Barker-like systems of sequences and their processing
New systems of binary sequences, that give the similar correlation properties after signal processing as that of the Barker sequences, are suggested and analyzed. The author considers processing of such systems, as well as ways of their application to radio systems and their comparison with compleme...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-563932014-02-18T03:16:08Z Barker-like systems of sequences and their processing Holubnychyi, A.G. Системы передачи и обработки сигналов New systems of binary sequences, that give the similar correlation properties after signal processing as that of the Barker sequences, are suggested and analyzed. The author considers processing of such systems, as well as ways of their application to radio systems and their comparison with complementary sequences. Предложены и проанализированы новые системы бинарных последовательностей, которые дают такие же свойства функции автокорреляции после обработки сигналов, что и последовательности Баркера. Рассмотрены принцип их обработки, пути использования в радиосистемах, выполнен их сравнительный анализ с комплементарными последовательностями. Запропоновано та проаналізовано нові системи бінарних послідовностей, які дають такі ж властивості функції автокореляції після обробки сигналів, що і послідовності Баркера. Розглянуто принцип їх обробки, шляхи використання в радіосистемах, виконано їх порівняльний аналіз з комплементарними послідовностями. 2013 Article Barker-like systems of sequences and their processing / A.G. Holubnychyi // Технология и конструирование в электронной аппаратуре. — 2013. — № 6. — С. 19-24. — Бібліогр.: 7 назв. — англ. 2225-5818 http://dspace.nbuv.gov.ua/handle/123456789/56393 621.396.9 en Технология и конструирование в электронной аппаратуре Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Системы передачи и обработки сигналов Системы передачи и обработки сигналов |
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Системы передачи и обработки сигналов Системы передачи и обработки сигналов Holubnychyi, A.G. Barker-like systems of sequences and their processing Технология и конструирование в электронной аппаратуре |
| description |
New systems of binary sequences, that give the similar correlation properties after signal processing as that of the Barker sequences, are suggested and analyzed. The author considers processing of such systems, as well as ways of their application to radio systems and their comparison with complementary sequences. |
| format |
Article |
| author |
Holubnychyi, A.G. |
| author_facet |
Holubnychyi, A.G. |
| author_sort |
Holubnychyi, A.G. |
| title |
Barker-like systems of sequences and their processing |
| title_short |
Barker-like systems of sequences and their processing |
| title_full |
Barker-like systems of sequences and their processing |
| title_fullStr |
Barker-like systems of sequences and their processing |
| title_full_unstemmed |
Barker-like systems of sequences and their processing |
| title_sort |
barker-like systems of sequences and their processing |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2013 |
| topic_facet |
Системы передачи и обработки сигналов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/56393 |
| citation_txt |
Barker-like systems of sequences and their processing / A.G. Holubnychyi // Технология и конструирование в электронной аппаратуре. — 2013. — № 6. — С. 19-24. — Бібліогр.: 7 назв. — англ. |
| series |
Технология и конструирование в электронной аппаратуре |
| work_keys_str_mv |
AT holubnychyiag barkerlikesystemsofsequencesandtheirprocessing |
| first_indexed |
2025-07-05T07:39:40Z |
| last_indexed |
2025-07-05T07:39:40Z |
| _version_ |
1836791817775874048 |
| fulltext |
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2013, ¹ 6
19
SIGNALS TRANSFER AND PROCESSING SYSTEMS
UDC 621.396.9
A. G. HOLUBNYCHYI, Ph.D. (Eng.)
Ukraine, Kyiv, National Aviation University
E-mail: a.holubnychyi@nau.edu.ua
BARKER-LIKE SYSTEMS OF SEQUENCES
AND THEIR PROCESSING
Barker sequences (codes) are generally well
known in telecommunication systems (DSSS
tech nology, synchronization) and radar techno-
logy. They are characterized by low sidelobes
of normalized autocorrelation function (ACF)
|RSL(τ)| ≤ 1/N (N is the length of sequence) [1].
Binary Barker sequences (elements ai ∈ {±1}) are
only known for lengths N = 2; 3; 4; 5; 7; 11; 13.
There are no these sequences for odd lengths
N > 13, but it is unknown about an existence
of these sequences for even lengths N > 4
[2, p. 109]. Ternary Barker sequences (elements
ai ∈ {0, ±1}) are also known up to length N = 31
[3, p. 23]. There is a known kind of sequences
called “Generalized Barker Sequences (Codes)”,
they are polyphase sequences which consist of ele-
ments ai ∈ {exp(j⋅2πk/M)}, k≤M [4]. Polyphase
Barker sequences are known up to length N = 77 [5].
The problem boils down to a controversy con-
cerning the existence of sequences or systems of
sequences with a greater length that would make
it possible to obtain after their processing an ACF-
signal with a low value of side lobes and a narrow
central-lobe (as that of the Barker sequences). One
of the solutions of this problem is the complemen-
tary sequences [6]. They are pairs of sequences of
lengths N = 2n⋅10k⋅26m (n ≥ 0, k ≥ 0, m ≥ 0; except
for the case n = k = m = 0) with a property that
the result of adding of sidelobes of their ACF is
equal to zero. Complementary sequences are used
in pulse compression radar systems, telecommu-
nication systems IEEE 802.11b/g and other ap-
plications. Although the complementary sequences
are a good technical solution, the optimal ACF
structure synthesis problem still hasn’t been solved
in general, and there may be some other kinds of
sequences that can, for instance, provide better
noise stability of radio systems and consistency
with some digital modulation techniques, e.g.,
QAM.
New systems of binary sequences, that give the similar correlation properties after signal processing
as that of the Barker sequences, are suggested and analyzed. The author considers processing of such
systems, as well as ways of their application to radio systems and their comparison with complementary
sequences.
Keywords: Barker sequences, complementary sequences, correlation properties, sidelobe suppression,
signal processing.
This research is a continuation of the research
described in [7] and concerns the problem of syn-
thesis of the optimal ACF structure. The goal of
this study is to suggest new systems of binary se-
quences (generation rules for such sequences) and
their application in signal processing, which would
make it possible to obtain ACF-signals with the
maximum normalized absolute value of sidelobes
1/Nmax (as that of the Barker sequences), where
Nmax is the maximum length of sequence in the
system of sequences.
Systems of Barker-like Sequences
Systems of sequences being suggested consist
of the Barker sequence a = {–1; 1; –1; –1; –1}
(length N = 5) and binary sequences of lengths
N = 20⋅2q, q = 0, 1, 2, 3, ..., which can be ob-
tained by using the generation rule
– , ;
(– ) , ;
(– ) , ;
,
– ,
– ;
– ,
,
– ;
,( / – ); , / .
for subtype
for subtype
for subtype
for subtype
a
i
i m
a i n
a A
a B
i N n
a A
a B
i N n
m N n N
1 1
1 2 1
1 2
1 2
2 2
1 4 1 1 4
–
–
–
i
m
n
n
n
n
n
n
2 1
2
2
2 1
2 1
=
=
= +
=
= +
= +
= =
Z
[
\
]
]
]
]
]]
]
]
]
]
]
)
)
(1)
Such systems of sequences allow to obtain, after
the joint signal processing (considered below), an
ACF-signal with the maximum normalized abso-
lute value of sidelobes 1/Nmax, which would match
to the maximum value of sidelobes in a Barker
sequence of length Nmax, if it existed. Therefore,
such systems of sequences may be called “Barker-
like systems of sequences”.
Such system can contain subtype A or subtype B
sequences (signs of some ACF sidelobes depend
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
on the subtype, but their absolute values do not
depend on the subtype).
Generation rule (1) generally gives binary se-
quences of lengths N = 4k, k = 1, 2, 3, ..., but in
our case only lengths N = 20⋅2q, q = 0, 1, 2, 3, ...
are required.
The ACF of sequences that have been obtained
by (1) is partly presented for certain τ values and
N ≥ 12 in
( )
, ;
1 ,
–1 ,
, ;
, , , – ;
– , .
for subtype
for subtypeR
N
A
B
m m N
N
0
1 3
0 2 4 0
4
1
16 4
τ
τ
τ
τ
τ
=
=
=
= + =
=
` j
Z
[
\
]
]
]
]
]]
)
(2)
It is important that the central lobe of ACF
R(0) = N is always separated from the first
high sidelobe by low sidelobes (0 or ±1); high
sidelobes are concentrated between zero sidelobes
(at τ = 2 + 4m, m = 0, 1, 2, 3, ...).
The structure of sequences which have been
obtained by (1) is shown in Fig. 1.
Signal processing and ways of using Barker-
like systems of sequences
Signal processing and ways of application of
the suggested systems of sequences are based on
the following property of the systems: the result
of multiplication of ACF-signals (at the outputs
of matched filters), that exist at the same time
interval and match to these sequences, constitutes
a signal that is similar to the ACF-signal with a
narrow central-lobe (equals to the duration of the
Fig. 1. Structure of suggested subtype А (a) and subtype В (b) sequences
a)
b)
Fig. 2. Signals in the considered example of
Barker-like system of sequences
U1(t)
1
–1
U2(t)
1
–1
0 4 8 12 16 t
0 2 4 6 8 10 12 14 16 18 t
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
Fig. 3. Matched filters for sequences
N1=5 (a) and N2=20 (b)
Fig. 4. Forms of signals after MF for sequences N1=5 (a) and N2=20 (b), and the result of their multiplication (c)
a)
b)
inverter
delay line
a)
b)
c)
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
Fig. 5. Signal processing of Barker-like systems
of sequences
shortest element of sequences in all the system)
and low sidelobes (normalized absolute values do
not exceed 1/Nmax).
Let us take a detailed look at signal processing
by using the example of the Barker-like system
consisting of two sequences {–1; 1; –1; –1; –1}
(N1 = 5, Barker sequence) and {–1; 1; –1; –1; 1;
–1; –1; –1; 1; –1; –1; –1; –1; 1; –1; –1; –1; 1;
1; 1} (N2 = 20, sequence of subtype A). In Fig. 2
are shown signals for those sequences (the same
time interval, e. g. 20 µs, is used). Matched filters
(MF) for signals of each sequence are shown in
Fig. 3, and ACF-signals after the matched filters
and the result of their multiplication (central-lobes
of ACF-signals are reduced to the zero moment)
are illustrated in Fig. 4.
In the case of the suggested systems of sequences
the multiplication of ACF makes it possible to
combine their advantages (low sidelobes of ACF1
and narrow central-lobe of ACF2) and neutralize
their disadvantages (wide central-lobe of ACF1 and
high sidelobes of ACF2). It is possible because the
ACF of suggested sequences has a comb structure:
the central-lobe is always separated from the first
high sidelobe by low sidelobes (0 or ±1) — this
results in the appearance of a narrow central-lobe
after multiplication. High sidelobes are also sepa-
rated from one another by zero sidelobes. Thus low
and high sidelobes of different ACFs are partially
or totally cross-thinned — this leads to partial or
total sidelobe suppression.
Characteristics of some of the suggested Barker-
like systems of sequences (data are confirmed by
computer modeling) are given in Table 1.
Theoretical justification of the fact, that the
maximum normalized absolute value of ACF
sidelobes is 1/Nmax for any possible number of
sequences in a system, is expected in further re-
search.
Using the Barker sequence of length N = 5 in the
suggested systems is important for the considered
case, because if other basic binary Barker sequence
(e.g., N=13) and system (e.g., N1=13 and N2=52)
were used, the resultant maximum normalized
absolute value of sidelobes wouldn’t be 1/Nmax
(for the case N1=13 and N2=52 it would be 7/169,
which is more than 1/Nmax = 1/N2 = 1/52).
In Fig. 5 is shown the signal processing system
for Barker-like systems of sequences. Delay lines
(2q+1 – 0.5)τq+2 in this system are used to align in
time the centers of the central lobes.
In Table 2 an example of possible modulation
scheme for DSSS technology is shown. In this
example we have used the system of sequences
N1=5 and N2=20 and QPSK modulation with
Gray coding.
In our example, for correct operation of the signal
processing system in a DSSS-transmitter, to transmit
Table 1
Characteristics of some of the suggested Barker-like systems of sequences
Sequences in the system
Central-lobe
width of ACF
in the result of
multiplication
Maximum normalized
value of sidelobes of
ACF in the result of
multiplication
Example of
the suitable
modulation
N1 = 5 and N2 = 20 TS /20 1/20 QPSK
N1 = 5, N2 = 20 and N3 = 40 TS /40 1/40 8-PSK
N1 = 5, N2 = 20, N3 = 40 and N4 = 80 TS /80 1/80
16-PSK or
16-QAM
N1 = 5, N2 = 20, N3 = 40, N4 = 80
and N5 = 160
TS /160 1/160 32-QAM
N1 = 5, N2 = 20, N3 = 40, N4 = 80,
N5 = 160 and N6 = 320
TS /320 1/320 64-QAM
TS – signal duration.
T
20 2 –w w
S
2$
τ =
(2q+1–0.5)τq+2
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
a zero bit, it is necessary to invert only one chip flow,
which matches to the sequence N2 = 20; the chip
flow for the sequence N1 = 5 is constant and doesn’t
depend on the bit value. Generally, in such systems
may be inverted a certain odd number of chip flows,
but not more than the total number of these flows.
Comparison of Barker-like systems of
sequences with complementary sequences
One of the nearest analogs of the suggested
Barker-like sequences are the complementary
sequences. Their comparison is shown in Table 3.
Thus, by the criterion of value of ACF sidelobes
after signal processing, the proposed systems of
sequences are slightly inferior to known comple-
mentary sequences and equal to Barker sequences.
Another important factor in the application of the
proposed systems of sequences is noise stability of
a telecommunication or radar system. In the case
of application of such systems of sequences, after
signal processing non-stationary noise tends to ap-
pear if stationary noise is at the input (unlike com-
plementary systems, where after signal processing
the stationary noise appears). However, multipli-
cation of useful signals during signal processing
(used for Barker-like systems of sequences) may
give better noise stability than adding of useful
signals (used for complementary systems). The
issue of noise stability when using the suggested
Barker-like systems of sequences will be addressed
in further research.
Conclusions
The proposed systems of binary sequences
and signal proces sing using such systems make
it possible to obtain ACF-signals with the same
maximum normalized absolute value of sidelobes
as the one for the Barker sequences. It has been
established using computer modeling, that the
maximum length of binary sequences of such
systems is at least 320.
Comparison of the suggested sequences with
the known complementary sequences shows that
Bit 1 0
Chips
(N1=5)
–1 1 –1 –1 –1 –1 1 –1 –1 –1
Chips
(N2=20)
-1 1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 -1 1 1 -1 1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1
QPSK
phases
π
4
3π
4
π
4
π
4
5π
4
7π
4
7π
4
7π
4
3π
4
π
4
π
4
π
4
π
4
3π
4
π
4
π
4
π
4
3π
4
3π
4
3π
4
3π
4
π
4
3π
4
3π
4
7π
4
5π
4
5π
4
5π
4
π
4
3π
4
3π
4
3π
4
3π
4
π
4
3π
4
3π
4
3π
4
π
4
π
4
π
4
Table 2
Example of using of the considered sequences in DSSS
Table 3
Comparison of Barker-like systems of sequences with complementary sequences
Parameter Barker-like systems of sequences Complementary sequences
The main principle of signal
processing
Multiplication
of results of matched filtering of
each sequence
Adding
of results of matched filtering
of each sequence
Quantity of sequences in
the system L ≥ 2 L = 2
Lengths of sequences in the
system
N1 = 5 (Barker)
Nw = 20⋅2w–2, w = 2, ..., L
N1 = N2 = 2n⋅10k⋅26m,
n ≥ 0, k ≥ 0, m ≥ 0, except for
the case n = k = m = 0
Maximum normalized value
of sidelobes of ACF in the
result of signal processing
1/(20⋅2L–2) 0
Central-lobe width in the
result of signal processing TS/NL TS/N1
Suitable modulation 2L-level shift keying modulation
(e.g., 64-QAM for L = 6)
4-level shift keying
modulation (e.g., QPSK)
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
both this types are complementary, but matched
filtering results in the case of the proposed
sequences are multiplied, while for the known
sequences the results are added.
Due to their properties after signal processing,
the suggested systems of sequences can be used
in the pulse-compression radar technology, in
synchro nizing system, and in DSSS technology
for wideband signal forming and data transfer.
REFERENCES
1. Barker R. H. Group synchronizing of binary digital
sequences. Communication Theory, London, Butterworth,
1953, pp. 273–287.
2. Babak V. P., Bilets`kii A. Ya. [Deterministic signals
and spectra] Kiev: Tekhnika, 2003. (in Russian) [Бàбàê
В. П., Білецький А. Я. Детерміновані сигнали і спектри.–
Кèїâ: Òåõíіêà, 2003]
3. Gantmakher V. E., Bystrov N. E., Chebotarev
D.V. Noise-like signals. Analysis, synthesis, processing,
St-Petersburg, Nauka i tekhnika, 2005. [Гàíòмàõåð В. Е.,
Быñòðîâ Н. Е., Чåбîòàðåâ Д.В. Шóмîïîдîбíыå ñèãíàëы.
Анализ, синтез, обработка.– Санкт-Петербург: Наука и
òåõíèêà, 2005]
4. Golomb S. W., Scholtz D. A. Generalized Barker
Sequences. IEEE Trans. on Inf. Theory, 1965, vol. 11,
no 4, pp. 533–537. DOI: 10.1109/TIT.1965.1053828
5. Nunn C. J., Coxson G. E. Polyphase pulse compression
codes with optimal peak and integrated sidelobes. IEEE Trans.
on Aerospace and Electronics Systems, 2009, vol. 45, no 2,
pp. 775–781. DOI: 10.1109/TAES.2009.5089560.
6. Turyn R. J. Hadamard matrices, Baumert-Hall units,
four-symbol sequences, pulse compression, and surface wave
encodings. Journal of Combinatorial Theory (Series A),
1974, vol. 16, no 3, pp. 313–333. DOI: 10.1016/0097-
3165(74)90056-9.
7. Holubnychyi A. Generalized binary Barker sequences
and their application to radar technology. Proc. of the Signal
Processing Symposium (SPS-2013), Poland, Jachranka, 2013,
pp. 1–9. DOI: 10.1109/SPS.2013.6623610.
Received 05.09 2013
О. Г. ГОЛУБНИЧИЙ
Уêðàїíà, м. Кèїâ, Нàціîíàëьíèé àâіàціéíèé óíі âåðñèòåò
E-mail: a.holubnychyi@nau.edu.ua
БАРКЕРОПОДІБНІ СИСТЕМИ ПОСЛІ ДОВНОСТЕЙ ТА ЇХ ОБРОБКА
Запропоновано та проаналізовано нові системи бінарних послідовностей, які дають такі ж властивості
функції автокореляції після обробки сигналів, що і послідовності Баркера. Розглянуто принцип їх об-
робки, шляхи використання в радіосистемах, виконано їх порівняльний аналіз з комплементарними
послідовностями.
Ключові слова: послідовності Баркера, комплементарні послідовності, кореляційні властивості, приду-
шення бічних пелюсток, обробка сигналів.
А. Г. ГОЛУБНИЧИЙ
Уêðàèíà, ã. Кèåâ, Нàцèîíàëьíыé àâèàцèîííыé óíèâåðñèòåò
E-mail: a.holubnychyi@nau.edu.ua
БАРКЕРОПОДОБНЫЕ СИСТЕМЫ ПОСЛЕДОВАТЕЛЬНОСТЕЙ
И ИХ ОБРАБОТКА
Предложены и проанализированы новые системы бинарных последовательностей, которые дают такие
же свойства функции автокорреляции после обработки сигналов, что и последовательности Баркера.
Рассмотрены принцип их обработки, пути использования в радиосистемах, выполнен их сравнительный
анализ с комплементарными последовательностями.
Ключевые слова: последовательности Баркера, комплементарные последовательности, корреляционные
свойства, подавление боковых лепестков, обработка сигналов.
|