Time measurement, synchronization and time-resolution |
159 |
|
|
6.5 Perfect periodic ACF: minimax binary sequences
Interest in sequences with good periodic ACF is not exhausted by their role as the basic material for designing good aperiodic sequences. Many applications exploit periodic discrete signals (CW-radar, navigation, pilot and synchronization channels of mobile radio etc.), making periodic ACF critically important for system performance. We will address as ‘perfect’ a periodic ACF which has only zero sidelobes, i.e. values between the periodic mainlobes repeating with the period N. Using normalized notation, we write this condition as:
pðmÞ ¼ E |
|
aiai m |
¼ |
( |
0; m |
¼ 0modN |
ð6:6Þ |
|
|
1 N 1 |
|
|
|
1; m |
0modN |
|
|
|
|
X |
|
|
|
|
6¼ |
|
|
|
i¼0 |
|
|
|
|
|
|
where the congruence m ¼ 0modN is read as m is divisible by N (is multiple of N). It is obvious that for the perfect ACF p, max ¼ 0. Figure 6.9 shows the ACF of a discrete baseband signal with rectangular chips, manipulated by the code with perfect periodic ACF. The practical benefits of perfect ACF are obvious from Figure 6.10, where
.
Rp(τ)
τ
∆ |
N∆ |
N∆ |
|
Figure 6.9 Perfect periodic ACF |
|
a0 |
a0 |
a0 |
(a)
t
∆ 
a0 |
N∆ |
a0 |
a0 |
(b)
t
τ
(c)
t
Figure 6.10 Resolution of signal replicas; perfect periodic ACF
160 |
Spread Spectrum and CDMA |
|
|
waveforms (a) and (b) show two time-shifted copies of the same bandpass periodic signal whose code ACF meets (6.6). When the superposition of these signals arrives at the input of the filter matched to a one-period segment of the signal, two time-shifted replicas of the signal ACF are observed at the output. If the time delay between signal copies is greater than ACF mainlobe duration 2D (but smaller than (N 2)D), the filter responses to both signals are entirely resolved with no corruption of each other (Figure 6.10c).
Let us investigate non-normalized periodic ACF of a binary sequence composed of elements f 1g:
N 1
X
RpðmÞ ¼ aiai m |
ð6:7Þ |
i¼0 |
|
where the conjugation asterisk is not required, since all ai ¼ 1. Summing both parts of (6.7) over the range m ¼ 0, 1, . . . , N 1 produces:
N 1 |
N 1 N 1 |
N 1 |
N 1 |
|
X |
X X |
X |
X |
ð6:8Þ |
RpðmÞ ¼ |
aiai m ¼ ai |
ai m ¼ja~0j2 |
||
m¼0 |
m¼0 i¼0 |
i¼0 |
m¼0 |
|
with: |
|
|
|
|
|
|
N 1 |
|
|
|
a~0 ¼ |
X |
|
|
|
ai |
|
|
|
i¼0
being the constant component (imbalance) of code sequence fa0, a1, . . . , aN 1g. Since the constant component of a binary sequence may take on only integer values, the sum in (6.8) is a squared integer. Suppose now that a binary code has perfect periodic ACF.
Then Rp(0) ¼ N 1 |
(ai)2 |
¼N and Rp(m) ¼ 0, m ¼ 1, 2, . . . , N 1, giving: |
|
P |
|
|
|
i¼0 |
|
|
|
|
|
N 1 |
|
|
|
X |
ð6:9Þ |
|
|
RpðmÞ ¼ N ¼ ja~0j2 |
|
|
|
m¼0 |
|
Assuming m 6¼0modN, let Ne and Nd be the numbers of products aiai m in the sum of (6.7) equalling þ1 and 1, respectively. Then Rp(m) ¼ Ne Nd ¼ 0 means Ne ¼ Nd and N ¼ Ne þ Nd ¼ 2Ne. Thus, according to (6.9) and the last result, length N is an even squared integer, i.e. the necessary condition for obtaining perfect ACF for a binary sequence is N ¼ 4h2, where h is integer. All of these lengths (4, 16, 36, 64, . . . ) were investigated in the early 1960s by Turin, who proved that the only binary code2 with perfect periodic PACF of length N 12 100 is a trivial one of length 4: þ1þ1þ1 1 [28]. Later, the nonexistence of such sequences was proved up to lengths N < 4 1652 ¼ 108 900 [29]. Their existence beyond this range looks quite improbable.
2 We do not consider as new (and it is universally adopted) sequences obtained from an initial one by a cyclic shift, mirror imaging or changing signs of all elements. These transforms do not change periodic ACF (Problem 5.5), and sequences obtainable from each other in this way are treated as trivially different or equivalent.