Polyphase sequences with low autocorrelation
Peter Borwein, Ron Ferguson
Abstract— Low autocorrelation for sequences is usually described in terms of low base energy, i.e., the sum of the sidelobe energies, or the maximum modulus of its autocorrelations, a Barker
sequence occurring when this value is
. We describe first an
algorithm combining stochastic methods and calculus to finding
polyphase sequences which are good local minima for the base
energy. Starting from these, a second algorithm uses calculus to
locate sequences which are local minima for the maximum modulus on autocorrelations. In our tabulation of smallest base energies
found at various lengths, statistical evidence suggests we have good
candidates for global minima or ground states up to length 45. We
extend the list of known polyphase Barker sequences to length 63.
I. I NTRODUCTION
The acyclic autocorrelation coefficients or sidelobes for a sequence
are given by
!"$#%& ')(*#,+-
with denoting the complex conjugate of . Binary sequences, where /.0# , with low sidelobe values have been
extensively studied. Barker sequences [1], where 1 2 1$3 or 1
depending on whether the number of terms in this sum is even
or odd, represent ideal cases and are known only for lengths
2,3,4,5,7,11 and 13. The Barker sequence condition was extended to
for sequences where the entries are roots
of unity of the form
where
, drawn
from a fixed alphabet of size
(see Golomb and Scholtz [9]
for a more complete reference, Golomb and Zhang [12] for examples. For
or 6 this still means
or 1, but
for other values of , it is possible to have
.
Barker sequences for alphabet sizes of 3,4 and 6 are known up
to lengths 15. These have been compiled through exhaustive
searches. However, Barker sequences for longer lengths require
larger alphabets and the possibility for exhaustive searches diminishes.
The more general setting allows for polyphase sequences,
where the entries of lie on the unit circle in the complex plane
(see Boemer and Antweiler [2], and later work by Friese and
Zottmann [7], Friese [6], Brenner [4]). Then, for
,
we have
for some unique
with
leading to an equivalent formulation of the sequence as
. The th acyclic autocorrelation coefficient
may then be written
1 1546# :9;&<=:>
8
7
B
BC/D
E
B
? @(A#
1 1F/3 2
3$GC1 1AGH#
L7 ;NMO<
P,PUVPU
P
7 ;XWYM < ZMO<\[]-^ #)4JIK4'
346P GQR
ST2 Simon Fraser University, Burnaby, B.C., Canada
Supported in part by grants from NSERC of Canada and MITACS Symbolic
Analysis Project
1 F1_6#
b , c- `Xa
`dae b +_ maxfg1 1gh
#i4jk4*'e(KQgli4m#n
Below, we tabulate the lowest ` a -norms we have found at
in any case, the Barker condition concerns
Since
the maximum value or -norm on the remaining coefficients
, requiring
lengths up to 63. These values tend to rise as the lengths, but,
because of the increasing complexity of the problem, it is difficult to speculate on a first length where a polyphase Barker
sequence may no longer exists.
Earlier authors speculated that they may not exist beyond
length 30. Friese [6] presented polyphase Barker sequences to
length 36 and stated his disbelief that they would exist at lengths
significantly greater. Brenner [4] gave sequences up to length
45. We extend this to length at least 63.
Other measures to describe low autocorrelation are the norms,
`Oo
g Z ot V=Uo
b
`Nog +_qp sr 1 1
where uv@# . The most tractable
b of these, from a computational standpoint, seems to be `d% + . Indeed, historically, low
autocorrelation for sequences has been described in terms of
low base energy w
s
1 1 x#zy{`On b + |
which is the sum of the sidelobe energies, or high merit factor
(see, for example, [8])
w
Q' For binary sequences, Mertens [11] has tabulated minimum
base energies up to length 48 (later extended to length 60),
obtained through exhaustive search. Stochastic methods have
been used to find longer sequences with low base energy, or
equivalently, high merit factors. We suggest that extension of
this effort to polyphase sequences may be the more useful approach. In any case, low
norm will mean low base energy.
We describe a method for converting sequences of low base energy to sequences with low
norm.
We tabulate the lowest base energies and
norm found for
sequences up to length 64. We present Barker sequences for
lengths 46 to 63 in terms of the smallest alphabets found. These,
naturally, are not sequences with the lowest
norm, but are
obtained through a rounding process using some stochastic optimization.
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II. A N
ALGORITHM FOR FINDING LOW BASE ENERGIES
1 1o
P
b
`No
+
is a sinusoidal function in each of the variWe note that
ables . Consequently, the number of minima for the functions
proliferate as the length increases, making the direct use
of calculus impractical for finding the best examples with low
autocorrelation. Some researchers have used special types, for
example Huffman sequences [6], as starting points and then
used stochastic methods for improvement. Others have used
stochastic methods alone to find sequences with low
or
norm [4]. The method used for this paper combines methods of
calculus with stochastic variation.
Starting from some fixed sequence, we choose a coordinate,
at random and regard the base energy as a function of the
single variable . This enable us to rewrite
I
` `O
P
7 ;XWYMO< ZM <\[] ^ !
"#
$ ! value. A study of this function shows that it can have up to two
local minima in this interval. Using gradients or other methods
may not proceed to the better one. Instead, we locate the minimum for % !
and the closer of the two minima
& ' for in this interval. We pick some intermediate point and use Newton’s method from elementary calculus
to find where the derivative is zero. This locates the minimum
point we seek.
This is one iteration in the optimization. Since we minimize
over the free free coordinate at each stage, the sequence of values for the base energy obtained by repeated iteration is decreasing. Because all values are positive, it will converge to a
local minimum for the base energy. Resetting a number of coordinates at random and repeat this process enables us to obtain
a selection of local minima for the base energy.
XP y +V+
\QP y +V+
c 1 1 o /
4 ' ( Q
\`&og b +V+ o u
`o
uF /
Q Gu )Gju b
b
o a `Nog _+ `dae +-
this is, effectively, a minimizing
process for the ` a -norm. Furb
1 1F4 # ,
thermore, whenever `&o
+ 4 # , we must have
so the Barker condition is satisfied.
The components of the gradient of the ` o norm at a point S
may be calculated from the formula
`Nog b + p c 1 1 o t =Uo
P
P # p Z 1 1 o t s Z u 1 1 F 1 1 Q
u P
c
1 1 o,c W osV^X=Uo 1 1 P
Q c 1 1 o
] ^ oZ
c
W
1
1
W
o
s
V
X
^
U
=
o
o
b
P
g
Z
Q 1 ` a +1 W ] ^
c
wt \
+ P 1 1 a weighted sum ofb the components of the gradient of the 1 , 1 .
o
Note that 1
`da) +1 # for at least one , while for others it
is 4# . Thus, the sum in the denominators of the weights never
approaches 0, in fact, it is always bounded between 1 and '0( Q
even when u is very large, so that, in general, these calculations
can be carried to high accuracy. For each , we obtain MO< 1 1 yQ 1 %1 dP y
by differentiating the formula 1 2 1 ,+V+ykQ 1 1 \QP y ++- derivable
from above.
Using gradients starting from a local minimum for the u norm, the algorithm used a line search method to find a local
minimum for the u norm. Using a formula such as
u x#n 3 u y{3
means the u values increase exponentially, though the initial
growth is fairly slow, which we found led to better final values. We stopped where u is approximately 300,000, which
involved finding minimum points for just over 200 u values.
b
At this value the ` o norm is essentially the ` a norm on .
u
-
+
/.
0
y 7 c;YM < y 7 ;YM <
) W c ^ 7 ;XWYMO<- M <![%c] ^ ;Y,M 7 ;NM < ] for
where k
I ( 3 and 0 otherwise, and 7 < [] for I y 4*'
and 0 otherwise. This gives
1 1 y 27 ;YM < y 7 ;NM < y 7 c;YM < y 7 c;NM <
where 1 1 y/1 1 y 1 1 , and , and enables us to write y
Z
Z
Z
`On b + y p t 7 ;NM < y p t 7 :;YM <
c
c
y p t 7 c;YM < y p t 7 Z:;YM <
y n 7 ;YM < y 7 :;NM < y n7 Z;NM < y 7 Z:;YM <
yiQ 1 nn1 XP y n2+V+ y*Q 1 1 \QP y Z +V+-
Over the interval 3 4 P G QR , we find to find the point
giving the smallest value of this function and reset P to this
as
1 14m#
uv8#
u
u
III. OTHER NORMS
The Barker condition, max
, means that
for all
, i.e., these -norm values
are small. This suggests that a Barker sequence should lie near
a minimum point for each of the -norms, and, in fact, that
these minimum points for different values should be close together. Starting from a minimum point for the base energy, we
use methods of calculus to find minimum points for -norms
for a sequence of increasing -values,
.
Since
(*),+
0
0
0
2
1
3
0
26
54
0
=>
78
9
78
= >
@
A
B,C *D
9
0
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<
;: 0
@
@ A
B,@ C *D
@
: @
@
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@
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I '
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Though effective, this algorithm is quite expensive computationally. However, it was used only for the relatively small
number of sequences found with sufficiently small norm, so
that the running time of the combined algorithms was not appreciably affected.
`
IV. BARKER SEQUENCES WITH SMALL ALPHABETS
For calculations in the algorithms described above, the entries for the polyphase sequences were rounded to millionth
roots of unity, which, practically, gave a fine enough grid of
points to adequately approximate continuous values. To test for
a Barker sequence with smaller alphabet size
near a Barker
sequence found for the alphabet of size one million, the coordinates were first rounded off to the coarser alphabet values.
Beyond some value for
this would always yield a Barker
sequence, but these are, in general, far from being the minimum alphabet size possible in this neighbourhood. To find if a
smaller alphabet was possible, a stochastic algorithm for minimizing the
norm was used after this rounding. Iteratively,
coordinates were chosen first at random, but for the final optimization, in sequence, and a switch of coordinate value made to
a neighbouring value in the alphabet if the norm was lowered.
To find the better alphabets, many repeated applications starting
each time from the rounded values was usually required. In the
spirit of submissions by other authors, we generally chose to
look for alphabets sizes divisible by 10. More effort would undoubtedly find smaller alphabet sizes at most lengths. This algorithm is fairly expensive as well, but the number of polyphase
Barker sequences to consider was not large. Figure 1 summarizes the optimal results found for sequences of lengths up to
64. Note that these entries are somewhat independent, i.e., the
sequence with the lowest base energy does not necessarily lead
to the sequence with the lowest -norm, which, in turn, may
not lead to a Barker sequence with the smallest alphabet.
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V. S TATISTICAL ANALYSIS
3
In collecting data on sequences with base energy less than
, we continued to run our programs at lengths up to 45 to
the point that we could have good expectation that the best examples were collected. In advance, we do not know the size of
the sample space for minimum points with energies in our range
of interest. However, at some point, the continual repetition of
previous examples suggests that we come close to exhausting
the sample space. We use the statistical model described below to give substance to this observation. Part of this may be
described as the inverse collector’s problem as described in [5]
and [10]. However, our problem is not specifically to establish
the most probable size of the sample space, but to estimate the
likelihood that we have found the example with the lowest base
energy.
Let be the size of the sample space, i.e., the total number
of sequences, unique up to normalization as described in [9],
with base energy
at a particular length . Let be the
number of trials in terms of examples collected, the number
of these examples with different energy. Where is the event
that we have collected the optimal example, what we seek to
evaluate is
4 3
P
/1 P- +-
P
the probability that we have the best example, given the values
and arising from our data. We make assumptions
(1) The the occurrence of the different examples are equilikely.
(2) There is no specific bias on the actual size of the sample
space in the range where this is significant.
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Fig. 1. Statistics of the three different sequences found at each length with
(1) lowest base energy , (2) lowest norm, (3) smallest alphabet size /1 P- + ; r /1 y ? + y ? 1 P-+ From our assumptions, we have /1 iy ? + y
? + . Using Baysian probability theory,
we then derive
K
y ? 1 P- + r c 1 c 1 i y ? y PV+ I%VPV+ For !
yj? , the number of possible outcomes for P trials
is iy{?U+ . Let "$#%XPV+ be the number of permutations of distinct objects taken P at a time, with each object appearing at
least once. The the total number b of possible outcomes with
different samples occurring is # ; #%"&#%dPV+ and we obtain
# ;
M t
# Z"& #%dPV+ iy ?U+
/1 P- + p
#
;' r iy # ? ; r # " # dPV+ iy I+ M
M ; r
# y ?U+
'; r # # ; y ?U+ M
Then, we use
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E
4
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n
40
41
42
43
44
45
46
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62
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t
126
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19
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11.9
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0.98
0.99
0.99+
0.99
0.99+
0.94
0.79
to locate low base energy examples for binary sequences, see
[3]. Repeated application again leads to repetition of the lower
base energy examples. At lengths up to, say, 55, for binary sequences, the examples found may be cross referenced to lists
generated by exhaustive search to confirm that we are locating
all of the sequences with base energy in this range.
We thank Josh Knauer for his assistance in conducting the
computations and Jonathan Jedwab for discussion concerning
stochastic methods. We thank the reviewers as well for their
suggestions on improving the exposition of the paper.
TABLE I
R ESULTS OF STATISTICAL ANALYSIS APPLIED AT
LENGTH
, WITH
SAMPLES COLLECTED , THE
EXPECTED SIZE OF SAMPLE SPACE , THE PROBABILITY GROUND
DIFFERENT BASE ENERGIES FOUND ,
STATE FOUND
In Table I we present results for data collected at lengths 40
through 46. For lengths up to 45 it was possible to run the
programs longer so that confidence in these results is strong.
At lengths beyond 45, the running times became prohibitive
for obtaining much repetition, so our results for lengths 46 –
63 should not be considered to be global optima to within any
reasonably high confidence level.
VI. N EW BARKER SEQUENCES
Figure 2 lists examples of polyphase Barker sequences found
at lengths not previously reported in the literature. The phases
E
are sequence coordinates in terms of powers of .
,
normalized according to equivalences described in [9].
'
F!QRs? B +
VII. C OMMENTS
In the process of optimizing the -norm in the neighbourhood of a small local minimum for the base energy, the base energy can increase considerably, and the end sequence may not
be a Barker sequence. The local minima for the base energy are
distinct; they are not related in terms of Golomb’s invariance
transformations [9]. This, in general, carries forward as well to
the Barker sequences found for different minima. The number
of these Barker sequences found increases from 1 at length 6 to
50 at length 22. Between lengths 22 and 45 at least 50 Barker
sequences were found in each case, while above this the number
decreased, so that only one example was found for each of the
lengths 57 through 63. However, we suspect that this is more a
factor the increasing difficulty in finding low base energy examples rather than their unavailability. At length 42, the algorithm
found approximately 10 Barker sequences per day per machine
(1.2 Gig). A rough analysis of running times versus data collected at different lengths suggests that at length 64, this might
extrapolate to finding approximately 1 Barker sequence every
3 weeks using 50 machines, which is of the order of our effort
expended at this length.
The analogy drawn between the coupon collector’s problem
and the collection of low base energy examples seems natural.
The first assumption, namely, that of each different example is
equilikely, we cannot expect to be strictly true. This method of
analysis needs further study and probably refinement. We compare our experience here with in using a stochastic algorithm
`a
R EFERENCES
[1] R. H. Barker, Group synchronization of binary digital systems, in Communication Theory, W. Jackson, Ed. London, U.K.: Butterworths, 1953, pp.
273-287.
[2] L. Boemer and M. Antweiler, Polyphase Barker sequences, Electron. Lett.,
25 (1989), no. 23, 1577-1579.
[3] P. Borwein, Ron Ferguson, J. Knauer, The merit factor of binary sequences,
preprint (2003), to appear.
[4] A. R. Brenner, Polyphase Barker sequences up to length 45 with small
alphabets, IEE Electronics Letters, vol. 34, no. 16, pp. 1576-1577, 1998.
[5] B. Dawkins, Siobhan’s Problem: The coupon collector revisited, Amer.
Statistician 45 (1991), 76-82.
[6] M. Friese, Polyphase Barker sequences up to length 36, IEEE Trans. Inform. Theory ,42 (1996), No. 4, S. 1248-1250.
[7] M. Friese and H. Zottmann, Polyphase Barker sequences up to length 31,
Electron. Lett., 30 (4), 1996,
[8] M. J. E. Golay, 1982 The merit factor of long low autocorrelation binary
sequences, IEEE Trans. Inf. Theory IT-28, 1982, 543-549
[9] S. W. Golomb and R. A. Scholtz, Generalized Barker sequences, IEEE
Trans. Inform. Theory, vol. IT-11, pp. 533-537, Oct. 1965.
[10] E. Langford and R. Langford, Solution of the inverse collector’s problem,
Math. Scientist 27 (2002), 32-35.
[11] S. Mertens, Exhaustive Search for Low Autocorrelation Binary Sequences, J. Phys. A: Math. Gen. 29 , 1996, L473-L481.
[12] N. Zhang and S. W. Golomb, Sixty-phase generalized Barker sequences,
IEEE Trans. Inform. Theory, vol. 35, pp. 911-912, July 1989.
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found.
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Barker sequences at lengths
with alphabet sizes
, the smallest