 Numerically controlled oscillator

A numerically controlled oscillator (NCO) is a digital signal generator which creates a synchronous (i.e. clocked), discretetime, discretevalued representation of a waveform, usually sinusoidal.^{[1]} NCOs are often used in conjunction with a digitaltoanalog converter (DAC) at the output to create a direct digital synthesizer (DDS).^{[3]}
Numerically controlled oscillators offer several advantages over other types of oscillators in terms of agility, accuracy, stability and reliability.^{[2]} 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 FSK/PSK modulators/demodulators.^{[2]}
Contents
Operation
An NCO generally consists of two parts:
 A phase accumulator (PA), which adds to the value held at its output a frequency control value at each clock sample.
 A phasetoamplitude converter (PAC), which uses the phase accumulator output word (phase word) usually as an index into a waveform lookup table (LUT) to provide a corresponding amplitude sample. Sometimes interpolation is used with the lookup table to provide better accuracy and reduce phase error noise. Other methods of converting phase to amplitude, including mathematical algorithms such as power series can be used, particularly in a software NCO.
When clocked, the phase accumulator (PA) creates a modulo2^{N} sawtooth waveform which is then converted by the phasetoamplitude 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 lookup table. If the PAC capacity is 2^{M}, 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 timevarying 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.^{[4]}
The frequency accuracy relative to the clock frequency is limited only by the precision of the arithmetic used to compute the phase.^{[4]} NCOs are phase and frequencyagile, and can be trivially modified to produce phasemodulated or frequencymodulated by summation at the appropriate node, or provide quadrature outputs as shown in the figure.
Phase accumulator
A binary phase accumulator consists of an Nbit binary adder and a register configured as shown in Figure 1.^{[5]} Each clock cycle produces a new Nbit output consisting of the previous output obtained from the register summed with the frequency control word (FCW) which is constant for a given output frequency. The resulting output waveform is a staircase with step size ΔF, the integer value of the FCW.^{[6]} In some configurations, the phase output is taken from the output of the register which introduces a one clock cycle latency but allows the adder to operate at a higher clock rate.^{[2]}
The adder is designed to overflow when the sum of the absolute value of its operands exceeds its capacity (2^{N}−1). The overflow bit is discarded so the output word width is always equal to its input word width. The remainder φ_{n}, called the residual, is stored in the register and the cycle repeats, starting this time from φ_{n} (see figure 2).^{[5]} Since a phase accumulator is a finite state machine, eventually the residual at some sample K must return to the initial value φ_{0}. The interval K is referred to as the grand repetition rate (GRR) given by
where GCD is the greatest common divisor function. The GRR represents the true periodicity for a given ΔF which for a high resolution NCO can be very long.^{[5]} Usually we are more interested in the operating frequency determined by the average overflow rate, given by^{[6]}
 (1)
The frequency resolution, defined as the smallest possible incremental change in frequency, is given by^{[6]}
 (2)
Equation (1) shows that the phase accumulator can be thought of as a programmable noninteger frequency divider of divide ratio ΔF / 2^{N}.^{[4]}
Phasetoamplitude converter
The phaseamplitude converter creates the sampledomain waveform from the truncated phase output word received from the PA. The PAC can be a simple read only memory containing 2^{M} contiguous samples of the desired output waveform which typically is a sinusoid. Oftentimes though, various tricks are employed to reduce the amount of memory required. This include various trigonometric expansions,^{[7]} trigonometric approximations^{[5]} and methods which take advantage of the quadrature symmetry exhibited by sinusoids.^{[8]} Alternatively, the PAC may consist of random access memory which can be filled as desired to create an arbitrary waveform generator.
Spurious products
Spurious products are the result of harmonic or nonharmonic distortion in the creation of the output waveform due to nonlinear numerical effects in the signal processing chain. Only numerical errors are covered here. For other distortion mechanisms created in the digitaltoanalog converter see the corresponding section in the directdigital synthesizer article.
Phase truncation spurs
The number of phase accumulator bits of an NCO, N is usually between 24 and 64. If the PA output word were used directly to index the PAC lookup table an untenably high storage capacity in the ROM would be required. As such, the PA output word must be truncated to span a reasonable memory space. Truncation of the phase word cause phase modulation of the output sinusoid which introduces nonharmonic distortion in proportion to the number of bits truncated. The number of spurious products created by this distortion is given by:
 (3)
where W is the number of bits truncated.
In calculating the Spuriousfree dynamic range, we are interested in the spurious product with the largest amplitude relative to the carrier output level given by:
where P is word width of the DAC. For W >4,
Another related spurious generation method is the slight modulation due to the GRR outlined above. The amplitude of these spurs is low for large N and their frequency is generally too low to be detectable but they may cause issues for some applications.^{[5]}
Amplitude truncation spurs
Another source of spurious products is the amplitude 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.^{[9]}
Mitigation techniques
Phase truncation spurs can be reduced substantially by the introduction of white gaussian noise prior to truncation. The socalled dither noise is summed into the lower W+1 bits of the PA output word to linearize the truncation operation. Often the improvement can be achieved without penalty because the DAC noise floor tends to dominate system performance. Amplitude truncation spurs can not be mitigated in this fashion. Introduction of noise into the static values held in the PAC ROMs would not eliminate the cyclicality of the trunction error terms and thus would not achieve the desired effect.^{[4]}
See also
References
 ^ ^{a} ^{b} Jane Radatz, The IEEE Standard Dictionary of Electrical and Electronics Terms, IEEE Standards Office, New York, NY, 1997
 ^ ^{a} ^{b} ^{c} ^{d} "Numerically Controlled Oscillator". Lattice Semiconductor Corporation. 2009. http://www.latticesemi.com/products/intellectualproperty/ipcores/numericallycontrolledosci/index.cfm.
 ^ While some authors use the terms DDS and NCO interchangeably,^{[2]} by convention an NCO refers to the digital (i.e. the discretetime, discrete amplitude) portion of a DDS^{[1]}
 ^ ^{a} ^{b} ^{c} ^{d} Kroupa,Venceslav F.,Direct Digital Frequency Synthesizers, IEEE Press, 1999, ISBN 0780334388
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Popek, Grzegorz; Kampik, Marian (October 2009), "LowSpur Numerically Controlled Oscillator Using Taylor Series Approximation", XI International PhD Workshop, Silesian University of Technology, Gliwice, Poland, http://mechatronika.polsl.pl/owd/pdf2009/030.pdf
 ^ ^{a} ^{b} ^{c} All About Direct Digital Synthesis, Analog Devices, Inc., http://www.analog.com/library/analogdialogue/archives/3808/dds.html
 ^ US 7437391, Miller, Brian M., "Numerically controlled oscillator and method of operation", issued October 14, 2008
 ^ US 4486846, Mccallister, Ronald D. & Daniel Shearer III, "Numerically controlled oscillator using quadrant replication and function decomposition", published 12/04/1984
 ^ "The NCO as a Stable, Accurate Synthesizer". Intersil Corporation. 1998. http://www.intersilsemi.com/data/tb/tb318.pdf.
Categories: Digital signal processing
 Synthesizers
 Oscillators
 Digital electronics
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