DIGITAL SPREAD SPECTRUM SYSTEMS

1
DIGITAL SPREAD SPECTRUM SYSTEMS
ENG-737

• Wright State University
• James P. Stephens

2
AUTOCORRELATION OF PN SEQUENCES

• A very important property of PN sequences is the
  periodic auto correlation function
• Recall that autocorrelation is defined as
• Simply stated, it is a comparison of a sequence
  to a time-shifted version of itself
• The equation asks For an offset of m chips, how
  similar is the sequence to inself?

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PN SEQUENCE AUTOCORRELATION FUNCTION
Normalized
Rbb(?)
NTc
1.0
1/N
?
Tc
4
POWER SPECTRAL DENSITY FOR M-SEQUENCE
Chip Duration Tc and Period NTc
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DISCRETE AUTOCORRELATION Example
n 3 N 2n 1 7 Initial Fill 0 0 1
Sequence ? 1 0 0 1 0 1 1 Convert to
-1, 1 ? 1 -1 -1 1 -1 1 1

• The periodic autocorrelation
• . . . 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1
  1 -1 1 1 1 -1 -1 1 -1 1 1 . .
• 1 -1 -1 1 -1 1 1
  m0
• 1 1 1 1 1 1
  1 7
• 1 -1 -1 1 -1 1 1 m1
• -1 1 -1-1 -1
  1 1 -1
• and so on . . . . . . .

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DISCRETE AUTOCORRELATION Example
m Sum of Products Rbbm
-6 -1 -1/7
-5 -1 -1/7
-4 -1 -1/7
-3 -1 -1/7
-2 -1 -1/7
-1 -1 -1/7
0 7 1
1 -1 -1/7
2 -1 -1/7
3 -1 -1/7
4 -1 -1/7
5 -1 -1/7
6 -1 -1/7
Discrete Time Plots
In the limit as m ? ? Approaches that of White
Noise
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BINARY SYMBOL CONVERSION

• Consider an m-sequence ck
• Where, ck ? 0, 1 for all k
• Now define the m-sequence bk ? -1, 1
• By letting bk 2ck -1, this translates 0 ? -1
  and 1 ? 1
• This is a useful MatLab conversion necessary for
  many MatLab computations
• Note In GF(2) arithmetic, multiplication of 1
  and -1 is the same as addition of 0 and 1.
• Example

a b
-1 -1 1
-1 1 -1
1 -1 -1
1 1 1
a b
0 0 0
0 1 1
1 0 1
1 1 0
Exclusive OR
Multiplication
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MATLAB IMPLEMENTATION

• The following files are found on my web site at
  http//www.cs.wright.edu/jstephen/
• pnf.m
• oct2bin.m
• masseyf.m
• Example
• pn1pnf(6,103,1,1500,1,1)
• pn2pnf(6,103,1,63,1,1)
• pn1(pn12)-1
• pn2(pn22)-1
• autoxcorr(pn2,pn1)
• plot(auto)

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MATLAB IMPLEMENTATION (Cont.)

• autoauto(100400)
• plot(auto)

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M-SEQUENCE SECURITY ISSUES

• How secure are m-sequences? Even if they do not
  repeat for years?
• Masseys Algorithm can determine the generating
  polynomial, at most, in 2 times n error free
  chips
• Example
• Generate m-sequence where N7, i.e. n3
• Fill up matrix using only 2n 6 chips
• Obtain system of equations that can be solved
  simultaneously
• Thus, a code that uses 89 stages that will not
  repeat for 73,000 years (1 Mcps chip rate) can be
  determined by only processing 2 89 178 chips

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CODE SEQUENCE DETERMINATION

• Berlekamp-Massey linear feedback shift register
  prediction algorithm
• Golds Frequency Agile Predictor
• B-M requires (at most) 2n error-free chips to
  determine generating polynomial
• Golds based on relative times of arrival of
  single frequency
• Demonstrated on Workstation with MATLAB

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MASSEY ALGORITHM EXAMPLE
a(k)

• Code sequence of period N7 (n3)
• a a(0), a(1), a(2), a(3), a(4), a(5), . . . .
  
• a 1 1 1 0 1 0 0 1 1 1 0 1 0 0 (14
  chips)
• a(k) C1 a(k-1) C2 a(k-2) C3 a(k-3)
• a(3) a(2) a(1) a(0) 0 1 1 1
• a(4) a(3) a(2) a(1) 1 0 1 1
• a(5) a(4) a(3) a(2) 0 1 0 1
• We obtain the system of equations
• (1) 0 1 C1 1 C2 1 C3
• (2) 1 0 C1 1 C2 1 C3
• (3) 0 1 C1 0 C2 1 C3

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MASSEY ALGORITHM EXAMPLE

• Add equations (1) and (2)
• 1 1 C1 0 C2 0 C3
• C1 1
• Substitute in equation (3)
• 0 1 (1) 0 C2 1 C3
• C3 1
• Substitute C1 1, C3 1, into (1) gives
• 0 1(1) 1 C2 1(1)
• C2 0
• Solution is X3 X 1

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MODEL FREQUENCY HOPPER
Code-of-the-day
Modulo-2 Adder
Binary Register
FH Output
Mapper
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GOLDS TIME-OF-ARRIVAL ALGORITHM
Code-of-the-day
Code Generator
Binary Output Register
Output Hopping Sequence
00 gt a 01 gt b 10 gt c 11 gt d
Frequency Mapper
Times-of-arrival 4,5,8,9,10,13,14,23,24,25,26,27
,31
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PERFORMANCE

• Importance of real-time implementation
• Rapidly changing battle scenario
• Typically short message bursts
• Ability to change code-of-the-day
• Data collection time
• Probability of correct synchronization
• Prediction algorithm run time

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DATA COLLECTION TIME
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PROBABILITY OF CORRECT SYNCHRONIZATION
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ALGORITHM TIME
Based on 32-stage shift register
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ALGORITHM TIME VERSUS NUMBER OF HOP FREQUENCIES
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PARTIAL CORRELATION

• The important random properties of m-sequences
  are the result of usually very long sequences
• However, it is not possible to perform full
  length correlation in a receiver
• Also, if the duration of a message bit is less
  than that of the period of the PN sequence, then
  to decode the message, we need to correlate over
  something less that a full code period
• In these cases, the correlation function we just
  looked at does not apply
• The Partial Correlation function is written

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PARTIAL CORRELATION (Cont.)

• We can treat the ams as random variables ? 1
• So,
• The variance is

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PARTIAL CORRELATION
M-sequence Autocorrelation
M-sequence Partial Correlation
50 chips
63 chips
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PARTIAL CORRELATION (Cont.)
50 chips
40 chips
30 chips
20 chips
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MATLAB IMPLEMENTATION OF PARTIAL CORRELATION

• To perform a full autocorrelation of an
  m-sequence of length 63
• pn1pnf(6,103,1,1500,1,1)
• pn2pnf(6,103,1,63,1,1)
• pn1(pn12) -1
• pn2(pn22) -1
• autoxcorr(pn2,pn1)
• autoauto(100400)
• plot(auto)
• To perform a partial autocorrelation for this
  sequence, change line 2
• pn2pnf(6,103,1,50,1,1)

Try different values here !
Note Each time you change line 2, be sure to
re-enter lines 4, 5, 6, and 7 since these values
must be re-computed.
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PERIODIC VS. APERIODIC CORRELATION
1
4
2
3
PN Code Reference
Periodic
1
PN Code Reference
Aperiodic
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APERIODIC AUTOCORRELATION OF AN M-SEQUENCE

• Determine the Aperiodic autocorrelation of
  m-sequence 45o
• n 5, N 31
• pn1pnf(5,45,1,31,1,1)
• pn2pnf(5,45,1,31,1,1)
• pn1(pn12)-1
• pn2(pn22)-1
• xxcorr(pn2,pn1)
• plot(x)

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APERIODIC AUTOCORRELATION
Peak Value
Approximately 15 side lobe values
Index of discrimination
G(x) 45o
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PARTIAL APERIODIC AUTOCORRELATION
G(x) 45o 15 Chips
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CROSSCORRELATION

• Whereas autocorrelation compares a code with
  itself, crosscorrelation compares one code to
  another
• Example Determine the periodic and aperiodic
  crosscorrelation of 45o and 75o
• Note the number of sidelobe values
• Periodic sidelobes are 3-valued
• Aperiodic is very random looking
• Refer to the handout from Robert Golds
  publication on Linear Code Sequences

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CROSSCORRELATION(45o and 75o)
Periodic
Aperiodic
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COMPOSITE CODES

• Although m-sequences are quite desirable for many
  applications, there is considerable research in
  finding new codes that offer improved performance
  in area
• Composite codes are generated by a combination of
  linear maximal sequences
• The following are examples of composite codes
• Gold Codes CDMA Applications
• JPL Codes Rapid synchronization
• Kasami Codes Length construction and CDMA
  Applications

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GOLD CODES

• Many details regarding Gold Codes are explained
  in Appendix 7 of Dixon text
• To be assured of good performance in CDMA
  applications, a system must have
• Low crosscorrelation properties among codes
• Low partial crosscorrelation properties among
  codes
• From Golds report
• Length 25 1 31 Fixed Polynomial
  45o
• Bound N Spectrum Polynomial
• 31 1 31(1) -1(30) 45
• 11 1 7(5) 3(10) 51
• -1(5) -5(5)
• -9(5)
• 9 4 7(10) -1(15) 75
• -9(6) 67
• 53 73

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REGARDING M-SEQUENCES FOR CDMA APPLICATIONS

• There is no general bound on the
  cross-correlation properties of m-sequences
• There is a limited number of available code for a
  large number of users within the family of
  m-sequences

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GOLD CODE CONSTRUCTION
LRS 1 and LRA 2 must have same length
Code 1
LRS 1

Clock
LRS 2
Code 2
Output Gold Code ( Code 1 xor Code 2)
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GOLD CODE CONSTRUCTION
x5 x3 1 51o
Gold Code
x5 x4 x3 x2 1 75o
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GOLD CODE CONSTRUCTION

• 51o xor 75o ( x5 x3 1)(x5 x4 x3
  x2 1)
• x10 x9 x8 x7
  x5
• x8
  x7 x6 x5 x3
• 
  x5 x4 x3 x2 1
• x10 x9 x6 x5
  x4 x2 1
• 3165o non-maximal

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MAXIMUM NUMBER OF GOLD CODES

• Normally, you fix one register and change the
  fill of the other to generate a family of Gold
  Codes
• Since there are 2n 1 initial fills, you can
  generate 2n 1 Gold Codes from each
  configuration
• The code length will be N 2n 1, where n is
  the length of the individual registers which must
  be of equal size
• For the 5-stage registers shown in the previous
  chart, N 31
• Since the generating polynomial for the Gold Code
  is degree-10, an m-sequence would be 210 1
  1023
• 1023 / 31 33 codes
• This comes from 31 initial fills plus the
  m-sequence generated from each separate register
  alone
• Recall from Table 3.3 in Dixons text, the number
  of possible m-sequences given n is tabulated in
  the next chart

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MAXIMUM NUMBER OF GOLD CODES

• n of m-sequences. of Gold
  Codes
• 5 6
  31 2
• 6 4
  63 2
• 7 18
  127 2
• 8 16
  255 2
• 9 48
  511 2
• . .
  .
• . .
  .
• . .
  .
• 15 1800
  32767 2

Not only do Gold Codes provide a larger ensemble
of codes, but they provide uniformly low-cross
correlation properties while exhibiting good
autocorrelation properties
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CHOSING A SET OF PREFERRED PAIRS

• M-sequences used to generate Gold Codes must be
  of the same length and chosen according to a
  process of preferred pairs selection (Details
  are in Appendix 7 of Dixon text)
• Table A7.3 says there are no preferred pairs for
  n6. Not true! 103o and 147o are a preferred
  pair
• Arbitrary selection of code pairs can result in
  very poor correlation performance
• Table A7.1, page 501, shows that for n5, a
  preferred pair yields a peak value of cross
  correlation of 9 for a preferred pair, but 11 for
  a non-preferred pair
• The selection of preferred pairs will always
  give 3-valued autocorrelation which will bound
  the correlation to

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Table A7.1 Performance of Preferred Pairs
Compared with Worst Case Pairs
Degree Period Worst Case Preferred Pair Difference (dB)
5 31 11 9 1.7
6 63 23 15 3.7
7 127 41 17 7.6
8 255 95 31 9.7
9 511 113 33 10.7
10 1023 383 63 15.7
11 2047 287 65 12.9
12 4095 1407 127 20.9
13 8191 703 127 14.9
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PREFERRED PAIRS OF GOLD CODES
Degree Preferred Pairs Bound
5 5,25,4,3,2 9
6 None
7 7,37,3,2,1 7,3,2,17,5,4,3,2,1 15
8 8,7,6,5,2,18,7,6,1 31
9 9,49,6,4,3 9,6,4,39,8,4,1 33
10 10,9,8,7,6,5,4,310,9,7,6,4,1 10,8,7,6,5,4,310,9,7,6,4,1 10,8,5,110,7,6,4,2,1 63
11 11,211,8,5,2 11,8,5,211,10,3,2 65
12 12,11,9,8,7,5,2,112,10,5,4,2,1 12,11,6,4,2,112,11,8,7,5,4,3,2 127
13 13,4,3,113,10,9,7,5,4 13,10,9,7,5,413,12,8,7,6,5 129
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CHOSING A SET OF PREFERRED PAIRS

• The algorithm for choosing preferred pairs is
  as follows
• Select n and a polynomial
• Find k associated with the polynomial (polynomial
  root)
• If n is odd, then calculate 2k 1, if n is even
  then calculate 2(n2)/2 k
• The number calculated is the polynomial root (k)
  associated with the second code that completes a
  preferred pair

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CHOSING A SET OF PREFERRED PAIRS

• Example
• Let n8, then choose 435(1)
• k1
• n is even therefore, 2(82)/2 1 33
• None listed
• Try n8, then choose 747(11)
• k11
• N is even therefore, 2(82)/2 11 43 ?
  703o

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JPL RANGING CODES

• JPL Codes are constructed by modulo-2 addition of
  2 or more m-sequences whose lengths are
  relatively prime to one another
• Advantages of JPL Codes
• Very long codes can be generated which are useful
  for unambiguous ranging over long distances
• May be generated easily with a small number of
  short registers
• Synchronization can be more easily accomplished
  by separate operation on each of the component
  codes
• Relatively prime means no common factors among
  numbers

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JPL CODE CONSTRUCTION
a
a xor b xor c
b
c