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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

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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

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

14


Signal Space summary

  • Inner Product

  • Norm ||x(t)||

  • Orthogonality

15


  • Corrolation Coefficient, 

  • Distance, d

16


Modulation

QAM

BPSK

QPSK

BFSK

17


  • Modulation

  • BPSK

  • QPSK

  • MPSK

  • QAM

  • Orthogonal FSK

  • Orthogonal MFSK

  • Noise

  • Probability of Error

Modulation

18


Binary Phase Shift Keying – (BPSK)

-

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

26


Quaternary PSK - QPSK

(00)

(10)

(11)

(01)

*

27


X(t)

28


(00)

(10)

(11)

(01)

29


Exrecise

30


MPSK

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

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


d

QAM (Quadrature Amplitude Modulation)

38


d

QAM=QASK=AM-PM

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.

42


The equivalent lowpass waveforms:

The vector presentation:

Which correspond to the signal space diagram

Note that

43


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

Where

44


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

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

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)

51


“0”

“1”

52


ORTHOGONAL MFSK

53


All signals are orthogonal to each other

54


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

61


What about Noise

  • White Gaussian Noise

T

T

  • The coefficients are random variables !

62


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 .

73


74


M=256

M=128

M=64

M=32

M=16

M=4

+

75


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