[sdiy] Frequency multipliers

[David G. Dixon]

> > Are there some relatively simple analog circuits that can take an
> > input sine frequency of x and output a multiple of x?
> 
> If you've got a single sine wave or a quadrature signal, you can use a
> multiplier to get double the frequency (or a square root circuit to
> half it).  Higher multiples are possible, but more cumbersome and less
> effective due to additional products you would have to cancel.  Or
> maybe you don't need to cancel them, since these by-products are also
> multiples of the input frequency.

The most straightforward way to get a sine wave of twice the frequency is to
take the square of the sine wave with a multiplier, and then apply a gain of
2 and a negative dc bias of half the original amplitude to the output.  You
will end up with a sine wave at twice the frequency.  This is an electronic
manifestation of the following trigonometric identity, known as a
"half-angle formula" (one of three):

cos(z/2) = +/- sqrt[(1 + cos(z))/2]

or, put another way,

cos(z) = 2*cos^2(z/2) - 1

This is equation 4.3.21 on page 72 of Abramowitz and Stegun, Handbook of
Mathematical Functions, Dover, New York, 1965 (just about the best $15 I
ever spent!).

Obviously, this operation may also be done in reverse using a square-root
circuit.

(Hey, this has the makings of an excellent final exam question for
second-year EE students!)

Now, if you're really clever, you can design a circuit to get a sine wave at
three times the frequency of the original.  Here's the trigonometric
identity (ibid., equation 4.3.28):

cos(3z) = -3*cos(z) + 4*cos^3(z)

Recognizing that cos^3 = cos*cos^2, this can be done with two multipliers
and a summer, or one additional multiplier and summer.  No dc level shifting
is involved.

This sounds like so much fun, I think I'll have to do it myself!