View source for Phase noise

{{Short description|Frequency domain representation of random fluctuations in the phase of a waveform}}
[[File:Phase Noise measured in ssa.png|thumb|250px|right|Phase noise measured by signal source analyzer (SSA). The SSA shows the positive part of the phase noise. In this picture there is a phase noise of the main carrier, 3 other signals and "noise hill".]]
[[File:Phasenrauschen(2).png|thumb|250px|right|A weak signal disappears in the phase noise of the stronger signal]]

In [[signal processing]], '''phase noise''' is the [[frequency-domain]] representation of random fluctuations in the [[phase (waves)|phase]] of a [[waveform]], corresponding to [[time-domain]] deviations from perfect periodicity ([[jitter]]).  Generally speaking, [[radio-frequency]] engineers speak of the phase noise of an [[oscillator]], whereas [[digital-system]] engineers work with the jitter of a clock.

==Definitions==
An ideal [[electronic oscillator|oscillator]] would generate a pure [[sine wave]].  In the frequency domain, this would be represented as a single pair of [[Dirac delta function]]s (positive and negative conjugates) at the oscillator's frequency; i.e., all the signal's [[power (physics)|power]] is at a single frequency.  All real oscillators have [[phase modulated]] [[Electronic noise|noise]] components.  The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise [[sidebands]].

Consider the following noise-free signal:

:<math>x(t)= A\cos(2 \pi f_0 t)</math>

Phase noise is added to this signal by adding a [[stochastic process]] represented by <math>\phi(t)</math> to the signal as follows:

:<math>x(t)= A\cos(2 \pi f_0 t + \phi (t))</math>

Different phase noise processes, <math>\phi(t)</math>, possess different power [[Spectral density]] (PSD). For example, a white noise PSD follows a <math>f^0</math> trend, a pink noise PSD follows a <math>f^{-1}</math> trend, and a brown noise PSD follows a <math>f^{-2}</math> trend.

<math> \operatorname{S}_{\phi}(f) </math> is the single-sided (f>0) '''phase noise PSD''' <math> \left[ \frac{rad^2}{Hz} \right] </math>, given by the [[Fourier transform]] of the [[Autocorrelation]] of the phase noise.<ref>{{Citation |last=Rubiola |first=Enrico |year=2008 |title=Phase Noise and Frequency Stability in Oscillators |publisher=Cambridge University Press |isbn=978-0-521-88677-2}}</ref>

:<math> \operatorname{S}_{\phi}(f) = \mathcal{F}\left[ \operatorname{E} \left[ \phi(t)\overline{\phi(t+\tau)} \right]\right]  </math>

The noise can also be represented at the single-sided (f>0) '''frequency noise PSD''', <math>\operatorname{S}_{\Delta \nu}(f) \left[ \frac{Hz^2}{Hz} \right] </math>, or the '''fractional frequency stability PSD''', <math>\operatorname{S}_{y}(f) \left[ \frac{1}{Hz} \right] </math>, which defines the frequency fluctuations in terms of the deviation from the carrier frequency, <math>f_0</math>.

:<math> \operatorname{S}_{\Delta \nu}(f) = f^2\operatorname{S}_{\phi}(f)  </math>

:<math> \operatorname{S}_{y}(f) = \frac{\operatorname{S}_{\Delta \nu}(f)}{f_0^2} = \frac{f^2\operatorname{S}_{\phi}(f)}{f_0^2}</math>

The phase noise can also be given as the '''spectral purity''', <math>\mathcal{L}\left(f\right) \left[ \frac{dBc}{Hz} \right]</math>, the single-sideband power in a 1&nbsp;Hz bandwidth at a frequency offset, f, from the carrier frequency, <math>f_0</math>, referenced to the carrier power.

:<math> \mathcal{L}\left(f\right) = 10\log_{10} \left( \frac{\operatorname{S}_{\phi}(f)}{2} \right) </math>

The stochastic dynamics of <math>\phi(t)</math> due to white noise can be understood as a random walk or diffusion process,<ref>{{cite journal |last=Lax |first=Melvin |title=Classical noise. V. noise in self-sustained oscillators |journal=Physical Review|volume = 160 | pages = 290–307 |doi=10.1103/PhysRev.160.290}}</ref><ref>{{cite journal |last1=Ham |first1=Donhee |last2=Hajimiri |first2=Ali |title=Virtual damping and Einstein relation in oscillators |journal=IEEE Journal of Solid-State Circuits|volume = 38 | issue = 3 | pages = 407–418 |doi=10.1109/JSSC.2002.808283}}</ref> with a characteristic diffusion coefficient <math>D</math>. The variance of <math>\phi(t)</math> grows with time <math>t</math> according to <math> \overline{\phi^2(t)} = 2 D t </math>, and the PSD of the signal <math>x(t)</math> yields what is described above in case of white noise perturbation. If phase diffusion is faster (larger <math>D</math>) for a given signal amplitude <math>A</math>, the PSD becomes wider and shorter, yielding larger phase noise. The problem of determining the PSD (phase noise) is commensurate with determining <math>D</math>.

==Jitter conversions==
Phase noise is sometimes also measured and expressed as a power obtained by integrating {{math|ℒ(''f'')}} over a certain range of offset frequencies. For example, the phase noise may be −40&nbsp;dBc integrated over the range of 1&nbsp;kHz to 100&nbsp;kHz. This integrated phase noise (expressed in degrees) can be converted to '''jitter''' (expressed in seconds) using the following formula:
:<math>\text{jitter (seconds}) = \frac{\text{phase error (} {}^\circ \text{)}}{360^\circ \times \text{frequency (hertz)}}</math>

In the absence of [[1/f noise]] in a region where the phase noise displays a –20{{nbsp}}dBc/decade slope ([[Leeson's equation]]), the [[root-mean-square|RMS]] cycle jitter can be related to the phase noise by:<ref>{{Citation|title=An Overview of Phase Noise and Jitter|date=May 17, 2001|url=http://literature.cdn.keysight.com/litweb/pdf/5990-3108EN.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://literature.cdn.keysight.com/litweb/pdf/5990-3108EN.pdf |archive-date=2022-10-09 |url-status=live|publisher=Keysight Technologies}}</ref>
: <math>\sigma^2_c = \frac{f^2 \mathcal{L}\left(f\right)}{f_\text{osc}^3}</math>

Likewise:
: <math>\mathcal{L}\left(f\right) = \frac{f_\text{osc}^3 \sigma_c^2}{f^2}</math>

==Measurement==
Phase noise can be measured using a [[spectrum analyzer]] if the phase noise of the [[device under test]] (DUT) is large with respect to the spectrum analyzer's [[local oscillator]]. Care should be taken that observed values are due to the measured signal and not the shape factor of the spectrum analyzer's filters. Spectrum analyzer based measurement can show the phase-noise power over many decades of frequency; e.g., 1&nbsp;Hz to 10&nbsp;MHz.  The slope with offset frequency in various offset frequency regions can provide clues as to the source of the noise; e.g., low frequency [[flicker noise]] decreasing at 30&nbsp;dB per decade (= 9&nbsp;dB per octave).<ref>{{Citation |last=Cerda |first=Ramon M. |title=Impact of ultralow phase noise oscillators on system performance |date=July 2006 |journal=[[RF Design]] |pages=28&ndash;34 |url=http://rfdesign.com/mag/607RFDF2.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://rfdesign.com/mag/607RFDF2.pdf |archive-date=2022-10-09 |url-status=live }}</ref>

Phase noise measurement systems are alternatives to spectrum analyzers. These systems may use internal and external references and allow measurement of both residual (additive) and absolute noise.  Additionally, these systems can make low-noise, close-to-the-carrier, measurements.

==Linewidths==
The sinusoidal output of an ideal [[electronic oscillator|oscillator]] is a [[Dirac delta function]] in the power spectral density centered at the frequency of the sinusoid. Such perfect spectral purity is not achievable in a practical oscillator. Spreading of the spectrum line caused by phase noise is characterized by the fundamental linewidth and the integral linewidth.<ref>{{cite thesis |last= Chauhan|first= Nitesh|date= 2024|title= Stabilized Sources in Visible for Atomic, Molecular and Quantum Applications.|url= https://escholarship.org/uc/item/3wk0q2g6 |degree= PhD|publisher=UC Santa Barbara}}</ref>

The '''fundamental linewidth''', also known as the [[White noise]]-limited linewidth or the intrinsic linewidth, is the linewidth of an oscillator's PSD in the presence of only white noise sources (noise with a PSD that follows a <math>f^0</math> trend, ie. equivalent across all frequencies). The fundamental linewidth takes Lorentzian [[spectral line shape]], and is given by twice the phase diffusion constant, <math>2D</math> (full width, half maximum). White noise provides a <math>1/\sqrt{\tau}</math> Allan Deviation plot at small averaging times.

The '''integral linewidth''', also known as the effective linewidth or the total linewidth, is the linewidth of an oscillator's PSD in the presence of both white noise sources (noise with a PSD that follows a <math>f^0</math> trend) and pink noise sources (noise with a PSD that follows a <math>f^{-1}</math> trend). Pink noise is sometimes called [[Flicker noise]], or simply 1/f noise. The integral linewidth takes Voigt lineshape, a convolution of the white noise-induced Lorentzian lineshape and the pink noise-induced Gaussian lineshape. Pink noise provides a <math>\tau^{0}</math> Allan Deviation plot at moderate averaging times. This flat line on the Allan Deviation plot is also known as the flicker floor.

Additionally, the oscillator might experience [[Frequency drift]] over long periods of time, slowly moving the center frequency of the Voigt lineshape. This drift is a brown noise source (noise with a PSD that follows a <math>f^{-2}</math> trend), and provides a <math>\sqrt{\tau}</math> Allan Deviation plot at large averaging times.

==Limiting System Performance==
A laser is a common oscillator that is characterized by its noise, and thus its [[Laser linewidth]]. The laser noise provides fundamental limitations of the systems that the laser is used in, such as loss of sensitivity in radar and communications systems, lack of definition in imaging systems, and a higher bit error rate in digital systems.

Lasers with a near-[[Infrared]] center wavelength are used in many [[atomic, molecular, and optical physics]] experiments to provide photons that interact with atoms. The requirements for the spectral purity at specific frequency offsets of the lasers used in qubit operation (such as clock transition lasers and state preparation lasers) are highly stringent because the coherence time of the qubit is directly related to the linewidth of the lasers.<ref>{{cite thesis |last= Chauhan|first= Nitesh|date= 2024|title= Stabilized Sources in Visible for Atomic, Molecular and Quantum Applications.|url= https://escholarship.org/uc/item/3wk0q2g6 |degree= PhD|publisher=UC Santa Barbara}}</ref>

==See also==
*[[Allan variance]]
*[[Flicker noise]]
*[[Leeson's equation]]
*[[Maximum time interval error]]
*[[Noise spectral density]]
*[[Spectral density]]
*[[Spectral phase]]
*[[Opto-electronic oscillator]]

==References==
{{reflist}}

==Further reading==
*{{Citation |last=Rubiola |first=Enrico |year=2008 |title=Phase Noise and Frequency Stability in Oscillators |publisher=Cambridge University Press |isbn=978-0-521-88677-2 |ref=none}}
*{{Citation |first1=E. |last1=Rubiola |first2=F. |last2=Vernotte |title=The Companion of Enrico's Chart for Phase Noise and Two-Sample Variances |journal=Trans. Microwave Theory Techniques |volume=71 |issue=7 |date=July 2023 |pages=2996–3025 |doi=10.1109/TMTT.2023.3238267}}
*{{Citation |last=Rubiola |first=Enrico |year= |title=The Enrico's Chart of Phase Noise and Two-Sample Variances |doi=10.5281/zenodo.4399218 |ref=none}}
*{{Citation |first1=U. L. |last1=Rohde |first2=E. |last2=Rubiola |first3=J. C. |last3= Whitaker |title=Microwave and Wireless Synthesizers (Chapter 2) |publisher=John Wiley & Sons |date=2020 |isbn=978-1-119-66600-4 |ref=none}}
*{{Citation |last=Wolaver |first=Dan H. |year=1991 |title=Phase-Locked Loop Circuit Design |publisher=Prentice Hall |isbn=978-0-13-662743-2 |ref=none}}
*{{Citation |first=M. |last=Lax |title=Classical noise. V. Noise in self-sustained oscillators |journal=[[Physical Review]] |volume=160 |issue=2 |pages=290&ndash;307 |date=August 1967 |doi=10.1103/PhysRev.160.290 |bibcode = 1967PhRv..160..290L |ref=none}}
*{{Citation |first1=D. |last1=Ham |first2=A. |last2=Hajimiri |title=Virtual damping and Einstein relation in oscillators |journal=IEEE Journal of Solid-State Circuits |volume=38 |issue=3 |date=March 2003 |pages=407–418 |doi=10.1109/JSSC.2002.808283}}
*{{Citation |first1=A. |last1=Hajimiri |first2=T. H. |last2=Lee |url=http://loveboat.stanford.edu/papers/JSSC98FEB-ali.pdf |title=A general theory of phase noise in electrical oscillators |archive-url=https://web.archive.org/web/20160305060323if_/http://loveboat.stanford.edu/papers/JSSC98FEB-ali.pdf |archive-date=2016-03-05 |access-date=2021-09-16 |journal=IEEE Journal of Solid-State Circuits |volume=33 |issue=2 |date=February 1998 |pages=179–194 |doi=10.1109/4.658619 |ref=none|bibcode=1998IJSSC..33..179H }}
* {{Citation |first=R. |last=Pulikkoonattu |title=Oscillator Phase Noise and Sampling Clock Jitter |date=June 12, 2007 |url=http://documents.epfl.ch/users/p/pu/pulikkoo/private/report_pn_jitter_oscillator_ratna.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://documents.epfl.ch/users/p/pu/pulikkoo/private/report_pn_jitter_oscillator_ratna.pdf |archive-date=2022-10-09 |url-status=live |publisher=ST Microelectronics |location=Bangalore, India |series=Tech Note |access-date=March 29, 2012 |ref=none}}
*{{Citation |first1=A. |last1=Chorti |first2=M. |last2=Brookes |title=A spectral model for RF oscillators with power-law phase noise |journal=IEEE Transactions on Circuits and Systems I: Regular Papers |volume=53 |issue=9 |date=September 2006 |pages=1989–1999 |doi=10.1109/TCSI.2006.881182 |ref=none|url=http://spiral.imperial.ac.uk/bitstream/10044/1/676/1/A%20spectral%20model%20for%20RF.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://spiral.imperial.ac.uk/bitstream/10044/1/676/1/A%20spectral%20model%20for%20RF.pdf |archive-date=2022-10-09 |url-status=live |hdl=10044/1/676 |s2cid=8855005 |hdl-access=free }}
*{{Citation |first1=Ulrich L. |last1=Rohde |first2=Ajay K. |last2=Poddar |first3=Georg |last3=Böck |title=The Design of Modern Microwave Oscillators for Wireless Applications |publisher=John Wiley & Sons |location=New York, NY |date=May 2005 |isbn=978-0-471-72342-4 |ref=none}}
*  Ulrich L. Rohde,  A New and Efficient Method of Designing Low Noise Microwave Oscillators, https://depositonce.tu-berlin.de/bitstream/11303/1306/1/Dokument_16.pdf
* Ajay Poddar, Ulrich Rohde, Anisha Apte, “ How Low Can They Go, Oscillator Phase noise model, Theoretical, Experimental Validation, and Phase Noise Measurements”, IEEE Microwave Magazine, Vol. 14, No. 6, pp.&nbsp;50–72, September/October 2013.  
* Ulrich Rohde, Ajay Poddar, Anisha Apte, “Getting Its Measure”, IEEE Microwave Magazine, Vol. 14, No. 6, pp.&nbsp;73–86, September/October 2013
* U. L. Rohde, A. K. Poddar, Anisha Apte, “Phase noise measurement and its limitations”, [[Microwave Journal]], pp.&nbsp;22–46, May 2013
* A. K. Poddar, U.L. Rohde,   “Technique to Minimize Phase Noise of Crystal Oscillators”, [[Microwave Journal]], pp.&nbsp;132–150,  May 2013.
* A. K. Poddar, U. L. Rohde, and E. Rubiola, “Phase noise measurement: Challenges and uncertainty”, 2014 IEEE IMaRC, Bangalore, Dec  2014.
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