An overview of Digital Modulation

8/13/2019 An overview of Digital Modulation

1/135

6.4 Digital Modulation-An overviewModern Mobile communication uses digital modulation techniques.

Advancements in DSP and VLSI have made digital modulation more

cost effective than analog transmission systems

Advantages:1. Greater noise immunity

2. Robustness to channel impairments

3. Easier multiplexing of (voice, data & video)

4. Security 1


2/135

5. Digital error- control codes detectand/orcorrect Tx errors

6. Complex signal conditioning &processing(encryption,equalization)

7. Programmable DSP Digital modulators,demodulators software

8. Modem design using embedded softwareimplementation(No redesign/replace Modem).

2


3/135

Modulating (message) signal

Symbols/pulses

m finite states

Each symbol n bits of information

n = log m bits/symbol

3


4/135

6.4.1 Factors that influence the choice

of Digital Modulation Desirable Modulation Scheme

low BER at low SNR

occupies min. BW

Easy & cost effective to implement

Performance of Modulation Scheme is measured interms of

power efficiency

BW efficiency

4


5/135

Power Efficiencyp

Ability of modulation technique to preservethe

fidelity[Bit Error Prob] of msg evenat low power

levels

To Noise immunity by

signal power

Amountby which signal power to have certain

fidelitytype of modulation is used.

5


6/135

Power efficiency : p: (Energy efficiency)

p= Signal energy per bit = Eb

Noise power spectral density No

Required at Rx to have certain Probability of

error

Power Efficiency

6


7/135

Bandwidth efficiency

Ability of mod scheme to accommodate data

within a limited BW

How efficiently allocated BW is utilized

B = Throughput data rate = R bits/s bps/Hz

Hz in a given BW B BW of modulated RF

System capacity B B Tx more data in a

given spectrum

7


8/135

Shannons Channel Coding theorem

For a given prob of error max possible BW

efficiency limited by noise in the channel

Channel capacity formula = Bmax = [C/B] =

log*1S/N+

C Channel capacity

B RF BW

8


9/135

6.4.2 Bandwidth and Power Spectral Density of Digital

Signals

PSD of a random signal w(t) is

WT(f) FT of WT(t)

WT(t) = w(t) forT/2


10/135

The PSD of a bandpass signal is related to PSD of

its baseband complex envelope

S(t) modulated (BandPass) signal

g(t) complex baseband envelope

S(t) = Re{g(t) exp(j2fct)}

Ps(f) = 0.25[Pg(f-fc)+Pg(-f-fc)]

Pg(f) = PSD of g(t)

10


11/135

Absolute BW

Range of freq over which signal has nonzeroPSD.

But for rectangular pulses , PSD extends overinfinite range of frequencies

Simplermeasure of BW null-to-null BW = Widthof main spectral lobe

11


12/135

Popular Measure of BW measures dispersion

of spectrum Half Power BW(3 dB Bandwidth)

HP BW(3 dB bandwidth) Interval b/w

frequencies at which PSD has dropped to one

half power or 3dB below the peak values.

12


13/135

6.5 Line Coding

Digital baseband signals use line codes to Provide

particular spectral characteristics of a pulse train

Common line Codes for mobile comm.

Return-to-zero(RZ) non-return-to-zero(NRZ)

Manchester codes

Unipolar v/g levels 0 or v

Bipolar v/g levels -v or v13


14/135

14


15/135

RZ pulse train returns to zero within every bitperiod

Spectral widening butimprovestimingsynchronization

NRZ dontreturn to zero during a bit period

signal constant throughout a bit period

more efficient than RZ but results in poorsynchronization

have large dc component

notused in dc blocking ckts such as

audio amplifiers or phone switching equipment15


16/135

Manchester code

Spl NRZ line code

no dc comp simple synch used in phonelines and dc blocking ckts

Use 2 pulses for each binary symbol

zero-crossings are guaranteed in every bit perid

provide easy clock recovery

16


17/135


18/135

Higher level mod (M-ary keying) signal set

more than 2 signals

Signal set size M No of bits/symbol logM

18


19/135

Elements of S viewed as points in vector space

Finite set of physically realizable w/fs in vectorspace expressed as linear combination of N

orthonormal w/fs Form the basisof the vector

space

Representing modulation signal on a vector space

Find set of signals that form basis for thatvector space.

19


20/135

Once basis is found any pt in vector space

linear combination of basis signals

The basis signals are orthogonaltoone another in

time such that

Each of the basis signalsis normalizedto have

unit energy, i.e.,

20

For BPSK scheme


21/135

For BPSK scheme,

the basis signal

then, the BPSK signal set can be represented as

Eb-Energy per bit

Tb-Bit period

21


22/135

constellation diagram

Graphical representation of the complex envelope of

each possible symbol state.

The X-axis represents the in-phase components

y-axisrepresents the quadrature component of

complex envelope22


23/135

The number of basis signals required to represent

the complete modulationsignalsetis called thedimensionof the vector space.

No of basis signals


24/135

Prob of bit error distance b/w closest point in

the constellation.

Modulation scheme densely packed less

energy efficient

Average probability of error for a particular

modulation signal,

NoNoise spectral density

dij Euclidean distance b/w ith& jthsignal points

Q-function

24


25/135

Linear Modulation Techniques

In linear modulation techniques, the amplitude of the

transmitted signals, s(t), varies linearly with the modulatingdigital signal, m(t)(Hence non constant envelope).

Linear modulation techniques are bandwidth efficient, butmust be transmitted using linear RF amplifiers which havepoor power efficiency.

Using the power efficient nonlinear RF amplifiers could

cause severe adjacent channel interference, and results inthe loss of all the spectral efficiency gained by linearmodulation.

25


26/135

Constant Envelop Modulation(Amplitude of carrier isconstant)

The constant envelop family of modulation has the

following advantages: Power efficient Class C amplifiers can be used without introducing

degradation in the spectrum occupancy.

Low out-of-band radiation(-60 to -70 dB)

Limiter-discriminator detection can be used, which simplifiesreceiver design and high immunity towards noise and fading effects.

But, they occupy a larger bandwidth than linearmodulation schemes.

Many practical mobile radio communication systems useNon linear modulation Schemes

26


27/135

Binary Frequency Shift Keying (BFSK)

The frequency of a constant amplitude carrier signal is

switched between two values according to the two

possible message states(binary 1 or 0).

where is a constant offset from the nominal carrier

frequency

f2

27

G ti f FSK(di ti FSK)


28/135

Generation of FSK(discontinuous FSK)

switchingb/w two independent oscillators

depending on databit is 1 or 0 and it isdiscontinuous at switching times (phase discontinuity)

Discontinuous FSK signal is represented as,

28


29/135

Phase discontinuities several problems

spectral spreading

spurious transmissions

hence not used in highly regulated wireless systems

29


30/135

30


31/135

Spectr m and BW of BFSK signals


32/135

Spectrum and BW of BFSK signals

Complex envelope of FSK signal nonlinear function of

m(t)

PSDof FSK has discrete frequency components at fc,

fc+nf, fc-nf n is integer

PSDof continuous phase FSK falls off as inversefourth power of frequency offset from fc.

PSDof discontinuous phase FSK falls off asinverse square of frequency offset from fc.

1/[ 2f ]4

1/[ 2f ]2

32

Transmission BW of FSK signal from Carsons rule


33/135

Transmission BW of FSKsignal, from Carsons rule,

BT= 2f + 2B

B BWof digital baseband signal

If first null BW is used, BW of rectangular pulses,is B=R. FSK transmission BW = 2f + 2R = 2(f+R)

If raised cosine pulse-shaping filter is used,B = (1 + ) R/2 BT= 2f + (1+)R

roll off factor of the filter33

Coherent Detection of Binary FSK


34/135


Two Correlator with locally generated coherent

reference signals

Diff of Correlator o/p compared with threshold diff>Th1

diff


35/135


35

Non coherent Detection of Binar FSK


36/135

Detect FSK signal without coherent carrier

reference

Pair of matched filters envelope detectors

Filter in upper path matched to FSK signal of

frequency fL

Filter in lower path matched to FSK signal of

frequency fH

Non coherent Detection of Binary FSK

36



37/135


37



38/135

Matched filters BPFcentered at fH& fL

Envelope detectors O/P sampled at t=kTb

compared

If envelope detector o/p> or < threshold

o/p 1/0


38


39/135

Probability of error of an FSK system non

coherent detection,

39

6 9 2 Minimum Shift Keying


40/135

6.9.2 Minimum Shift Keying

Special type of Continuous phasefrequency shift keying(CPFSK)

Peak frequency deviation = (Bit rate)/4

Frequency difference between the logical one and logical

zero statesequal to half the data rate MSKContinuous phase FSK with modulation index = 0.5

Modulation Index of FSK = FMmodulation index

F peak RF frequency deviation

Rb Bit rate

40

Modulation index = 0 5


41/135

Modulation index = 0.5

min frequency spacing b/w fH& fL

that allows two FSK signals to be coherently

orthogonal

Two FSK signals vH(t) & vL(t) are orthogonaliff,

MSK fast FSK frequency spacing used =1/2of[used in noncoherent FSK]

41


42/135

MSK used in mobile radio comm system

constant envelope

spectral efficiency

good BER performance

selfsynchronizing capacity

42


43/135

43


44/135

MI(t) & MQ(t)

odd & evenbits of bipolar data stream

1

feed the In-phase& Quadrature arms of

modulator at a rate of Rb/2

44


45/135

MSK special type of continuous phase FSK of

above SMSK(t) is rewritten using trigonometric

identities as,

If k 0 or mI(t) is 1 or -1

MSK constant amplitude

45

h i i b d d


46/135

Phase continuity at bit transition period is ensured

by choosingcarrier frequency fc integral

multiple of one fourth the bit period 1/4T

Compare SFSK& SMSK MSKsignal is an FSKsignal

with binary signaling frequencies fc+1/4T & fc-1/4T

phaseof MSKsignal varies linearly during each

bit period

46


47/135

N li d PSD i


48/135

Normalized PSD is,

From PSD of MSK

MSK has lower side

lobethan QPSK & OQPSK

99%of MSK power is

contained within B = 1.2/T

48


49/135

99% of QPSK & OQPSK is contained within B = 8/T

MSKhas faster rolloff due to smoother pulse

functions

Main lobe of MSK is wider than QPSK & OQPSK

MSK is less spectrally efficient than PSK techniqueswhen compared in terms of first null bandwidth

49


50/135

No change in phase at bit transition periods

MSKhas continuous phase property

Envelopeis constanteven after band limiting

50


51/135

Small variations in envelope removed by hard

limiting at the Rx without raising the out-of-

band radiation levels

Amplitudeis constant MSK signals are amplified

using nonlinear amplifiers

MSK simple demodulation & synchronization

ckts. Hence popular among mobile radiocommunications

51

MSK Transmitter


52/135

MSK Transmitter

52

MSK Transmitter and Receiver


53/135

MSK Transmitter and Receiver

CarrierXlied with cos[t/2T] produce two

phase-coherent signals at fc+1/4T & fc-1/4T

2 FSK signals separated using 2 NBPF combinedto form in-phase x(t) & quadrature

y(t) carrier components

These carriers are Xliedwith odd & even bit

streams mi(t) & mq(t) produce MSK signal

SMSK(t)53

MSK Receiver


54/135

MSK Receiver

54

S (t) Xlied by in phase x(t) & quadrature y(t)


55/135

SMSK(t) Xliedby in-phase x(t)& quadrature y(t)

carriers

o/p of Xliersare integratedover two bit periods

dumped to decision ckt at the end of each two

bit periods

Based on o/p of integrator threshold detector

o/p is 0/1

o/p data streams mi(t)/mq(t) offset combined

to get demodulated signal.55

6 9 3 Gaussian Minimum Shift Keying (GMSK)


56/135

6.9.3 Gaussian Minimum Shift Keying (GMSK)

Simple Binary modulation scheme derivative of

MSK

Sidelobe levels by passing modulating NRZ

data waveform through premodulation Gaussianpulseshaping filter

NRZ data w/f Gaussian Filter Smoothened o/p

stabilizesthe instantaneous frequency variationsover time which reduces the sidelobe levels in

Transmitted spectrum 56

GMSKcan be


57/135

detected using coherent detector(as MSK)

detectedusing noncoherent detector (as FSK)

Has excellent power efficiency(due to constant

envelope) & spectral efficiency

Premodulation gaussian filtering Introduces ISIin

the Tx signal but degradationis not severe if 3dBbandwidthbit duration product (BT)of filter

is greater than 0.5

57

GMSK d l ti filt h i l


58/135

GMSK premodulation filter has impulse response,

Transfer function,

GMSK filter defined by B & baseband symbol

duration T.

Therefore GMSK is defined by its BT product58


59/135


60/135

GMSK Bit E R t


61/135

GMSK Bit Error Rate

Bit error probability for GMSK is,

61

GMSK Transmitter & Receiver


62/135

GMSK Transmitter :

Pass a NRZ message bit stream through aGaussian baseband filter having impulse

response

followed by FM modulator 62


63/135

Usedin US Cellular Digital Packet Data (CDPD)

& Global System for mobile (GSM) system.

63

GMSK Receiver :


64/135

GMSK Receiver :

64

GMSK Receiver :


65/135

GMSK Receiver :

Orthogonal coherent detectors

Or

Use simple noncoherent detectors FMdiscriminators

65


66/135

67


67/135

69


68/135

Pbm

70

6.10 Combined Linear & Constant Envelope Modulation


69/135

Techniques

Digital BB data is sent by varying both

envelope & phase (or frequency) of carrier

M-ary modulation

71

M i li h


70/135

M-ary signaling scheme

two or more bits are grouped together

symbols

one of M signals S1(t), S2(t),,SM(t) is Tx during

each symbol period of duration Ts

No. of possible signals= M = 2 ; n integer

amplitude, phase or frequency of carrier is

varied M-ary ASK, M-ary PSK or M-ary FSK72

M-ary Modulation Schemes


71/135

y

Achieve better BW efficiency than PowerEfficiency

Eg: 8-PSK BW = log8 = 3 times smaller thanBPSK

BER performance worse than BPSK

73

6.10.1 M-ary Phase Shift Keying (MPSK)


72/135

M-ary PSK carrier phase takes on one of M

possible values

Modulated waveform,

Energy per symbol

symbol period

74

Si(t) can be rewritten in quadrature form as,


73/135

( ) q ,

i = 1,2,..,M

Choose the basis signals,

Over 0 t Ts

75

M-ary PSK signal is


74/135

M-ary PSK signal is,

i=1,2,..,M

There are 2 basis signals(therefore constellation is

2 dimensional)

M-ary msg points equally spaced on a circleof

radius Es centered at origin

76

Fig 6 45 constellation diagram of 8-ary PSK


75/135

Fig 6.45 constellation diagram of 8-ary PSK

M-ary PSK ampof Tx signal const circular

constellation

MPSK const envelope signal when no pulse shaping

used

77


76/135

78

= 6.62


77/135

Fig 6.45 distance b/w adjacent symbols =2Es(sin(/M)) &

Average symbol error probability ofcoherent M-aryPSK is,

79


78/135

Power Spectra of M-ary PSK


79/135

Ts symbol duration

Tb Bit duration

Ts= TblogM

PSD of M-ary PSK signal with rectangular pulses is,

81

Fig 6.46A M fi t ll


80/135

As M first nullBW Rb = const

M BWefficiency

Fixed Rb M B B

M constellationisdensely packed power efficiency

82


81/135

6 10 2 M ary Quadrature Amplitude Modulation (QAM)


82/135

6.10.2 M-ary Quadrature Amplitude Modulation (QAM)

QAM Hybrid modulation Technique: Vary both A& Phase

Fig 6.47 constellation diagram

16ary QAM

square lattice of signal points

84


83/135

85

General M-ary QAM is,


84/135

y

Emin energy of signal with lowest amplitude

Closest point has min energy

ai & bi pair of independent integers

M-ary QAM

doesnthave const energy per symbol,

doesnt have const dist b/w symbol states

86

Si(t)in terms of 2 basis fns,


85/135

(ai,bi) element of L by L matrix, gives location

87

Eg: 16-QAM signal constellation Fig 6.47,


86/135

LXL matrix is,

L=16=4

88

Average prob of error M-ary QAM, using


87/135

g p y Q , gcoherent detection,

In terms of average signal energy, Eav

Power spectrum & BW efficiency of QAM identical to M-ary PSK modulation

Power efficiency of QAM superior to M-ary PSK89


88/135

6.10.3 M-ary Frequency Shift Keying (MFSK) and OFDM


89/135

M-FSK signals,the transmitted signal is defined as

M Tx signals

equal energy

equal duration signal frequencies are separated by 1/2Ts Hz

& signals are orthogonal

91

Coherent detection of M-ary FSK:


90/135

has bank of M correlators

matched filters tuned to M distinct carriers Avg prob of error =

Noncoherent detection: Using matched filters followed by envelope

detectors,

Avg probability of error is,

92

Using only leading terms of binomial expansion,


91/135

the prob of error,

BW of coherentM-ary FSK signal,

BW of noncoherentMSK is,

93

BW efficiency of M-ary FSK with M


92/135

M-FSK BW inefficient

All M signals are orthogonal no crowding insignal space power effi with M , It can be

amplified using nonlinear amplifiers

Orthogonality characteristic of MFSK led toOFDM power efficient signaling for a large no. ofusers on the same channel.

94

6.11 Spread spectrum modulation techniques.


93/135

Modulation & demodulation

techniques BW efficiency

power efficiency

Spread spectrum

pseudorandom

BW =>BW inefficient (disadvantage)

noiselike properties Advantage :multiple users use same BW

simultaneously.

95

Spreading of waveform is controlledby:

P d i (PN)


94/135

o Pseudo noise(PN) sequence or

o Pseudo noise code.

At Rx-> SS signals demodulatedusing locally

generated pseudo random carrier.

cross correlation with PN sequence->

despreads the spread spectrum signal=> restores

the message.

96

Advantage:

I t f j ti bilit


95/135

Interference rejection capability

o Each user is given a unique PN code-> orthogonalto

the code of other users.

o Rx separates each user based on their codes, even

though they occupy the same spectrum at all times.

Narrowband interference-> removed using notch

filtering.

97

Advantage:


96/135

Eliminates freq planning since all cells can use the

same channels.

SS signals have uniform energy over a large BW->

hence at a given time only a small portion of the

spectrum will undergo fading.

98


97/135

99

6.11.1 pseudo-noise (PN) sequences.


98/135

Binary sequence with an autocorrelationthat

resembles the autocorrelation of a random binary

sequence.

A.C->also resemblesthe A.Cof band limited white

noise

100

Characteristics of PN sequences:


99/135

Characteristics of PN sequences:

Equal number of 0s & 1s.

Very low correlation b/w shifted versions of the

sequence.

Very low cross correlation b/w any two sequences.

101

PN sequence is generated using-> sequential logiccircuits


100/135

circuits.

Has consecutive stages of two state memory.

Feedback logic.102

Binary sequence are shiftedthrough shift registers


101/135

y q g g

wrt clock pulse

Output of various stages are logically combined&

fedback as the i/pto the first stage.

If feedback logic->has ex-or gates-> shift register is

called a linear PN sequence generator.

103

Initial contentsof memory stages and the feed


102/135

back logic -> decides>successive contents ofmemory.

If linear S.R reaches zero state at some time->always remain in the zero state-> o/p all 0s.

There are (2^m)-1 non zero states for m-stage F/Bregisters.

Sequence ->linear F/B register-> maximal length(ML) sequence.

104

6.11.2 direct sequence spread spectrum(DS-SS).


103/135

DS-SS->system->spreadsthe baseband data bymultiplying BB data pulseswith PN sequence.

Single pulse/symbol of PN waveform-> chip.

105


104/135

106

Data symbols Rectangular pulses v=+1/-1.

information bits


105/135

information bits

added in modulo 2fashion.

chips before being phase modulated

At Rx-> use coherent/differentially coherent PSKdemodulation.

The received spread spectrum signal for singleuser:

Diff user have diff p(t)107

m(t)rectangular pulses of amplitude+1/-1each symbol duration=Ts


106/135

each symbol duration Ts

P(t) PN sequence each pulsechip(Narrower than the bit) rectangular pulse+1/-1and duration=Tc.[Tc BW of spread spectrum.

B->BWof m(t)cos2 fct-> Bw of modulatedsignal.108


107/135

109

Rx signal passes through wideband filter


108/135

Multipliedby local replica of PN code sequence p(t).

If p(t)=+1/-1=>P^2(t)=1

multiplication-=>yields despread signal s(t). BWNB Signal

-at input of demodulator.

Coherent PSK/differential PSK demodulator -> givesm(t).

110


109/135

WBF O/P NB interference

Rx Correlator o/p after despreading->Signal BW=B

Interference get spread

Interference BW >Bss

Strong interference/weak signal STRONG SIGNAL & WEAK

INTERFERENCE111

Filteringaction of demodulator-> removes most


110/135

of interference spectrum.

Interference rejection ratio=Bss/B.

Greater the PG=> greater the ability to suppress

in-band interference.

112

Direct Sequence Spread Spectrum

E l


111/135

Example

113

Approximate Spectrum of DSSS Signal


112/135

114

6.11.3 frequency hopped spread spectrum.


113/135

It involves a periodic change of transmission frequency

hopset-set of possible carrier frequencies.

Hopping occurs over a frequency band that includes a

number of channels.

BW of channel used in hopset-> instantaneous BW. B

BW of spectrum over which hopping occurs-> total

hopping BW. Bss

115

If single carrier frequency(single channel) is usedon each hop >single channel modulation


114/135

on each hop-> single channel modulation

Time duration b/w hop->hop duration/hoppingperiod(Th).

Bss->total hopping BW.

B->instantaneous BW

PG=Bss/B.

116

6.11.3 frequency hopped spread spectrum.


115/135

Carrier frequency abruptly change(hop)in

accordance with PN code sequence.

The set of possible carrier frequencies is called the

hopset

Hopping occurs over a frequency band that

includes a number of channels

117

The frequency of the carrier is periodically modified(hopped) following a specific sequence of frequencies


116/135

(hopped) following a specific sequence of frequencies.

In FHSS systems, the spreading code is this list offrequencies to be used for the carrier signal, a.k.a. thehopping sequence

The amount of time spent on each hop is known as dwelltime and is typically in the range of 100 ms.

Redundancy is achieved through the possibility to

execute re-transmissions on different carrier frequencies(hops).

119


117/135

120


118/135

121

If only a single carrier frequency is used on

h h d l i i ll d i l h l


119/135

122

each hop, Modulation is called single channel

modulation

The time duration between hops is called the

hop duration or the hopping period and is

denoted by Th

The total hopping bandwidth and the

instantaneous bandwidth are denoted by Bssand B respectively.

The processing gain = Bss/B for FHSS systems

Dehopped signal->hoppingis removedfrom Rx

signal


120/135

signal.

If the Frequency synthesizer produced by the

receiver synthesizer is synchronized with the

frequency pattern of the received signal ,then the

mixer output is a dehopped signal.

It is possible to have collisions in an FH system

where an undesired user transmits in the channel

at the same time as the desired user

123

Frequency hopping-fast/slow.


121/135

fast frequency hopping

more than one frequency hop during eachtransmitted symbol.

hopping rate information symbol rate.

124


122/135


123/135

6.11.4 performance of direct sequence spreadspectrum.


124/135

direct sequence spread spectrum with k users.

128


125/135

129

Each user-> PN sequence->N chips/ message symbol period T

NTc=T


126/135

Transmitted signal of Kth user,

Pk(t)->PN code sequence of Kth user.

mk(t)->data sequence of Kth user.

Rx signal will consist of sum of k different transmitted signals->[one desired user & k-1 undesired user]+additive noise.

130

Reception Decision variable of ith Tx bit of user 1


127/135

if m1,i=-1=> bitis received in error if Zi(1)>0.

Probability of error=

Rx signal r(t)-> linear combination of signals plusadditive noise

131

Response of receiver due to the desired signal from user

1:


128/135

1:

Gaussian random variable representing noise with zero

mean & variance: with n(t) Additive Gaussian Noise

Multiple access interferencefrom user k,

132

Ik-> has cummulative effects of N random chipsfrom kthinterferer over the integration period Tof one bit


129/135

of one bit.

From central limit theorem-> sum of these effectswill tend toward gaussian distribution

(k-1)->users which serve as identically distributedinterferers=> total multiple access interference

Average probability of error is,

133

Single user=> k=1=> Peexpression=expression for


130/135

g e p p

BPSK modulation.

Interference limited case-> No ->Eb/No->

Therefore

134

6.11.5 Performance of frequency hopping spreadspectrum.


131/135

FH-SS systems-> many users hop their carrier frequencies

using BFSK modulation.

If 2 users are notsimultaneously utilizing the same

band=> prob of error for BFSKis

If 2 users Tx simultaneously in the same frequency

band-> collision/ hitoccurs.

Overall prob of error,

Ph->prob of a hit. Prob that hit doesnt occur135

If there are M hopping channels(slots)=>

h / b h f ll b


132/135

There is 1/M prob that a given interferer will bepresent in the desired users slot.

If there are (k-1)interfering users=> prob that atleast one is present in the desired frequency slot

When M is too large.

136

If k=1,


133/135

If Eb/No->,

Till now assumption:

All users hoptheir carrier frequencies

synchronously. =>slotted frequency hopping

137

Asynchronous case when,


134/135

Radio signals will not arrive synchronously[sametime] to each user due to various propagationdelays.

Prob of hit for asynchronous case is,

->Nb: number of bits per hop.

-> prob of hit is increased.

138


135/135

Documents

An overview of Digital Modulation