Barker code

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A Barker code is a sequence of N values of +1 and −1,

a_j for

such that

for all ^[1].

Here is a table of all Barker codes, where negations and reversals of the codes have been omitted. A Barker code has a maximum autocorrelation of 1 (when codes are not aligned). Longer Barker-like codes exist; there is a 28 baud sequence which has sidelobes no larger than 2, and which thus has better RMS performance than the codes below. The table below shows all known Barker codes, it is conjectured that no other perfect binary phase codes exist^[2].

All Barker Codes

Length	Codes
2	+1 −1	+1 +1
3	+1 +1 −1
4	+1 −1 +1 +1	+1 −1 −1 −1
5	+1 +1 +1 −1 +1
7	+1 +1 +1 −1 −1 +1 −1
11	+1 +1 +1 −1 −1 −1 +1 −1 −1 +1 −1
13	+1 +1 +1 +1 +1 −1 −1 +1 +1 −1 +1 −1 +1

Barker codes of length 11 and 13 are used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties.

The +ve and -ve amplitudes of the pulses forming the Barker codes imply the use of biphase modulation; that is, the change of phase in the carrier wave is 180 degrees.

A Barker code resembles a discrete version of a continuous chirp, another low-autocorrelation signal used in other pulse compression radars.

Pseudorandom number sequences can be thought of as cyclic Barker Codes, having perfect (and uniform) cyclic autocorrelation sidelobes. Very long pseudorandom number sequences can be constructed.

Similar to the Barker Codes are the complementary sequences, which cancel sidelobes exactly; the pair of 4 baud Barker Codes in the table form a complementary pair. There is a simple constructive method to create arbitrarily long complementary sequences but codes created in this way are not necessarily useful.

[edit] References

1. ^ Barker, R. H. "Group Synchronizing of Binary Digital Sequences." In Communication Theory. London: Butterworth, pp. 273-287, 1953.
2. ^ http://mathworld.wolfram.com/BarkerCode.html Barker Code at Mathworld