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Overview on Phase Noise and Jitter

Technical Overviews

Phase noise and jitter are two related quantities associated with a noisy oscillator. Phase noise is a frequency-domain view of the noise spectrum around the oscillator signal, while jitter is a timedomain measure of the timing accuracy of the oscillator period. This paper discusses the relationship between phase noise and jitter in free-running oscillators. The reader is referred to three other papers for a more detailed look at the theory: Demir [1], Hajimiri [2] and Herzel [3].

The phase noise and jitter theory discussed here is mainly taken from these three papers. The jitter in a closed-loop phase-locked loop is not discussed here; see [4, 9, 10].

Phase Noise as a Lorentzian Spectrum

We usually observe the asymptotic behavior of phase noise £ measured at an offset frequency f from the carrier

where a is some constant. But this implies that phase noise goes to infinity at f=0. This is obviously wrong, as it implies that there is infinite noise power at f=0. For very noisy oscillators, it could also suggest that £ > 0 dBc/Hz at small enough offsets

where c is a scalar constant that describes the phase noise of the oscillator (in the absence of 1/f noise and ignoring any noise floor). This choice of expressing the characteristic constant as cfosc2πwill become clear when the constant c is reused in the equations for jitter.

The Lorentzian spectrum nicely avoids any singularities at f=0 while maintaining the same asymptotic behavior. It also has the property that the total power in £ from minus infinity to plus infinity is 1. This means that phase noise doesn’t change the total power of the oscillator, it merely broadens its spectral peak. If we borrow some terminology from laser technology (lasers are just optical oscillators), we can talk about the spectral line width of an oscillator

For quiet oscillators, the phase noise at offset frequencies greater than 1 Hz will always be less than 0 dBc/Hz, avoiding the embarrassing question of what a phase noise greater than 0 dBc/Hz means. At some very small frequency less than 1 Hz, the phase noise will be greater than 0 dBc/Hz, but only within a bandwidth of a fraction of a Hertz. This is what we should expect, to see something representing the carrier signal itself at f=0, which in the absence of noise would have a spectrum of d (f). The plot to the right shows the phase noise for f2–310×=ππcosc. We do see phase noise greater than 0 dBc/Hz, but only at an offset frequency less than 0.03 Hz. The total power integrated over a 1 Hz bandwidth still unity.

For noisy oscillators, the phase noise will again always be less than 0 dBc/Hz, but this gives us a spectral line width that is greater than 1 Hz. The phase noise goes flat below fHW , showing the wide spectral line width of the oscillator due to high phase noise. (Note: this does not equate to the problem we have observed in the ADS phase noise simulation where the pnmx phase noise goes flat below some small offset frequency.) The plot to the right shows the phase noise for f2–310×=ππcosc. Note if we were to extrapolate from the phase noise at 100 kHz, we would get a phase noise of +30 dBc/Hz at an offset frequency of 1 Hz. But this doesn’t make sense that the noise is stronger than the carrier. Instead the carrier is broadened over a 6.3 kHz (two-sided) bandwidth.

Jitter

We intuitively know that jitter is a variation in the zero-crossing times of a signal, or a variation in the period of the signal. But a variation with respect to what? There are several different types of jitter than can be defined [1,3]; they will subscripted them to distinguish between them.

Jitter is a statistical measure of a noisy oscillation process. The period of each cycle of the oscillator is different, due to the noise-induced jitter. We will refer to τn as the period of cycle n. For a free-running oscillator with noise, the oscillation period will have a Gaussian distribution. This distribution has a mean τavg, whose inverse can be defined as the average frequency of oscillation avgoscτ=f1. The distribution also has a standard deviation, which we will later define as the cycle-to-cycle jitter sc. Jitter is defined as an rms quantity.

Phase noise

Phase noise no longer displays a Lorentzian spectrum when 1/f noise is present. There is no closed-form expression for the phase noise spectrum in the presence of 1/f noise. The discussions so far have concentrated on the case where the transistor models do not have any flicker noise. The analytical models of [1, 2, 3] and the Lorentzian spectrum were derived in the absence of flicker noise. But any physical transistor exhibits flicker noise, and this contributes to an increase in oscillator phase noise at small offset frequencies. Flicker noise is several orders of magnitude worse for MOSFETs than for BJTs. A BJT will typically display a flicker noise corner frequency around 1 kHz, while in a MOSFET it will be around 1 MHz.

Only in the last few years have analytical models for phase noise with flicker noise been proposed [5 to 8]. Herzel [5] decomposes the model into a Lorentzian for the white noise sources and Gaussian for the flicker noise source. The final spectrum is then a convolution of a Lorentzian with a Gaussian spectrum.

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