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71-654: These properties are used for clock synchronization, demodulation , frequency synthesis , clock multipliers , and signal recovery from a noisy communication channel. Since 1969, a single integrated circuit can provide a complete PLL building block, and nowadays have output frequencies from a fraction of a hertz up to many gigahertz . Thus, PLLs are widely employed in radio , telecommunications , computers (e.g. to distribute precisely timed clock signals in microprocessors ), grid-tie inverters (electronic power converters used to integrate DC renewable resources and storage elements such as photovoltaics and batteries with

142-484: A carrier signal which is used to carry it through a telephone line , coaxial cable , or optical fiber . Demodulation was first used in radio receivers . In the wireless telegraphy radio systems used during the first 3 decades of radio (1884–1914) the transmitter did not communicate audio (sound) but transmitted information in the form of pulses of radio waves that represented text messages in Morse code . Therefore,

213-502: A Grade II* listed building and serves as a regional European Documentation Centre . The Wills Memorial Building was commissioned in 1912 by George Alfred Wills and Henry Herbert Wills , the magnates of the Bristol tobacco company W. D. & H. O. Wills , in honour of their father, Henry Overton Wills III, benefactor and first Chancellor of the university who donated £100,000 to the university. Sir George Oatley, who also worked on

284-434: A PLL is used to synthesize new frequencies which are a multiple of a reference frequency, with the same stability as the reference frequency. Other applications include: Some data streams, especially high-speed serial data streams (such as the raw stream of data from the magnetic head of a disk drive), are sent without an accompanying clock. The receiver generates a clock from an approximate frequency reference, and then uses

355-496: A PLL to phase-align it to the data stream's signal edges . This process is referred to as clock recovery . For this scheme to work, the data stream must have edges frequently-enough to correct any drift in the PLL's oscillator. Thus a line code with a hard upper bound on the maximum time between edges (e.g. 8b/10b encoding ) is typically used to encode the data. If a clock is sent in parallel with data, that clock can be used to sample

426-430: A complex process juggling the interactions of the two. The typical trade-off of increasing the bandwidth is degraded stability. Conversely, the tradeoff of extra damping for better stability is reduced speed and increased settling time. Often the phase-noise is also affected. All phase-locked loops employ an oscillator element with variable frequency capability. This can be an analog VCO either driven by analog circuitry in

497-644: A continuous or intermittent pilot signal. Wills Memorial Building 51°27′22″N 2°36′16″W  /  51.45611°N 2.60444°W  / 51.45611; -2.60444 The Wills Memorial Building (also known as the Wills Memorial Tower or simply the Wills Tower ) is a neo-Gothic building in Bristol, England, designed by Sir George Oatley and built as a memorial to Henry Overton Wills III by his sons George and Henry Wills . Begun in 1915 and not opened until 1925, it

568-421: A demodulator may represent sound (an analog audio signal ), images (an analog video signal ) or binary data (a digital signal ). These terms are traditionally used in connection with radio receivers , but many other systems use many kinds of demodulators. For example, in a modem , which is a contraction of the terms modulator /demodulator, a demodulator is used to extract a serial digital data stream from

639-538: A demodulator, if they pass the radio waves on nonlinearly . An AM signal encodes the information into the carrier wave by varying its amplitude in direct sympathy with the analogue signal to be sent. There are two methods used to demodulate AM signals : SSB is a form of AM in which the carrier is reduced or suppressed entirely , which require coherent demodulation. For further reading, see sideband . Frequency modulation (FM) has numerous advantages over AM such as better fidelity and noise immunity. However, it

710-486: A divider following a mixer allows the divider to operate at a much lower frequency than the VCO without a loss in loop gain. The equations governing a phase-locked loop with an analog multiplier as the phase detector and linear filter may be derived as follows. Let the input to the phase detector be f 1 ( θ 1 ( t ) ) {\displaystyle f_{1}(\theta _{1}(t))} and

781-520: A few seconds per hour compared to a reference atomic clock (such as the NIST-F2 ). That time difference becomes substantial over time. Instead, the owner can synchronize their mechanical clock (with varying degrees of accuracy) by phase-locking it to a reference clock. An inefficient synchronization method involves the owner resetting their clock to that more accurate clock's time every week. But, left alone, their clock will still continue to diverge from

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852-438: A large number of frequencies can be produced from a single stable, accurate, quartz crystal–controlled reference oscillator (which were expensive before commercial-scale hydrothermal synthesis provided cheap synthetic quartz). Some PLLs also include a divider between the reference clock and the reference input to the phase detector. If the divider in the feedback path divides by N {\displaystyle N} and

923-412: A little slower (if their clock was fast) or faster (if their clock was slow). If they don't overcompensate, then their clock will be more accurate than before. Over a series of such weekly adjustments, their clock's notion of a second would agree close enough with the reference clock, so they could be said to be locked both in frequency and phase. An early electromechanical version of a phase-locked loop

994-447: A lower-frequency reference clock (usually 50 or 100 MHz) up to the operating frequency of the processor. The multiplication factor can be quite large in cases where the operating frequency is multiple gigahertz and the reference crystal is just tens or hundreds of megahertz. All electronic systems emit some unwanted radio frequency energy. Various regulatory agencies (such as the FCC in

1065-533: A number of other buildings for the university, was chosen as architect and told to "build to last". He produced a design in the Perpendicular Gothic style, to evoke the famous university buildings of Oxford and Cambridge . The building was funded through the fortunes which the Wills family made through tobacco. Oatley later claimed that his inspiration for the building came from a dream where he saw

1136-545: A resonant circuit would soon oscillate at the same frequency. Automatic synchronization of electronic oscillators was described in 1923 by Edward Victor Appleton . In 1925, David Robertson , first professor of electrical engineering at the University ;of Bristol , introduced phase locking in his clock design to control the striking of the bell Great George in the new Wills Memorial Building . Robertson’s clock incorporated an electromechanical device that could vary

1207-569: A tower on a hill, with shields around it. Construction started in 1915 but was halted in 1916 due to the continuation of the First World War . Work resumed in 1919, and the Wills Memorial Building was finally opened on 9 June 1925 by King George V and Queen Mary , having cost a total of £501,566 19s 10d. The building was opened with a Royal Salute of 21 chimes from 'Great George', the nine-and-a-half ton bell within

1278-699: Is considered one of the last great Gothic buildings to be built in England. Standing near the top of Park Street on Queens Road, it is a landmark building of the University of Bristol that currently houses the School of Law and the Department of Earth Sciences , as well as the Law and Earth Sciences libraries. It is the fourth highest structure in Bristol , standing at 215 ft (65.5 m). Many regard

1349-402: Is frequently required to know when the loop is out of lock. The more complex digital phase-frequency detectors usually have an output that allows a reliable indication of an out of lock condition. An XOR gate is often used for digital PLLs as an effective yet simple phase detector. It can also be used in an analog sense with only slight modification to the circuitry. The block commonly called

1420-404: Is locked to the phase of the input. Analog phase locked loops are generally built with an analog phase detector, low-pass filter and VCO placed in a negative feedback configuration. A digital phase locked loop uses a digital phase detector; it may also have a divider in the feedback path or in the reference path, or both, in order to make the PLL's output signal frequency a rational multiple of

1491-437: Is much more complex to both modulate and demodulate a carrier wave with FM, and AM predates it by several decades. There are several common types of FM demodulators: QAM demodulation requires a coherent receiver. It uses two product detectors whose local reference signals are a quarter cycle apart in phase: one for the in-phase component and one for the quadrature component. The demodulator keeps these product detectors tuned to

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1562-519: Is over twice the height of the nearby Cabot Tower . It is 16 metres square and ornamented with heraldic shields. It is topped by an octagonal lantern which houses Great George (England's ninth-largest bell , weighing over 9.5 tonnes) which strikes on the hour. In addition to the Great Hall there is a General Library, Reception Room and Council Chamber, and a further 50 rooms including some teaching space such as seminar rooms and lecture theatres. In

1633-414: Is proportional to their phase difference. The error signal is then low-pass filtered and used to drive a VCO which creates an output phase. The output is fed through an optional divider back to the input of the system, producing a negative feedback loop . If the output phase drifts, the error signal will increase, driving the VCO phase in the opposite direction so as to reduce the error. Thus the output phase

1704-431: Is that the phase and frequency of the generated clock be unaffected by rapid changes in the voltages of the power and ground supply lines, as well as the voltage of the substrate on which the PLL circuits are fabricated. This is called substrate and supply noise rejection . The higher the noise rejection, the better. To further improve the phase noise of the output, an injection locked oscillator can be employed following

1775-467: The PID controller are used to design this function. The second common consideration is limiting the amount of reference frequency energy (ripple) appearing at the phase detector output that is then applied to the VCO control input. This frequency modulates the VCO and produces FM sidebands commonly called "reference spurs". The design of this block can be dominated by either of these considerations, or can be

1846-517: The transistor circuits for phase/frequency detectors not seen until the 1970s.  Robertson’s work predated research towards what was later named the phase-lock loop in 1932, when British researchers developed an alternative to Edwin Armstrong 's superheterodyne receiver , the Homodyne or direct-conversion receiver . In the homodyne or synchrodyne system, a local oscillator was tuned to

1917-505: The Entrance Hall are two ceremonial staircases. The building is also used as a conference venue. In 2006, cleaning work began on the Wills Memorial Building costing £750,000. Cleaning on the building revealed the engraving "IO TRIVMPHE" intended as a tribute to the architect of the building Sir George Oatley. The engraving had remained hidden for over 80 years and recognises the role of Sir Isambard Owen (then Vice-Chancellor) in

1988-552: The NE565 using bipolar transistors , that were complete phase-locked loop systems on a chip, and applications for the technique multiplied. A few years later, RCA introduced the CD4046 Micropower Phase-Locked Loop using CMOS , which also became a popular integrated circuit building block. Phase-locked loop mechanisms may be implemented as either analog or digital circuits. Both implementations use

2059-431: The PLL loop filter (usually a low-pass filter) generally has two distinct functions. The primary function is to determine loop dynamics, also called stability . This is how the loop responds to disturbances, such as changes in the reference frequency, changes of the feedback divider, or at startup. Common considerations are the range over which the loop can achieve lock (pull-in range, lock range or capture range), how fast

2130-463: The United States) put limits on the emitted energy and any interference caused by it. The emitted noise generally appears at sharp spectral peaks (usually at the operating frequency of the device, and a few harmonics). A system designer can use a spread-spectrum PLL to reduce interference with high-Q receivers by spreading the energy over a larger portion of the spectrum. For example, by changing

2201-419: The VCO in the PLL. In digital wireless communication systems (GSM, CDMA etc.), PLLs are used to provide the local oscillator up-conversion during transmission and down-conversion during reception. In most cellular handsets this function has been largely integrated into a single integrated circuit to reduce the cost and size of the handset. However, due to the high performance required of base station terminals,

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2272-419: The VCO output to the N th harmonic of the reference signal. Instead of a simple phase detector, the design uses a harmonic mixer (sampling mixer). The harmonic mixer turns the reference signal into an impulse train that is rich in harmonics. The VCO output is coarse tuned to be close to one of those harmonics. Consequently, the desired harmonic mixer output (representing the difference between the N harmonic and

2343-435: The VCO output) falls within the loop filter passband. It should also be noted that the feedback is not limited to a frequency divider. This element can be other elements such as a frequency multiplier, or a mixer. The multiplier will make the VCO output a sub-multiple (rather than a multiple) of the reference frequency. A mixer can translate the VCO frequency by a fixed offset. It may also be a combination of these. For example,

2414-409: The VCO output, a PLL AM detector exhibits a high degree of selectivity and noise immunity which is not possible with conventional peak type AM demodulators. However, the loop may lose lock where AM signals have 100% modulation depth. One desirable property of all PLLs is that the reference and feedback clock edges be brought into very close alignment. The average difference in time between the phases of

2485-464: The VCO's output signal with the phase of periodic input reference signal V i and outputs a voltage (stabilized by the filter) to adjust the oscillator's frequency to match the phase of V o to the phase of V i . Phase can be proportional to time , so a phase difference can correspond to a time difference. Left alone, different clocks will mark time at slightly different rates. A mechanical clock , for example, might be fast or slow by

2556-486: The VCO. The PD output voltage is used to control the VCO such that the phase difference between the two inputs is held constant, making it a negative feedback system. Different types of phase detectors have different performance characteristics. For instance, the frequency mixer produces harmonics that adds complexity in applications where spectral purity of the VCO signal is important. The resulting unwanted (spurious) sidebands, also called " reference spurs " can dominate

2627-569: The adjoining wing was enlarged by Ralph Brentnall. The alleged connection of the Wills family to historical slavery via the U.S. tobacco industry attracted controversy in the 2010s, with some students petitioning the University of Bristol to rename the building in March 2017. In 2018, the university published a response and consultation on how to address the issue, including (amongst others) the suggestion of renaming university buildings and changing

2698-813: The base-band signal such as amplitude, frequency or phase are transmitted in the carrier signal. For example, for a signal modulated with a linear modulation like AM ( amplitude modulation ), we can use a synchronous detector . On the other hand, for a signal modulated with an angular modulation, we must use an FM ( frequency modulation ) demodulator or a PM ( phase modulation ) demodulator. Different kinds of circuits perform these functions. Many techniques such as carrier recovery , clock recovery , bit slip , frame synchronization , rake receiver , pulse compression , Received Signal Strength Indication , error detection and correction , etc., are only performed by demodulators, although any specific demodulator may perform only some or none of these techniques. Many things can act as

2769-475: The battery discharges into the grid. The block diagram shown in the figure shows an input signal, F I , which is used to generate an output, F O . The input signal is often called the reference signal (also abbreviated F REF ). At the input, a phase detector (shown as the Phase frequency detector and Charge pump blocks in the figure) compares two input signals, producing an error signal which

2840-527: The building as synonymous with the University of Bristol. It is the centrepiece building of the university precinct and is used by the university for degree ceremonies and examinations, which take place in the Great Hall. Architecture commentator Nikolaus Pevsner described it as: " a tour de force in Gothic Revival, so convinced, so vast, and so competent that one cannot help feeling respect for it. " It has been designated by English Heritage as

2911-518: The case of an APLL or driven digitally through the use of a digital-to-analog converter as is the case for some DPLL designs. Pure digital oscillators such as a numerically controlled oscillator are used in ADPLLs. PLLs may include a divider between the oscillator and the feedback input to the phase detector to produce a frequency synthesizer . A programmable divider is particularly useful in radio transmitter applications and for computer clocking, since

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2982-455: The clock at each data flip-flop is phase-matched to the received clock. In that type of application, a special form of a PLL called a delay-locked loop (DLL) is frequently used. Many electronic systems include processors of various sorts that operate at hundreds of megahertz to gigahertz, well above the practical frequencies of crystal oscillators . Typically, the clocks supplied to these processors come from clock generator PLLs, which multiply

3053-660: The coat of arms. On 28 February 2022 the Great Hall of the Wills Memorial Building was occupied by a group of students. The student occupiers barricaded themselves in the Great Hall in solidarity with ongoing industrial action by the University and Colleges Union which represents many teaching and professional services staff at the University. The building's dominant feature is the Wills Tower, built in reinforced concrete faced with Bath and Clipsham stone , with carving designed in collaboration with Jean Hahn of King's Heath Guild, Birmingham. At 215 ft (65.5 m) high it

3124-407: The data. Because the clock must be received and amplified before it can drive the flip-flops which sample the data, there will be a finite, and process-, temperature-, and voltage-dependent delay between the detected clock edge and the received data window. This delay limits the frequency at which data can be sent. One way of eliminating this delay is to include a deskew PLL on the receive side, so that

3195-408: The desired input frequency and multiplied with the input signal. The resulting output signal included the original modulation information. The intent was to develop an alternative receiver circuit that required fewer tuned circuits than the superheterodyne receiver. Since the local oscillator would rapidly drift in frequency, an automatic correction signal was applied to the oscillator, maintaining it in

3266-405: The filter requirements and reduce the capture range well below or increase the lock time beyond the requirements. In these applications the more complex digital phase detectors are used which do not have as severe a reference spur component on their output. Also, when in lock, the steady-state phase difference at the inputs using this type of phase detector is near 90 degrees. In PLL applications it

3337-401: The grid's voltage phase angle, which is measured using a PLL. In photovoltaic applications, the more the sine wave produced leads the grid voltage wave, the more power is injected into the grid. For battery applications, the more the sine wave produced lags the grid voltage wave, the more the battery charges from the grid, and the more the sine wave produced leads the grid voltage wave, the more

3408-400: The lock-in range of allowable inputs. Depending on the application, either the output of the controlled oscillator, or the control signal to the oscillator, provides the useful output of the PLL system. A phase detector (PD) generates a voltage, which represents the phase difference between two signals. In a PLL, the two inputs of the phase detector are the reference input and the feedback from

3479-423: The loop achieves lock (lock time, lock-up time or settling time ) and damping behavior. Depending on the application, this may require one or more of the following: a simple proportion (gain or attenuation), an integral (low-pass filter) and/or derivative ( high-pass filter ). Loop parameters commonly examined for this are the loop's gain margin and phase margin . Common concepts in control theory including

3550-486: The modulating audio signal, so it can drive an earphone or an audio amplifier. Fessendon invented the first AM demodulator in 1904 called the electrolytic detector , consisting of a short needle dipping into a cup of dilute acid. The same year John Ambrose Fleming invented the Fleming valve or thermionic diode which could also rectify an AM signal. There are several ways of demodulation depending on how parameters of

3621-533: The octagonal belfry of the tower, which is tolled on the death of a monarch or chancellor. Oatley received a knighthood that year in recognition of his work on the building. In 1940, during the Bristol Blitz of the Second World War , the Great Hall with its hammerbeam roof was badly damaged by a German bomb-blast. It was restored in the 1960s to Oatley's original design; at the same time

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3692-419: The operating frequency up and down by a small amount (about 1%), a device running at hundreds of megahertz can spread its interference evenly over a few megahertz of spectrum, which drastically reduces the amount of noise seen on broadcast FM radio channels, which have a bandwidth of several tens of kilohertz. Typically, the reference clock enters the chip and drives a phase locked loop (PLL), which then drives

3763-425: The oscillator's frequency control circuit. Some technologies are known to perform better than others in this regard. The best digital PLLs are constructed with emitter-coupled logic ( ECL ) elements, at the expense of high power consumption. To keep phase noise low in PLL circuits, it is best to avoid saturating logic families such as transistor-transistor logic ( TTL ) or CMOS . Another desirable property of all PLLs

3834-607: The output of the VCO is f 2 ( θ 2 ( t ) ) {\displaystyle f_{2}(\theta _{2}(t))} with phases θ 1 ( t ) {\displaystyle \theta _{1}(t)} and θ 2 ( t ) {\displaystyle \theta _{2}(t)} . The functions f 1 ( θ ) {\displaystyle f_{1}(\theta )} and f 2 ( θ ) {\displaystyle f_{2}(\theta )} describe waveforms of signals. Then

3905-524: The output of the phase detector φ ( t ) {\displaystyle \varphi (t)} is given by Demodulation Demodulation is extracting the original information-bearing signal from a carrier wave . A demodulator is an electronic circuit (or computer program in a software-defined radio ) that is used to recover the information content from the modulated carrier wave. There are many types of modulation so there are many types of demodulators. The signal output from

3976-424: The phase and frequency of the incoming AM signal's carrier. The recovered phase at the VCO differs from the carrier's by 90°, so it is shifted in phase to match, and then fed to a multiplier. The output of the multiplier contains both the sum and the difference frequency signals, and the demodulated output is obtained by low-pass filtering . Since the PLL responds only to the carrier frequencies which are very close to

4047-406: The power grid), and other electronic applications. A simple analog PLL is an electronic circuit consisting of a variable frequency oscillator and a phase detector in a feedback loop (Figure 1). The oscillator generates a periodic signal V o with frequency proportional to an applied voltage, hence the term voltage-controlled oscillator (VCO). The phase detector compares the phase of

4118-400: The rate of oscillation of the pendulum, and derived correction signals from a circuit that compared the pendulum phase with that of an incoming telegraph pulse from Greenwich Observatory every morning at 10:00 GMT. Including equivalents of every element of a modern electronic PLL, Robertson’s system was notably ahead of its time in that its phase detector was a relay logic implementation of

4189-457: The receiver merely had to detect the presence or absence of the radio signal, and produce a click sound. The device that did this was called a detector . The first detectors were coherers , simple devices that acted as a switch. The term detector stuck, was used for other types of demodulators and continues to be used to the present day for a demodulator in a radio receiver. The first type of modulation used to transmit sound over radio waves

4260-495: The reference clock at the same few seconds per hour rate. A more efficient synchronization method (analogous to the simple PLL in Figure 1) utilizes the fast-slow timing adjust control (analogous to how the VCO's frequency can be adjusted) available on some clocks. Analogously to the phase comparator, the owner could notice their clock's misalignment and turn its timing adjust a small proportional amount to make their clock's frequency

4331-460: The reference clock may not actually be a pure clock at all, but rather a data stream with enough transitions that the PLL is able to recover a regular clock from that stream. Sometimes the reference clock is the same frequency as the clock driven through the clock distribution, other times the distributed clock may be some rational multiple of the reference. A PLL may be used to synchronously demodulate amplitude modulated (AM) signals. The PLL recovers

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4402-403: The reference frequency. A non-integer multiple of the reference frequency can also be created by replacing the simple divide-by- N counter in the feedback path with a programmable pulse swallowing counter . This technique is usually referred to as a fractional-N synthesizer or fractional-N PLL. The oscillator generates a periodic output signal. Assume that initially the oscillator is at nearly

4473-446: The reference input divider divides by M {\displaystyle M} , it allows the PLL to multiply the reference frequency by N / M {\displaystyle N/M} . It might seem simpler to just feed the PLL a lower frequency, but in some cases the reference frequency may be constrained by other issues, and then the reference divider is useful. Frequency multiplication can also be attained by locking

4544-661: The same basic structure. Analog PLL circuits include four basic elements: There are several variations of PLLs. Some terms that are used are "analog phase-locked loop" (APLL), also referred to as a linear phase-locked loop" (LPLL), "digital phase-locked loop" (DPLL), "all digital phase-locked loop" (ADPLL), and "software phase-locked loop" (SPLL). Phase-locked loops are widely used for synchronization purposes; in space communications for coherent demodulation and threshold extension , bit synchronization , and symbol synchronization. Phase-locked loops can also be used to demodulate frequency-modulated signals. In radio transmitters,

4615-488: The same frequency as the reference signal. If the phase from the oscillator falls behind that of the reference, the phase detector changes the control voltage of the oscillator so that it speeds up. Likewise, if the phase creeps ahead of the reference, the phase detector changes the control voltage to slow down the oscillator. Since initially the oscillator may be far from the reference frequency, practical phase detectors may also respond to frequency differences, so as to increase

4686-493: The same phase and frequency of the desired signal. The technique was described in 1932, in a paper by Henri de Bellescize, in the French journal L'Onde Électrique . In analog television receivers since at least the late 1930s, phase-locked-loop horizontal and vertical sweep circuits are locked to synchronization pulses in the broadcast signal. In 1969, Signetics introduced a line of low-cost monolithic integrated circuits like

4757-538: The system's clock distribution. The clock distribution is usually balanced so that the clock arrives at every endpoint simultaneously. One of those endpoints is the PLL's feedback input. The function of the PLL is to compare the distributed clock to the incoming reference clock, and vary the phase and frequency of its output until the reference and feedback clocks are phase and frequency matched. PLLs are ubiquitous—they tune clocks in systems several feet across, as well as clocks in small portions of individual chips. Sometimes

4828-480: The transmission and reception circuits are built with discrete components to achieve the levels of performance required. GSM local oscillator modules are typically built with a frequency synthesizer integrated circuit and discrete resonator VCOs. Grid-tie inverters based on voltage source inverters source or sink real power into the AC electric grid as a function of the phase angle of the voltage they generate relative to

4899-470: The two signals when the PLL has achieved lock is called the static phase offset (also called the steady-state phase error ). The variance between these phases is called tracking jitter . Ideally, the static phase offset should be zero, and the tracking jitter should be as low as possible. Phase noise is another type of jitter observed in PLLs, and is caused by the oscillator itself and by elements used in

4970-410: Was amplitude modulation (AM), invented by Reginald Fessenden around 1900. An AM radio signal can be demodulated by rectifying it to remove one side of the carrier, and then filtering to remove the radio-frequency component, leaving only the modulating audio component. This is equivalent to peak detection with a suitably long time constant. The amplitude of the recovered audio frequency varies with

5041-609: Was used in 1921 in the Shortt-Synchronome clock . Spontaneous synchronization of weakly coupled pendulum clocks was noted by the Dutch physicist Christiaan Huygens as early as 1673. Around the turn of the 19th century, Lord Rayleigh observed synchronization of weakly coupled organ pipes and tuning forks . In 1919, W. H. Eccles and J. H. Vincent found that two electronic oscillators that had been tuned to oscillate at slightly different frequencies but that were coupled to

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