טכניקות בתקשורת מרחיבת סרט (Spread Spectrum) Chapter 1c

Dr. Moshe Ran- Spread Spectrum1

טכניקות בתקשורת מרחיבת סרט

(Spread Spectrum) Chapter 1c

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Dr. Moshe Ran / Spread Spectrum2

נושאי לימוד

הרחבת ספקטרום – Spread Spectrumמבוא הסטורי לטכניקות לשם מה? חזרה- מושגי יסוד ועקרונות של מערכות תקשורת ספרתיות; רעשים והפרעות במערכות תקשורת, דרישות מערכתיות על התקשורת,

השוואת שיטות אפנון ספרתיות, יעילות ספקטרלית.

1פרק

: קונספט ( -Spread Spectrum)מבוא למערכות מרחיבות סרט מרחיבות סרט; שיטות הרחבת סרט המבוססות על ומודלים למערכות

(FH)( דילוגים בתדר ( THדילוגים בזמן ((DSהרחבה ישירה

2פרק

3פרק LFSR, Gold Sequence, Walsh - סדרות קוד למערכות מרחיבות סרט

; ביצועים של מערכות עם (DS) ביצועים של מערכות עם הרחבת סרט ישירה Spread ; שיטות גילוי, עקיבה וסנכרון של אותות (FH)דילוגי תדר

Spectrum

4פרק

עם קודים Spread Spectrum קודים לתיקון שגיאות, ביצועים של מערכות Viterbi לתיקון שגיאות, אלגוריתם

5 פרק

6פרק בתקשורת תאית CDMAעקרונות

7פרק Spread Spectrumשימושים ואפליקציות של מערכות

שו"ת 8

שו"ת8

שו"ת8

שו"ת8

שו"ת8

שו"ת8

שו"ת4


3. Pseudo- Noise (PN) Sequence

1. Definition

2. PN Implementation

3. ,,

4. ,,

5. ,,

6. ,,

7. ,,

8. ,,


3.1 PN Sequence Definition

A PN sequence is a deterministic sequence known to the receiver and transmitter which has features of a random sequence.

- Spectrum- Correlation- Frequency of occurrence of subsequences.

WHY PN and not True Random?!

True random = sample of a sequence of independent r.v uniformly distributed on the alphabet

True random SpSp is like one-time pad in cryptographic system.

Generation, recording and distribution of “sample random sequences” at very high rates to provide PG is not feasible.


3.2 PN Sequence Implementation

Methods implementation of a finite pseudo-noise sequence of length N or a periodic pseudo-noise sequence with a period N .

Memory of N cells. Suitable only for short sequence.

Counter with additional logic. No simple logic can be found.

Linear feedback shift register (LFSR) or equivalently

Pseudo- Random Binary Sequence (PRBS ).

Most useful method.


PN based on Memory of N cells

Example of ROM –based generator

naAddress generator

ROMnb

1010,1100,1011,0010 .Can we replace the ROM with Boolean function!?

Assume – period N=16 is desired, and the specified bn should be

(0) (1) (2) (3)

2

(0) 0 (1) 1 (2) 2 (3) 32 2 2 2

n n n n n

n n n n

n a a a a a

a a a a

That is – address is 4-bits binary counter producing consecutive numbers in the range {0,…, 15}

k bits 2k bits


ROM based - cont.

Possible bn

(0) (1) (2) (2) (3)

Exclusive-OR or i.e., modulo-2 sum

AND logic

n n n n n nb a a a a a

NOTES:

This mapping is a “replacement” function: every input an is mapped to bn.

I.E., an address-to-bit mapping – specified by a table.

Need deep understanding of Finite Fields theory to design PN generators. The mapping in the example above – RM)1,m(


n110001

201000

311001

400100

510101

601101

711100

800010

910011

1001010

1111011

1200111

1310110

1401110

1511111

1600000

(0)na

(1)na

(2)na

(3)na nb


Counter Based PN generator

This solution can be described by

(0)na (1)

na(2)na

(3)na

4-bit counter

+

(0) (1) (2) (2) (3)n n n n n nb a a a a a


Complexity issues

Number of operation: linear function of ~2k )exponential in k(

Are counters good for implementing PN?

Probably NO. Since the sequences do not look “random”.

While PN -on the average- changes every other bit

The counter sequences are changing “much slower” Linear recursive relations are much better choice.

( )ina


3.3 Linear Feedback Shift Register Configurations

Fibonacci configuration

Galois configuration

The sequence is detemined by:Number of cells, Feedback, Initial state of the shift register.

The all zero state produces a sequence with period one.The order of a sequence is the length of the shortest LFSR which may generate the sequence.

5nb 4nb 3nb 2nb 1nb nb

6nr

5nr

4nr 3

nr 2nr 1

nr 1nb


3.4 M-sequence

An m- sequence or a maximal length sequence is defined as a sequence generated by a linear feedback shift register with m cells and with a period of .

While generating the m-sequence, the generator passes through all possible states of the register besides the all zero state.

The number of different binary m-sequence of order .

Where are the prime number in the decomposition of

N

2 1mN

is pL N L

1

12 1L ji

pi

i

pN L

L p

2 1L 2 1L

L

1

i

jei

ip

ip


56

1060

151800

202400

pL N L


3.5 Statistical Properties of Binary M-sequences

a. Balance Property

The number of “ones” in one period of the m-sequence exceeds the number of “zeros” by 1.

b. Events counting Every times except the all zero J-tuple which occurs times.

c. shift-and-Add Property The sum of the sequence and a shifted version of the sequence is another shifted version of the sequence.

d. Periodic autocorrelation We refer to the sequence

Is replaced by 1 and 1 is replaced by –1. The periodic autocorrelation of is defined

tuple ( ) occurs 2L JJ J L 2 1L J

'n n nb b b

1 i.e. 0nb

nc

nc

1

1 N

c n r nn

R c cN


1 =0 mod

10 mod c

NR

NN


3.6 Statistical Properties of Random Binary Sequence

a.Balance Property

The probability of one equals probability of zero.

b. Events counting the probability of any

c. Autocorrelation we refer to the sequence I.e. 0 us replaced by 1 and 1 is replaced by –1. The autocorrelation of a sequence is defined

And is equal to

tuple is 2 JJ

1 nb

nc

nc

*R c n r nE c c

1 =0 mod

0 0 modc

NR

N


3.7 Autocorrelation of Continuous Sequence

The periodic autocorrelation of the continuous waveform is defined by

The continuous autocorrelation of can be obtained by connecting the discrete autocorrelation of by straight lines.

c t

*

0R

cNT

c c t c t dt c t

nc

1

N

cT

1.0

cR

cNT

?


3.8 Spectrum of an M-sequence

The spectrum of a continuous M-Sequence is the Fourier transform of the periodic autocorrelation of .

The power spectrum of direct sequence spread spectrum signal is continuous and has deviations from the sinc form.

c t

f

1

cT1

cT

cS f

1

cNT

2

cT

2

cT

2sinc ( )1 c

NfT

N


3.9 Selected M-sequences

The reversed order sequence is called

the complementary sequence and is

another m-sequence. If the original

sequence is generated by a LFSR of

order with the taps and , the

complementary sequence is generated

by a LFSR of order with the taps

and .

mtaps

3)1(

4)1(

5)2(

6)1(

7)1),(3(

8)4,3,2),(6,5,1(

9)4(

10)3(

11)2(

12)6,4,1),(7,4,3(

13)4,3,1(

14)5,3,1),(12,11,1(

15)1),(4),(7(

20)3(

25)3),(7(

m 1 2( , ,...) m

m

1 2( , ,...)m m m


3.10 Linear Span of a Sequence

The linear span of a sequence is defined as the length of the shortest LFSR which generates the sequence.

If there are known consecutive bits of a sequence with a leaner span of , all the sequence can be calculated. In the binary case only known consecutive bits are required.

For example, suppose that a binary sequence has a linear span of 5 and a portion of the sequence contains

…1011101100011…

Where the sequence index is increasing from left to right. The binary sequence satisfy the linear recursion

4

51

n i n i ni

b a b b

2LL

2 1L nb

nb


3.11 Linear Span Of A Sequence (Cont.)

…1011101100011…

However we know that , then

The solution is

4

51

n i n i ni

b a b b

1 2 3 5

2 3 4

1 3 4 5

1 2 4 5

2 3 5

0

1

1

0

0

a a a a

a a a

a a a a

a a a a

a a a

5 1a

1 2 3

2 3 4

1 3 4

1 2 4

1

1

0

1

a a a

a a a

a a a

a a a

1 2 3 40, 1, 0, 0.a a a a


3.12 Sequence With Large Linear Span

There are sequence with a period of length with a linear span

larger then

The most common approach to obtain a large linear span is the LFSR with feedforward logic.

2 1m

L i.e .m L m

g

f

1n Na 2n Na 1na na

nbNON-LINEAR FUNCTION

MEMORY

LINEAR FUNCTION


Analytical derived sequence with a large linear span are known such as:

a. GMW

b. Bent

L


3.13 Sequence for CDMA Systems

In CDMA system many users share the same frequency band with different sequence with a small crosscorrelation.

A popular family of sequence is the Gold sequences. In a family of Gold sequences of length there are sequences, and maximum crosscorrelation is approximately

The maximum crosscorrelation is

2 1m 2 1m 22 .m

1

2

1

2

2 for odd

2 for 2mod4

m

m

m

m

+

+

+

31

2 32

1

1

f x x x

f x x x


3.14 Sequences for FH/CDMA Systems

In a FH/CDMA system many users share the same frequency band with different sequences with a minimal probability of hit.

nb1nb 1n Jb n Jb 1n Lb

L JC 1LC LC

1JV 1V 0V

vnf

FREQUENCY SYNTHESIZER

)one-to-one mapping(SEQUENCE

SELECTOR


3.15 Aperiodic And Odd Autocorrelation

The Aperiodic autocorrelation of a sequence cn with a period N is defined as

Applications: where only one waveform is transmitted such as radar.

The odd autocorrelation of a sequence cn with a period N is defined as

The odd autocorrelation is useful in data communication where the one symbol time is equal to one period. It represent the effect of changes in data.

The odd and the Aperiodic autocorrelation depend on the initial state of the sequence generator.

1

,1

0

1 for 0

1 for <0

N

n nn

a cN

n nn

c cNR

c cN


7.18 Best Odd Autocorrelation M- Sequences

The optimization was performed over all initial states of all m-sequences of period .N

317

6311

12717

25525

51137

102351

204785

N Ro

טכניקות בתקשורת מרחיבת סרט (Spread Spectrum) Chapter 1c

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Transcript of טכניקות בתקשורת מרחיבת סרט (Spread Spectrum) Chapter 1c