Spreading codes

1
Project IEEE P802.15 Working Group for Wireless
Personal Area Networks (WPANs) Submission Title
Codes for preamble and data Date Submitted
7 June, 2005 Source Michael Mc Laughlin
Company Decawave Ltd. Address 25 Meadowfield,
Sandyford, Dublin 18, Ireland Voice353-1-29549
37 , FAX Whats a FAX?, E-Mail
michael_at_decawave.com Re 802.15.4a. Abstra
ct Discusses the desirable properties of
spreading sequences Purpose To promote
discussion in 802.15.4a. Notice This document
has been prepared to assist the IEEE P802.15. It
is offered as a basis for discussion and is not
binding on the contributing individual(s) or
organization(s). The material in this document is
subject to change in form and content after
further study. The contributor(s) reserve(s) the
right to add, amend or withdraw material
contained herein. Release The contributor
acknowledges and accepts that this contribution
becomes the property of IEEE and may be made
publicly available by P802.15.
2
Spreading sequencesDesirable properties
3
Five KEY properties of spreading sequences

• Sequence Length
• Pulse Repetition Frequency
• Autocorrelation properties
• Periodic autocorrelation (Channel sounding)
• Aperiodic autocorrelation (Data mode)
• Spectral peak to average ratio (SPAR)
• FCC requirements
• Temporal peak to average ratio (TPAR)
• Power supply requirements

4
Periodic Autocorrelation (1)

• For channel sounding, a repeated sequence is
  appropriate.
• Periodic autocorrelation function is the
  important property for a channel sounding
  sequence
• Ipatov ternary sequences have perfect periodic
  autocorrelation i.e. all side lobes are zero
• PBTS codes (from WBA/I2R) also have perfect
  periodic autocorrelation

5
Periodic Autocorrelation (2)

• m-sequences have ideal periodic autocorrelation,
  i.e. their autocorrelation function is N (the
  sequence length) at one sample period and -1
  everywhere else.
• Correlator output operating on repeated, periodic
  sequences with perfect periodic autocorrelation
  is exactly, the channel impulse response, also
  repeated, plus noise.

6
Example Correlator Outputs
7
Aperiodic Autocorrelation

• For transmitting data, aperiodic autocorrelation
  function (AACF) is appropriate
• Previous and next sequences may not be the same
• Good AACF means low ISI
• Golay Merit Factor (GMF) is a common measure of
  goodness of AACF (Golay 1977)

8
Golay Merit Factor

• GMF is defined as
• where ac is the aperiodic auto correlation
  function of a length n sequence
• The average GMF of binary sequences is 1.0
• Best known GMF for binary sequences is 14.08 for
  the Barker 13 sequence, next is 12.1 for the
  Barker 11 sequence.
• The mean Golay merit factor of the length 32
  Walsh codes is 0.194.
• GMF greater than 6.0 is rare

9
Autocorrelation High GMF
10
Autocorrelation Low GMF
11
Matched Filter Output High GMF
12
Matched Filter Output Low GMF
13
Spectral Peak to Average ratio (SPAR)

• In absence of ITU recommendations, use the FCC
  requirements.
• Spectrum measured in 1MHz frequency bins for 1ms
  intervals.
• Need Low SPAR.
• SPAR in dBs converts to power backoff required.
• Best known SPAR for ternary sequences known to
  author is 1.17 dB for the Barker 11 and next 1.32
  for Barker 13.
• (aside, of course, from a single impulse)

14
Temporal Peak to Average Ratio

• Lower TPAR allows low voltage silicon
• Best GMF (Infinite) is a single impulse.
• Impulse also has 0dB SPAR
• TPAR of an impulse is worst
• Need to balance sequence length and PRF to get a
  good SPAR and a good TPAR.

15
Example sequences

• One of the Ipatov length 57 sequences
• -00--0--------0-0-----0--0
  0--
• GMF is 3.75
• A Length 63 m sequence
• -----------------------------
  ---
• GMF is 3.52
• Both of these sequences, if transmitted
  repeatedly back to back, have a flat spectrum
• Ipatov sequences are available at the following
  lengths
• 7,13,21,31,57,73,91,127,133,183,273,307,381,512,5
  53,651,757,871,993,1057,1407,1723

16
Sequence length and PRF

• If sequence is repeated, spectral lines spaced at
  the 1/sequence length apart.
• Want these to be lt 2MHz apart for FCC
  compliance and low SPAR
• Needs to be longer than Channel Impulse Response
• e.g. CM8 has significant energy to 850ns.
• For a 1000ns duration sequence, a length 553
  sequence requires 10 times lower TPAR than
  length 57, but 10 times larger PRF.

17
TG4a CM1 Magnitudes
18
TG4a CM8 Magnitudes
19
TG4a CM6 Magnitudes
20
Basic Difference sets used for length 31 Ipatov
Ternary Sequences

• Fewest zeros
• Parameters L31,k6, ?1
• Difference set 1 5 11 24 25 27
• Balanced zeros
• Parameters L31,k15, ?7
• Difference set 1 2 3 4 6 8 12 15
  16 17 23 24 27 29 30

21
Auto correlation. Fewest zeros ipatov sequence
22
Auto correlation. Balanced zero ipatov sequence
23
Autocorrelation of magnitude. Balanced zero codes
24
Autocorrelation of magnitude. Fewest zero codes
25
Cross correlation of fewest zeros ipatov with
modified magnitude sequence
Cross correlation of 0 1 1 0 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 0 1 with -4 1 1 -4 1 1 1 1 1
1 -4 -4 1 1 1 1 1 1 1 1 1 1 1 1 1 -4
1 1 1 -4 1 i.e. 0 replaced by -4
26
Cross correlation of balanced zeros ipatov with
modified magnitude sequence
Cross correlation of 0 0 1 1 0 1 0 1 0
0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0
1 1 1 0 0 with -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 i.e. 0 replaced by -1
27
12 Length 31 codesBalanced Ipatov Sequences
(BITS)
6 Combination of 6 codes with best cross
correlation 3 Combination of 3 codes with best
cross correlation
28
Best 20 of Length 31 Fewest zero codes
29
SPAR, L31 balanced codes
Lower is better
30
Autocorrelation Golay Merit FactorL31 balanced
codes
Higher is better
31
Cross Correlation
Coherent cross-correlation matrix 16 6
4 4 6 4 6 16 6 6 6 4 4 6
16 6 4 4 4 6 6 16 6 6 6
6 4 6 16 6 4 4 4 6 6
16 Non-coherent cross-correlation matrix
16 4 4 4 6 4 4 16 6 4 4
4 4 6 16 4 4 4 4 4 4 16 4
6 6 4 4 4 16 4 4 4 4 6
4 16
32
Preamble PSD for BITSat 30.875MHz PRF
33
Preamble Spectrum Analyzer OutputBITS 30.875MHz
PRF
34
SPAR vs Data mode PSDBITS- Codeword No. 10
Codeword No. 10 SPAR 3.26dB
35
SPAR vs Data mode SpectrumBITS- Codeword No. 10
Codeword No. 10 SPAR 3.26dB
36
Aperiodic PSD 30.85MHz PRF
Codeword No. 10 SPAR 3.26dB
37
Aperiodic PSD 15.4MHz PRF
Codeword No. 10 SPAR 3.26dB
38
Using these codes for dataBPSK with PPM for
non-coherent compatibility
bi-1 0, bi 0
bi-1 0, bi 1
bi-1 1, bi 0
bi-1 1, bi 1
39
Conclusion

• Recommendations
• Preamble
• Use periodic BITS codes at 30.875 MHz PRF for
  Preamble
• Data Transmission
• Use BITS codes
• Use BPSK with PPM for non-coherent compatibility
  at variable PRF

40
References

• Ipatov V. P. Ipatov, Ternary sequences with
  ideal autocorrelation properties Radio Eng.
  Electron. Phys., vol. 24, pp. 75-79, Oct. 1979.
• Høholdt et al Tom Høholdt and Jørn Justesen,
  Ternary sequences with Perfect Periodic
  Autocorrelation, IEEE Transactions on
  information theory.