Spread Spectrum Ppt

7/30/2019 Spread Spectrum Ppt

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Spread Spectrum

The spread spectrum techniques was

developed initially for military and intelligence

requirements.

The essential idea is to spread the information

signal over a wider bandwidth to make

jamming and interception more difficult.

In this chapter, we will look some spread

spectrum techniques and multiple access

technique based on spread spectrum.


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Outline

5.1 Concept of Spread Spectrum

5.2 Frequency-Hopping Spread Spectrum

5.3 Direct Sequence Spread Spectrum

5.4 Code Division Multiple Access5.5 Generation of Spreading Sequences


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Concept of Spread Spectrum

General model of spread spectrum digital

communication system.

Analog signal with

a relatively narrow

bandwidth

To increase the

bandwidth of the

signal


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Input is fed into a channel encoder

Produce an analog signal with a relatively narrow bandwidth around

some center frequency.

Further modulated using a sequence of digits known as a

spreading code or spreading sequence.

The spreading code is generated by a pseudonoise, or pseudorandom

number generator.

The effect of this modulation is to increase significantly the bandwidth

(spread the spectrum) of the signal to be transmitted. At the receiver, the same digit sequence is used to demodulate

the spread spectrum signal.

The signal is fed into a channel decoder to recover the data.


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What can we gain from this apparent waste of

spectrum?

Gain immunity from various kinds of noise and

multipath distortion

Can be used for hiding and encrypting signals.

Only a recipient who knows the spreading code

can recover the encoded information.Several users can independently use the same

higher bandwidth with very little interference

(CDMA).


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Outline






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Frequency-Hopping Spread Spectrum

Basic Approach

FHSS Using MFSK

FHSS Performance


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Basic Approach

Frequency-Hopping Spread Spectrum -- FHSS

Signal is broadcast over a seemingly random series

of radio frequencies, hopping from frequency to

frequency at fixed intervals.

A receiver, hopping between frequencies in

synchronization with the transmitter, picks up the

message.


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Basic Approach

Advantages

Attempts to jam the signal on one frequency only

knock out a few bits of it.

Would-be eavesdroppers hear only unintelligible

blips


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Basic Approach

An example of frequency-hopping signal

A number of channels

are allocated for the

FH signal.

Typically, there are 2k

carrier frequencies

forming 2k channels.

The width of each

channel corresponds to

the bandwidth of the

input signal.


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Basic Approach

The transmitter operates inone channel at a time for a

fixed interval.

During that interval, some

number of bits are transmittedusing some encoding scheme.

The sequence of channels

used is dictated by a spreading

code. Both transmitter and

receiver use the same code to

tune into a sequence of

channels in synchronization.


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Basic Approach


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Basic Approach

Typical block diagram for a FH system

Generate

spreading code


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Basic Approach

Binary data are fed into a modulator using some modulation

scheme (FSK or BPSK)

Spreading code (generate by PN sourse) serves as an index

into a table of frequencies

Each kbits of the spreading code specifies one of the 2kcarrier

frequencies. At each k PN bits, a new carrier frequency is

selected.

The signal produced from the initial modulator is modulated

by this frequency.

Produce a new signal with the same shape but now centered on

the selected carrier frequency.


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[Example] An FHSS system employs a total

bandwidth of Ws=400MHz and an individual

channel bandwidth of 100Hz. What is the

minimum number of PN bits required for each

frequency hop?


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Basic Approach

The question is: How does the FH spreaderwork? (How it move the signal frequency to

the selected frequency?)


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Basic Approach

Assuming we are using BFSK as the data

modulation scheme.

Then the output signal from modulator or the

input signal to the FH spreader can be defined

as:

BFSKoffrequencycarrier,

signalofamplitude

0)2cos(

1)2cos()(

21

2

1

ff

A

where

binarytfA

binarytfAtsd


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Basic Approach

The frequency synthesizer generates a constant-

frequency tone.

The frequency hops among a set of2kfrequencies.

The hopping pattern determined by k bits from the PNsequence.

Assume the duration of one hop is the same as

the duration of one bit.

Ignore phase differences between sd(t) and the

spreading signal c(t), or chipping signal.


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Basic Approach

If the frequency of the signal generated by the

frequency synthesizer during the ith hop is fi, the

product signalp(t) during the ith hop is:

Using the trigonometric identity:

0)2cos()2cos(

1)2cos()2cos()()()(i2

i1

binarytftfA

binarytftfAtctstp d

))cos())(cos(21()cos()cos( yxyxyx


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Basic Approach

We can easily get

The bandpass filter is used to block the difference

frequency and pass the sum frequency, then the

spread spectrum signal is

0))(2cos())(2cos(5.0

1))(2cos())(2cos(5.0)(

i2i2

i1i1

binarytfftffA

binarytfftffAtp

0))(2cos(5.0

1))(2cos(5.0)(

i2

i1

binarytffA

binarytffAts


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Basic Approach

Thus, during the ith bit interval, the frequency

of the spread spectrum signal isf1+fi if the data

bit is 1 and f2+fiif the data bit is 0.

The central frequency is now moved tof2+fi or

f1+fi.


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Problems

[Problems]1. How does the FH spreader work when we use

BPSK modulation? (How it move the signal

frequency to the selected frequency when BPSK is

used?)


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Basic Approach


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Basic Approach

On reception, a signal of the form s(t) just defined

will be received.

The spread spectrum signal is demodulated using the

same sequence of PN-derived frequencies. Then demodulated to produce the output data.

How does the receiver work?


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Basic Approach

Signal s(t) is multiplied by a replica of the spreading

signal to yield a product signal of the form:

Again using the trigonometric identity

0)2cos())(2cos(5.01)2cos())(2cos(5.0)()()(

i2

i1'

binarytftffAbinarytftffAtctstp

i

i

0)2cos())2(2cos(25.0

1)2cos())2(2cos(25.0)(

2i2

1i1'

binarytftffA

binarytftffAtp


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Basic Approach

At the receiver, the bandpass filter is used to

block the sum frequency and pass the

difference frequency, then a signal of the form

of is yielded.

0)2cos(25.0

1)2cos(25.0)(

2

1'

binarytfA

binarytfAtsd

)(' tsd


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Problems

[Problems]2. How does the FH despreader work when we use

BPSK modulation? (How it move the signal

frequency back to the carrier frequency when BPSK

is used?)


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Basic Approach

FHSS Using MFSK

FHSS Performance


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FHSS using MFSK

MFSK is a common modulation technique used inconjunction with FHSS.

Recall: MFSK uses M=2L different frequencies to

encode the digital inputL bits at a time.

elementsignalperbitsofnumber

2elementssignaldifferentofnumber

sfrequenciecarrierthe

1)2cos()(MFSK i

L

M

f

where

MitfAts

L

i


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FHSS using MFSK

If Tc is greater than or equal to Ts the

spreading modulation is referred to as slow-

frequency-hop spread spectrum

Otherwise it is known as fast-frequency-hop

spread spectrum.


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FHSS using MFSK

[Example 1] If we use the MFSK example from

last chapter.

M=4, which means that four different frequencies are

used to encode the data input 2 bits at a time.


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FHSS using MFSK

Each signal element is a discrete frequency tone. The

total MFSK bandwidth is Wd=Mfd.

If we use an FHSS scheme with k=2, there are 4=2k

different channels, each of width Wd. The total FHSSbandwidth is Ws=2

kWd.

Each 2 bits of the PN sequence is used to select one

of the four channels. That channel is held for a

duration of two signal elements, or four bits

(Tc=2Ts=4T)


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FHSS using MFSKPN sequence

Input binary data

Slow FHSS

Frequency


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[Example 2] Using the same MFSK example.

M=4 and k=2. Wd=MfdandWs=2kWd.

However in this case, each signal element is

represented by two frequency tones. Then

Ts=2Tc=2T.


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Frequency

PN sequence

Input binary data

Fast FHSS


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Another two example


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time

Fr

equency


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time

Frequency


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Basic Approach

FHSS Using MFSK

FHSS Performance


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FHSS Performance

Typically, a large number of frequencies are used in

FHSS so that Ws is much larger than Wd.

Large value of k results in a system that is quite

resistant to jamming. Suppose MFSK has bandwidth Wd and noise jammer

has the same bandwidth. If the jammer has a fixed

power Sj on the signal carrier frequency, then the ratio

of signal energy per bit to noise power density is:

j

db

dj

b

j

b

S

WE

WS

E

N

E

/


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If frequency hopping is used, the jammer must jam all

2kfrequencies.

For fixed noise power, since Ws=2kWd, this reduces

the jamming power density to Sj/(2kWd). Then

The gain in signal-to-noise ratio (the processing gain)is

j

k

db

d

k

j

b

j

b

S

WE

WS

E

N

E

2)2/('

k

d

sP

W

WG 2


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[Example] An FHSS system using MFSK with

M=4 employs 1000 different frequencies.

What is the processing gain?


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[Example] The following table illustrates the operationof an FHSS system for one complete period of thePN sequence.

a. What is the period of the PN sequence?

b. The system makes use of a form of FSK, what form of FSKis it?

c. What is the number of bits per symbol?

d. What is the number of FSK frequencies?

e. What is the length of PN sequence per hop?

f. Is this a slow or fast FH system?

g. What is the total number of possible hops?


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Time 12 13 14 15 16 17 18 19

Input data0 1 1 1 1 0 1 0

Frequency f1 f3 f2 f2

PN sequence 001 001

Time 0 1 2 3 4 5 6 7 8 9 10 11

Input data 0 1 1 1 1 1 1 0 0 0 1 0

Frequency f1 f3 f27 f26 f8 f10

PN sequence 001 110 011


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[Example] The following table illustrates the operationof an FHSS system for one complete period of thePN sequence.

a. What is the period of the PN sequence?

b. The system makes use of a form of FSK, what form of FSKis it?

c. What is the number of bits per symbol?

d. What is the number of FSK frequencies?

e. What is the length of PN sequence per hop?

f. Is this a slow or fast FH system?

g. What is the total number of possible hops?


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Time 12 13 14 15 16 17 18 19

Input data 0 1 1 1 1 0 1 0

Frequency f9 f1 f3 f3 f22 f11 f3 f3

PN sequence 011 001 001 001 110 011 001 001

Time 0 1 2 3 4 5 6 7 8 9 10 11

Input data 0 1 1 1 1 1 1 0 0 0 1 0

Frequency f1 f21 f11 f3 f3 f3 f22 f10 f0 f0 f1 f22

PN sequence 001 110 011 001 001 001 110 011 001 001 001 110


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Outline






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Direct Sequence Spread Spectrum

DSSS Using BPSK

DSSS Performance


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DSSS Using BPSK

Direct Sequence Spread Spectrum DSSS

Each bit in the original signal is represented by

multiple bits in the transmitted signal, using a

spreading code.The spreading code spreads the signal across a

wider frequency band in direct proportion to the

number of bits used.

A 10-bit spreading code spreads the signal across a

frequency band that is 10 times greater than a 1-bit

spreading code.


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DSSS Using BPSK

One technique with DSSS is to combine the digital

information stream with the spreading code bit stream

using an exclusive-OR (XOR).

The XOR obeys the following rules:

101110011000


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DSSS Using BPSK


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DSSS Using BPSK

This example shows that

An information bit of one inverts the spreading

code bits in the combination

An information bit of zero causes the spreadingcode bits to be transmitted without inversion.

The combination bit stream has the data rate of the

original spreading code sequence, so it has a widerbandwidth than the information stream.

In this example, the rate of spreading code bit

stream is four times the information rate.

DSSS U i BPSK


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DSSS Using BPSK

A BPSK signal can be expressed as:

To produce the DSSS signal, we multiply the

above by c(t), which is the PN sequence taking

on values of +1 and -1.

0binary1

1binary1s(t)

signalNRZbipolaris)(

)2cos()(BPSK

tswhere

tfAtse cBPSK

)2cos()()( tftctAse cDSSS

DSSS U i BPSK


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DSSS Using BPSK

At the receiver, the incoming signal is

multiplied by c(t). Because c(t)c(t)=1, the

original signal is recovered:

BPSKcDSSS etctftctAstce )()2cos()()()(

DSSS U i BPSK


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DSSS Using BPSK

The DSSS signal expression can be interpreted

in two ways, leading two different

implementations.

First multiply s(t) and c(t) together and thenperform the BPSK modulation.

Alternatively, we can first perform the BPSK

modulation on the data stream s(t) to generate thedata signal eBPSK. This signal can then be multiplied

by c(t).

DSSS U i BPSK


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DSSS Using BPSK

Transmitter using the second interpretation:

DSSS U i BPSK


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DSSS Using BPSK

Receiver using the second interpretation:

DSSS U i BPSK


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DSSS Using BPSK

[Example]

DSSS U i BPSK


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DSSS Using BPSK

Di t S S d S t


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Direct Sequence Spread Spectrum

DSSS Using BPSK

DSSS Performance

DSSS P f


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DSSS Performance

The information signal has a bit width of T,

which is equivalent to a data rate of 1/T. the

spectrum of the signal depending on the

encoding technique, is roughly 2/T.

Similarly, the spectrum of the PN signal is

2/Tc.

The amount of spreading that is achieved is adirect result of the data rate of the PN stream.

DSSS P f


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DSSS Performance

DSSS P f


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DSSS Performance

Effectiveness against jammingAssume a simple jamming signal at the center

frequency of the DSSS system. The jamming

signal is:

The received signal is

)2cos(2)( tfSts cjj

noisewhiteadditive)(

signalreceived)(

powersignaljammer

)()()()(

tn

te

Swhere

tntstete

r

j

jDSSSr

DSSS Performance


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DSSS Performance

The despreader at the receiver multiplies er(t) by c(t), so the

signal component due to the jamming signal is:

This is simply a BPSK modulation of the carrier tone. Thus

the carrier power Sj is spread over a bandwidth of

approximately 2/Tc.

The BPSK demodulator following the DSSS despreader

includes a bandpass filter matched to the BPSK with

bandwidth of2/T.

Most of the jamming power is filtered.

)2cos()(2)( tftcSty cji

DSSS Performance


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DSSS Performance

As an approximation, the jamming power passed by

the filter is

The jamming power has been reduced by a factor of(Tc/T) through the use of spread sprectrum.

The gain in signal-to-noise ratio is:

)/()/2/()/2( TTSTTSFS cjcjj

bandwidthsignalspectrumspread

bandwidthsignal

ratebitspreadingratedata

c

d

c

d

cc

c

P

W

W

RRwhere

W

W

R

R

T

TG

Outline


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Outline





Code Division Multiple Access


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Basic Principles

CDMA for DSSS



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CDMA is a multiplexing technique used with

spread spectrum.

Assuming data signal with rate D, break each bit

into k chips according to a fixed pattern that isspecific to each user, called the users code.

The new channel has a chip data rate of kD chips

per second.



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A simple example with k=6. It is simplest tocharacterize a code as a sequence of 1s and -1s.



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Three users A, B and C, each of which is

communicating with the same base station

receiverR. The code for each user:

CA= < 1 -1 -1 1 -1 1>

CB= < 1 1 -1 -1 1 1>

CC= < 1 1 -1 1 1 -1>

The base station receiver is assumed to knowuserA B Cs code.



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At the receiver

If the receiver receives a chip pattern d=

The receiver has user us code Cu=< c1 c2 c3 c4 c5c6>

The receiver performs electronically the following

decoding function

665544332211)( cdcdcdcdcdcddSu



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User A:

If A wants to send a 1 bit, A transmits its code as

a chip pattern d=< 1 -1 -1 1 -1 1>.

If A wants to send a 0 bit, A transmits thecomplement of its code (1s and -1s reversed) as a

chip pattern d=.

The code for user A is c=CA

= < 1 -1 -1 1 -1 1>



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User A:

If A sends a 1 bit,

If A sends a 0 bit

Note that it is always the case that -6SA(d)6no

matter what dis.

611)1()1(11

)1()1()1()1(11)1,1,1,1,1,1( AS

61)1()1(11)1(

)1(1)1(111-)1,1,1,1,1,1( AS


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User A:

The only ds resulting in the extreme values of

6 and -6 are As code and its complement

respectively.

If SA produces a +6, received a 1 bit from A

If SA produces a -6, received a 0 bit from A

Otherwise we assume that someone else is sendinginformation or there is an error.



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If userB is sending and we try to received it with SA,that means we are decoding the received chip pattern

with the wrong code (CA).

If B sends a 1 bit, then d= < 1 1 -1 -1 1 1>, c=CA= < 1 -

1 -1 1 -1 1>

Since SA=0, we know that someone else apart from A is

sending information (in this case is user B).

If B had sent a 0 bit, the decoder would produce a value of

0 for SA again.

01)1()1()1(11

)1(1)1()1(1)1()1,1,1-,1,1,1( AS


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Transmit (data bit = 0) -1 -1 1 1 -1 -1

Receiver codeword 1 -1 -1 1 -1 1

Multiplication -1 1 -1 1 1 -1 = 0



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If the decoder is linear and if A and B transmit

signals dA and dB respectively, at the same

time, then since

equal to 0 when it is decoded by As code.

The code of A and B that have the property

that are called orthogonal

code.

)()()()( AABAAABAA dSdSdSddS )( BA dS

0)()( ABBA CSCS



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Combined signal 2 0 -2 0 0 2

Receiver codeword 1 1 -1 -1 1 1

Multiplication 2 0 2 0 0 2 = 6

Transmission from A and B, receiver attempts to recover As transmissionA (data bit = 1) 1 -1 -1 1 -1 1

B (data bit = 1) 1 1 -1 -1 1 1




Transmission from A and B, receiver attempts to recover Bs transmission



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Combined signal 0 -2 0 2 -2 0


Multiplication 0 -2 0 -2 -2 0 = -6

Transmission from A and B, receiver attempts to recover As transmissionA (data bit = 1) 1 -1 -1 1 -1 1

B (data bit = 0) -1 -1 1 1 -1 -1

Combined signal 0 -2 0 2 -2 0



Transmission from A and B, receiver attempts to recover Bs transmission



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Such code are very nice but there are not many

of them. More common is the case that is

small in absolute value when XY.

Then it is easy to distinguish between the twocases when X=Y and when XY.

In our example, but

, in the latter case the C

signal would make a small contribution to the

decoded signal.

)( YX CS

0)()( ACCA CSCS

2)()( BCCB CSCS



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Transmit (data bit = 0) -1 -1 1 -1 -1 1


Multiplication -1 -1 -1 1 -1 1 = -2



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Receiver codeword 1 1 -1 1 1 -1


Transmission from B and C, receiver attempts to recover Cs transmission



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B (data bit = 1) 1 1 -1 -1 1 1

C (data bit = 0) -1 -1 1 -1 -1 1

Combined signal 0 0 0 -2 0 2

Receiver codeword 1 1 -1 1 1 -1

Multiplication 0 0 0 -2 0 -2 = -4

(f) Transmission from B and C, receiver attempts to recover Bs transmissionB (data bit = 1) 1 1 -1 -1 1 1

C (data bit = 0) -1 -1 1 -1 -1 1

Combined signal 0 0 0 -2 0 2



Transmission from B and C, receiver attempts to recover Cs transmission



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Using the decoder Su, the receiver can sort outtransmission from u even when there may be other

users broadcasting in the same cell.

In practice, the CDMA receiver can filter out the

contribution from unwanted users or they appear aslow-level noise.

However the system breaks down:

if there are many users competing for the channel with theuser the receiver is trying to listen to

if the signal power of one or more competing signals is too

high (perhaps because it is very near the receiver)



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Basic Principles

CDMA for DSSS

CDMA for DSSS


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CDMA for DSSS

Look at CDMA from the viewpoint of a DSSS system usingBPSK.

CDMA for DSSS


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CDMA for DSSS

There are n users, each transmitting using a differentorthogonal PN sequence.

For each user, the data stream di(t) is BPSK

modulated to produce a signal with a bandwidth ofWsand then multiplied by the spreading code ci(t) for

that user.

All of the signals, plus noise, are received at the

receivers antenna.

CDMA for DSSS


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C o SSS

Suppose that the receiver is attempting to recover thedata of user 1. The incoming signal is multiplied by

the spreading code of user 1 and them demodulated.

The effect of this is to narrow the bandwidth of thatportion of the incoming signal corresponding to user

1 to the original bandwidth of unspread signal, which

is proportional to the data rate.

Because the remainder of the incoming signal isorthogonal to the spreading code of user 1, that

remainder still has the bandwidth Ws.

CDMA for DSSS


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Thus the unwanted signal energy remainsspread over a large bandwidth and the wanted

signal is concentrated in a narrow bandwidth.

The bandpass filter at the demodulator cantherefore recover the desired signal.

Outline


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Generation of Spreading Sequences


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p g q

Spreading consists of multiplying the inputdata by the spreading sequence, where the bit

rate of the spreading sequence is higher than

that of the input data. When the signal is received, the spreading is

removed by multiplying with the same

spreading code, exactly synchronized with thereceived signal.



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p g q

Spreading code:There should be an approximately equal number of

ones and zeros in the spreading code.

Few or no repeated patterns In CDMA application, further requirement of lack

of correlation

Two general categories of spreading

sequences:

PN sequences

Orthogonal codes



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p g q

PN Sequences

Orthogonal Code

Multiple Spreading



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p g q

PN Sequences

PN Properties

LFSR implementation

M-Sequences Properties

PN Sequences


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q

Ideal spreading sequence would be a randomsequence of binary ones and zeros.

It is required that transmitter and receiver must

have a copy of the random bit stream.

A predictable way is needed to generate the

same bit stream at the transmitter and receiver

and also retain the desirable properties of arandom bit stream.

A PN generatorcan meet this requirement.

PN Sequences


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q

A PN generator will produce a periodic sequence thateventually repeats but that appears to be random.

PN sequences are generated by an algorithm using

some initial value called the seed.

The algorithm is deterministic and therefore producessequences of numbers that are not statistically random.

But if the algorithm is good, the resulting sequences will

pass many reasonable tests of randomness.

Such numbers are often referred to as pseudorandomnumbers, orpseudorandom sequences.

Unless you know the algorithm and the seed, it is

impractival to predict the sequence.

PN Sequences


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q

PN properties:

Randomness

Unpredictability

Two criteria are used to validate that a

sequence of numbers is random

Uniform distribution

Independence

PN Sequences


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q

Uniform distribution:The distribution of numbers in the sequence should

be uniform

The frequency of occurrence of each of the

numbers should be approximately the same.

For a stream of binary digits, we need to expand

on this definition.

PN Sequences


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For a stream of binary digits, two properties aredesired:

Balance property: in a long sequence the fraction of

binary ones should approach .

Run property: a run is defined as a sequence of all 1s or

a sequence of all 0s. The appearance of the alternate

digit is the beginning of a new run. About of the runs

should be of length 1, of length 2, 1/8 of length3, and

so on.

000100110101111

PN Sequences


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Independence:

No one value in the sequence can be inferred from

the others.

No such test to prove independenceA number of tests can be applied to demonstrate

that a sequence does not exhibit independence.

General strategy is to apply a number of such testsuntil confidence that independence exists is

sufficiently strong.

PN Sequences


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In applications such as spread spectrum,Correlation property is required.

If a period of the sequence is compared term by

term with any cycle shift of itself, the number ofterms that are the same differs from those that are

different by at most 1.



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PN Sequences

PN Properties

LFSR implementation


LFSR implementation


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Linear Feedback Shift Register Implementation: acircuit consisting of XOR gates and a shift register

implementing the PN generator for spread spectrum

A string of 1-bit storage devices.

Each device has an output line, which indicates the value

currently stored, and an input line.

At discrete time instants, known as clock times, the value in

the storage device is replaced by the value indicated by its

input line.

The entire LFSR is clocked simultaneously, causing a 1-bit

shift along the entire register.

LFSR implementation


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The circuit is implemented as follows:

The LFSR contains n bits.

There are from 1 to (n-1) XOR gates.

The presence or absence of a gate corresponds tothe presence or absence of a term in the generator

polynomial (explained subsequently), P(X),

excluding theXn term.

LFSR implementation


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Two equivalent ways of characterizing the PNLFSR:

A sum of XOR terms

Generator polynomial

11221100 nnn BABABABAB

nn

n XXAXAXAXAXP1

1

2

2

1

1

0

0)(

LFSR implementation


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Fig: Binary Linear Feedback Shift Register Sequence Generator

LFSR implementation


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An actual implementation would not have themultiple circuits; instead, forAi=0, the corresponding

XOR circuit is eliminated.

[Example] A 4-bit LFSR:

103 BBB

LFSR implementation


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Advantages of shift register technique:The sequences generated can be nearly random

with long periods

LFSRs are easy to implement in hardware and canrun at high speeds (this is important because the

spreading rate is higher than the data rate)

LFSR implementation


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Output ofn-bit LFSR: Is periodic with maximum periodN=2n-1.

All-zeros sequence occurs

if the initial contents of the LFSR are all zero or the coefficients ofBn are all zero (no feedback)

When a period of N is given, a feedback

configuration can always be found, the resulting

sequence are called maximal-length sequences, or

m-sequences.

LFSR implementation


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Step-by-step operation ofthe 4-bit LFSR with an

initial state of 1000

(B3=1, B2=0, B1=0,

B1=0).

The values currently

stored in the four shift

register elements

The values that

appear at the output

of the XOR circuit

The value of the

output bit, which is B0

in this example.


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LFSR implementation


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The period of the sequence, or the length of them-sequence: 24-1=15. the output repeats after

15 bits.

The same periodic m-sequence is generatedregardless of the initial state of the LFSR

(except for 0000).

With each different initial state, the m-sequence begins at a different point in its

cycle.


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LFSR implementation


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For any given size of LFSR, a number ofdifferent unique m-sequences can be generated

by using different values for the Ai.

The table next page shows the sequence lengthand number of unique m-sequences that can be

generated for LFSRs of various sizes, also an

example generating polynomial.

LFSR implementation


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Bn for

LFSR implementation


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One useful attribute of the generatorpolynomial is that it can be used to find

sequence generated by the corresponding

LFSR, by taking the reciprocal of thepolynomial.

[Example] A three bit LSFR with

P(X)=1+X+X3

, we perform the division1/(1+X+X3).

LFSR implementation


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Watch out: The

subtractions are done

modulo 2, or using

the XOR function, in

this case, subtraction

produces the same

result as addition.

LFSR implementation


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So the result of 1/(1+X+X3) is

1+X+X2 +(0X3)+X4+(0X5) +(0X6)

after which the pattern repeats.

This means that the shift register output is

1110100

Because the period of this sequence is 7=23-1,

this is an m-sequence.



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PN Sequences

PN Properties

LFSR implementation


M-Sequences


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M-sequences have several properties that makethem attractive for spread spectrum

applications:

Property 1. m-sequence has 2n-1 ones and 2n-1-1zeros.

Property 2. If we slide a window of length n along

the output sequence for N shifts (where N=2n-1),

each n-tuple appears exactly once.

1 1 1 0 1 0 0 1 1 1 0 1 0

111, 110, 101, 010, 100, 001, 011

M-Sequences


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Property 3. There is one run of ones of length n;one run of zeros of length n-1; one run of ones and

one run of zeros of length n-2; two runs of ones

and two runs of zeros of length n-3; and in general,

2n-3 runs of ones and 2n-3 runs of zeros of length 1.

When n=4,N=15, 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1

1 run of1s of length 4

1 run of0s of length 3 1 run of1s and 1 run of0s of length 2

2 runs of1s and 2 runs of0s of length 1 (2n-3 =2)

M-Sequences


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Correlation is the concept of determining how muchsimilarity one set of data has with another.

Correlation is defined with a range between -1 and 1

with the following meanings:

Value 1, the second sequence matches the first sequenceexactly

Value 0, there is no relation at all

Value -1, the two sequences are mirror images of each

other.

Other value indicate a partial degree of correlation.


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Autocorrelation is the correlation of a sequence withall phase shifts of itself.

Cross correlation function: the comparison is made

between two sequences from different sources ranther

than a shifted copy of a sequence with itself. It is

defined as: N

k

kkBAN

R1

1)(

N

k

kkBB

N

R1

1)(

M-Sequences


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Property 4. For many applications, the 0,1sequence is changed to a 1 sequence. Tthe

periodic autocorrelation of the resulting sequence

is:

The periodic autocorrelation of a 1 m-sequence

is:

N

k

kkBBN

R1

1)(

otherwise

...2N,N,0,

11

N

R

M-Sequences


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M-Sequences


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The cross correlation between an m-sequence andnoise is low

This property is useful to the receiver in filtering out noise

The cross correlation between two different m-

sequences is low

This property is useful for CDMA applications

Enables a receiver to discriminate among spread spectrum

signals generated by different m-sequences



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PN Sequences

Orthogonal Code

Multiple Spreading

Orthogonal Code


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Orthogonal Code: a ser of sequences in which all

pairwise cross correlations are zero.

An orthogonal set of sequences is characterized by

the following equality:

For CDMA, each user uses one of the sequences in

the set as a spreading code, providing zero cross

correlation among all users.

durationbit

settheofmembersthandththeare,

settheinsequenceeachoflengththe

01

0

ji

Mwhere

jikk

ji

M

k

ji



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Orthogonal Code

Walsh Codes

Variable-length Orthogonal code

Walsh Codes


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Walsh codes: the most common orthogonalcodes used in CDMA applications.

A set of Walsh codes of length n consists of the nrows of an nn Walsh matrix.

The matrix is defined recursively as:

Where n is the dimension of the matrix and theoverscore denotes the logical NOT of the bits inthe matrix.

nn

nn

n

WW

WWW0W 21

Walsh Codes


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The Walsh matrix has the property that everyrow is orthogonal to every other row and to the

logical NOT of every other row.

The next figure shows the Walsh matrices ofdimensions 2, 4 and 8. Recall that to compute

the cross correlation, we replace 1 with +1 and

0 with -1.

Walsh Codes


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Orthogonal Code

Walsh Codes




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3G mobile CDMA systems are designed tosupport users at a number of different datarates.

Effective support can be provided by usingspreading codes at different rates whilemaintaining orthogonality.

Suppose that the minimum data rate to be

supported is Rmin and that all other data rate arerelated by power of 2.



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If a spreading sequence of lengthNis used for theRmin data rate, such that each bit of data is spread byN=2nbits of the spreading sequence, then thetransmitted data rate isNRmin.

For a data rate of2Rmin

, a spreading sequence oflengthN/2=2n-1 will produce the same output rate of

NRmin.

In general, a code length of2n-kis needed for a bitrate of2kR

min.

E.H. Dinan and B. Jabbari, Spreading codes for directsequence CDMA and wideband CDMA cellular networks,

IEEE Comm. Mag. 36 (September 1998)



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PN Sequences

Orthogonal Code

Multiple Spreading

Multiple Spreading


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When sufficient bandwidth is available, a multiplespreading techniques can prove highly effective. A

typical approach is:

Spread the data rate by an orthogonal code to provide

mutual orthogonality among all users in the same cell

To further spread the result by a PN sequence to provide

mutual randomness between users in different cells.

In such a two-stage spreading, the orthogonal codes are

referred to as channelization codes, and the PN codes arereferred to as scrambling codes.

Stallings W. Wireless Communications and Networkschapter 10.

Review Questions


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[Review Questions]

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