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# Digital technique improves on phase-noise measurements

Martin Rowe, Senior Technical Editor- June 9, 2009

Dr. Sam Stein is the VP of engineering at Symmetricom Timing Test and Measurement Division. Symmetricom engineers have recently developed a digital sampling technique for measuring phase noise in oscillators. I spoke with Stein about this technique by phone from his office in Boulder, CO.

**Q: Why is phase noise measurement important?
A:** Phase noise determines the performance of many systems such
as the CW radar systems used to detect moving targets. The phase noise
reflected by stationary objects competes with the Doppler shifted
carrier from a moving target. This competition generally sets the lowest
velocity target that may be identified. In chirped-FM radar such as
weather radar, the reflected signal is compared to currently broadcast
signal. The difference in frequency between the reflected signal and the
currently transmitted carrier is proportional to the distance from a
reflecting object such as a cloud. The Allan deviation of the reference
oscillator corresponding to the round trip time to the reflecting object
and back quantifies the reference oscillator’s contribution to the
noise or jitter of the radar.

For precision oscillators, manufacturers publish specifications in both the frequency domain (phase noise) and in the time domain (Allan deviation). For application-specific oscillators, manufacturers may only publish specs relevant to that application. For example, a 155.2-MHz oscillator is typically only used in SONET applications. Those users need jitter and phase-noise specs as opposed to Allan deviation. For general-purpose oscillators, manufacturers will publish a wide range of specifications.

**Q: How is phase noise measured?
A:** We wanted a way to easily measure phase noise and phase
changes in high-precision oscillators. Traditional measurements use
analog heterodyne techniques, that is, a double-balanced mixer
translates the phase difference between an unknown clock and a reference
clock into voltage. But, the measurements are frequency, level, line
length, and impedance dependent. The unknown and reference oscillators
need to run precisely the same frequency because you have to phase-lock
them with a phase-locked loop (PLL)in order that the variations in the
output of the double-balanced mixer be proportional to the phase
deviations between the two oscillators.

**Q: What is the difference between phase noise and jitter?
A:** Phase noise is the component of oscillator noise that
represents variations in timing of the signal. This should be contrasted
with amplitude noise, which is the component of noise that represents
variations in the level of the signal. Jitter is a statistic of phase
noise. The jitter is the integral of the phase noise over some
bandwidth.

**Q: What is the important parameter in phase-noise measurements?
A:** When we talk about phase noise, we refer to the power
density per unit of bandwidth relative to the noise of the carrier. The
carrier has a value of 0 dBc by definition. The single-sideband phase
noise at some frequency offset from the carrier will be lower. We wanted
to measure the phase noise 10 kHz from a 10-MHz carrier down to level
of —–170 dBc/Hz using a completely automated technique that would
require no expertise on the part of the user. We knew that we had to do
something that hadn’t done before. Our engineers started by using the
traditional analog heterodyne techniques but found that we were 10 dB
off from where we wanted. A year later, we were still 10 dB away.

**Q: What made you try a digital sampling technique?
A:** At the beginning of the project, one of my engineers said
“Why don’t we just do this digitally? We can sample the signal and apply
digital-signal processing to the sampled signal.” I didn’t think the
technique would work for measuring phase noise. It’s OK for software
radio where you’re just trying to decode symbols. Nobody had ever made
measurements at 170 dB below the carrier using digital techniques.

At about the same time, NIST in Boulder, CO (www.boulder.nist.gov) advertised that it wanted to fund the development of a digital software-based phase-noise measurement system. With the government backing, we found that we could perform all of the traditional measurements—frequency conversion, phase detection, computation of the spectrum—in firmware and software. Instrument architecture is an ADC„³FPGA„³DSP„³x86 processor. The system measures phase noise by digitizing the signal in question and compares it to a reference signal.

**Q: How does the technique work?
A:** We digitally measure the phase difference between an unknown
signal and a reference signal and compute a time series of phase. We
generate 2.5 million phase samples per second, which lets us compute
single-sideband phase noise out to 1 MHz from the carrier. We low-pass
filter the data and decimate it to produce phase noise at all points
from the carrier (see

**figure**).

**Q: What is the math behind the calculations?
A:** The arctangent calculation is the key. First, we convert the
RF signals to baseband, then calculate the in-phase (I) and quadrature
(Q) components of the signal, which is sin(ωt+Φ) and cos(ωt+Φ),
respectively. From that, we calculate their ratio to get the tangent of
the phase difference between the unknown signal and the reference
signal. We then must find the angle that has that tangent.

We get the angle by calculating the arctangent. But, there are an infinite number of angles between the input and the reference signal that have the same arctangent. We want to find only the angle of the arctangent that falls within one half cycle of the signal. We must arbitrarily pick a phase difference that’s small, less than π radians, to start. At 2.5 Msamples/s, we get enough samples between the points where the tangent function reaches plus or minus infinity. At those points, we’ve gone outside the half cycle of interest. We can detect that each time we see a plus or minus infinity, we know we’ve gone around π radians and we can tell in which direction (positive or negative) the phase noise is from the reference carrier frequency. We can track the total phase change by counting the cycles with the DSP.

**Q: How does that compare to using the analog heterodyne technique.
A:** Using a double-balanced mixer, you have to stay within
approximately 45° of the zero crossing, which requires a PLL
(phase-locked loop). The PLL eliminates the multiple zero crossings and
makes it impossible to use the measurements to compute Allan deviation.
We don’t require a PLL and you can put in a 5-MHz or 10-MHz reference
signal, your company’s gold standard, and measure any oscillator. You
can’t do that with analog techniques.

The digital technique is purely ratiometric, which cancels the amplitude noise of the signal, and doesn’t require calibration either. The analog techniques requires the user to calibrate the phase to voltage transfer function of the double-balanced mixer. For the digital technique, the linearity of the ADC is important, but not the absolute value of the ADC’s voltage reference. The noise density of the ADC is equivalent to about -150 dBc/Hz, which would be the limitation of the instrument, so we use two digital phase difference measurement engines within the instrument and compute the cross spectrum and cross Allan deviation to get better performance.