Quote:
Originally Posted by Terry_VK5TM
Had a look at that link and am now completely confused (then again, maths anything over addition and subtraction was always a mystery to me  ). |
Ah, now this is neat. Maths it may be but it's so neat you have to admire it.
The process is referred to as "Bessel Nulls"
If I take a carrier and FM it, I will usually get a large number of sidebands, all at the modulation frequency spacing around the carrier. It's a lot more complicated than AM. You would need a narrow receiver to tune to individual sidebands in order to resolve them and see them as individuals... or a spectrum analyser with narrow enough filters to resolve them.
Now, if I vary the deviation setting on my signal generator, the sidebands all stay where they are, frequency-wise, but they change in amplitude, but not all together. Some go up, some go down, some pass through nulls... hell! even the carrier itself nulls at some points.
THe nice bit is that the ratios of deviation to modulating frequencies where these nulls happen can be calculated. So, as long as your modulating frequency is accurate, there are certain deviation values where you can be pretty certain of the deviation value.
Sneaky, huh?
To understand what's going on, you need to know about the phasor diagram representation of modulation. A carrier can be represented by an arrow spinning around its blunt end. Spinning at the carrier frequency. The height of its pointy end represents the instantaneous voltage, and by spinning it traces out a sinewave in time, at the carrier frequency.
To amplitude modulate it, we need to be able to change the arrow lenght, without mucking about with its direction, that has to keep spinning at the sme old rate.
Someone did the maths for one sinusoid multiplied by another one and came out with an equation which ALSO fitted three sinusoids. One at the carrier frequency, one at the carrier plus the modulation frequencies, and one at carrier minus the modulation frequencies.
Look at the signal on a scope and you see AM. Look with a spectrum analyser, and you see the carrier with the sidebands. Both views are truthful and just alternative viewpoints of equal validity.
Now, for FM, we want to rotate the arrow so it swings a bit ahead o itself to a bit behind. The amplitude/length must not change while we do this or we'd get AM as well.
The difficulty is that the pointy end of our arrow mus swing precisely along a circular arc.
What we did with AM shows that signal components are
additive and while they can do straight line things, curves are a whole different world. You can't make a perfect curve, but you can approximate to one. So you have one pair of sidebands, not in phase like the AM ones, but 90 degrees shifted. So they add wag, not length... well they do affect the length because an arc isn't straight. So we add a pair of AM sidebands at twice Fmodto try to cancel most of the unwanted AM. But this affects the wag, so we add another pair of wagging-phased sidebands at three times Fmod, and that needs amplitude correction, and that needs wag correction and so on.
We wind up with a series of sidebands making a sort of iterative approximation tending to the right place on the arc. If we allowed infinite number of sidebands, and infinite bandwidth to fit them in, it would be perfect.
All mathematical, all seeming abstract, but it touches on the real world of soldering irons in a couple of ways:
1) If you can't give an FM signal infinite bandwidth, you have to live with some distortion.
2) That bit about sidebands being at 90 degrees to what you'd expect for AM means that if you mix (multiply) an FM signal with itself shifted through 90 degrees, you get a nice FM demodulator. It's called the quadrature detector and features in a zillion FM receivers near you.
The addition of a lot of (usually) progressively smaller vectors to make a staircase approximating to a circular arc can be done with a mathematical series, the Bessel function. Ns the bessel functions describe the amoplitudes of our different sidebands. Bessel functions pass through zero several times, and reverse phase as they do so. The zero crossings are the 'nulls' and are superb ways to calibrate signal generators and deviation meters. All yoiu need is an accurate audio generator and something to let you see the individual sidebands and the carrier.
It's a whole world you wouldn't suspect existed, but you can do useful tricks there.
Squaring the circle? or circling the square?
David