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DIGITAL CARRIER MODULATION SCHEMES. Dr.Uri Mahlab. 1. Dr. Uri Mahlab. תוכן עניינים :. Introduction of Binary Digital Modulation Schemes 2-10 Probability of error 11-21 Transfer function of the optimum filter 22-26 Matched filter receiver 27-29 Correlation receiver 30-32

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DIGITAL

CARRIER

MODULATION

SCHEMES

Dr.Uri Mahlab

1

Dr. Uri Mahlab


תוכן עניינים :

Introduction of Binary Digital Modulation Schemes 2-10

Probability of error 11-21

Transfer function of the optimum filter 22-26

Matched filter receiver 27-29

Correlation receiver 30-32

Example (the BER average of PSK) 33-35

Binary ASK Signaling Schemes 36-40

Coherent ASK 41-43

Noncoherent ASK 44-49

Binary PSK signaling schemes 50-51

Coherent PSK 52-54

Differentially Coherent PSK 55-57

Binary FSK signaling schemes 58-59

Coherent FSK 60-62

Noncoherent FSK 63-64

Comparison of digital modulation schemes 65

M-ary signaling schemes 66-85

Probability of error of M-ary orthogonal signaling scheme 86-88

Synchronization Methods 89-90

1.א

Dr. Uri Mahlab


INTRODUCTION

In order to transmit digital information over *

bandpass channels, we have to transfer

the information to a carrier wave of

.appropriate frequency

We will study some of the most commonly *

used digital modulation techniques wherein

the digital information modifies the amplitude

the phase, or the frequency of the carrier in

.discrete steps

2

Dr. Uri Mahlab


The modulation waveforms fortransmitting :binary information over bandpass channels

3

Dr. Uri Mahlab


OPTIMUM RECEIVER FOR BINARY

:DIGITAL MODULATION SCHEMS

The function of a receiver in a binary communication *

system is to distinguish between two transmitted signals

.S1(t) and S2(t) in the presence of noise

The performance of the receiver is usually measured *

in terms of the probability of error and the receiver

is said to be optimum if it yields the minimum

.probability of error

In this section, we will derive the structure of an optimum *

receiver that can be used for demodulating binary

.ASK,PSK,and FSK signals

4

Dr. Uri Mahlab


Description of binary ASK,PSK, and

: FSK schemes

-Bandpass binary data transmission system

+

Input

ּ+

+

5

Dr. Uri Mahlab


:Explanation *The input of the system is a binary bit sequence {bk} with a *

.bit rate r b and bit duration Tb

The output of the modulator during the Kth bit interval *

.depends on the Kth input bit bk

The modulator output Z(t) during the Kth bit interval is *

a shifted version of one of two basic waveforms S1(t) or S2(t) and

:Z(t) is a random process defined by

.1

6

Dr. Uri Mahlab


The waveforms S1(t) and S2(t) have a duration *

of Tb and have finite energy,that is,S1(t) and S2(t) =0

if

and

Energy

:Term

7

Dr. Uri Mahlab



Choice of signaling waveforms for various types of digital*

modulation schemes

S1(t),S2(t)=0 for

.The frequency of the carrier fc is assumed to be a multiple of rb

Type of

modulation

ASK

PSK

FSK

Dr. Uri Mahlab


:Receiver structure

output

10

Dr. Uri Mahlab


:{Probability of Error-{Pe*

The measure of performance used for comparing *

!!!digital modulation schemes is the probability of error

The receiver makes errors in the decoding process *

!!! due to the noise present at its input

The receiver parameters as H(f) and threshold setting are *

!!!chosen to minimize the probability of error

11

Dr. Uri Mahlab


:The output of the filter at t=kTb can be written as *

12

Dr. Uri Mahlab


:The signal component in the output at t=kTb

h( ) is the impulse response of the receiver filter*

ISI=0*

13

Dr. Uri Mahlab


Substituting Z(t) from equation 1 and making*

change of the variable, the signal component

:will look like that

14

Dr. Uri Mahlab


:The noise component n0(kTb) is given by *

.The output noise n0(t) is a stationary zero mean Gaussian random process

:The variance of n0(t) is*

:The probability density function of n0(t) is*

15

Dr. Uri Mahlab



:The conditional pdf of V0 given bk = 0 is given by*

.3

:It is similarly when bk is 1*

17

Dr. Uri Mahlab


Combining equation 2 and 3 , we obtain an*

:expression for the probability of error- Pe as

.4

18

Dr. Uri Mahlab


:Conditional pdf of V0 given bk

:The optimum value of the threshold T0* is*

19

Dr. Uri Mahlab


Substituting the value of T*0 for T0 in equation 4*

we can rewrite the expression for the probability

:of error as

20

Dr. Uri Mahlab


The optimum filter is the filter that maximizes*

the ratio or the square of the ratio

(maximizing eliminates the requirement S01<S02)

21

Dr. Uri Mahlab


:Transfer Function of the Optimum Filter*

The probability of error is minimized by an *

appropriate choice of h(t) which maximizes

Where

And

22

Dr. Uri Mahlab


If we let P(t) =S2(t)-S1(t), then the numerator of the*

:quantity to be maximized is

Since P(t)=0 for t<0 and h( )=0 for <0*

:the Fourier transform of P0 is

23

Dr. Uri Mahlab


:Hence can be written as*

(*)

We can maximize by applying Schwarz’s*

:inequality which has the form

(**)

24

Dr. Uri Mahlab


Applying Schwarz’s inequality to Equation(**) with-

and

We see that H(f), which maximizes ,is given by-

(***)

!!! Where K is an arbitrary constant

25

Dr. Uri Mahlab


Substituting equation (***) in(*) , we obtain-

:the maximum value of as

:And the minimum probability of error is given by-

26

Dr. Uri Mahlab


:Matched Filter Receiver*

If the channel noise is white, that is, Gn(f)= /2 ,then the transfer -

:function of the optimum receiver is given by

From Equation (***) with the arbitrary constant K set equal to /2-

:The impulse response of the optimum filter is

27

Dr. Uri Mahlab


Recognizing the fact that the inverse Fourier *

of P*(f) is P(-t) and that exp(-2 jfTb) represent

:a delay of Tb we obtain h(t) as

:Since p(t)=S1(t)-S2(t) , we have*

28

Dr. Uri Mahlab


:Impulse response of the Matched Filter *

1

t

0

2 \Tb

(a)

0

2 \Tb

t

1-

(b)

2

2 \Tb

0

t

Tb

2

(c)

t

(d)

0

2

2 \Tb

0

t

(e)

Tb

Dr. Uri Mahlab


:Correlation Receiver*

The output of the receiver at t=Tb*

Where V( ) is the noisy input to the receiver

Substituting and noting *

: that we can rewrite the preceding expression as

(# #)

30

Dr. Uri Mahlab


Equation(# #) suggested that the optimum receiver can be implemented *

as shown in Figure 1 .This form of the receiver is called

A Correlation Receiver

-

+

31

Dr. Uri Mahlab


In actual practice, the receiver shown in Figure 1 is actually *

.implemented as shown in Figure 2

In this implementation, the integrator has to be reset at the

- (end of each signaling interval in order to ovoid (I.S.I

+

c

Figure 2

The bandwidth of the filter preceding the integrator is assumed *

!!! to be wide enough to pass z(t) without distortion

32

Dr. Uri Mahlab


Example actually *: A band pass data transmission scheme

uses a PSK signaling scheme with

The carrier amplitude at the receiver input is 1 mvolt and

the psd of the A.W.G.N at input is watt/Hz. Assume

that an ideal correlation receiver is used. Calculate the

.average bit error rate of the receiver

33

Dr. Uri Mahlab


:Solution actually *

Data rate =5000 bit/sec

Receiver impulse response

Threshold setting is 0 and

34

Dr. Uri Mahlab


:Solution Continue actually *

=Probability of error = Pe *

35

Dr. Uri Mahlab


* Binary ASK signaling schemes: actually *

The binary ASK waveform can be described as

Where and

We can represent :Z(t) as

36

Dr. Uri Mahlab


Where D(t) is a lowpass pulse waveform consisting of actually *

.rectangular pulses

:The model for D(t) is

37

Dr. Uri Mahlab


:The power spectral density is given by actually *

The autocorrelation function and the power spectral density

:is given by

38

Dr. Uri Mahlab


:The psd of Z(t) is given by actually *

39

Dr. Uri Mahlab


If we use a pulse waveform D(t) in which the individual pulses

g(t) have the shape

40

Dr. Uri Mahlab


Coherent ask
Coherent ASK pulses

We start with

The signal components of the receiver output at the

:of a signaling interval are

41

Dr. Uri Mahlab


:The optimum threshold setting in the receiver is pulses

:The probability of error can be computed as

42

Dr. Uri Mahlab


:The average signal power at the receiver input is given by pulses

We can express the probability of error in terms of the

:average signal power

The probability of error is sometimes expressed in *

: terms of the average signal energy per bit , as

43

Dr. Uri Mahlab


Noncoherent ask
Noncoherent ASK pulses

:The input to the receiver is *

44

Dr. Uri Mahlab


Non-coherent ASK Receiver pulses

45

Dr. Uri Mahlab


:The pdf is pulses

46

Dr. Uri Mahlab


pdf’s of the envelope of the noise and the signal * pulses

:pulse noise

47

Dr. Uri Mahlab


:The probability of error is given by pulses

48

Dr. Uri Mahlab


49 pulses

Dr. Uri Mahlab


Binary psk signaling schemes
BINARY PSK SIGNALING pulsesSCHEMES

:The waveforms are *

:The binary PSK waveform Z(t) can be described by *

.D(t) - random binary waveform *

50

Dr. Uri Mahlab



Coherent psk
Coherent PSK pulses

:The signal components of the receiver output are

52

Dr. Uri Mahlab


:The probability of error is given by pulses

53

Dr. Uri Mahlab


54 pulses

Dr. Uri Mahlab


DIFFERENTIALLY COHERENT * pulses

:PSK

DPSK modulator

55

Dr. Uri Mahlab


DPSK demodulator pulses

Filter to

limit noise

power

Lowpass

filter or

integrator

Threshold

device

(A/D)

Delay

56

Dr. Uri Mahlab


Differential encoding decoding
Differential encoding & decoding pulses

57

Dr. Uri Mahlab


Binary fsk signaling schemes
* BINARY FSK SIGNALING SCHEMES : pulses

:The waveforms of FSK signaling

:Mathematically it can be represented as

58

Dr. Uri Mahlab


Power spectral density of FSK signals pulses

Power spectral density of a binary FSK signal

with

59

Dr. Uri Mahlab


Coherent fsk
Coherent FSK pulses

:The local carrier signal required is

The input to the A/D converter at sampling time

60

Dr. Uri Mahlab



.Which are usually encountered in practical system pulses

:We now have

:When

62

Dr. Uri Mahlab


Noncoherent fsk
Noncoherent FSK pulses

63

Dr. Uri Mahlab


Noncoharenr demodulator of binary FSK pulses

ENVELOPE

DETECTOR

+

THRESHOLD

DEVICE

(A/D)

-

ENVELOPE

DETECTOR

64

Dr. Uri Mahlab


Probability of error for binary digital modulation * pulses

:schemes

65

Dr. Uri Mahlab


M-ARY SIGNALING SCHEMES pulses

:M-ARY coherent PSK

The M possible signalsthat would be transmitted

:during each signaling interval of duration Ts are

:The digital M-ary PSK waveform can be represented

66

Dr. Uri Mahlab



Phasor diagram for QPSK pulses

That are derived from a coherent local carrier

reference

68

Dr. Uri Mahlab


If we assume that S pulses 1 was the transmitted signal

:during the signaling interval (0,Ts),then we have

69

Dr. Uri Mahlab


QPSK receiver scheme pulses

70

Dr. Uri Mahlab


:The outputs of the correlators at time t=T pulsesS are

71

Dr. Uri Mahlab


Probability of error of QPSK: pulses

72

Dr. Uri Mahlab


73 pulses

Dr. Uri Mahlab


Phasor diagram for M-ary PSK ; M=8 pulses

74

Dr. Uri Mahlab


The average power requirement of a binary PSK pulses

:scheme are given by

75

Dr. Uri Mahlab


* COMPARISION OF POWER-BANDWIDTH pulses

:FOR M-ARY PSK

Value

of M

4

8

16

32

0.5

0.333

0.25

0.2

0.34 dB

3.91 dB

8.52 dB

13.52 dB

76

Dr. Uri Mahlab

Dr. Uri Mahlab


* M-ary for four-phase pulsesDifferential PSK:

RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK

Integrate

and dump

filter

Z(t)

Integrate

and dump

filter

77

Dr. Uri Mahlab


:The probability of error in M-ary differential PSK pulses

:The differential PSK waveform is

78

Dr. Uri Mahlab


:Transmitter for differential PSK* pulses

Serial to

parallel

converter

Diff

phase

mod.

Envelope

modulator

BPF

(Z(t

79

Dr. Uri Mahlab


* M-ary Wideband FSK pulsesSchemas:

Let us consider an FSK scheme witch have the

: following properties

80

Dr. Uri Mahlab


:Orthogonal Wideband FSK receiver pulses

MAXIMUM

SELECTOR

Z(t)

81

Dr. Uri Mahlab


:The filter outputs are pulses

82

Dr. Uri Mahlab


:N pulses0 is given by

:The probability of correct decoding as

:In the preceding step we made use of the identity

83

Dr. Uri Mahlab


The joint pdf of Y2 ,Y3 ,…,YM * pulses

:is given by

84

Dr. Uri Mahlab


where pulses

85

Dr. Uri Mahlab


Probability of error for M-ary orthogonal * pulses

: signaling scheme

86

Dr. Uri Mahlab


The probability that the receiver incorrectly * pulses

decoded the incoming signal S1(t) is

Pe1 = 1-Pe1

The probability that the receiver makes *

an error in decoding is

Pe = Pe1

We assume that , and

We can see that increasing values of M lead to smaller power

requirements and also to more complex transmitting

receiving equipment.

87

Dr. Uri Mahlab


In the limiting case as M the probability of error Pe satisfies

The maximum errorless rb at W data can be transmitted

using an M- ary orthogonal FSK signaling scheme

The bandwidth of the signal set as M

88

Dr. Uri Mahlab


:Synchronization Methods error P

For optimum demodulation of ASK ,FSK ,and PSK waveforms

timing information is needed at the receiver

There are three general methods used for synchronization in

:digital nodulation schemes

.Use of primary or secondary time standard

.Utilization of a separate synchronization signal

Extraction of clock information from the modulated waveform

.itself , referred to as self - synchronization

.1

.2

.3

89

Dr. Uri Mahlab


(Extraction of local carrier for coherent demodulation of PSK signals)

Open loop carrier recovery scheme

Squaring

circuit

BPF

Frequency

divider

Closed loop carrier recovery scheme

Squaring

circuit

Loop

Filter

VCO

Frequency

doubler

Recovered carrier

cos (wct)

90

Dr. Uri Mahlab


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