DIGITAL SPREAD SPECTRUM SYSTEMS

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DIGITAL SPREAD SPECTRUM SYSTEMS
ENG-737

• Wright State University
• James P. Stephens

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DIRECT SEQUENCE IMPLEMENTATION
(A)
x(t)
BPSK modulator
BPSK modulator
Data
s(t) A x(t) g(t) cos wt
sx(t) A x(t) cos wt
A cos wt
g(t)
Carrier
(B)
(C)
r(t) AA x(t - td) g(t - td) cos w(t td)
f
BPSK demod
Filter

s(t) AA x(t - td)
Output of correlator

g(t - td)
Correlator
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SPREADING / DESPREADING
Locally generated PN bit stream B
Code Inversion
Locally generated PN bit stream D same as B
above
Embedded Ref.
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SPREADING CODES
IMPORTANT CODE PROPERTIES

• Must be easily generated at the transmitter and
  receiver (i.e. deterministic)
• Behave as much as possible like a random sequence
• Be difficult to exploit
• Support multiple access
• Provide ease of synchronization

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SIMPLE SHIFT REGISTER GENERATOR (SSRG)

• Data at first stage is shifted once toward the
  right each time clock pulse occurs
• Clock input are normally not shown

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SIMPLE SHIFT REGISTER GENERATOR (SSRG)
N 2n - 1
Where, N length of sequence n number of
stages
Repeats
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M-SEQUENCES

• If sequence goes through all possible states
  except all zeros, it is maximal

If not maximal, the output sequence will be
dependent on the initial fill
State Diagram
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GENERAL FORM OF A LINEAR SSRG

• Let
• Be the output sequence
• The general form for a linear SSRG is

The fundamental recursion relationship describing
the sequence element ck and the contents of the
register is
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GENERAL FORM OF A LINEAR SSRG (Cont)

• Where,
• The sum is modulo-2 addition
• ai is either 0 or 1
• i 1, 2, 3, . . . . n-1 and an 1
• The values are called
• initial conditions or initial fill

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5-STAGE SIMPLE SHIFT REGISTER GENERATOR
x0
x1
x2
x3
x4
x5
Clock

• n 5 number of register stages
• N length of sequence generated (maximal only if
  correct taps are chose)
• Generator described by a polynomial over a field
  of two symbols, called Galois Field of 2 (GF2)
• Generator g(x) is written

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5-STAGE SIMPLE SHIFT REGISTER GENERATOR (Cont)

• ai are the tap coefficients, for the example
  shown
• a0 a1 a3 a4 a5 1
• a2 0
• Therefore, the generator polynomial is written
• g(x) x5 x4 x3 x 1
• This is called the generators characteristic
  polynomial

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OCTAL REPRESENTATION CONVERSION

• Example Convert 103o to generator polynomial
• Solution
• 1 0 3
• 0 0 1 0 0
  0 0 1 1
• X8 X7 X6 X5 X4
  X3 X2 X1 X0 X6 X 1

3
6
1
2
4
5
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MAXIMAL LENGTH SEQUENCES

• When the sequence goes through all 2n 1
  possible states, it is called an m-sequence
• The periodic cycles of a linear SSRG sequence
  depend on
• Register length, n
• Feedback taps
• Initial conditions
• Only if feedback taps are chosen properly will
  the sequence be an m-sequence
• A non-maximal sequence is one in which the
  register does not go through all possible states,
  i.e. the sequence will be initial fill dependent

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EXAMPLE OF A NON-MAXIMAL SEQUENCE
24 1 15
Null Seq Seq A Seq B
0 0 0 0 - 0 0 0 0 0 - R
1 0 0 0 - 8 1 1 0 0 -12 0 1 1 0 - 6 0 0 1 1 - 3 0
0 0 1 -1 1 0 0 0 - R
1 1 1 1 - 15 0 1 1 1 - 7 1 0 1 1 - 11 1 1 0 1 -
13 1 1 1 0 - 14 1 1 1 1 - R
State Diagrams
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PRIMITIVE POLYNOMIALS
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NUMBER OF M-SEQUENCESfrom register length 3
through 16
Register Length n
Number of sequences

• gt 2
• 4 gt 4
• 5 gt 6
• 6 gt 4
• 7 gt 18
• 8 gt 16
• 9 gt 48
• 10 gt 60
• 11 gt 176
• 12 gt 96
• 13 gt 630
• 14 gt 756
• 15 gt 1800
• 16 gt 2048

Where ?(2n 1) is a Euler number i.e. the number
of positive integers including 1 that are
relatively prime to and less than 2n 1
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COMPARISON OF SSRG WITH MSRG
(a) Simple Shift Register Generator (SSRG)
Configuration of x5 x2 1
(b) Equivalent Modular Shift Register Generator
(MSRG) Configuration of x5 x3 1
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RECIPROCAL CODES
Time reversed replicas
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CODE SEQUENCE PERIODS FOR M-SEQUENCES
Code Sequence Periods for Various m-sequence
Lengths at 1 Mcps

• 127 1.27 x 10-4 sec
• 255 2.55 x 10-4 sec
• 511 5.11 x 10-4 sec
• 1,023 1.023 x 10-3 sec
• 2,047 2.047 x 10-3 sec
• 4,095 4.095 x 10-3 sec
• 8,191 9.191 x 10-3 sec
• 131,071 1.31 x 10-1 sec
• 524,287 5.24 x 10-1 sec
• 8,388,607 8.388 sec
• 27 134,217,727 13.421 sec
• 2,147,483,647 35.8 min
• 879,609,302,207 101.7 days
• 2,305,843,009,213,693,951 7.3 x 104 years
• 89 618,970,019,642,690,137,449,562,111 1.95 x
  109 years

Source Spread Spectrum Systems R.C. Dixon
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GENERATING PN SEQUENCES IN MATLAB

• Requires
• pnf.m
• oct2bin.m

• x pnf(8,435,1,500,1,1)
• of stages
• Octal polynomial
• Initial Fill in Octal
• Length of sequence
• Sampling Frequency
• Bit rate

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GENERATING PN SEQUENCES IN MATLAB (Cont.)
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PROPERTIES OF M-SEQUENCES

• M-sequences must have an even number of taps
• Reciprocal codes can be generated by reversing
  the order of the taps
• Exhibit the shift and add property
• If an m-sequence is added to a phase shifted
  version of itself, then the resulting sequence is
  another shift of the original sequence
• Example
• 1 1 1 0 1 0
  0 ? Original sequence
• 1 0 1 0 0 1
  1 ? Shifted by 2
• 0 1 0 0 1 1
  1 ? Same sequence shifted by 4
• Balance Property
• In one period of an m-sequence, there is one more
  1 than 0 (result of not having an all zeros state)

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RUN PROPERTIES OF M-SEQUENCES (Cont.)

• 5. In one period there a 2(n-1) runs of
  consecutive 1s or 0s.
• ½ are of length 1
• ¼ are of length 2
• 1/2l are of length l (Ex 1/8 of length 3,
  1/16 of length 4, . . .)
• Or equivalently
• 1 run of 1s of length n
• 1 run of 0s of length n-1
• 1 run of 1s and 1 run of 0s of length n-2
• 2 runs of 1s and 2 runs of 0s of length n-3
• 4 runs of 1s and 4 runs of 0s of length n-4
• 2n-3 runs of 1s and 2n-3 runs of 0s of length. 1

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BALANCE PROPERTYExample

• M-sequence where n5, therefore l4
• N 25 1 31

Total runs 2n-1 25-1 24 16 Total number
of bits 5 4 3 3 4 4 4 4 31
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BALANCE PROPERTYExample

• Generating polynomial 67o (n 5 and N 31)
• The m-sequence is given as
• 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0
  0 0 0 1 1 1 0 0 1 1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Runs Present Number of Each Computation
0 4 ½(2n-1) ½(16) 8
1 4 Included in above
00 2 ¼(16) 4
11 2 Included in above
000 1 1/8(16) 2
111 1 Included in above
0000 1 1/16(16) 1
11111 1 1/16(16) 1
Total 16
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PROPERTIES OF M-SEQUENCES (Cont.)

• Number of maximal sequences possible
• Number of sequences 1/n ?(2n 1)
• Where
• n the number of SSRG stages
• ?(k) is Eulers number
• ?(k) equals the number of positive integers less
  than k and relatively prime to k
• If k is not prime, use
• Where pfi s are the prime factors of k

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NUMBER OF M-SEQUENCES Example

• Let n 5, N 25 -1 31
• 31 is a prime number, therefore
• of sequences 1/n?(2n 1) 1/530 6
• Note 30 is the number of positive integers
  less than k
• This number does not include reciprocal codes
  since they are time-reversed replicas of the same
  sequence
• Let n 4, N 24 -1 15
• 15 is not a prime number, therefore prime factors
  are 3 and 5
• of sequences
• (1/4)(24 1)(1 1/3)(1 1/5)
  (1/4)(15)(2/3)(4/5) 2
• The table 3.3 on page 72 of the Dixon text is
  incorrect for n4 and is verified from
  Error-Correcting Codes by Peterson and Weldon