Time measurement, synchronization and time-resolution |
173 |
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Certainly, we are interested in estimating only the sidelobes, i.e. shifts m, which are not
multiples |
of |
N ¼ p. Exploiting the multiplication |
property |
(6.20) |
gives |
( m) ¼ |
( 1) |
(m) and (i m) ¼ [i(1 mi 1)] ¼ (i) |
(1 mi 1), |
where |
the |
inverse always makes sense, since zero i is removed from the sum in (6.24). As a result:
p 1
X
RpðmÞ ¼ ðmÞ½1 þ ð 1Þ& þ |
2ðiÞ ð1 mi 1Þ |
ð6:25Þ |
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i¼1 |
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i6¼m |
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Turn now to (6.22). For any length N ¼ p ¼ 1 mod4 ( 1) ¼ 1 and the first term in the expression above equals 2 (m) ¼ 2 for any m 6¼0 modN. On the other hand, for
lengths of the type N ¼ 3 mod4 |
( 1) ¼ |
1 and the same term of (6.25) vanishes. To |
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cope |
with |
the |
second term in |
(6.25), |
note first that for any non-zero i from |
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GF(p) |
2(i) ¼ 1. Second, if i ran over all non-zero elements of GF(p) then i 1 as well |
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as |
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6¼ |
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mi 1 |
(m |
0 modp) would both run over the same range in some other orders. |
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Therefore, 1 mi 1 would run over p 1 elements of the field including zero but excluding 1, since mi 1 cannot take zero value. Actually, however, the zero element should also be eliminated from the possible values of 1 mi 1, since i in the second term of (6.24) does not take on the value i ¼ m, corresponding to 1 mi 1 ¼ 0, and the whole
range of values 1 |
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mi 1 |
is from 2 to p |
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1. Summarizing all these reasons: |
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p 1 |
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p 1 |
p 1 |
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X |
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X |
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2ðiÞ ð1 mi 1Þ ¼ |
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¼ |
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ðxÞ ¼ |
¼ |
ðxÞ ð1Þ ¼ 1 |
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i m |
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x |
2 |
x |
1 |
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i¼1 |
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6¼ |
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where the final step follows from the character properties (6.19) and (6.21). Now the periodic ACF of a Legendre sequence appears to be one of two types (h below is natural):
1. If length is of the form N ¼ 4h þ 1 (i.e. N ¼ 1 mod4), then:
RpðmÞ ¼ ( N3; |
or |
¼ |
1; m 0 modN |
ð6:26Þ |
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m |
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0 modN |
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þ |
6¼ |
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2. If length is of the form N ¼ 4h þ 3 (i.e. N ¼ 3 mod4), then:
ð |
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Þ ¼ |
1; |
m 6¼0 modN |
ð |
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Þ |
Rp |
m |
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N; |
m ¼ 0 modN |
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6:27 |
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The last result, reproducing |
(6.12), |
shows that Legendre sequences |
of lengths |
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N ¼ 4h þ 3 are minimax, i.e. possess optimal periodic correlation properties possible for binary sequences of odd lengths.
Example 6.9.1. Length N ¼ 7 falls within the set N ¼ 4h þ 3. The element 3 is primitive in GF(7), since raising it to the powers 0, 1, . . . , 5 gives all different non-zero elements: 30 ¼ 1, 31 ¼ 3, 32 ¼ 2, 33 ¼ 6, 34 ¼ 4, 35 ¼ 5. As is directly seen from this series, logarithms
of 1, 2 and 4 are even, while those of 3, 5 and 6 are odd. Hence, (1) ¼ (2) ¼ |
(4) ¼ 1, and |
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(3) ¼ (5) ¼ (6) ¼ 1. Now, |
according to |
(6.23), placing |
plus |
ones in |
the positions |
i ¼ 0, 1, 2, 4 and minus ones in |
the positions |
i ¼ 3, 5, 6 gives |
the |
Legendre |
sequence of |
174 |
Spread Spectrum and CDMA |
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length 7. The computation illustrated by Table 6.4 confirms the optimality of the correlation properties of the binary sequence obtained.
Table 6.4 Calculating periodic ACF of the Legendre sequence fþ þ þ þ g
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m |
a0 |
a1 |
a2 |
a3 |
a4 |
a5 |
a6 |
Rp(m) |
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0 |
þ |
þ |
þ |
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þ |
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þ7 |
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1 |
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þ |
þ |
þ |
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þ |
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1 |
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2 |
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þ |
þ |
þ |
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þ |
1 |
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3 |
þ |
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þ |
þ |
þ |
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1 |
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Legendre sequences form a very powerful class of binary codes with minimax periodic ACF. The condition of their existence (any prime length of the form N ¼ 4h þ 3) is significantly looser than that of m-sequences (N ¼ 2n 1), which means that a lot more Legendre sequences are available as compared to m-sequences. For example, within the range 50–1500 binary m-sequences exist of only 5 lengths, while the number of Legendre sequences is 114.
6.10 Binary codes with good aperiodic ACF: revisited
After collecting necessary knowledge on binary sequences with good periodic ACF, we may return to the idea formulated in Section 6.4 and consisting in utilizing these sequences as a starting point for finding codes with attractive aperiodic ACF. Consider some sequence a0, a1, . . . , aN 1 of length N. Any of its cyclic shifts as, asþ1, . . . , aN 1, a0, . . . , as 1, where 0 s N 1, has the same periodic ACF as the original code, since the periodic ACF is invariant to a cyclic shift (see Problem 5.5). Yet the aperiodic ACF of the cyclically shifted replica may differ from the initial one. Along with the bound (6.5), this fact sets up the basis for a popular algorithm for searching codes with an acceptable aperiodic ACF described below.
At the first step a set of candidate sequences with good periodic ACF, length N preassigned, is somehow collected. It may include all known sequences [34–37] of a given N, whose periodic ACF sidelobes—consistent with (6.5)—allow one to hope for a low level of a, max, or be limited according to the designer’s technological preferences. For example, if a binary code of length N ¼ 63 is sought, the initial set may be limited to only all m-sequences of this length (no Legendre sequence of this length exists, since N is not prime), or include more sequences with promising periodic ACF; with necessary length N ¼ 127 it may cover all m-sequences along with Legendre sequences5 or, again, contain other sequences with sufficiently low periodic sidelobes.
5 A score of different primitive polynomials of the same degree may exist, each generating a specific m-sequence of the same length. Therefore, a quantity of m-sequences of the fixed length exists and all of them are appropriate for the search for good aperiodic codes. Unlike this, there are only two possible Legendre sequences of the same length differing by the first symbol (þ1 in one of them and 1 in the other, see footnote 4 on p. 172).
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At the second step the exhaustive search is performed over all one-period segments of candidate sequences by the criterion of the least maximal aperiodic ACF sidelobe. Specifically, one period segment of the first candidate sequence is taken, its aperiodic ACF is computed and its maximal sidelobe is stored in memory along with the numbers of the candidate sequence and its shift. Then the segment is cyclically shifted by one position and all the calculations are repeated. If an updated maximal aperiodic sidelobe is lower than the previous one, its value and new shift number replace the previous data in memory; otherwise registered data are kept unchanged. These iterations are repeated N times, i.e. for all cyclic shifts of the first candidate sequence, after which the next candidate sequence is tried in the same way, etc. The outcome of the search is the sequence with minimal value of a, max among all sequences picked at the first step. Of course, there is no guarantee that the result will be the best possible among all binary sequences of a given length.
This procedure, which was first proposed in the early 1960s, has been used by many authors, gradually covering wider and wider sets of candidate binary sequences. One of the most detailed lists of binary codes synthesized in this way may be found in [34].
Example 6.10.1. Length N ¼ 23 1 ¼ 7 meets the condition of existence of m-sequences.
There are two |
primitive binary polynomials |
of degree 3 |
f (x) ¼ x3 þ x þ 1 |
and |
f (x) ¼ x3 þ x2 þ 1. |
An immediate test shows that |
m-sequences |
generated by them |
are |
just mirror images of each other, i.e. one of them is the other read from left to right. Such a transform does not affect either periodic or aperiodic ACF (Problem 5.5). Therefore it is enough to include in the candidate set only a single m-sequence: that of Example 6.7.1: 1, þ1, þ1, 1, þ1, 1, 1. In addition, N ¼ 7 is prime of the sort N ¼ 4h þ 3, i.e. minimax Legendre sequences of this length exist, too: the one of Example 6.9.1 (þ1, þ1, þ1, 1, þ1, 1, 1) and its replica with the first element changed to 1. The latter completely repeats the m-sequence selected, while the former—after changing the signs of all elements—coincides with a cyclically shifted skipped m-sequence. Since the polarity change again does not affect either periodic or aperiodic ACF (Problem 5.5), only one minimax sequence out of the four analysed is sufficient to enter the candidate set. Let it be the Legendre one starting with þ1. Calculating its aperiodic ACF gives the following values of Ra(m), m ¼ 1, 2, . . . , 6: 0, þ1, 0, 1, 2, 1, and a, max ¼ 2/7. After one cyclic left shifting the sequence becomes þ1, þ 1, 1, þ1, 1, 1, þ1. For this Ra(4) ¼ 3, a, max ¼ 3/7, i.e. the maximal aperiodic sidelobe is worse than for the original one. The next cyclic shift is þ1, 1, þ1, 1, 1, þ1, þ1 with Ra (1) ¼ 2, a, max ¼ 2/7, i.e. no better than for the initial sequence. At the next shift we come to the sequence 1, þ 1, 1, 1, þ1, þ1, þ1, having aperiodic ACF with sidelobes Ra(m) ¼ 0, 1; m 6¼0, i.e. a, max ¼ 1/7. This sequence is globally optimal among all PSK codes, since no such code can have smaller maximal aperiodic sidelobe (see (6.4)). Actually, the Barker code of length 7 is found which is a mirror replica of that of Table 6.1.
Example 6.10.2. Assume N ¼ 257 ¼ 64 4 þ 1. Since the number 257 is prime, two Legendre sequences of this length exist differing in only their first symbol. However, N is of the form 4h þ 1, and hence their PACF has maximal sidelobe p, max ¼ 3/N ¼ 3/257 (see (6.26)).