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Baseband Receiver. Unipolar vs. Polar Signaling Signal Space Representation. Unipolar vs. Polar Signaling. æ. ö. æ. ö. E. æ. ö. 2. E. A. T. ç. ÷. ç. ÷. ç. ÷. =. =. =. b. Q. d. P. Q. Q. ç. ÷. ç. ÷. ç. ÷. b. 2. N. 2. N. 2N. è. ø. è. ø. è. ø. 0. 0. 0. - PowerPoint PPT Presentation

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Baseband Receiver Unipolar vs. Polar Signaling Signal Space Representation

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Baseband ReceiverUnipolar vs. Polar SignalingSignal Space Representation

Unipolar vs. Polar Signaling

Error Probability of Binary Signals Unipolar Signaling

0,0,0)(1,0,)(

0

1

binaryforTttsbinaryforTtAts

dttstsE

dttstsdttsdtts

dttstsE

T

b

TTT

T

d

0 01

0 01

2

0 0

2

0 1

2

0 01

)()(22

)()(2)()(

)()(

00

002

TA

Eb

0

2

0 22 NTAQN

EQP db

02NEQ b

Polar Signaling (antipodal)

0,0,)(1,0,)(

0

1

binaryforTtAtsbinaryforTtAts

00

2

0

224

2 NEQ

NTAQ

NEQP bd

b

Bipolar signals require a factor of 2 increase in energy compared to Orthagonal

Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than Orthagonal

0

2NEQP b

b

0

)(

NEQP

AntipodalBipolarOrthogonal

bb

6

Error Performance Degradation

Two Types of Error-Performance DegradationPower degradationSignal distortion

(a) Loss in Eb/No. (b) Irreducible PB caused by distortion.

Signal Space Analysis

8 Level PAM

S0(t) S1(t) S2(t) S3(t) S4(t) S5(t) S6(t) S7(t)

8 Level PAM

Then we can represent

(t)

tastS 4

tastS 35

tastS 56

tastS 77

tastS 13 tastS 32

tastS 70 tastS 51

8 Level PAM

We can have a single unit height window (t) as the receive filter

And do the decisions based on the value of z(T)

We can have 7 different threshold values for our decision (we have one

threshold value for PCM detection)

In this way we can cluster more & more bits together and transmit them as

single pulse

But if we want to maintain the error rate then the

transmitted power = f (clustered no of bitsn)

(t) as a unit vector

One dimensional vector space (a Line)

If we assume (t) as a unit vector i, then we can represent signals s0(t) to s7(t) as points on a line (one dimensional vector space)

S1(t) S2(t) S3(t) S4(t) S5(t) S6(t) S7(t)S0(t)

(t)

0

12

A Complete Orthonormal Basis

Signal space What is a signal space?

Vector representations of signals in an N-dimensional orthogonal space

Why do we need a signal space? It is a means to convert signals to vectors and vice versa. It is a means to calculate signals energy and Euclidean

distances between signals. Why are we interested in Euclidean distances

between signals? For detection purposes: The received signal is

transformed to a received vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal.

Signal space To form a signal space, first we need to

know the inner product between two signals (functions): Inner (scalar) product:

Properties of inner product:

dttytxtytx )()()(),( *

= cross-correlation between x(t) and y(t) )(),()(),( tytxatytax

)(),()(),( * tytxataytx

)(),()(),()(),()( tztytztxtztytx

Signal space – cont’d The distance in signal space is measure by

calculating the norm. What is norm?

Norm of a signal:

Norm between two signals:

We refer to the norm between two signals as the Euclidean distance between two signals.

xEdttxtxtxtx

2)()(),()(

)()( txatax

)()(, tytxd yx

= “length” of x(t)

Signal space - cont’d N-dimensional orthogonal signal space is

characterized by N linearly independent functions called basis functions. The basis functions must satisfy the orthogonality condition

where

If all , the signal space is orthonormal.

Orthonormal basis Gram-Schmidt procedure

N

jj t1

)(

jiij

T

iji Kdttttt )()()(),( *

0

Tt 0Nij ,...,1,

jiji

ij 01

1iK

Signal space – cont’d Any arbitrary finite set of waveforms where each member of the set is of duration T,

can be expressed as a linear combination of N orthonogal waveforms

where .

where

Mii ts 1)(

N

jj t1

)(

MN

N

jjiji tats

1

)()( Mi ,...,1MN

dtttsK

ttsK

aT

jij

jij

ij )()(1)(),(1

0

* Tt 0Mi ,...,1Nj ,...,1

),...,,( 21 iNiii aaas2

1ij

N

jji aKE

Vector representation of waveform Waveform energy

Signal space - cont’d

N

jjiji tats

1

)()( ),...,,( 21 iNiii aaas

iN

i

a

a

1

)(1 t

)(tN

1ia

iNa

)(tsi

T

0

)(1 t

T

0

)(tN

iN

i

a

a

1

ms)(tsi

1ia

iNa

ms

Waveform to vector conversion Vector to waveform conversion

Schematic example of a signal space

),()()()(),()()()(),()()()(

),()()()(

212211

323132321313

222122221212

121112121111

zztztztzaatatatsaatatatsaatatats

zsss

)(1 t

)(2 t ),( 12111 aas

),( 22212 aas

),( 32313 aas

),( 21 zzz

Transmitted signal alternatives

Received signal at matched filter output

Example of distances in signal space

)(1 t

)(2 t),( 12111 aas

),( 22212 aas

),( 32313 aas

),( 21 zzz

zsd ,1

zsd ,2zsd ,3

The Euclidean distance between signals z(t) and s(t):

3,2,1

)()()()( 222

211,

i

zazatztsd iiizsi

1E

3E

2E

Example of an ortho-normal basis functions Example: 2-dimensional

orthonormal signal space

Example: 1-dimensional orthonornal signal space

1)()(

0)()()(),(

0)/2sin(2)(

0)/2cos(2)(

21

20

121

2

1

tt

dttttt

TtTtT

t

TtTtT

t

T

)(1 t

)(2 t

0

T t

)(1 t

T1

0

1)(1 t

)(1 t0

Example of projecting signals to an orthonormal signal space

),()()()(),()()()(

),()()()(

323132321313

222122221212

121112121111

aatatatsaatatatsaatatats

sss

)(1 t

)(2 t),( 12111 aas

),( 22212 aas

),( 32313 aas

Transmitted signal alternatives

dtttsaT

jiij )()(0 Tt 0Mi ,...,1Nj ,...,1

Implementation of matched filter receiver

)(tr

1z)(1 tT

)( tTN Nz

Bank of N matched filters

Observationvector

)()( tTtrz jj Nj ,...,1

),...,,( 21 Nzzzz

N

jjiji tats

1

)()(

MN Mi ,...,1

Nz

z1

z z

Implementation of correlator receiver

),...,,( 21 Nzzzz

Nj ,...,1dtttrz j

T

j )()(0

T

0

)(1 t

T

0

)(tN

Nr

r1

z)(tr

1z

Nz

z

Bank of N correlators

Observationvector

N

jjiji tats

1

)()( Mi ,...,1

MN

Example of matched filter receivers using basic functions

Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal.

Reduced number of filters (or correlators)

T t

)(1 ts

T t

)(2 ts

T t

)(1 t

T1

0

1z z)(tr z

1 matched filter

T t

)(1 t

T1

0

1z

TA

TA0

0

26

Example 1.

27

(continued)

28

(continued)

29

Signal Constellation for Ex.1

30

Notes on Signal Space Two entirely different signal sets can have the same geometric

representation. The underlying geometry will determine the performance and the

receiver structure for a signal set. In both of these cases we were fortunate enough to guess the correct

basis functions. Is there a general method to find a complete orthonormal basis for an

arbitrary signal set? Gram-Schmidt Procedure

31

Gram-Schmidt Procedure

Suppose we are given a signal set:

We would like to find a complete orthonormal basis for this signal set.

The Gram-Schmidt procedure is an iterative procedure for creating an orthonormal basis.

1{ ( ),..........., ( )}Ms t s t

1{ ( ),.........., ( )}, Kf t f t K M

32

Step 1: Construct the first basis function

33

Step 2: Construct the second basis function

34

Procedure for the successive signals

35

Summary of Gram-Schmidt Prodcedure

36

Example of Gram-Schmidt Procedure

37

Step 1:

38

Step 2:

39

Step 3:

* No new basis function

40

Step 4:

* No new basis function. Procedure is complete

41

Final Step:

42

Signal Constellation Diagram

Bandpass SignalsRepresentation

44

Representation of Bandpass Signals

Bandpass signals (signals with small bandwidth compared to carrier frequency) can be represented in any of three standard formats:

1. Quadrature Notation s(t) = x(t) cos(2πfct) − y(t) sin(2πfct) where x(t) and y(t) are real-valued baseband signals called the in-

phase and quadrature components of s(t)

45

(continued)

2. Complex Envelope Notation

where is the complex baseband or envelope of . 3. Magnitude and Phase

where is the magnitude of , and is the phase of .

2 2( ) Re[( ( ) ( )) ) Re[ ( ) ]c cj f t j f tls t x t jy t e s t e

( )ls t ( )s t

( ) ( )cos(2 ( ))cs t a t f t t 2 2( ) ( ) ( )a t x t y t ( )s t

1 ( )( ) tan [ ]( )

y ttx t

( )s t

46

Key Ideas from I/QRepresentation of Signals We can represent bandpass signals independent of carrier frequency. The idea of quadrature sets up a coordinate system for looking at common modulation types. The coordinate system is sometimes called a signal constellation diagram. Real part of complex baseband maps to x-axis and imaginary part of complex baseband maps to the y-axis

47

Constellation Diagrams

Interpretation of Signal Constellation Diagram Axis are labeled with x(t) and y(t)

In-phase/quadrature or real/imaginary Possible signals are plotted as points Symbol amplitude is proportional to distance from

origin Probability of mistaking one signal for another is

related to the distance between signal points Decisions are made on the received signal based

on the distance of the received signal (in the I/Q plane) to the signal points in the constellation

For PAM signals How???

)(1 t0

1s2s

bEbE

Binary PAM

)(1 t0

2s3s

52 bE

56 bE

56 bE

5

2 bE

4s 1s4-ary PAM

T t

)(1 t

T1

0

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