Phase noise

Phase noise is the frequency domain representation of rapid, short-term, random fluctuations in the phase of a waveform, caused by time domain instabilities ("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

Historically there have been two conflicting yet widely used definitions for phase noise. The definition used by some authors defines phase noise to be the power spectral density (PSD) of a signal's phase, while the other definition is based on the PSD of the signal itself. Both definitions yield the same result at offset frequencies well removed from the carrier. At close-in offsets however, the two definitions differ.

The IEEE defines phase noise as $\mathcal{L}\left(f\right)=S_{\phi}(f)/2$ where the "phase instability" S_φ(f) is the one-sided spectral density of a signal's phase deviation. Although S_φ(f) is a one-sided function, it represents "the double-sideband spectral density of phase fluctuation". The phase noise expression ℒ(f) is pronounced "script ell of f".

Background

An ideal oscillator would generate a pure sine wave. In the frequency domain, this would be represented as a single pair of Dirac delta functions (positive and negative conjugates) at the oscillator's frequency, i.e., all the signal's power is at a single frequency. All real oscillators have phase modulated noise components. The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise sidebands. Oscillator phase noise often includes low frequency flicker noise and may include white noise.

Consider the following noise free signal:

v(t) = Acos(2πf₀t).

Phase noise is added to this signal by adding a stochastic process represented by φ to the signal as follows:

v(t) = Acos(2πf₀t + φ(t)).

Phase noise is a type of cyclostationary noise and is closely related to jitter. A particularly important type of phase noise is that produced by oscillators.

Phase noise (ℒ(f)) is typically expressed in units of dBc/Hz, and it represents the noise power relative to the carrier contained in a 1 Hz bandwidth centered at a certain offsets from the carrier. For example, a certain signal may have a phase noise of -80 dBc/Hz at an offset of 10 kHz and -95 dBc/Hz at an offset of 100 kHz. Phase noise can be measured and expressed as single sideband or double sideband values, but as noted earlier, the IEEE has adopted the definition as one-half of the double sideband PSD.

Jitter conversions

Phase noise is sometimes also measured and expressed as a power obtained by integrating ℒ(f) over a certain range of offset frequencies. For example, the phase noise may be -40 dBc integrated over the range of 1 kHz to 100 kHz. This Integrated phase noise (expressed in degrees) can be converted to jitter (expressed in seconds) using the following formula: $\text{Jitter (seconds}) = \text{Phase error (degrees)}/(360\times \text{Frequency (Hertz)})$

In the absence of 1/f noise in a region where the phase noise displays a –20 dBc/decade slope, the rms cycle jitter can be related to the phase noise by:

$\sigma^2_c = \frac{f^2 \mathcal{L}\left(f\right)}{f_{osc}^3}$

Likewise:

$\mathcal{L}\left(f\right) = \frac{f_{osc}^3 \sigma^2_c}{f^2}$

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 Hz to 10 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 dB per decade (=9 dB per octave).

Spectral purity

The sinewave output of an ideal oscillator is a single line in the frequency spectrum. Such perfect spectral purity is not achievable in a practical oscillator. Spreading of the spectrum line caused by phase noise must be minimised in the local oscillator for a superheterodyne receiver because it defeats the aim of restricting the receiver frequency range by filters in the IF (intermediate frequency) amplifier.

See also

• Flicker noise
• Leeson's equation
• Noise spectral density
• Spectral density

References

• National Institute of Standards and Technology Time and Frequency Metrology Group

Further reading

• Rubiola, Enrico (2008), Phase Noise and Frequency Stability in Oscillators, Cambridge University Press, ISBN 978-0-521-88677-2 
• Wolaver, Dan H. (1991), Phase-Locked Loop Circuit Design, Prentice Hall, ISBN 0-13-662743-9 
• Lax, M. (August 1967), "Classical noise. V. Noise in self-sustained oscillators", Physical Review 160 (2): 290–307, Bibcode:1967PhRv..160..290L, doi:10.1103/PhysRev.160.290 
• Hajimiri, A.; Lee, T. H. (February 1998), "A general theory of phase noise in electrical oscillators", IEEE Journal of Solid-State Circuits 33 (2): 179–194, doi:10.1109/4.658619 
• Pulikkoonattu, R. (June 12, 2007), Oscillator Phase Noise and Sampling Clock Jitter, Tech Note, Bangalore, India: ST Microelectronics, retrieved March 29, 2012 
• Chorti, A.; Brookes, M. (September 2006), "A spectral model for RF oscillators with power-law phase noise", IEEE Trans. on Circuits and Systems I: Regular Papers 53 (9): 1989–1999, doi:10.1109/TCSI.2006.881182 
• Rohde, Ulrich L.; Poddar, Ajay K.; Böck, Georg (May 2005), The Design of Modern Microwave Oscillators for Wireless Applications, New York, NY: John Wiley & Sons, ISBN 0-471-72342-8 

External links

• The site http://rubiola.org contains a bunch of information and literature about phase noise, frequency stability, experimental techniques, etc.
• A technical article about phase noise in signal sources due to phase modulation.
• Phase-noise measurement software for various GPIB-equipped spectrum analyzers (freeware, includes Win32 C++ source)
• Clock (CLK) Jitter and Phase Noise Conversion
• Phase noise in frequency synthesizers
• Phase Noise measurement using the phase lock technique
• Noise in Mixers, Oscillators, Samplers, and Logic: An Introduction to Cyclostationary Noise by Joel Phillips and Ken Kundert
• Predicting the phase noise and jitter of PLL-based frequency synthesizers by Ken Kundert
• Y. T. Su, K. T. Wong & K.-P. Ho, "Linear MMSE Estimation of Large-Magnitude Symmetric Levy-Process Phase-Noise", IEEE Transactions on Communications.