are two dimensional signals and, hence, they are represented by the vectors

The distance between a pair of signal vectors is

k,m=1,2,…,M

When the signal amplitudes take the discrete values

In this case the minimum

distance is

37

d

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d

Exrecise

39

M=256

M=128

M=64

M=32

M=16

M=4

+

40

For an M - ary QAM Square Constellation

In general for large M - adding one bit requires 6dB more energy to maintain same d .

41

### Binary orthogonal signals

Consider the two signals

Where either fc=1/T or fc>>1/T, so that

Since Re(p12)=0, the two signals are orthogonal.

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The equivalent lowpass waveforms:

The vector presentation:

Which correspond to the signal space diagram

Note that

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We observe that the vector representation for the equivalent lowpass signals is

Where

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### M-ary Orthogonal Signal

Let us consider the set of M FSK signals

m=1,2,….,M

This waveform are characterized as having equal energy and cross-correlation coefficients

45

The real part of is

0

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First, we observe that =0 when and .

Since |m-k|=1 corresponds to adjacent frequency slots ,

represent the minimum frequency separation between adjacent signals for orthogonality of the M signals.

47

For the case in which ,the FSK signals

are equivalent to the N-dimensional vectors

=( ,0,0,…,0)

=(0, ,0,…,0)

Orthogonal signals for M=N=3

signal space diagram

=(0,0,…,0, )

Where N=M. The distance between pairs of signals is

all m,k

Which is also the minimum distance.

48

51

“0”

“1”

52

### ORTHOGONAL MFSK

53

All signals are orthogonal to each other

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### How togeneratesignals

55

0 T 2T 3T 4T 5T 6T

+

0 T 2T 3T 4T 5T 6T

56

0 T 2T 3T 4T 5T 6T

+

0 T 2T 3T 4T 5T 6T

57

0 T 2T 3T 4T 5T 6T

+

0 T 2T 3T 4T 5T 6T

58

IQ Modulator

+

59

IQ Modulator

Pulse shaping filter

+

60

### NOISE

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• White Gaussian Noise

T

T

• The coefficients are random variables !

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### WHITE GAUSSIAN NOISE (WGN)

We write

• All are gaussian variables

• All are independent

63

• All have same probability distribution

64

• White Gaussian Noise has energy in every dimension

65

Exrecise

### Probability of Error for Binary Signaling

The two signal waveforms are given as

These waveforms are assumed to have equal energy E and their equivalent lowpass um(t), m=1,2 are characterized by the complex-valued correlation coefficient ρ12 .

66

The optimum demodulator forms the decision variables

Or,equivalently

And decides in favor of the signal corresponding to the larger decision variable .

67

Lets see that the two expressions yields the same probability of error .

Suppose the signal s1(t) is transmitted in the interval 0tT . The equivalent low-pass received signal is

Substituting it into Um expression obtain

Where Nm, m=1,2, represent the noise components in the decision variables,given by

68

And .

The probability of error is just the probability that the decision variable U2 exceeds the decision variable u1 . But

Lets define variable V as

N1r and N2r are gaussian, so N1r-N2r is also gaussian-distributed and, hence, V is gaussian-distributed with mean value

69

And variance

Where N0 is the power spectral density of z(t) .

The probability of error is now

70

Where erfc(x) is the complementary error function, defined as

It can be easily shown that

71

### Distance, d

• For equal energy signals

• =-1 (antipodal)

• =0 (orthogonal)

• 3dB “better” then orthogonal signals

72

It is interesting to note that the probability of error P2 is expressed as

Where d12 is the distance of the two signals . Hence,we observe that an increase in the distance between the two signals reduces the probability of error .

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M=256

M=128

M=64

M=32

M=16

M=4

+

75