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A '''numerically controlled oscillator''' ('''NCO''') is a digital [[signal generator]] which creates a [[Synchronous circuit|synchronous]] (i.e. clocked), discrete-time, discrete-valued representation of a [[waveform]], usually [[sinusoidal]].<ref name="IEEE">
Numerically controlled oscillators offer several advantages over other types of oscillators in terms of agility, accuracy, stability and reliability.<ref name="latticeSC">{{ cite web | title = Numerically Controlled Oscillator | url = http://www.latticesemi.com/products/intellectualproperty/ipcores/numericallycontrolledosci/index.cfm | publisher = Lattice Semiconductor Corporation | year = 2009 }}</ref> NCOs are used in many communications systems including digital up/down converters used in 3G wireless and software radio systems, digital PLLs, radar systems, drivers for optical or acoustic transmissions, and multilevel [[Frequency-shift keying|FSK]]/[[Phase-shift keying|PSK]] modulators/demodulators.<ref name="latticeSC"/>
==Operation==
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[[Image:Generic NCO.png|frame|Figure 1: Numerically controlled oscillator with optional quadrature output]]
When clocked, the phase accumulator (PA) creates a [[modulo operation|modulo]]-2<sup>N</sup> [[sawtooth wave]]form which is then converted by the phase-to-amplitude converter (PAC) to a sampled sinusoid, where N is the number of bits carried in the phase accumulator. N sets the NCO frequency resolution and is normally much larger than the number of bits defining the memory space of the PAC [[look-up table]]. If the PAC capacity is 2<sup>M</sup>, the PA output word must be truncated to M bits as shown in Figure 1. However, the truncated bits can be used for interpolation. The truncation of the phase output word does not affect the frequency accuracy but produces a time-varying periodic phase error which is a primary source of spurious products. Another spurious product generation mechanism is finite word length effects of the PAC output (amplitude) word.<ref name="kroupa">{{ cite book | author = Kroupa,
The frequency accuracy relative to the clock frequency is limited only by the precision of the arithmetic used to compute the phase.<ref name="kroupa"/> NCOs are phase- and frequency-agile, and can be trivially modified to produce a [[phase modulation|phase-modulated]] or [[frequency modulation|frequency-modulated]] output by summation at the appropriate node, or provide [[quadrature phase|quadrature]] outputs as shown in the figure.
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:<math>\mbox{GRR}=\frac{2^N}{\mbox{GCD}(\Delta F,2^N)}</math>
where GCD is the [[greatest common divisor]] function. The GRR represents the true periodicity for a given <math>\Delta F</math> which for a high resolution NCO can be very long.<ref name="Grzeg"/> Usually we are more interested in the ''operating frequency'' determined by the average overflow rate, given by<ref name="ADI">{{ citation | url = http://www.analog.com/library/analogdialogue/archives/38-08/dds.html | title = All About Direct Digital Synthesis | publisher = Analog Devices }}</ref>
:<math>F_{out} = \frac{\Delta F}{2^N}F_{clock} </math> (1)
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==Phase-to-amplitude converter==
The phase-amplitude converter creates the sample-___domain waveform from the truncated phase output word received from the PA. The PAC can be a simple [[read only memory]] containing 2<sup>M</sup> contiguous samples of the desired output waveform which typically is a sinusoid. Often though, various tricks are employed to reduce the amount of memory required. This include various trigonometric expansions,<ref>{{ Citation | inventor-last = Miller | inventor-first = B. M. | issue-date = October 14, 2008 | title = Numerically controlled oscillator and method of operation | country-code = US | patent-number = 7437391 }}</ref> trigonometric approximations<ref name="Grzeg">{{ cite web | first1 = G. | last1 = Popek | first2 = Marian | last2 = Kampik | title = Low-Spur Numerically Controlled Oscillator Using Taylor Series Approximation | url = http://mechatronika.polsl.pl/owd/pdf2009/030.pdf | date = October 2009 | work = XI International PhD Workshop OWD 2009 | publisher = Silesian University of Technology | ___location = Gliwice, Poland }}</ref> and methods which take advantage of the quadrature symmetry exhibited by sinusoids.<ref>{{ Citation | inventor-last = McCallister | inventor-first = R. D. | inventor2-last = Shearer | inventor2-first = Daniel III | publication-date = 12/04/1984 | title = Numerically controlled oscillator using quadrant replication and function decomposition | country-code = US | patent-number = 4486846 }}</ref> Alternatively, the PAC may consist of [[random access memory]] which can be filled as desired to create an [[arbitrary waveform generator]].
==Spurious products==
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===Amplitude truncation spurs===
Another source of spurious products is the amplitude [[Quantization (signal processing)|quantization]] of the sampled waveform contained in the PAC look up table(s). If the number of DAC bits is P, the
AM spur level is approximately equal to −6.02 P − 1.76 [[dBc]].<ref name="intersil">{{ cite web | title = The NCO as a Stable, Accurate Synthesizer | url = http://www.intersil.com/content/dam/Intersil/documents/tb31/tb318.pdf | publisher = Intersil Corporation | year = 1998 }}</ref>
===Mitigation techniques===
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