# Numbers in Generating Phase Coherent Electronic Systems

I am reading the following paper 'Störmer’s numbers in generating phase coherent electronic systems' written in 1969.

More or less I understand that

in microwave applications there is clearly a use for electronic systems which generate a relatively high carrier frequency, and a local oscillator frequency differing by a relatively low IF or modulating frequency, with all three locked in phase.

However the following statement is not very clear for me:

If two large numbers are both factorizable into convenient factors and differ by 1, they are ideal for generating phase coherent electronic systems.

Why the situation is ideal for generating phase coherent systems? Also, I am not very sure how these numbers seem to have direct applications to Doppler navigation in types of CW radar and what were the technology limitations in 1969 and what is by now to use the numbers?

Any basic explanations are very welcomed.

Why the situation is ideal for generating phase coherent systems?

Imagine this situation:

You want to mix two frequencies $$\f_1< f_2\$$ in order translate the band of interest onto an IF, $$\f_{IF} = f_2 - f_1\$$. You want to do this in a manner that is phase-coherent, so you need to use the same oscillator you used to generate the frequencies to generate the frequencies used mixing back down.

Remember that physically a mixer exploits the intermodulation due to a nonlinear effect; for example, when you apply the sum of two pure sine to the base of a transistor (with an $$\e^x\$$ output/input relation), then the Taylor expansion of the output/input relation gives you one component that is constant, one that is linear to the input (which is what you want to use when you want to use that transistor as low-distortion amplifier), one that is quadratic (~ x²), one that is cubic (~ x³), one that has fourth-power relation … and so on. The quadratic part is what gives you the difference frequency (because $$\(\sin(ax) + \sin(bx))^2 = \sin(ax)^2 + \sin(bx)^2 + 2\sin(ax)\sin(bx)\$$, and by trigonometric identities, that $$\2\sin(ax)\sin(bx)\$$ contains difference and sum frequencies: $$\\sin(ax) \sin(bx) = \frac12(\cos((a-b)x) -\cos((b+a)x))\$$.)

You will hence optimize the transistor and its operational point such that you get a response that looks as quadratic as possible.

Now, sadly, you don't only get the quadratic term, and hence not only the simple difference $$\f_2-f_1\$$, but also higher-order intermodulation terms like $$\2f_2-f_1, -(f_2-2f_1), 2f_2-3f_1, \ldots\$$. Now, some of these will fall into your bandwidth around $$\f_{IF}\$$! Bad! But: in general, the higher the order, the lower the power. While $$\ 2f_2-3f_1\$$ might still be a problem, it's unlikely that $$\127f_2-128f_1\$$ is – these terms should have a very low factor in the Taylor series expansion of the transfer function to begin with.

However, if you construct $$\f_1\$$ and $$\f_2\$$ such that the least common multiple is very large, then it can't happen that $$\Mf_1- Nf_2 \approx f_{IF}\$$ for "small" $$\M, N\in \mathbb N\$$.

How do you construct numbers that way? You let them have no common prime factors. Putting the 1 apart is an easy way to ensure that (simple proof: if $$\K\$$ can be divided by $$\P>1\$$, then $\K+1$ can not).

Still, you want these numbers to be synthesizable from "convenient" factors. Why? Because you can't buy a precise GHz-range oscillator, but you can buy a precise 16 MHz oscillator, and square it to get 32 MHz, filter that to be clean of higher-order intermodulation products, square that again to get 64 MHz, mix that with the original 16 MHz to get 80 MHz, square that to get 160 MHz… and so on. So, you have a way of building M·2 MHz oscillators pretty easily.

Use a 27 MHz (=3³ MHz) oscillator, and mix as desired and you can now generate frequencies in a M·2 MHz + N·3 MHz raster.

And what the author observes is that numbers of the form $$\P^n\$$ paired with $$\P^n -1\$$ are good choices, and so are the pairs $$\(P^n -1)^2\$$ & $$\(P^n)^2\left((P^n)^2-2n\right)\$$.

what were the technology limitations in 1969 and what is by now to use the numbers?

This is all timeless math – it hasn't changed since 1969. What has changed:

the problem in the last case being to find a p² - 2 which is factorizable.

Every dishwasher these days has the computational power to try and factorize integers that are large enough for any realistic use. This is not an exaggeration.

have direct applications to Doppler navigation in types of CW radar

CW radars are exactly the kinds of applications where you need exquisitely clean mixing products without unwanted harmonics appearing in the band of interest.

• Now, tt's clear, thank you. But how you would estimate the P numbers for ZHz processing or higher? Could you add an extra example with suitable M & N for it please? Commented Jul 12 at 19:23
• you don't estimate – estimation involves a random variable. You just pick a set / range of oscillator frequencies you have available and find polynomials of that that fulfill one of these equations, simply by trying out. Commented Jul 12 at 19:26

If you generate two high-frequency signals by multiplying a low-frequency oscillator by two integer factors that differ by 1, their "beat" frequency will be exactly the same as the frequency of the original oscillator, and there will be no systemic errors in that result.

For example, you could transmit one of the two high-frequency signals as a CW radar signal, and mix the return signal with the other one. Then, any variations in that return signal relative to the original reference oscillator are due to whatever objects the signal encountered. A Doppler shift would indicate motion toward or away from the radar, and the phase coherence of the system allows you to determine the direction.