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56-457: The Selfoss non-directional beacon (Ident: SE ) is located on the field. [REDACTED] Media related to Selfoss Airport at Wikimedia Commons This article about a European airport is a stub . You can help Misplaced Pages by expanding it . This article about an Icelandic building or structure is a stub . You can help Misplaced Pages by expanding it . This article about transport in Iceland

112-549: A mod n {\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}} , which is what we wanted to prove. It remains to be shown that d ≤ x ≤ d + n − 1 . Let k and r be the integers such that a − d = kn + r with 0 ≤ r ≤ n − 1 (see Euclidean division ). Then ⌊ a − d n ⌋ = k {\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k} , thus x =

168-408: A − n ⌊ a − d n ⌋ = a − n k = d + r {\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r} . Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1 . But we've seen that x = d + r , so we are done. The modulo with offset

224-402: A divmod functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture 's IDIV instruction, the C programming language's div() function, and Python 's divmod() function. Sometimes it is useful for the result of a modulo n to lie not between 0 and n − 1 , but between some number d and d + n − 1 . In that case, d

280-459: A . Many implementations use truncated division , for which the quotient is defined by where trunc {\displaystyle \operatorname {trunc} } is the integral part function ( rounding toward zero ), i.e. the truncation to zero significant digits. Thus according to equation ( 1 ), the remainder has the same sign as the dividend a so can take 2| n | − 1 values: Donald Knuth promotes floored division , for which

336-485: A fix of their geographic location on the surface of the Earth. Fixes are computed by extending lines through known navigational reference points until they intersect. For visual reference points, the angles of these lines can be determined by compass ; the bearings of NDB radio signals are found using radio direction finder (RDF) equipment. Plotting fixes in this manner allow crews to determine their position. This usage

392-493: A locator outer marker , or LOM); in Canada, low-powered NDBs have replaced marker beacons entirely. Marker beacons on ILS approaches are now being phased out worldwide with DME ranges or GPS signals used, instead, to delineate the different segments of the approach. German Navy U-boats during World War II were equipped with a Telefunken Spez 2113S homing beacon. This transmitter could operate on 100 kHz to 1500 kHz with

448-506: A mod d n is implemented in Mathematica as Mod[a, n, d]  . Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division: For both cases, the remainder can be calculated independently of

504-616: A mod n ) is the remainder of the Euclidean division of a by n , where a is the dividend and n is the divisor . For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers , many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n

560-474: A DME in a similar installation for the ILS as the outer marker, only in this case, they function as the inner marker. NDB owners are mostly governmental agencies and airport authorities. NDB radiators are vertically polarised. NDB antennas are usually too short for resonance at the frequency they operate – typically perhaps 20 metres length compared to a wavelength around 1000 m. Therefore, they require

616-488: A language where modulo has the sign of the dividend, that is incorrect, because when n (the dividend) is negative and odd, n mod 2 returns −1, and the function returns false. One correct alternative is to test that the remainder is not 0 (because remainder 0 is the same regardless of the signs): Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example,

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672-419: A modulo or remainder operator , such as a % n or a mod n . For environments lacking a similar function, any of the three definitions above can be used. When the result of a modulo operation has the sign of the dividend (truncated definition), it can lead to surprising mistakes. For example, to test if an integer is odd , one might be inclined to test if the remainder by 2 is equal to 1: But in

728-404: A power of 150 W. It was used to send the submarine's location to other submarines or aircraft, which were equipped with DF receivers and loop antennas. NDBs typically operate in the frequency range from 190 kHz to 535 kHz (although they are allocated frequencies from 190 to 1750 kHz) and transmit a carrier modulated by either 400 or 1020 Hz. NDBs can also be collocated with

784-433: A suitable matching network that may consist of an inductor and a capacitor to "tune" the antenna. Vertical NDB antennas may also have a T-antenna , nicknamed a top hat , which is an umbrella-like structure designed to add loading at the end and improve its radiating efficiency. Usually a ground plane or counterpoise is connected underneath the antenna. Apart from Morse code identity of either 400 Hz or 1020 Hz,

840-406: Is 0 to n − 1 ( a mod 1 is always 0; a mod 0 is undefined, being a division by zero ). When exactly one of a or n is negative, the basic definition breaks down, and programming languages differ in how these values are defined. In mathematics , the result of the modulo operation is an equivalence class , and any member of the class may be chosen as representative ; however,

896-566: Is a stub . You can help Misplaced Pages by expanding it . Non-directional beacon A non-directional beacon ( NDB ) or non-directional radio beacon is a radio beacon which does not include inherent directional information. Radio beacons are radio transmitters at a known location, used as an aviation or marine navigational aid . NDB are in contrast to directional radio beacons and other navigational aids, such as low-frequency radio range , VHF omnidirectional range (VOR) and tactical air navigation system (TACAN). NDB signals follow

952-400: Is a power of two and automatically implement them as expression & (constant-1) , allowing the programmer to write clearer code without compromising performance. This simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend (including C ), unless the dividend is of an unsigned integer type. This is because, if the dividend

1008-524: Is at the position. However, using a separate RBI and compass, this requires considerable mental calculation to determine the appropriate relative bearing. To simplify this task, a compass card driven by the aircraft's magnetic compass is added to the RBI to form a radio magnetic indicator (RMI). The ADF needle is then referenced immediately to the aircraft's magnetic heading, which reduces the necessity for mental calculation. Many RMIs used for aviation also allow

1064-454: Is called an offset and d = 1 is particularly common. There does not seem to be a standard notation for this operation, so let us tentatively use a mod d n . We thus have the following definition: x = a mod d n just in case d ≤ x ≤ d + n − 1 and x mod n = a mod n . Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod 0 n . The operation of modulo with offset

1120-486: Is equipped with ILS, it is called a precision approach runway. NDBs are most commonly used as markers or "locators" for an instrument landing system (ILS) approach or standard approach. NDBs may designate the starting area for an ILS approach or a path to follow for a standard terminal arrival route , or STAR. In the United States, an NDB is often combined with the outer marker beacon in the ILS approach (called

1176-434: Is important in situations where other navigational equipment, such as VORs with distance measuring equipment (DME), have failed. In marine navigation, NDBs may still be useful should Global Positioning System (GPS) reception fail. To determine the distance to an NDB station, the pilot uses this method: A runway equipped with NDB or VOR (or both) as the only navigation aid is called a non-precision approach runway; if it

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1232-415: Is less than 50 watts ; "medium" from 50 W to 2,000 W; and "high" at more than 2,000 W. There are four types of non-directional beacons in the aeronautical navigation service: The last two types are used in conjunction with an instrument landing system (ILS). NDB navigation consists of two parts — the automatic direction finder (ADF) equipment on the aircraft that detects an NDB's signal, and

1288-401: Is negative, the modulo will be negative, whereas expression & (constant-1) will always be positive. For these languages, the equivalence x % 2 == x < 0 ? x | ~(2 - 1) : x & (2 - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations. Optimizations for general constant-modulus operations also exist by calculating the division first using

1344-413: Is related to the floor function as follows: To see this, let x = a − n ⌊ a − d n ⌋ {\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor } . We first show that x mod n = a mod n . It is in general true that ( a + bn ) mod n = a mod n for all integers b ; thus, this is true also in

1400-435: Is the floor function ( rounding down ), and ⌈ ⌉ {\displaystyle \lceil \,\rceil } is the ceiling function ( rounding up ). Thus according to equation ( 1 ), the remainder is non negative : Common Lisp and IEEE 754 use rounded division , for which the quotient is defined by where round is the round function ( rounding half to even ). Thus according to equation ( 1 ),

1456-478: Is the case, Flight inspection organizations periodically check critical parameters with properly equipped aircraft to calibrate and certify NDB precision. The ICAO minimum accuracy for NDBs is ±5° Besides their use in aircraft navigation, NDBs are also popular with long-distance radio enthusiasts ( DXers ). Because NDBs are generally low-power (usually 25 watts, some can be up to 5 kW), they normally cannot be heard over long distances, but favorable conditions in

1512-654: The DXing hobby. In North America, the NDB band is from 190 to 435 kHz and from 510 to 530 kHz. In Europe, there is a longwave broadcasting band from 150 to 280 kHz, so the European NDB band is from 280 kHz to 530 kHz with a gap between 495 and 505 kHz because 500 kHz was the international maritime distress (emergency) frequency . The beacons that transmit between 510 kHz and 530 kHz can sometimes be heard on AM radios that can tune below

1568-466: The constant-divisor optimization . Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange . The properties involving multiplication, division, and exponentiation generally require that a and n are integers. @mod , @rem In addition, many computer systems provide

1624-436: The ionosphere can allow NDB signals to travel much farther than normal. Because of this, radio DXers interested in picking up distant signals enjoy listening to faraway NDBs. Also, since the band allocated to NDBs is free of broadcast stations and their associated interference, and because most NDBs do little more than transmit their Morse code callsign, they are very easy to identify, making NDB monitoring an active niche within

1680-544: The NDB and are charted in brown on sectional charts. Green and red airways are plotted east and west, while amber and blue airways are plotted north and south. As of September 2022, only one colored airway is left in the continental United States, located off the coast of North Carolina and is called G13 or Green 13. Alaska is the only other state in the United States to make use of the colored airway systems. Pilots follow these routes by tracking bearings across various navigation stations, and turning at some. While most airways in

1736-479: The NDB may broadcast: Navigation using an ADF to track NDBs is subject to several common effects: While pilots study these effects during initial training, trying to compensate for them in flight is very difficult; instead, pilots generally simply choose a heading that seems to average out any fluctuations. Radio-navigation aids must keep a certain degree of accuracy, given by international standards, Federal Aviation Administration (FAA), ICAO, etc.; to assure this

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1792-492: The NDB transmitter. The ADF can also locate transmitters in the standard AM medium wave broadcast band (530 kHz to 1700 kHz at 10 kHz increments in the Americas, 531 kHz to 1602 kHz at 9 kHz increments in the rest of the world). ADF equipment determines the direction or bearing to the NDB station relative to the aircraft by using a combination of directional and non-directional antennae to sense

1848-408: The NDB. Similarly, the aircraft will track directly away from the NDB if the needle is maintained on the 180 degree mark. With a crosswind, the needle must be maintained to the left or right of the 0 or 180 position by an amount corresponding to the drift due to the crosswind. The formula to determine the compass heading to an NDB station (in a no wind situation) is to take the relative bearing between

1904-674: The United States are based on VORs, NDB airways are common elsewhere, especially in the developing world and in lightly populated areas of developed countries, like the Canadian Arctic , since they can have a long range and are much less expensive to operate than VORs. All standard airways are plotted on aeronautical charts , such as the United States sectional charts , issued by the National Oceanic and Atmospheric Administration (NOAA). NDBs have long been used by aircraft navigators , and previously mariners, to help obtain

1960-423: The aircraft and the station, and add the magnetic heading of the aircraft; if the total is greater than 360 degrees, then 360 must be subtracted. This gives the magnetic bearing that must be flown: (RB + MH) mod 360 = MB. When tracking to or from an NDB, it is also usual that the aircraft track on a specific bearing. To do this it is necessary to correlate the RBI reading with the compass heading. Having determined

2016-710: The aviation industry. Airservices Australia began shutting down a number of ground-based navigation aids in May 2016, including NDBs, VORs and DMEs. In the United States as of 2017, there were more than 1,300 NDBs, of which fewer than 300 were owned by the Federal Government. The FAA had begun decommissioning stand-alone NDBs. As of April 2018, the FAA had disabled 23 ground-based navaids including NDBs, and plans to shut down more than 300 by 2025. The FAA has no sustaining or acquisition system for NDBs and plans to phase out

2072-403: The beginning of the medium wave (MW) broadcast band. However, reception of NDBs generally requires a radio receiver that can receive frequencies below 530 kHz. Often "general coverage" shortwave radios receive all frequencies from 150 kHz to 30 MHz, and so can tune to the frequencies of NDBs. Specialized techniques (receiver preselectors, noise limiters and filters) are required for

2128-485: The curvature of the Earth , so they can be received at much greater distances at lower altitudes, a major advantage over VOR. However, NDB signals are also affected more by atmospheric conditions, mountainous terrain, coastal refraction and electrical storms, particularly at long range. The system, developed by United States Army Air Corps (USAAC) Captain Albert Francis Hegenberger , was used to fly

2184-513: The device to display information from a second radio tuned to a VOR station; the aircraft can then fly directly between VOR stations (so-called "Victor" routes) while using the NDBs to triangulate their position along the radial, without the need for the VOR station to have a collocated distance measuring equipment (DME). This display, along with the omni bearing indicator (OBI) for VOR/ILS information,

2240-459: The direction in which the combined signal is strongest. This bearing may be displayed on a relative bearing indicator (RBI). This display looks like a compass card with a needle superimposed, except that the card is fixed with the 0 degree position corresponding to the centreline of the aircraft. In order to track toward an NDB (with no wind), the aircraft is flown so that the needle points to the 0 degree position. The aircraft will then fly directly to

2296-404: The drift, the aircraft must be flown so that the compass heading is the required bearing adjusted for drift at the same time as the RBI reading is 0 or 180 adjusted for drift. An NDB may also be used to locate a position along the aircraft's current track (such as a radial path from a second NDB or a VOR). When the needle reaches an RBI reading corresponding to the required bearing, then the aircraft

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2352-416: The existing NDBs through attrition, citing decreased pilot reliance on NDBs as more pilots use VOR and GPS navigation. Modulo operation In computing , the modulo operation returns the remainder or signed remainder of a division , after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n , a modulo n (often abbreviated as

2408-441: The identity ( − a ) / b = − ( a / b ) = a / ( − b ) {\displaystyle ({-a})/b={-(a/b)}=a/({-b})} . Some calculators have a mod() function button, and many programming languages have a similar function, expressed as mod( a , n ) , for example. Some also support expressions that use "%", "mod", or "Mod" as

2464-416: The language and the signs of a or n . Standard Pascal and ALGOL 68 , for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of n or a is negative (see the table under § In programming languages for details). a modulo 0 is undefined in most systems, although some do define it as

2520-441: The modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition): Examples: In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations. Compiler optimizations may recognize expressions of the form expression % constant where constant

2576-397: The particular case when b = − ⌊ a − d n ⌋ {\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor } ; but that means that x mod n = ( a − n ⌊ a − d n ⌋ ) mod n =

2632-471: The quotient q and the remainder r of a divided by n satisfy the following conditions: This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive; that choice determines which of the two consecutive quotients must be used to satisfy equation (1). In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on

2688-461: The quotient is defined by where ⌊ ⌋ {\displaystyle \lfloor \,\rfloor } is the floor function ( rounding down ). Thus according to equation ( 1 ), the remainder has the same sign as the divisor n : Raymond T. Boute promotes Euclidean division , for which the quotient is defined by where sgn is the sign function , ⌊ ⌋ {\displaystyle \lfloor \,\rfloor }

2744-555: The reception of very weak signals from remote beacons. The best time to hear NDBs that are very far away is the last three hours before sunrise. Reception of NDBs is also usually best during the fall and winter because during the spring and summer, there is more atmospheric noise on the LF and MF bands. As the adoption of satellite navigation systems such as GPS progressed, several countries began to decommission beacon installations such as NDBs and VOR. The policy has caused controversy in

2800-444: The remainder falls between − n 2 {\displaystyle -{\frac {n}{2}}} and n 2 {\displaystyle {\frac {n}{2}}} , and its sign depends on which side of zero it falls to be within these boundaries: Common Lisp also uses ceiling division , for which the quotient is defined by where ⌈⌉ is the ceiling function ( rounding up ). Thus according to equation ( 1 ),

2856-440: The remainder has the opposite sign of that of the divisor : If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree. If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree. If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree. If both the dividend and divisor are negative, then

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2912-420: The station that points in a specific direction, such as 270 degrees (due west). NDB bearings provide a charted, consistent method for defining paths aircraft can fly. In this fashion, NDBs can, like VORs, define airways in the sky. Aircraft follow these pre-defined routes to complete a flight plan . Airways are numbered and standardized on charts. Colored airways are used for low to medium frequency stations like

2968-407: The truncated and floored definitions agree. As described by Leijen, Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions. However, truncated division satisfies

3024-441: The usual representative is the least positive residue , the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division ). However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware . In nearly all computing systems,

3080-767: The world's first instrument approach on May 9, 1932. NDBs used for aviation are standardised by the International Civil Aviation Organization (ICAO) Annex 10 which specifies that NDBs be operated on a frequency between 190  kHz and 1750 kHz, although normally all NDBs in North America operate between 190 kHz and 535 kHz. Each NDB is identified by a one, two, or three-letter Morse code callsign. In Canada, privately owned NDB identifiers consist of one letter and one number. Non-directional beacons in North America are classified by power output: "low" power rating

3136-411: Was one of the primary radio navigation instruments prior to the introduction of the horizontal situation indicator (HSI) and subsequent digital displays used in glass cockpits . The principles of ADFs are not limited to NDB usage; such systems are also used to detect the locations of broadcast signals for many other purposes, such as finding emergency beacons. A bearing is a line passing through

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