A Barker code is a binary code with k^th autocorrelation <= 1 for all nonzero k.



At the workshop, the Barker code group split into four non-disjoint subgroups:



- An "algebra group", who explored symmetries of the search space that preserve the autocorrelations' magnitude.

- A "computing group", who explored methods for quickly finding binary codes with very good autocorrelation properties.

- A "statistics group", who explored ways to quantify what has been empirically observed about autocorrelation in the search space S_2^N.

- A "continuous group", who explored a non-discrete analogue of the problem of finding sequences with good autocorrelations

Altalli, R.

Cao, L.

Chen, F.

Cumberbatch, E.

Cummings, L.J.

Ferguson, P.

Li, S.

Liang, H.

Liu, Y.

Mercer, I.

Miller, J.

Nguyen, L.

Please, C.P.

Salem, M.

Tilley, B.

Vu, P.D.

Watt, J.

Yang, Y.

English

Mathematics in Industry Information Service Eprints Archive

Do the Barker Codes End?
Final Report
MPI 2008, Worcester, MA
1 February 2009
Working Group: Industrial Representative: G. Coxson, (Technology Ser-
vice Corporation); Faculty: E. Cumberbatch (Claremont), L.J. Cummings
(Nottingham), P. Ferguson (Simon Fraser), I. Mercer (York), C.P. Please
(Southampton), and B. Tilley (Olin); Students: R. Altalli (Northeastern),
L. Cao (NJIT), F. Chen (Purdue), S. Li (UMBC), H. Liang (WPI), Y. Liu
(Clemson), J. Miller (Delaware), L. Nguyen (Delaware), M. Salem (Arizona
State), P.D. Vu (SUNY Buffalo), J. Watt (Indiana), and Y. Yang (UMBC)
For convenience, we briefly summarize some notation and terminology.
A binary code is an n-tuple x = [x1, . . . , xN ] where xi ∈ S2 := {+1,−1}
for each i. Then the Cartesian product SN2 is the set of all 2
N binary codes
of length N . For 1 ≤ k ≤ N − 1, the kth autocorrelation of x is
ACFx(k) :=
N−k∑
i=1
xixi+k.
We have ACFx(0) = N (the “peak”), and if k 6= 0, we refer to ACFx(k) as
a “sidelobe”. Two measures of the collective sidelobe magnitude of a binary
code x are the peak sidelobe level,
PSLx = max
1≤k≤N−1
|ACFx(k)|,
and the energy,
Ex =
N−1∑
k=1
(ACFx(k))
2.
A Barker code is a binary code x with |ACFx(k)| ≤ 1 for all k 6= 0. If x is
a Barker code, then PSLx = 1, and by a parity argument, Ex = d(n−1)/2e.
We can also replace the code “alphabet” S2 with the set of mth roots of
unity for some m > 2, or the set of all complex numbers of unit modulus.
The definition of autocorrelation then changes to
∑
xix
∗
i+k where the star
denotes complex conjugation. We then call x a generalized Barker code
if |ACFx(k)| ≤ 1 for all k 6= 0. Note that in this case, the autocorrelations
are no longer restricted to be integers.
At the workshop, the Barker code group split into four non-disjoint sub-
groups:
1
• An “algebra group”, who explored symmetries of the search space that
preserve the autocorrelations’ magnitude
• A “computing group”, who explored methods for quickly finding bi-
nary codes with very good autocorrelation properties
• A “statistics group”, who explored ways to quantify what has been
empirically observed about autocorrelation in the search space SN2
• A “continuous group”, who explored a non-discrete analogue of the
problem of finding sequences with good autocorrelations.
What follows is a summary of what each of the four subgroups were able to
accomplish at the workshop.
1 Algebraic structure
Both in the case of binary sequences and sequences consisting of mth roots
of unity, there are certain operations that preserve the absolute value of
the autocorrelations and hence preserve the PSL and the energy. Note that
our “search space” grows exponentially (there are 2N binary sequences of
length N , and mN sequences of length N consisting of mth roots of unity).
There are some methods known for finding the best sequence of length N
that improve somewhat upon naive brute force search, by taking advantage
of symmetries of the search space as well as using techniques such as “branch
and bound” search. For example, some known algorithms for finding the
minimum energy or minimum PSL in the binary case have running times
of roughly 1.85N and 1.4N respectively, as opposed to 2N . See [3, 4] and
[1, 2] respectively. Further understanding of the symmetries of the search
space could result in further reductions in search time, both in the binary
and nonbinary case. For example, the problem presenter mentioned at the
workshop that it was not known to him whether the structure of the group
of PSL-preserving symmetries is known in the case of mth roots of unity for
even length N .
2 Computational exploration
It can be valuable not only to develop exhaustive algorithms that find the
minimum PSL or energy among all binary sequences of length N , but also to
find practical methods to quickly find sequences with close to optimal PSL
2
or energy. Often in real-world applications, it suffices to have sequences with
good PSL or energy, as opposed to best possible PSL or energy among all
sequences. Thus, one is sometimes willing to sacrifice exhaustiveness if one
can find good sequences quickly.
There are theoretical results giving necessary conditions on the ratio
between the number of positive entries and negative entries in a binary
Barker code (if they exist). We can also observe the ratio of positive to
negative entries in tables of known sequences with optimum or near-optimum
PSL or energy, and then decide to search through only those sequences
having a certain ratio of positive to negative entries.
At the workshop, we experimented with such searches. Moreover, rather
than performing an exhaustive search of all binary sequences with a par-
ticular ratio of positive to negative entries, we attempted to conduct more
“intelligent” searches using heuristic methods to quickly “zero in” on good
sequences, by setting a particular ratio of positive to negative entries at the
outset and then permuting entries.
3 Statistical analysis
For values of N up to a certain point, it is possible to exhaustively compute
the energy for all binary sequences of length N . These lengths can be plotted
in a histogram. When we do so, a pattern is immediately clear. We observed
during the workshop that these histograms conform astonishly well to a well-
known probability distribution called the Gumbel distribution or extreme
value distribution.
In fact, the data for energy of binary sequences fits this probability
distribution so well that by the standards of empirical science, there is no
practical doubt that it is the “correct” distribution. However, providing a
rigorous analytic proof of this statement may be difficult.
The left tail of the Gumbel distribution approaches 0 like e−e−x . One
could prove analytically that there are no Barker codes of length N if one
could show that the probability of a binary code having energy d(N − 1)/2e
is bounded above by some quantity less than 2−N (since there are 2N binary
sequences of length N). The rapid decrease of the left tail of the Gumbel
distribution, together with the excellence of the fit of the distribution to the
data, is strong evidence for the nonexistence of Barker codes for large N ,
although an analytic proof continues to elude us.
3
4 Continuum model
The connection between communications systems and binary codes dates
back to the middle of the 20th century. Since most communications systems
rely on wave propagation through a continuous medium, the means by which
information is transmitted is based on the solutions of linear partial differen-
tial equations. These equations, which rely on a balance between spatial and
temporal gradients, are solved using the classical techniques whose basis is
found in Fourier analysis. Further, since the original communications prob-
lems were based on electrical signals, the fundamental spatial component of
the equation could be represented as a known quantity (the carrier wave),
and the resulting mathematical problems for information transmission de-
pend on the modulation of the signal, either through amplitude modulation
or through phase modulation. These classical techniques can be found in
any text in continuous and discrete signals processing, such as Oppenheim
et al. [5].
The connection between the continuous Fourier representation described
above and the discrete binary sequences discussed in the workshop has been
outlined in [6]. We outline the discussion here based on the continuum
approach used at the workshop. Consider the real-valued signal
f(t∗) = A(t∗/T1)ei[φ(t
∗/T2)] + c.c
where t∗ is the dimensional time of interest, A is the real amplitude modula-
tion, a function that is 2piT1-periodic, and φ is the phase modulation that is
2piT2-periodic. If this signal is sent out from a transmitter, then an example
of a received signal is given by g,
g(t∗) = f(t∗ + τ∗) = A((t∗ + τ∗s )/T1)e
i φ((t∗+τ∗s )/T2) + c.c
where τ∗s is the time shift of interest. The autocorrelation function of f is
given by
I(τ∗ − τ∗s ) =
1
2piT1
∫ piT1
−piT1
f(t∗)f¯(t∗ + τ∗ − τ∗s ) dt∗ (1)
where I is the continuous form of the autocorrelation function that is desired.
For simplicity, let us scale time t∗ = T1t, where t is now a dimensionless time
to arrive at the autocorrelation function
I(τ) =
1
2pi
∫ pi
−pi
A(t)A(t+ τ)ei [φ(` t)−φ(`(t+τ))] dt (2)
where ` = T1/T2 ∈ Z is the ratio of the amplitude modulation period to the
phase modulation period, and we have centered τ to the desired time shift
4
−1
−0.5
0
0.5
1
t*
Si
gn
al
 
 
− π T1 π T1−πTc π Tc
Figure 1: Cartoon of sample signal along one period. The red dots corre-
spond to the actual signal, while the dashed line corresponds to the ampli-
tude modulation in t∗, which has an extent Tc/T1 of the fundamental period.
The phase modulation φ is conveyed by the change in sign of the signal with
its own periodicity of 2piT2 (not labelled).
(i.e. τ = 0 corresponds to the proper return signal time). In the classical
connection of discrete codes to the Fourier context, A(t) = pµ(t), where
µ = Tc/T1 is the fraction of time when A 6= 0, and ` = 1 (see [6]). A
pictorial representation of this setup is shown in Figure 1. In the problem
presented by TSC at the workshop, ` is the code length.
The goal is to determine the phase φ and the amplitude A such that the
difference between the convolution I and a desired autocorrelation function
is minimized. The ideal autocorrelation function would be unity at the signal
return time and zero at other times. This would correspond to an infinite
peak-to-peak amplitude ratio, since the off-peak amplitudes are zero. We
can define this function σ(τ) in terms of a Cauchy sequence
σ(τ) = lim
µ→0 limn→∞ In(τ) = limµ→0 limn→∞
(
sin (τ/µ)
τ/µ
)2n
.
A sampling of these In is shown in Figure 2. Note that I1/2 corresponds to
the inverse Fourier transform of the spectrum p1/µ for µ fixed. Although this
function can be approximated by many Cauchy sequences, for the purposes
of this problem this choice is instructive, since the peak-to-peak amplitude
of In decreases with increasing n.
5
−2 0 2
0
0.5
1
μ = 0.5 ,   n = 1
τ
I n
−2 0 2
0
0.5
1
μ = 0.5 ,   n = 2
τ
I n
−2 0 2
0
0.5
1
μ = 0.5 ,   n = 3
τ
I n
−2 0 2
0
0.5
1
μ = 0.25 ,   n = 1
τ
I n
−2 0 2
0
0.5
1
μ = 0.25 ,   n = 2
τ
I n
−2 0 2
0
0.5
1
μ = 0.25 ,   n = 3
τ
I n
−2 0 2
0
0.5
1
μ = 0.125 ,   n = 1
τ
I n
−2 0 2
0
0.5
1
μ = 0.125 ,   n = 2
τ
I n
−2 0 2
0
0.5
1
μ = 0.125 ,   n = 3
τ
I n
Figure 2: Sample autocorrelation functions In for a range of exponents
n and periodic box sizes L. N = 256 collocation points were used in the
discretization. Note that the peak-to-peak ratio increases with the exponent
n
6
To find A, φ, we take advantage of properties of Fourier transforms. Note
that In is an even function of τ , and the integral (2) is a convolution integral
for τ → −τ . Hence if F denotes the Fourier transform operator in τ , then
F {In} = F {f ∗ f} = [F {f}]2 .
Hence, the solution to f is given by
f = F−1
{√
F {In}
}
, (3)
provided that In(−τ) = In(τ). Note that we have chosen the positive signed
square root. Since the negative branch is also a solution, the transformation
φ→ φ± pi results in solutions corresponding to this branch.
To find f based on these In, we then apply the Fast Fourier Trans-
form using Matlab to perform the forward and inverse Fourier transforms.
Note that this computational technique requires a maximum number of col-
location points at which to evaluate the autocorrelation function In. The
number of these points N provides an upper limit for the length of the code
of interest (i.e. ` ≤ N). We anticipate that the length of the code is to be
determined by the solution. All of the results presented below have been
verified to satisfy the autocorrelation relation to within roundoff error.
Figure 3 shows the amplitude modulation A for the same autocorrela-
tion functions given in Figure 2. Notice that the amplitude modulation is
typically localized near the center τ = 0.
Figure 4 shows the phase modulation φ given by the solution for the same
autocorrelation functions shown in Figure 2. Notice that the phase shifts
are binary: either φ = 0 or φ = ±pi. Since the Barker code corresponds to
cosφ, the sign of the φ = ±pi cases is irrelevant. Figure 5 shows the Barker
code for each of these autocorrelation functions. Although the mapping is
not known between these codes and what the equivalent binary Barker code
would be, it appears that each of these cases could be construed as a simple
binary Barker code [1,−1] or [−1, 1].
As a final note, if we increase the size of n and decrease the size of µ,
we can find codes that do not correspond clearly with a binary Barker code.
Figure 6 shows a situation where the code appears to deviate significantly
from one case to a second. A close-in look at this last case (see Figure 7a)
shows that this sequence does not match cleanly with any of the known
Barker codes. Figure 7b shows the amplitude modulation A for this par-
ticular solution. Notice that the resolution of the amplitude modulation is
poor, but the choice of the phase modulation corrects this to produce an
accurate autocorrelation function.
7
−2 0 2
−0.1
0
0.1
μ = 0.5 ,   n = 1
t
A
−2 0 2
−0.1
0
0.1
μ = 0.5 ,   n = 2
t
A
−2 0 2
−0.1
0
0.1
μ = 0.5 ,   n = 3
t
A
−2 0 2
−0.1
0
0.1
μ = 0.25 ,   n = 1
t
A
−2 0 2
−0.1
0
0.1
μ = 0.25 ,   n = 2
t
A
−2 0 2
−0.1
0
0.1
0.2
μ = 0.25 ,   n = 3
t
A
−2 0 2
−0.1
0
0.1
0.2
μ = 0.125 ,   n = 1
t
A
−2 0 2
−0.1
0
0.1
0.2
μ = 0.125 ,   n = 2
t
A
−2 0 2
−0.1
0
0.1
0.2
0.3
μ = 0.125 ,   n = 3
t
A
Figure 3: Amplitude modulation of f for a given autocorrelation function
shown in Figure 2.
−2 0 2
−2
0
2
μ = 0.5 ,   n = 1
t
φ
−2 0 2
−2
0
2
μ = 0.5 ,   n = 2
t
φ
−2 0 2
−2
0
2
μ = 0.5 ,   n = 3
t
φ
−2 0 2
−2
0
2
μ = 0.25 ,   n = 1
t
φ
−2 0 2
−2
0
2
μ = 0.25 ,   n = 2
t
φ
−2 0 2
−2
0
2
μ = 0.25 ,   n = 3
t
φ
−2 0 2
−2
0
2
μ = 0.125 ,   n = 1
t
φ
−2 0 2
−2
0
2
μ = 0.125 ,   n = 2
t
φ
−2 0 2
−2
0
2
μ = 0.125 ,   n = 3
t
φ
Figure 4: Phase modulation φ of f for a given autocorrelation function
shown in Figure 2
8
−2 0 2
−1
0
1
μ = 0.5 ,   n = 1
t
φ
−2 0 2
−1
0
1
μ = 0.5 ,   n = 2
t
φ
−2 0 2
−1
0
1
μ = 0.5 ,   n = 3
t
φ
−2 0 2
−1
0
1
μ = 0.25 ,   n = 1
t
φ
−2 0 2
−1
0
1
μ = 0.25 ,   n = 2
t
φ
−2 0 2
−1
0
1
μ = 0.25 ,   n = 3
t
φ
−2 0 2
−1
0
1
μ = 0.125 ,   n = 1
t
φ
−2 0 2
−1
0
1
μ = 0.125 ,   n = 2
t
φ
−2 0 2
−1
0
1
μ = 0.125 ,   n = 3
t
φ
Figure 5: Barker code representation cosφ based on the phase modulation
results shown in Figure 4. Notice that the mapping of these codes to the
binary Barker codes is an open question.
9
−2 0 2
−1
0
1
μ = 1/2 ,   n =1
t
−2 0 2
−1
0
1
μ = 1/2 ,   n =4
t
−2 0 2
−1
0
1
μ = 1/2 ,   n =7
t
−2 0 2
−1
0
1
μ =1/22 ,   n =1
t
−2 0 2
−1
0
1
μ =1/22 ,   n =4
t
−2 0 2
−1
0
1
μ =1/22 ,   n =7
t
−2 0 2
−1
0
1
μ =1/46 ,   n =1
t
−2 0 2
−1
0
1
μ =1/46 ,   n =4
t
−2 0 2
−1
0
1
μ =1/46 ,   n =7
t
Figure 6: Note that the increasing values n and decreasing values of µ can
result in nontrivial Barker codes, as noted in the lower right figure.
In conclusion, we have found through a direct Fourier representation a
way to generate codes which depend on both amplitude and on phase. The
classical formulation of the code assumes ab initio that the amplitude mod-
ulation is piecewise constant. Were one to impose this amplitude restriction,
the set of phases φ that would best mimic the autocorrelation function for
amplitude modulation A is the solution of a constrained optimization prob-
lem. It is possible that a formulation based on the Fourier representation
could be implemented to provide insight into the existence of different codes,
not necessarily binary Barker codes, that may be useful for TSC to consider
for its applications.
References
[1] M.N. Cohen, M.R. Fox and J.M. Baden, “Minimum peak sidelobe pulse
compression codes”, in IEEE International Radar Conference, pages 633–
638. IEEE, 1990.
[2] G.E. Coxson, A. Hirschel and M.N. Cohen, “New results on minimum-
PSL binary codes”, in IEEE Radar Conference, pages 153–156, IEEE,
10
−0.2 −0.1 0 0.1 0.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
−0.2 −0.1 0 0.1 0.2
0
0.1
0.2
0.3
0.4
0.5
t
A
(a) (b)
Figure 7: (a) A close-in look at the code for µ = 1/46, n = 7 with N = 256.
Notice that the sequence of the code does not resemble the known binary
Barker codes. (b) The corresponding amplitude modulation A.
11
2001.
[3] S. Mertens, “Exhaustive search for low-autocorrelation binary se-
quences”, J. Phys. A: Math. Gen., 29: L473–L481 (1996).
[4] S. Mertens and H. Bauke, “Ground states of the Bernasconi
model with open boundary conditions”, available at
<http://odysseus.nat.uni-magdeburg.de/∼mertens/bernasconi/open.dat>
(2004).
[5] A.V. Oppenheim and A.S. Willsky and S.H. Nawab, Signals and Systems,
2nd edition, Prentice-Hall (1997).
[6] D.V. Sarwate and M.B. Pursley, “Crosscorrelation properties of pseudo-
random and related sequences”, Proc. IEEE, 68 (5) 593-619: (1980).
12


Minimum peak sidelobe pulse compression codes”,

Do the Barker Codes End?

Do the Barker Codes End?

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