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Barker code

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</style><div role="note" class="hatnote navigation-not-searchable">"Barker sequence" redirects here. For the sequence of <a href="Generalized_Wieferich_prime" class="mw-redirect" title="Generalized Wieferich prime">generalized Wieferich primes</a>, see <a href="Wieferich_pair#Barker_sequence" title="Wieferich pair">Wieferich pair § Barker sequence</a>.</div>

In <a href="Telecommunication_technology" class="mw-redirect" title="Telecommunication technology">telecommunication technology</a>, a Barker code or Barker sequence is a finite sequence of digital values with the ideal <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> property. It is used as a synchronising pattern between the sender and receiver of a stream of bits.

<div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2></div>

<a href="Binary_digits" class="mw-redirect" title="Binary digits">Binary digits</a> have very little meaning unless the significance of the individual digits is known. The transmission of a pre-arranged synchronising pattern of digits can enable a <a href="Signal" title="Signal">signal</a> to be regenerated by a <a href="Receiver_(radio)" class="mw-redirect" title="Receiver (radio)">receiver</a> with a low probability of error. In simple terms it is equivalent to tying a label to one digit after which others may be related by counting. This is achieved by transmitting a special pattern of digits which is unambiguously recognised by the receiver. The longer the pattern the more accurately the data can be <a href="Synchronization" title="Synchronization">synchronised</a> and errors due to <a href="Distortion" title="Distortion">distortion</a> omitted. These patterns are called Barker sequences or Barker codes, after the inventor <a href="Ronald_Hugh_Barker" title="Ronald Hugh Barker">Ronald Hugh Barker</a>. The process is described in "Group Synchronisation of Binary Digital Systems" published in 1953.<a href="#cite_note-Barker_comms_theory-1">[1]</a> These sequences were initially developed for <a href="Radar" title="Radar">radar</a>, <a href="Telemetry" title="Telemetry">telemetry</a>, and digital speech encryption in the 1940s and 1950s.

<div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2></div>

During and after WWII digital technology became a key subject for research e.g. for radar, missile and gun fire control and encryption. In the 1950s scientists were trying various methods around the world to reduce errors in transmissions using code and to synchronise the received data. The problem being transmission noise, time delay and accuracy of received data. In 1948 the mathematician <a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a> published an article '"A Mathematical Theory of Communication"' which laid out the basic elements of <a href="Communication" title="Communication">communication</a>. In it he discusses the problems of <a href="Noise" title="Noise">noise</a>.

Shannon realised that “communication signals must be treated in isolation from the meaning of the messages that they transmit” and laid down the theoretical foundations for <a href="Digital_circuits" class="mw-redirect" title="Digital circuits">digital circuits</a>. “The problem of communication was primarily viewed as a deterministic signal-reconstruction problem: how to transform a received signal, distorted by the physical medium, to reconstruct the original as accurately as possible” <a href="#cite_note-2">[2]</a> or see original.<a href="#cite_note-3">[3]</a>

In 1948 electronics was advancing fast but the problem of receiving accurate data had not. This is demonstrated in an article on Frequency Shift Keying published by Wireless World.<a href="#cite_note-4">[4]</a>

In 1953 R. H. Barker published a paper demonstrating how this problem to synchronise the data in transmissions could be overcome. The process is described in “Group Synchronisation of Binary Digital Systems”. When used in data transmissions the receiver can read and if necessary correct the data to be error free by <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> and <a href="Cross_correlation" class="mw-redirect" title="Cross correlation">cross correlation</a> by achieving zero autocorrelation except at the incidence position using specific codes. The Barker sequence process at the time produced great interest, particularly in the United States as his method solved the problem, initiating a huge leap forward in <a href="Telecommunications" title="Telecommunications">telecommunications</a>. The process has remained at the forefront of radar, <a href="Data_transmission" class="mw-redirect" title="Data transmission">data transmission</a> and telemetry and is now a very well known industry standard, still being researched in many technology fields.

“In a pioneering examination of group synchronization of binary digital systems, Barker reasoned it would be desirable to start with an autocorrelation function having very low sidelobes. The governing code pattern, he insisted, could be unambiguously recognized by the <a href="Detector" class="mw-redirect" title="Detector">detector</a>. To assure this premise, Barker contended the selected pattern should be sufficiently unlikely to occur by chance, in a random series of noise generated bits”<a href="#cite_note-Siegel1971-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A Barker code or Barker sequence is a finite sequence of N values of +1 and −1,

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<annotation encoding="application/x-tex">{\displaystyle a_{j}{\text{ for }}j=1,2,\dots ,N}</annotation>

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with the ideal autocorrelation property, such that the off-peak (non-cyclic) <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> coefficients

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<annotation encoding="application/x-tex">{\displaystyle c_{v}=\sum _{j=1}^{N-v}a_{j}a_{j+v}}</annotation>

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are as small as possible:

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<annotation encoding="application/x-tex">{\displaystyle |c_{v}|\leq 1\,}</annotation>

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</math><img src="./d18049481c82a72845fa58cb48704c847bb1aefd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.978ex; height:2.843ex;" alt="{\displaystyle |c_{v}|\leq 1\,}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq v<N}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq v<N}</annotation>

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</math><img src="./2be03d717b143bf2e9ba5afc9df7e6cf0b0143f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.551ex; height:2.343ex;" alt="{\displaystyle 1\leq v<N}" loading="lazy">.<a href="#cite_note-Barker_comms_theory-1">[1]</a>

Only nine Barker sequences<a href="#cite_note-6">[6]</a> are known, all of length N at most 13.<a href="#cite_note-BM-7">[7]</a> <a href="Ronald_Hugh_Barker" title="Ronald Hugh Barker">Barker</a>'s 1953 paper asked for sequences with the stronger condition

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<annotation encoding="application/x-tex">{\displaystyle c_{v}\in \{-1,0\}.}</annotation>

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</math><img src="./03232416e1083dc08ec23cd52c179a8ebce8b45c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.016ex; height:2.843ex;" alt="{\displaystyle c_{v}\in \{-1,0\}.}" loading="lazy"></dd></dl>

Only four such sequences are known, shown in bold in the table below.<a href="#cite_note-JEACE-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Known_Barker_codes">Known Barker codes</h2></div>

Here is a table of all known Barker codes, where negations and reversals of the codes have been omitted. A Barker code has a maximum autocorrelation sequence which has sidelobes no larger than 1. It is generally accepted that no other perfect binary phase codes exist.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> (It has been proven that there are no further odd-length codes,<a href="#cite_note-11">[11]</a> nor even-length codes of N < 1022.<a href="#cite_note-12">[12]</a>)

<caption>Known Barker codes

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<th>Length

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<th colspan="2">Codes</th>

<th>Sidelobe level ratio<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a>

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<td>−6 dB

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<td>−9.5 dB

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<td>−12 dB

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<td>−14 dB

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<td>−16.9 dB

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<td>−20.8 dB

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<td>−22.3 dB

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Barker codes of length N equal to 11 and 13 are used in <a href="Direct-sequence_spread_spectrum" title="Direct-sequence spread spectrum">direct-sequence spread spectrum</a> and <a href="Pulse_compression" title="Pulse compression">pulse compression radar</a> systems because of their low autocorrelation properties (the sidelobe level of amplitude of the Barker codes is 1/N that of the peak signal).<a href="#cite_note-15">[15]</a> A Barker code resembles a discrete version of a continuous <a href="Chirp" title="Chirp">chirp</a>, another low-autocorrelation signal used in other pulse compression radars.

The positive and negative amplitudes of the pulses forming the Barker codes imply the use of biphase modulation or binary <a href="Phase-shift_keying" title="Phase-shift keying">phase-shift keying</a>; that is, the <a href="Phase_shifting" class="mw-redirect" title="Phase shifting">change of phase</a> in the <a href="Carrier_wave" title="Carrier wave">carrier wave</a> is 180 degrees.

Similar to the Barker codes are the <a href="Complementary_sequence" class="mw-redirect" title="Complementary sequence">complementary sequences</a>, which cancel sidelobes exactly when summed; the even-length Barker code pairs are also complementary pairs. There is a simple constructive method to create arbitrarily long complementary sequences.

For the case of cyclic autocorrelation, other sequences have the same property of having perfect (and uniform) sidelobes, such as prime-length <a href="Legendre_symbol" title="Legendre symbol">Legendre sequences</a>, <a href="Zadoff%E2%80%93Chu_sequence" title="Zadoff–Chu sequence">Zadoff–Chu sequences</a> (used in 3rd- and 4th-generation cellular radio) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}-1}">

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</math><img src="./51e4bd4ef2f9549d026cbf643a91c0d12a8c6794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}" loading="lazy"> <a href="Maximum_length_sequence" title="Maximum length sequence">maximum length sequences</a> (MLS). Arbitrarily long cyclic sequences can be constructed.

<div class="mw-heading mw-heading2"><h2 id="Barker_modulation">Barker modulation</h2></div>

In wireless communications, sequences are usually chosen for their spectral properties and for low cross correlation with other sequences likely to interfere. In the 802.11 standard, an 11-chip Barker sequence is used for the 1 and 2 Mbit/s rates. The value of the autocorrelation function for the Barker sequence is 0 or −1 at all offsets except zero, where it is +11. This makes for a more uniform spectrum, and better performance in the receivers.<a href="#cite_note-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples_of_applications">Examples of applications</h2></div>

Applications of Barker codes are found in <a href="Radar" title="Radar">radar</a>,<a href="#cite_note-17">[17]</a> <a href="Mobile_phone" title="Mobile phone">mobile phone</a>,<a href="#cite_note-18">[18]</a> <a href="Telemetry" title="Telemetry">telemetry</a>,<a href="#cite_note-Siegel1971-5">[5]</a> <a href="Ultrasound" title="Ultrasound">ultrasound</a> imaging and testing,<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> <a href="GPS" class="mw-redirect" title="GPS">GPS</a>,<a href="#cite_note-21">[21]</a> and <a href="Wi-Fi" title="Wi-Fi">Wi-Fi</a>.<a href="#cite_note-22">[22]</a>

Many of these technologies use <a href="DSSS" class="mw-redirect" title="DSSS">DSSS</a>. This technique incorporates Barker code to improve the received signal quality and improve security.<a href="#cite_note-23">[23]</a>

These codes are also used in radio frequency identification, or <a href="RFID" class="mw-redirect" title="RFID">RFID</a>. Some examples where Barker code is used are: pet and livestock tracking, bar code scanners, inventory management, vehicle, parcel, asset and equipment tracking, inventory control, cargo and supply chain logistics.<a href="#cite_note-24">[24]</a> It is also used extensively for <a href="Intelligent_Transport_Systems" class="mw-redirect" title="Intelligent Transport Systems">Intelligent Transport Systems</a> (ITS) i.e. for vehicle guidance<a href="#cite_note-25">[25]</a>

<div class="mw-heading mw-heading2"><h2 id="Acceptance_probability">Acceptance probability</h2></div>

Barker's algorithm is an alternative to Metropolis–Hastings, which doesn't satisfy the detailed balance condition. Barker's algorithm does converge to the target distribution. Given the current state, x, and the proposed state, x', the acceptance probability is defined as:

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<annotation encoding="application/x-tex">{\displaystyle \alpha \left(x\rightarrow x^{\prime }\right)={\frac {P\left(x^{\prime }\right)}{P\left(x\right)+P\left(x^{\prime }\right)}}}</annotation>

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The formula doesn't satisfy detailed balance, but makes sure that the balanced condition is met.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBarker1953" class="citation book cs1"><a href="Ronald_Hugh_Barker" title="Ronald Hugh Barker">Barker, Ronald Hugh</a> (1953). "Group Synchronizing of Binary Digital Systems". Communication Theory. London: Butterworth. pp. 273–287.</cite>

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<li id="cite_note-JEACE-8"><a href="#cite_ref-JEACE_8-0">^</a> Using different pulse shape in Barker code also improves certain autocorrelation properties.

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