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139-458: The Decca Navigator System was a hyperbolic radio navigation system that allowed ships and aircraft to determine their position by using radio signals from a dedicated system of static radio transmitters. The system used phase comparison between pairs of low frequency signals between 70 and 129  kHz , as opposed to pulse timing systems like Gee and LORAN . This made it much easier to design receivers using 1940s electronics, and operation

278-412: A = ∏ p p a p {\textstyle a=\prod _{p}p^{a_{p}}} and b = ∏ p p b p {\textstyle b=\prod _{p}p^{b_{p}}} , their least common multiple and greatest common divisor are given by the formulas and Since this gives In fact, every rational number can be written uniquely as

417-419: A Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram. Here is an example: sharing two "2"s and a "3" in common: This also works for the greatest common divisor (gcd), except that instead of multiplying all of

556-426: A and b divide m (that is, there exist elements x and y of R such that ax = m and by = m ). A least common multiple of a and b is a common multiple that is minimal, in the sense that for any other common multiple n of a and b , m divides  n . In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of

695-406: A and b is denoted as lcm( a , b ). Some older textbooks use [ a , b ]. Multiples of 4 are: Multiples of 6 are: Common multiples of 4 and 6 are the numbers that are in both lists: In this list, the smallest number is 12. Hence, the least common multiple is 12. When adding, subtracting, or comparing simple fractions , the least common multiple of the denominators (often called

834-416: A , since 0 is the only common multiple of a and 0. The least common multiple of the denominators of two fractions is the " lowest common denominator " (lcd), and can be used for adding, subtracting or comparing the fractions. The least common multiple of more than two integers a , b , c , . . . , usually denoted by lcm( a ,  b ,  c , . . .) , is defined as the smallest positive integer that

973-469: A blip on the display in the same fashion as a target on the radar, and the exact delay between the primary and secondary easily determined. Consider the same examples as our original absolute-timed cases. If the receiver is located on the midpoint of the baseline the two signals will be received at exactly the same time, so the delay between them will be zero. However, the delay will be zero not only if they are located 150 km from both stations and thus in

1112-433: A clock of the required accuracy was difficult enough to build in fixed form, let alone portable. A high-quality crystal oscillator , for instance, drifts about 1 to 2 seconds in a month, or 1.4 × 10  seconds per hour . This may sound small, but as light travels 300 million metres per second (190,000 miles per second), this represents a drift of 420 km each hour. Only a few hours of flight time would render such

1251-402: A complete set of 8 stations in 1983. Omega would also prove to be one of the shortest-lived systems, shutting down on 20 September 1997. Omega stations broadcast a continuous-wave signal in a specific time-slot. The atomic clocks also ensured that their signals were sent out with the right frequency and phase; unlike previous systems, Omega did not need to have a primary/secondary arrangement as

1390-721: A computing center to plot the location. These systems were used into World War II . The first hyperbolic radio navigation system was the World War II -era Gee , introduced by the Royal Air Force for use by RAF Bomber Command . This was followed by the Decca Navigator System in 1944 by the Royal Navy , along with LORAN by the US Navy for long-range navigation at sea. Post war examples including

1529-446: A conventional radio receiver hooked to an oscilloscope. Because a secondary could not instantaneously transmit its signal pulse on receipt of the primary signal, a fixed delay was built into the signal. No matter what delay is selected, there will be some locations where the signal from two secondary would be received at the same time, and thus make them difficult to see on the display. Some method of identifying one secondary from another

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1668-466: A development effort known as Project 3 that was similar to Gee. Only halting progress had been made by the time they were introduced to Gee, which was already entering production. Gee was immediately selected for the 8th Air Force and the Project 3 team turned their attention to other uses, eventually considering convoy navigation in particular. The new concept relied on the use of skywaves to allow

1807-517: A few meters on the baseline up to a nautical mile at the edge of coverage. At night, skywave errors were greater and, on receivers without multipulse capabilities, it was not unusual for the position to jump a lane, sometimes without the navigator knowing. Although in the days of differential GPS this range and accuracy may appear poor, in its day the Decca system was one of the few, if not the only, position fixing system available to many mariners. Since

1946-472: A hyperbolic curve centred on the baseline. Navigational charts can be drawn with the curves for selected delays, say every 0.1 ms. The operator can then determine which of these lines they lie on by measuring the delay and looking at the chart. A single measurement reveals a range of possible locations, not a single fix. The solution to this problem is to simply add another secondary station at some other location. In this case two delays will be measured, one

2085-558: A large number of unique signals for widespread coverage. The Decca Navigation System was originally developed in the US, but eventually deployed by the Decca Radio company in the UK and commonly referred to as a British system. Initially developed for the Royal Navy as an accurate adjunct to naval versions of Gee, Decca was first used on 5 June 1944 to guide minesweepers in preparation for

2224-514: A line in Krasnodar, Revda and Novosibirsk, the later being the primary station. In 1991 two additional stations came online at Khabarovsk and Seyda. The stations use frequencies between 11 and 14 kHz. Two complicating factors for satnav systems are: (1) the transmitter stations (satellites) are moving; and (2) GPS satellite transmissions are synchronized with UTC (with a published offset), thus providing precise time. Item (1) necessitates that

2363-437: A microprocessor and displayed a position in latitude and longitude. Multipulse provided an automatic method of lane and zone identification by using the same phase comparison techniques described above on lower frequency signals. The nominally continuous wave transmissions were in fact divided into a 20-second cycle, with each station in turn simultaneously transmitting all four Decca frequencies (5 f , 6 f , 8 f and 9 f ) in

2502-691: A more or less distinct location could be identified. The lanes were grouped into zones , with 18 green, 24 red, or 30 purple lanes in each zone. This meant that on the baseline (the straight line between the Master and its Slave) the zone width was the same for all three patterns of a given chain. Typical lane and zone widths on the baseline are shown in the table below (for chain 5B): The lanes were numbered 0 to 23 for red, 30 to 47 for green and 50 to 79 for purple. The zones were labelled A to J, repeating after J. A Decca position coordinate could thus be written: Red I 16.30; Green D 35.80. Later receivers incorporated

2641-516: A new system using a 14 kHz inter-signal spacing was selected. This led to the common 5, 6, 8 and 9 f frequencies, used throughout the life of the Decca system. 7 f was reserved for a Loran-C -like extension, but never developed. A follow-up test was carried out in the Irish Sea in January 1944 to test a wide variety of upgrades and production equipment. By this time the competing Gee system

2780-411: A phase-coherent relationship for a brief period of 0.45 seconds each cycle. This transmission, known as Multipulse, allowed the receiver to extract the 1 f frequency and so to identify the lane that the receiver was in (to a resolution of a zone). As well as transmitting the Decca frequencies of 5 f , 6 f , 8 f and 9 f , an 8.2 f signal, known as Orange, was also transmitted. The beat frequency between

2919-425: A product of prime numbers . The prime numbers can be considered as the atomic elements which, when combined, make up a composite number . For example: Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5. This fact can be used to find the lcm of a set of numbers. Example: lcm(8,9,21) Factor each number and express it as

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3058-431: A product of prime number powers . The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 2 , 3 , and 7 , respectively. Thus, This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization . The same method can also be illustrated with

3197-409: A second such curve. The two curves will normally intersect at two locations, so some other navigation system or a third measurement is needed to determine the exact location. Hyperbolic location systems were first used during World War I in acoustic location systems for locating enemy artillery . The sound of a shell being fired was received by several microphones, and the time of reception sent to

3336-499: A set distance from each other, say 300 km so that they are nearly exactly 1 ms apart at light speed . Both stations are equipped with identical transmitters set to broadcast a short pulse at a specific frequency. One of these stations, called the "secondary" is also equipped with a radio receiver . When this receiver hears the signal from the other station, referred to as the "primary", it triggers its own broadcast. The primary station can then broadcast any series of pulses, with

3475-411: A signal that is highly correlated in time. Typical systems broadcast either short pulses at the same time, or continual signals that are identical in phase . A receiver located at the midpoint between the two stations will receive the signals at the same time or have identical phase, but at any other location the signal from the closer station will be received first or have a different phase. Determining

3614-526: A significant effect on pulse timing, but much less so for phase changes. Decca thus found itself in great demand for helicopter use, where runway approach aids like ILS and VOR were not suitable for the small airfields and essentially random locations the aircraft were used. One serious disadvantage to Decca was that it was susceptible to noise, especially from lightning . This was not a serious concern for ships, who could afford to wait out storms, but made it unsuitable for long-range air navigation where time

3753-550: A single transmitter instead of Decca's four, and Decca's frequencies proved susceptible to interference from static due to lightning, while VOR's higher frequencies were not quite as sensitive. Decca continued to propose that Dectra be used for the long-range role. In 1967 they installed another transmitter in Iceland to provide ranging along the Scotland-Newfoundland track, with a second proposed to be installed on

3892-497: A solution to the long-distance problem. Additionally, as the Decca system provided an X and Y location, as opposed to the angle-and-range VOR/DME, Decca proposed offering it with their Decca Flight Log moving map display to further improve ease of navigation. In spite of these advantages, the RTCA ultimately chose VOR/DME for two primary reasons; VOR offered coverage over about the same range as Decca, about 200 miles, but did so with

4031-406: A system unusable, a situation that remained in force until the introduction of commercial atomic clocks in the 1960s. However, it is possible to accurately measure the difference between two signals. Much of the development of suitable equipment had been carried out between 1935 and 1938 as part of the efforts to deploy radar systems. The UK , in particular, had invested considerable effort in

4170-484: Is 0 , since gcd( a , 0) = | a | . However, if both a and b are 0 , these formulas would cause division by zero ; so, lcm(0, 0) = 0 must be considered as a special case. To return to the example above, There are fast algorithms , such as the Euclidean algorithm for computing the gcd that do not require the numbers to be factored . For very large integers, there are even faster algorithms for

4309-406: Is distributive ; that is, lcm distributes over gcd and gcd distributes over lcm: This identity is self-dual: Then where the absolute bars || denote the cardinality of a set. The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R . A common multiple of a and b is an element m of R such that both

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4448-408: Is a set of hyperbolic lines of position called a pattern . As there were three secondaries there were three patterns, also termed Red, Green and Purple. The patterns were drawn on nautical charts as a set of hyperbolic lines in the appropriate colour. Navigators determined their location by reading the phase difference from two or more of the patterns from the displays. They could then look at

4587-450: Is divisible by each of a , b , c , . . . A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well. The least common multiple of two integers

4726-509: Is needed to reduce this ambiguity. LORAN-C achieved this by sending unique details in the pulses so each station could be uniquely identified. The signal was started off when the primary broadcast a sequence of nine pulses, with the precise timing between each pulse being used to identify the station. Each of the Secondary stations then sent out its own signals, consisting of eight pulses in similar identifying patterns. The receivers could use

4865-569: Is the Soviet Union 's counterpart to LORAN-C, and operates on similar principles and the same frequency. It differs primarily in details of the pulse envelopes. There are five CHAYKA chains distributed around the former Soviet Union, each with a primary and between two and four secondaries. Alpha, more correctly known by its Soviet name, RSDN-20, is essentially a version of Omega deployed in the former Soviet Union starting in 1962. The initial system used only three transmitters running roughly in

5004-714: The Azores . They also installed Dectra receivers with Omnitrac computers and a lightweight version of the Flight Log on a number of commercial airliners, notably a BOAC Vickers VC10 . The Omnitrac could take inputs from Decca (and Dectra), Loran-C, VOR/DME, an air data computer and doppler radars and combine them all to produce a latitude/longitude output along with bearing, distance-to-go, bearing and an autopilot coupling. Their efforts to standardize this were eventually abandoned as inertial navigation systems began to be installed for these needs. A more accurate system named Hi-Fix

5143-466: The D-Day invasions. The system was developed post-war and competed with GEE and other systems for civilian use. A variety of reasons, notably its ease-of-use, kept it in widespread use into the 1990s, with a total 42 chains around the world. A number of stations were updated in the 1990s, but the widespread use of GPS led to Decca being turned off at midnight on 31 March 2000. Decca was based on comparing

5282-626: The General Post Office 's POPI system, was introduced in 1954, proposing 28 stations that provided worldwide coverage. The system was predicted to offer 10 miles (16,000 m) accuracy at 2,000 miles (3,200 km) range 95% of the time. Further development was ended in favour of the Dectra system. In the early 1960s the Radio Technical Commission for Aeronautics (RTCA), as part of a wider ICAO effort, began

5421-831: The International Field Year for the Great Lakes . The last Canadian chain shut down in 1986, after Loran-C became widespread. In the late 1950s an experimental Decca chain was set up in the United States, in the New York area, to be used for navigating the Vertol 107 helicopters of New York Airways . These helicopters were operating from the principal local airports— Idlewild Airport on Long Island, Newark Airport in New Jersey, LaGuardia Airport in

5560-559: The Kingston By-Pass area to verify receiver accuracy. In the car installation, it was found possible to navigate within an individual traffic lane. The company entertained high hopes that the system could be used in aircraft, to permit much more precise navigation in the critical airspace around airports and urban centres where traffic density was highest. After the end of World War II the Decca Navigator Co. Ltd.

5699-940: The Persian Gulf (1 chain with stations in Qatar and the United Arab Emirates and a second chain in the north of the Gulf with stations in Iran) and the Bahamas (1 chain). Four chains were planned for Nigeria but only two were built and these did not enter into public service. Two chains in Vietnam were used during the Vietnam War for helicopter navigation, with limited success. During the Cold War period, following WWII,

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5838-418: The least common multiple , lowest common multiple , or smallest common multiple of two integers a and b , usually denoted by lcm( a ,  b ) , is the smallest positive integer that is divisible by both a and b . Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm( a , 0) as 0 for all

5977-410: The lowest common denominator ) is used, because each of the fractions can be expressed as a fraction with this denominator. For example, where the denominator 42 was used, because it is the least common multiple of 21 and 6. Suppose there are two meshing gears in a machine , having m and n teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to

6116-490: The 1950s. A PLL produces a steady output signal with the same frequency and phase as an input signal, even if that input is periodic or poorly received. In this case the important feature was that the PLL allowed the re-construction of a continuous signal from a number of short pulses. A system using PLLs could receive a single pulsed signal, like Gee, and then re-construct a continuous tone for phase measurement, like Decca. Re-using

6255-543: The 8.0 f (Red) and 8.2 f (Orange) signals allowed a 0.2 f signal to be derived and so resulted in a hyperbolic pattern in which one cycle (360°) of phase difference equates to 5 zones. Assuming that one's position was known to this accuracy, this gave an effectively unique position. During daylight, ranges of around 400 nautical miles (740 km) could be obtained, reducing at night to 200 to 250 nautical miles (460 km), depending on propagation conditions. The accuracy depended on: By day these errors could range from

6394-555: The Borough of Queens, nearer to Manhattan, and a site on the top of the (then) PanAm Building on Park Avenue. Use of Decca was essential because its signals could be received down to sea level, were not subject to the line-of-sight limitations of VOR / DME and did not suffer the slant-range errors that create problems with VOR/DME close to the transmitters. The Decca installations in the New York Airways helicopters included

6533-629: The British Admiralty Signal Establishment (ASE) became interested in the system, which was then classified as Admiralty Outfit QM . The first marine trials were conducted between Anglesey and the Isle of Man , at frequencies of 305/610 kHz, on 16 September 1942. Further trials were conducted in the northern Irish Sea in April 1943 at 70/130 kHz. It was decided that the original frequencies were not ideal, and

6672-538: The Cyclan transmitters, the US Navy started experiments with such a system in the mid-1950s, and turned the system on permanently in 1957. Numerous chains followed, eventually providing around-the-world coverage near US allies and assets. Although less accurate that Decca, it offered the combination of reasonable accuracy and long ranges, a combination that obsoleted almost all other systems then in use and led to their gradual withdrawal. LORAN-C remained in service well into

6811-507: The Decca VLF signal discovered on BOAC, later British Airways, test flights to Moscow, was that the carrier switching could not be detected even though the carrier could be received with sufficient strength to provide navigation. Such testing, involving civilian aircraft, is quite common and may well not be in the knowledge of a pilot. The 'low frequency' signalling of the Decca system also permitted its use on submarines. One 'enhancement' of

6950-418: The Decca phase-comparison system. He imagined a system specifically for medium-accuracy global navigation, and thus selected the extremely low frequency of 10 kHz as the basis for the signal. However, the problem with phase ambiguity, as in the case of Decca, meant that the system was not practical at the time. The primary problem was synchronizing the stations. Gee and LORAN stations were close enough that

7089-429: The Decca system was to offer the potential of keying the signal, using Morse code, to signal the onset of nuclear war. This option was never taken up by the UK government. Messages were clandestinely sent, however, between Decca stations thereby bypassing international telephone calls, especially in non-UK chains. Hyperbolic navigation Such systems rely on the ability of two widely separated stations to broadcast

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7228-505: The Decca-like Cyclan and Navarho concepts. None of those proved to offer any real advance over Decca; they either offered marginally improved range, or better range but too little accuracy to be useful. Gee and LORAN-A became possible due to the development of the oscilloscope – before this the accurate measurement of time was not possible. LORAN-C became possible due to the development of the low-cost phase-locked loop (PLL) in

7367-484: The East Newfoundland and Scottish chains, which were equipped with larger antennas and high-power transmitters, broadcasting 20 times as much energy as normal chain stations. Given that the length of the chain baselines did not change, and were relatively short, at long distance the signal offered almost no accuracy. Instead, Dectra operated as a track system; aircraft would navigate by keeping themselves within

7506-591: The Germans and thus free of jamming. After the war, it came off the secret list and was commercially developed by the Decca Company and deployed around UK and later used in many areas around the world. At its peak there were about 180 transmitting stations using "chains" of three or four transmitters each to allow position fixing by plotting intersecting electronic lines. Decca's primary use was for ship navigation in coastal waters, offering much better accuracy than

7645-457: The Kingston by-pass. There was a Decca School, at Brixham , Devon , where employees were sent on courses from time to time. Racal , the UK weapons and communications company, acquired Decca in 1980. Merging Decca's radar assets with their own, Racal began selling off the other portions of the company, including avionics and Decca Navigator. A significant amount of income from the Decca system

7784-451: The LCM frequency. Early Decca receivers were fitted with three rotating Decometers that indicated the phase difference for each pattern. Each Decometer, which could be read to a resolution of a centilane, drove a second indicator that counted the number of lanes traversed – each 360 degrees of phase difference was one lane traversed. In this way, assuming the point of departure was known,

7923-590: The R.A.F. established a confidential chain in Germany. The Master station was in Bad Iburg near Osnabrück and there were two Slaves. The purpose of this chain was to provide accurate air navigation for the corridor between Western Germany and Berlin in the event that a mass evacuation of allied personnel may be required. In order to maintain secrecy, frequencies were changed at irregular intervals. The headquarters of Decca Navigator were at New Malden, Surrey, just off

8062-691: The aircraft from their control locations. An example of the problem, cited by experts, was the collision of a Douglas DC8 and a Lockheed Constellation over Staten Island, New York , that—according to some experts—could have been avoided if the aircraft had been Decca-equipped and could not only have determined their positions more precisely but would not have suffered from the rho-theta position errors inherent in VOR/DME. Other chains were established in Japan (6 chains); Namibia and South Africa (5 chains); India and Bangladesh (4 chains); North-West Australia (2 chains);

8201-473: The basic system. Another advantage is that it is easy to display the relative phase of two signals using simple electromechanical gauges. In contrast to Gee and LORAN, which required the use of oscilloscopes to measure the signal timings, Decca used a series of three mechanical pointers which were a fraction of the cost, took up less room, and allowed simultaneous examination of three signals. This made Decca both much less expensive and easier to use. Decca had

8340-450: The center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using lcm ⁡ ( m , n ) {\displaystyle \operatorname {lcm} (m,n)} . The first gear must complete lcm ⁡ ( m , n ) m {\displaystyle \operatorname {lcm} (m,n) \over m} rotations for

8479-412: The chart to find where the two closest charted hyperbolas crossed. The accuracy of this measurement was improved by choosing the set of two patterns that resulted in the lines crossing at as close to a right angle as possible. When two stations transmit at the phase-locked frequency , the difference in phase between the two signals is constant along a hyperbolic locus. However, if two stations transmit on

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8618-554: The clocks were accurate enough to trigger the signals without an external reference. To start the sequence, the station in Norway would initially broadcast on 10.2 kHz for 0.9 seconds, then turned off for 0.2 seconds, then broadcast on 13.6 kHz for 1.0 seconds, repeating this pattern. Each station broadcast a series of four such signals lasting about a second each, and then stood silent while other stations took their turn. At any given instant, three stations would be broadcasting at

8757-421: The common frequency was small compared with the distance between the Master and Slave stations there were many possible lines of position for a given phase difference, and so a unique position could not be arrived at by this method. Other receivers, typically for aeronautical applications, divided the transmitted frequencies down to the basic frequency (1 f ) for phase comparison, rather than multiplying them up to

8896-459: The competing LORAN system. Fishing vessels were major post-war users, but it was also used on some aircraft, including a very early (1949) application of moving map displays . The system was deployed extensively in the North Sea and was used by helicopters operating to oil platforms . The opening of the more accurate Loran-C system to civilian use in 1974 offered stiff competition, but Decca

9035-425: The current Decca counterparts. The "ap" versions directly output the longitude and latitude to two decimals (originally in datum ED50 only) instead of using the "deco meter" displays, offering accuracy better than ±9.3 m, much better than the Decca units. This also eliminated the need for the special charts printed with Decca lanes and zones. Decca sued ap for infringement and, in the ensuing court battle, Decca lost

9174-426: The current dial reading allowed the navigator to directly read the current delay and look it up on a chart, a far easier process than Gee or LORAN. It was so much easier to use that Decca later added an automatic charting feature that formed a moving map display . Later additions to the signal chain allowed the zone and lane to be calculated directly, eliminating the need for manually setting the lane counters and making

9313-509: The development of their Chain Home system. The radar display systems for Chain Home were based on oscilloscopes (or oscillographs as they were known at time) triggered to start their sweep when the broadcast signal was sent. Return signals were amplified and sent into the display, producing a "blip". By measuring the distance along the face of the oscilloscope of any blips, the time between broadcast and reception could be measured, thus revealing

9452-403: The difference between the primary and secondary "A", and the other between the primary and secondary "B". By looking up both delay curves on the chart, two intersections will be found, and one of these can be selected as the likely location of the receiver. This is a similar determination as in the case with direct timing/distance measurements, but the hyperbolic system consists of nothing more than

9591-427: The display until they could see the varying signal within the blips, and then use phase comparison to accurately line up the timing. At low frequencies and long ranges, it would be difficult to know whether you are looking at the current phase of the signals directly from the stations, or comparing one direct signal to one from a cycle ago, or perhaps one reflected off the ionosphere . Some form of secondary information

9730-426: The display. To distinguish the chains in these cases, a second "A" signal, the "A1" or "ghost A", was periodically keyed in, and the pattern of flashing on the display could be used to identify the chain. The operator initially tuned in their receiver to see a stream of pulses on the display, sometimes including those of other chains which were nearby in frequency. They would then tune a local oscillator that triggered

9869-812: The early 1980s. An experimental chain was installed with coverage of central London and receivers placed in London buses and other vehicles to demonstrate an early vehicle location and tracking system. Each vehicle would report its location automatically via a conventional VHF two-way radio link, the data added to a voice channel. Another application was developed by the Bendix Pacific division of Bendix Corporation, with offices in North Hollywood, California, but not deployed: PFNS—Personal Field Navigation System—that would enable individual soldiers to ascertain their geographic position, long before this capability

10008-415: The end. In every period, one of the two secondaries would respond, alternating their "B" and "C" signals. The resulting pattern was "ABD…ACD…ABD…" A wide-band receiver was used to tune in chain and the output sent to the operator's oscilloscope . As the chains were closely spaced in frequency to allow them to be received by a single tuner, this sometimes resulted in the signals from several chains appearing on

10147-509: The ground stations. During this time he demanded that an airborne version of the receivers be made, and should be interchangeable with Gee. The resulting system emerged as LORAN , for LOng RAnge Navigation, and the first chain of two stations went live in June 1942. LORAN became LORAN-A when the design of its replacement started, this was initially the LORAN-B concept, but eventually replaced by

10286-748: The idea too complicated and work ended in 1939. O’Brien's friend, Harvey F. Schwarz, was chief engineer of the Decca Record company in England. In 1939 O’Brien sent him details of the system so it could be put forward to the British military. Initially Robert Watson-Watt reviewed the system but he did not follow it up, deeming it too easily jammed (and likely due to the existing work on the Gee system, being carried out by Watt's group). However, in October 1941

10425-411: The inherent disadvantage that the signal could only vary by as much as 360 degrees, and that pattern repeated in a circle around the stations. That meant there were a large number of locations that met any particular phase measurement, a problem known as "phase ambiguity". Whereas Gee and LORAN fixed you in one of two locations, Decca fixed you to one in hundreds. As the ambiguous regions radiated away from

10564-614: The invasion would be focussed on the Calais area. 21 minesweepers and other vessels were fitted with Admiralty Outfit QM and, on 5 June 1944, 17 of these ships used it to accurately navigate across the English Channel and to sweep the minefields in the planned areas. The swept areas were marked with buoys in preparation for the Normandy Landings . After the initial ship tests, Decca conducted tests in cars, driving in

10703-563: The last Decca chain in operation. In the immediate post-war era, Decca began studying a long-range system like Decca, but using much lower frequencies to enable reception of skywaves at long distances. In February 1946 the company proposed a system with two main stations located at Shannon Airport in Ireland and Gander International Airport in Newfoundland (today part of Canada). Together, these stations would provide navigation over

10842-445: The late 1950s offered accuracy within a few miles, which was enough to determine the lane. Experiments on the concept continued throughout the 1950s and 60s, in parallel with Decca's development of their almost identical DELRAC system. It was not until the 1960s, when ice-breaking ballistic submarines became a main deterrent force, that there was a pressing need for such a system. The US Navy authorized full deployment in 1968, reaching

10981-414: The location of a receiver requires that the two synchronized stations be tuned in at the same time so the signals can be compared. This reveals a difference in time, corresponding to a relative distance closer to one station or the other. Plotting all the locations where this time difference may occur produces a hyperbolic line on a chart. To take a "fix", a second station pair is also turned in to produce

11120-464: The main great circle route between London and New York. A third station in Bermuda would provide general ranging information to measure progress along the main track. Work on this concept continued, and in 1951 a modified version was presented that offered navigation over very wide areas. This was known as Delrac , short for "Decca Long Range Area Cover". A further development, including features of

11259-610: The middle of the baseline, but also if they are located 200 km from both stations, and 300 km, and so forth. So in this case the receiver cannot determine their exact location, only that their location lies somewhere along a line perpendicular to the baseline. In the second example the receivers determined the timing to be 0.25 and 0.75 ms, so this would produce a measured delay of 0.5 ms. There are many locations that can produce this difference - 0.25 and 0.75 ms, but also 0.3 and 0.8 ms, 0.5 and 1 ms, etc. If all of these possible locations are plotted, they form

11398-702: The monopoly. That signalled the beginning of the end for the company. Income dwindled and eventually, the UK Ministry of Transport stepped in, having the lighthouse authorities take responsibility for operating the system in the early 1990s. A ruling from the European Union forced the UK government to withdraw funding. The general lighthouse authority ceased Decca transmissions at midnight on 31 March 2000. The Irish chain provided by Bórd Iascaigh Mhara continued transmitting until 19 May 2000. Japan continued operating their Hokkaidō chain until March 2001,

11537-483: The need for an accurate position is less when the vessel is further from land, the reduced accuracy at long ranges was not a great problem. In 1936 William J. O'Brien, an engineer, contracted tuberculosis that put his career on hold for a period of two years. During this period he had the idea of position fixing by means of phase comparison of continuous wave transmissions. This was not the first such system, but O'Brien apparently developed his version without knowledge of

11676-417: The need to re-synchronize. Much development was needed before these became practical, but these issues were mostly solved by the 1960s. This left one other problem; phase comparison systems of this sort are ambiguous and need some other system to resolve which lane they are within. This was also in the process of being solved through the development of inertial navigation systems (INS). Even early models from

11815-550: The numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12. According to the fundamental theorem of arithmetic , every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors: where the exponents n 2 , n 3 , ... are non-negative integers; for example, 84 = 2 3 5 7 11 13 ... Given two positive integers

11954-405: The operator to pick the pair of signals on the display that were sent from stations as close to right angles to the receiver as possible, further improving accuracy. Maximum accuracy was normally quoted as 200 yards, although that was subject to operational errors. In addition to greater accuracy and ease of use, Decca was also more suitable for use over land. Delays due to refraction can have

12093-399: The oscilloscope's trace so that it matched the clock at the primary station (which could, and did, change over time). Next, they would use a variable delay that was added to the local oscillators signal to move the entire display back or forth so one of the "A" pulses was at the very left side of the 'scope (the action is identical to the "horizontal hold" dial on an analog television). Finally,

12232-475: The others, and made several advancements in the art that would prove useful. He initially imagined the system being used for aircraft testing, specifically the accurate calculation of ground speed. Some experiments were carried out in California in 1938, selecting frequencies with harmonic "beats" that would allow for station identification in a network of transmitters. Both the U.S. Army and Navy considered

12371-507: The phases of continuous signals instead of the timing of their pulses. This was more accurate, as the phase of a pair of signals could be measured to within a few degrees, four degrees in the case of Decca. This greatly improved inherent accuracy allowed Decca to use much longer wavelengths than Gee or LORAN while still offering the same level of accuracy. The use of longer wavelengths gave better propagation than either Gee or LORAN, although ranges were generally limited to around 500 miles for

12510-458: The positive integers become a lattice , with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them: The following pairs of dual formulas are special cases of general lattice-theoretic identities. It can also be shown that this lattice

12649-520: The process of introducing a standard long-range radio navigation system for aviation use. Decca proposed a system that could offer both high accuracy at short ranges and transatlantic navigation with less accuracy, using a single receiver. The system was known as Dectra , short for "Decca Track". Unlike the Delrac system, Dectra was essentially the normal Decca Navigator system with the modification of several existing transmitter sites. These were located at

12788-401: The product of primes, if negative exponents are allowed. When this is done, the above formulas remain valid. For example: The positive integers may be partially ordered by divisibility: if a divides b (that is, if b is an integer multiple of a ) write a ≤ b (or equivalently, b ≥ a ). (Note that the usual magnitude-based definition of ≤ is not used here.) Under this ordering,

12927-426: The pulses to be received over very long ranges. This produced considerably more complex received signals than with Gee's line-of-sight system, and was more difficult to interpret. With that exception, however, the two systems were very similar in concept, and differed largely in frequency selections and the details of the pulse timing. Robert J. Dippy , inventor of Gee, moved to the US in mid-1942 to help with details of

13066-454: The range of the system. However, the accuracy of the fix is a function of the wavelength of the signal, which increases at lower frequencies - in other words, using a lower frequency would necessarily lower the accuracy of the system. Hoping for the best, early experiments with "LF Loran" instead proved that accuracy was far worse than predicted, and efforts along these lines were dropped. Several halting low-frequency efforts followed, including

13205-421: The range to the target. With very slight modification, the same display could be used to time the difference between two arbitrary signals. For navigational use, any number of identifying characteristics could be used to differentiate the primary from secondary signals. In this case, the portable receiver triggered its trace when it received the primary signal. As the signals from secondary arrived they would cause

13344-426: The realignment. By that time, the second gear will have made lcm ⁡ ( m , n ) n {\displaystyle \operatorname {lcm} (m,n) \over n} rotations. Suppose there are three planets revolving around a star which take l , m and n units of time, respectively, to complete their orbits. Assume that l , m and n are integers. Assuming the planets started moving around

13483-446: The receiver is 300 km from one station and 450 from the other. If one draws circles of 300 and 450 km radius around the two stations on a chart, the circles will intersect at two points. With any additional source of navigation information, one of these two intersections can be eliminated as a possibility, and thus reveal their exact location, or "fix". There is a serious practical problem with this approach - in order to measure

13622-415: The receiver moves to another location along the line, the timing of the signals would change. For instance, if they time the signals at 0.25 and 0.75 ms, they are 75 km from the closer station and 225 from the further. If the receiver moves to the side of the baseline, the delay from both stations will grow. At some point, for instance, they will measure a delay of 1 and 1.5 ms, which implies

13761-479: The same frequency, it is impossible for the receiver to separate them. Instead, each chain was allocated a nominal frequency, known as 1 f , and each station in the chain transmitted at a harmonic of this base frequency, as follows: The frequencies given are those for Chain 5B, known as the English Chain, but all chains used similar frequencies between 70 kHz and 129 kHz. Decca receivers multiplied

13900-446: The same time on different frequencies. Receivers would select the set of stations that were most suitable for their given location, and then wait for the signals for those stations to appear during the 10 second chain. Calculation of the fix then proceeded in precisely the same fashion as Decca, although the much lower operating frequency led to much less accuracy. Omega's charts quote accuracies of 2 to 4 nautical miles. CHAYKA

14039-408: The satellite coordinates be known as a function of time (included in the broadcast messages). Item (2) enables satnav systems to provide timing as well as position information, but requires a more complex solution algorithm. However, these are technical differences from earth-fixed hyperbolic systems, but not fundamental differences. Least common multiple In arithmetic and number theory ,

14178-447: The satellite navigation era, until GPS finally led to its shutdown on 8 February 2010. In basic operation, measurement was a two-step process. The signals would first be tuned in and lined up on the screen in a fashion similar to Gee, with the position of the blips being used to produce a rough estimate of the location. This measurement was accurate enough to place the vehicle within a specific lane. The operator would then greatly magnify

14317-440: The secondaries could trigger when they heard the signal from the primary, but for a global system, the stations might not be visible to each other, especially when the atmosphere was not cooperative. The solution to this was introduced in 1955 in the form of the caesium atomic clock . These offered enough accuracy that they could be synchronized at their factory, shipped to the transmitter locations, and left running for years without

14456-407: The secondaries would be positioned at the vertices of an equilateral triangle with the master at the centre. The baseline length, that is, the master–secondary distance, was typically 60–120 nautical miles (110–220 km). Each station transmitted a continuous wave signal; comparing the relative phases of the signals from the master and one of the secondaries produced a relative phase measure that

14595-442: The secondary hearing these and generating the same series after a 1 ms delay. Consider a portable receiver located on the midpoint of the line drawn between the two stations, known as the baseline . In this case, the signals will, necessarily, take 0.5 ms to reach the receiver. By measuring this time, they could determine that they are precisely 150 km from both stations, and thereby exactly determine their location. If

14734-515: The signal defined by a particular Decca lane. The main advantage of Dectra compared to other systems being proposed for the RTCA solution was that it could be used for both medium-range navigation over land, as well as long-range navigation over the Atlantic. In comparison, the VOR/DME system that ultimately won the competition offered navigation over perhaps a 200-mile radius, and could not offer

14873-479: The signal timings to select chains, identify secondaries, and reject signals bounced off the ionosphere. LORAN-C chains were organized into the Master station, M, and up to five Secondary stations, V, W, X, Y, Z. All were broadcast at 100 kHz, a much lower frequency than earlier systems. The result was a signal that offered a daytime ground wave range of 2,250 miles, nighttime ground wave of 1,650 miles and skywaves out to 3,000 miles. Timing accuracy

15012-400: The signals received from the Master and each Slave by different values to arrive at a common frequency ( least common multiple , LCM) for each Master/Slave pair, as follows: It was phase comparison at this common frequency that resulted in the hyperbolic lines of position. The interval between two adjacent hyperbolas on which the signals are in phase was called a lane . Since the wavelength of

15151-417: The speed of the trace across the display would be tuned so the D pulse was just visible on the right. The distance of the B or C pulse from the A pulse could now be measured with an attached scale. The resulting delays could then be looked up on a navigational chart. The display was relatively small, which limited resolution, and thus the determination of the delay. A measurement accuracy of 1 microsecond

15290-739: The standard aviation navigation systems, the Montreal system was moved eastward to cover the Anticosti Island area of the Gulf of St. Lawrence , and the western Newfoundland chain was later repositioned to better cover the Cabot Strait . A series of chains was also proposed to cover the Northwest Passage had oil tanker traffic used the area, but this never came to be. Another was briefly set up covering Lake Ontario in 1971 for

15429-754: The star after an initial linear alignment, all the planets attain a linear alignment again after lcm ⁡ ( l , m , n ) {\displaystyle \operatorname {lcm} (l,m,n)} units of time. At this time, the first, second and third planet will have completed lcm ⁡ ( l , m , n ) l {\displaystyle \operatorname {lcm} (l,m,n) \over l} , lcm ⁡ ( l , m , n ) m {\displaystyle \operatorname {lcm} (l,m,n) \over m} and lcm ⁡ ( l , m , n ) n {\displaystyle \operatorname {lcm} (l,m,n) \over n} orbits, respectively, around

15568-408: The star. There are several ways to compute least common multiples. The least common multiple can be computed from the greatest common divisor (gcd) with the formula To avoid introducing integers that are larger than the result, it is convenient to use the equivalent formulas where the result of the division is always an integer. These formulas are also valid when exactly one of a and b

15707-414: The stations and had a finite width, these became known as "lanes". Decca solved this problem through the use of an odometer -like display known as "decometers". Prior to leaving on a trip, the navigator would set the decometer's lane counter to their known position. As the craft moved the dial's hand would rotate, and increment or decrement the counter when it passed zero. The combination of this number and

15846-508: The system even easier to use. As each primary and secondary signal was sent at a different frequency, any number of delays could be measured at the same time; in practice, a single primary and three secondaries were used to produce three outputs. As each signal was sent on a different frequency, all three, known as "green", "red" and "purple", were simultaneously decoded and displayed on three decometers. The secondaries were physically distributed at 120 degree angles from each other, allowing

15985-694: The system might create a de facto standard (as it had become in other areas of the world). It also served to protect the marketing interests of the Hoffman Electronics division of ITT, a principal supplier of VOR/DME systems, that Decca might have been poised to usurp. This situation was exacerbated by the workload problems of the Air Traffic Controllers Association (ATCA), under its executive director Francis McDermott, whose members were forced to use radar data on aircraft positions, relaying those positions by radio to

16124-479: The system resulted in Decca receiving an order for 27 Admiralty Outfit QM receivers. The receiver consisted of an electronics unit with two dials and was known to its operators as the "Blue Gasmeter Job". A Decca chain was set up, consisting of a master station at Chichester and slaves at Swanage and Beachy Head . A fourth decoy transmitter was located in the Thames Estuary as part of the deception that

16263-403: The three involved operations (multiplication, gcd, and division); see Fast multiplication . As these algorithms are more efficient with factors of similar size, it is more efficient to divide the largest argument of the lcm by the gcd of the arguments, as in the example above. The unique factorization theorem indicates that every positive integer greater than 1 can be written in only one way as

16402-432: The time it took for the signals to reach the receiver, the receiver must know the precise time that the signal was originally sent. This is not possible in the case of uncooperative signal sources (like enemy artillery) and until the 2000s, widespread clock distribution was an unsolved problem until the widespread introduction of inexpensive GPS receivers. In the 1930s, such precise time measurements simply weren't possible;

16541-493: The unique Decca 'roller map' displays that enabled the pilot to see his or her position at a glance, a concept infeasible with VOR/DME. This chain installation was considered highly controversial at the time, for political reasons. This led to the U.S. Coast Guard, under instructions from the Treasury Department to which it reported, banning the use of Decca receivers in ships entering New York harbour for fear that

16680-404: The very long-range LORAN-C starting in 1957. LORAN eventually selected 1.950 MHz as its primary operating frequency. 7.5 MHz was selected for daytime use as an additional channel, but never used operationally. In comparison to Gee's 450 miles (720 km) range through air, LORAN had a range of about 1,500 miles (2,400 km) over water, and 600 miles (970 km) over land. Operation

16819-632: The well-known US Coast Guard Loran-C , the international Omega system, and the Soviet Alpha and CHAYKA . All of these systems saw use until their wholesale replacement by satellite navigation systems like the Global Positioning System (GPS) in the 1990s. In 2023 a prototype navigation system was tested based on detection of muon subatomic particles coming with cosmic rays , which would work underground and underwater. Consider two ground-based radio stations located at

16958-592: Was UK's Gee , first used experimentally by RAF Bomber Command in 1941. Gee was used both for bombing over Germany as well as navigation in the area of the UK, especially for landing at night. Several Gee chains were built in the UK, and after the war this expanded for four chains in the UK, two in France, and one in northern Germany. For a period following the formation of the International Civil Aviation Organization in 1946, Gee

17097-585: Was another early user, with branch offices set up in Toronto in 1953. The first chain was installed in southwest Newfoundland in 1956 as part of a joint Canada-US Navy surveying program. This led to commercial deployments the next year in Nova Scotia and an inland system for air traffic in the busy Quebec City - Montreal area. A fourth chain covering eastern Newfoundland was added in 1958. When meetings in Montreal in 1958 led to VOR and DME being selected as

17236-472: Was considered as the basis for a worldwide standard for navigation, but the VHF omnidirectional range (VOR) system was selected instead, and the last Gee chain was eventually shut down in 1970. Gee signals from a given chain were all sent on a single frequency. The primary station sent two signals, the "A" signal that marked the beginning of a timing period, and the "D" signal which was essentially two "A"s to mark

17375-405: Was developed using signalling in the 1.6 MHz range. It was used for specialised applications such as precision measurements involved with oil-drilling and by the Royal Navy for detailed mapping and surveying of coasts and harbours. The Hi-Fix equipment was leased for a period with temporary chains established to provide coverage of the area required, Hi-Fix was commercialised by Racal Survey in

17514-408: Was due to the receivers being leased to users, not sold outright. This guaranteed predictable annual income. When the patents on the original technology lapsed in the early 1980s, new receivers were quickly built by a number of companies. In particular, Aktieselskabet Dansk Philips ('Danish Philips', ap ) introduced receivers that could be purchased outright, and were much smaller and easier to use than

17653-525: Was estimated at 0.15 microseconds, offering accuracies on the order of 50 to 100 meters. In real-world use, the Coast Guard quoted absolute accuracy of 0.25 nautical miles, or better. One of the last hyperbolic navigation systems to enter operational use was one of the earliest to be developed; Omega traces its history to work by John Alvin Pierce in the 1940s, working on the same basic idea as

17792-728: Was formed (1945) and the system expanded rapidly, particularly in areas of British influence ; at its peak it was deployed in many of the world's major shipping areas. More than 15,000 receiving sets were in use aboard ships in 1970. There were 4 chains around England, 1 in Ireland and 2 in Scotland, 12 in Scandinavia (5 each in Norway and Sweden and 1 each in Denmark and Finland), a further 4 elsewhere in northern Europe and 2 in Spain. Canada

17931-411: Was generally similar to Gee, but only one of the secondary signals was displayed at a time. A fix required the operator to measure one delay, then the other, and then look up the resulting delays on the charts. This was a time-consuming process that could take several minutes, during which time the vehicle was moving. The accuracy was quoted as 1% of range. LORAN used two methods to identify a chain. One

18070-580: Was known to the Admiralty and the two systems were tested head-to-head under the code names QM and QH. QM was found to have better sea-level range and accuracy, which led to its adoption. A three-station trial was held in conjunction with a large-scale assault and landing exercise in the Moray Firth in February/March 1944. The success of the trials and the relative ease of use and accuracy of

18209-574: Was made possible by the satellite-based GPS (Global Positioning System). A further application of the Decca system was implemented by the U.S. Navy in the late 1950s and early 1960s for use in the Tongue of the Ocean/Eleuthera Sound area near The Bahamas, separating the islands of Andros and New Providence. The application was for sonar studies made possible by the unique characteristics of the ocean floor. An interesting characteristic of

18348-423: Was needed. Common methods included transmitting from the secondary only at certain times, using different frequencies, adjusting the envelope of the burst of signal, or broadcasting several bursts in a particular pattern. A set of stations, primary and secondaries, was known as a "chain". Similar methods are used to identify chains in the case where more than one chain may be received in a given location. Meint Harms

18487-418: Was of the essence. Several versions of Decca were introduced for this role, notably DECTRA and DELRAC, but these did not see widespread use. LORAN-A was designed to be quickly built on the basis of Gee, and selected its operating frequency based on the combination of the need for long over-water range and a selected minimum accuracy. Using much lower frequencies, in the kHz instead of MHz, would greatly extend

18626-405: Was presented on a clock-like display. The phase difference was caused by the relative distance between the stations as seen by the receiver. As the receiver moves these distances change and those changes are represented by the movement of the hands on the displays. If one selects a particular phase difference, say 30 degrees, and plots all the locations where that phase difference occurs, the result

18765-511: Was quoted, which resulted in an accuracy of the determination of the correct hyperbolic to about 150 meters, and when two such measurements were combined the resulting fix accuracy was around 210 m. At longer ranges, 350 miles for example, the error ellipse was about 6 miles by 1 mile. The maximum range was about 450 miles, although several long-range fixes were made under unusual circumstances. The US had also considered hyperbolic navigation as early as 1940, and started

18904-399: Was simplified by giving a direct readout of Decca coordinates without the complexity of a cathode-ray tube and highly skilled operator. The system was invented and developed by Decca in the UK. It was first deployed by the Royal Navy during World War II for the vital task of clearing the minefields to enable the D-Day landings. The Allied forces needed an accurate system not known to

19043-648: Was the first to have attempted the construction of a hyperbolic navigation systems, starting with musings on the topic in 1931 as part of his master's examination at Seefahrtschule Lübeck (Navigation College). After taking the position of Professor for Mathematics, Physics and Navigation at the Kaisertor in Lübeck, Harms tried to demonstrate hyperbolic navigation making use of simple transmitters and receivers. On 18 February 1932 he received Reichspatent-Nr. 546000 for his invention. The first operational hyperbolic navigation

19182-486: Was the operational frequency, with four "channels", as in Gee. The second was the rate at which the pulses were repeated, with "high", "low" and "slow" rates. This allowed for up to 12 chains in any given area. Additionally, the originally steady repetition of the pulses was later modified to create another eight unique patterns, allowing a total of 96 station pairs. Any given chain could use one or more pairs of stations, demanding

19321-534: Was well established by this time and continued operations to 2000. Decca Navigator, along with Loran and similar systems, was eventually replaced by the GPS in 2000, when that became available for public use. The Decca Navigator System consisted of individual groups of land-based radio transmitters organised into chains of three or four stations. Each chain consisted of a master station and three (occasionally two) secondary stations, termed Red, Green and Purple. Ideally,

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