# Phase Noise, Frequency Multiplication, and Intuition

Blog Post created by benz on Oct 13, 2016

Originally posted Jul 13, 2015

This is why we can’t have nice things at microwave and millimeter frequencies

Well, of course, we can have nice things at very high frequencies, but it’s more difficult and gets progressively harder as frequencies increase. I just couldn’t resist invoking the “can’t have nice things” meme, and to parents everywhere it has a certain resonance.

In many applications, the operating frequencies of the systems we design and test are increasing as part of our endless quest for available bandwidth. From direct improvements in data throughput to increasing resolution in radar-based synthetic vision, the requisite tradeoffs apply equally to test equipment and DUTs.

An intuitive understanding of life at higher frequencies was on my mind recently after reading an email that mentioned a classic rule of thumb: A perfect frequency doubler increases phase noise by 6 dB. Here’s an example of a synthesizer output at successively doubled frequencies. Successive doubling of the output of a frequency synthesizer increases phase noise by 6 dB for each step.

Of course, if the doubler is not a perfect device, then the increase will be larger than 6 dB because the doubler adds noise or undesirable phase deviation.

Why 6 dB? Perhaps that’s where different intuitive approaches can help. Years ago, when I first heard it, the 6 dB figure made sense from a time-domain perspective. If a deviation is constant in terms of time, the phase deviation at twice the frequency will be twice as large. Doubling the phase deviation—a linear term—will increase sideband power by 6 dB.

Heuristically, this intuitive approach feels correct, but I’ve learned to be cautious about relying too much on my intuition. Fortunately, more rigorous calculations—albeit based on approximations and simplifications—yield the same answer. Until I wrote this post, I didn’t realize that my approach also involved a version of the small-angle approximation.

A more general expression of this relationship applies to multipliers other than two:

20 log10 (N) dB where N is the multiplier constant

In practical microwave and millimeter systems, multipliers greater than two are common, placing a real premium on the phase noise performance of the fundamental oscillators. This applies equally to microwave and millimeter test equipment, in which the choice of local oscillator frequencies is a balance between performance at fundamental frequencies and required range of multipliers or harmonic numbers.

That balance can indeed yield nice things at very high frequencies. Here’s an example of the phase noise of a signal analyzer at 67 GHz using external mixing. This measurement of a low-noise millimeter source reveals the phase noise of a Keysight PXA X-series signal analyzer using a V-band smart external mixer at 67 GHz. The DUT, a PXG signal generator with a low-noise option, has even lower phase noise.

Frequency dividers are another example of this relationship, and can be treated as multipliers with a constant less than one. For example, a divide-by-two circuit (N = 0.5) yields an improvement of 6 dB, making it a practical and effective way to reduce phase noise.

Where do you get your insight into relationships such as this? Do you lean on visual approaches, mathematical calculations or something else altogether? Feel free to add a comment and share your perspective.