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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

1


Signal space

  • Signal Space

  • Inner Product

  • Norm

  • Orthogonality

  • Equal Energy Signals

  • Distance

  • Orthonormal Basis

  • Vector Representation

  • Signal Space Summary

2


Signal space
Signal Space

S(t)

S=(s1,s2,…)

  • Inner Product (Correlation)

  • Norm (Energy)

  • Orthogonality

  • Distance (Euclidean Distance)

  • Orthogonal Basis

3



Inner product x t y t
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
Norm - ||x(t)||

Similar to norm of vector

A

T

-A

7


Orthogonality
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
Distance, d

  • For equal energy signals

  • =-1 (antipodal)

  • =0 (orthogonal)

  • 3dB “better” then orthogonal signals

12


Equal energy signals
Equal Energy Signals

  • To maximize d

(antipodal signals)

  • PSK(phase Shift Keying)

13


PSK (Orthogonal Phase Shift Keying)

(Orthogonal if

14


Signal space summary
Signal Space summary

  • Inner Product

  • Norm ||x(t)||

  • Orthogonality

15


  • Distance, d

16


Modulation
Modulation

QAM

BPSK

QPSK

BFSK

17


Modulation1

  • Modulation

  • BPSK

  • QPSK

  • MPSK

  • QAM

  • Orthogonal FSK

  • Orthogonal MFSK

  • Noise

  • Probability of Error

Modulation

18



Binary antipodal signals vector presentation
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
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

26


Quaternary psk qpsk
Quaternary PSK - QPSK

(00)

(10)

(11)

(01)

*

27


X(t)

28


(00)

(10)

(11)

(01)

29



MPSK

31



Multi amplitude signal
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 hence, are represented by the scalar components

35


Multi amplitude multiphase signals qam signals
Multi-Amplitude MultiPhase signals hence, are represented by the scalar components QAM Signals

A quadrature amplitude-modulated (QAM) signal

or a quadrature-amplitude-shift-keying (QASK) is represented as

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


Qam quadrature amplitude modulation

d represented by the vectors

QAM (Quadrature Amplitude Modulation)

38


d represented by the vectors

QAM=QASK=AM-PM

Exrecise

39


M=256 represented by the vectors

M=128

M=64

M=32

M=16

M=4

+

40


For an represented by the vectorsM - ary QAM Square Constellation

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

41


Binary orthogonal signals
Binary orthogonal signals represented by the vectors

Consider the two signals

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

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

42


The equivalent lowpass waveforms: represented by the vectors

The vector presentation:

Which correspond to the signal space diagram

Note that

43



M ary orthogonal signal
M-ary Orthogonal Signal lowpass signals is

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 lowpass signals is

0

46


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 and .

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


Orthogonal fsk orthogonal frequency shift keying
Orthogonal FSK and . (Orthogonal Frequency Shift Keying)

51


“0” and .

“1”

52


Orthogonal mfsk
ORTHOGONAL MFSK and .

53



How to generate signals
How to and . generatesignals

55


0 and . T 2T 3T 4T 5T 6T

+

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

56


0 and . T 2T 3T 4T 5T 6T

+

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

57


0 and . T 2T 3T 4T 5T 6T

+

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

58


IQ Modulator and .

+

59


IQ Modulator and .

Pulse shaping filter

+

60


Noise
NOISE and .

61


What about noise
What about Noise and .

  • White Gaussian Noise

T

T

  • The coefficients are random variables !

62


White gaussian noise wgn
WHITE GAUSSIAN NOISE (WGN) and .

We write

  • All are gaussian variables

  • All are independent

63



65


Probability of error for binary signaling

Exrecise and .

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 and .

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 . probability of error .

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 probability of error .

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

The probability of error is now

70



Distance d1
Distance, d as

  • 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 .

73


74 is expressed as


M=256 is expressed as

M=128

M=64

M=32

M=16

M=4

+

75


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