sine/cos encoder to digital

[FvM]

It is not easy to use a LUT for tan(x).

Right. Actually it's atan(y/x), respectively atan2(x,y). A reasonable implementation uses a half quadrant LUT (0 .. 45 degree) and applies the identity
atan(y/x) = sgn*pi/2 - atan(x/y)

 

[c_mitra]

A reasonable implementation uses a half quadrant LUT (0 .. 45 degree) and ...

Actually, the formal expansion for atan(x) is:

atan(x)= x -x*x*x/3 +x*x*x*x*x/5 -x*x*x*x*x*x*x/7 +....

This series expansion converges slowly (to get 4 decimal accuracy, you will need perhaps 50 terms ...)

But more importantly, the expansion is valid only for x >=1; (for x=1, the left hand side is pi/4 or 45 deg).

For x>1, you will be forced to use the identity you cite.

In the present case, we need the atan2(x,y) function which in turn calls the atan(x) function after fixing the quadrant.

 

[doraemon]

Hello!

That's why we have atan2() function. It's already handling the atan() discontinuities.

I know, but in any case, you're going to end up with very large values when approaching from
the discontinuities when you calculate the ratio. That's why in all the encoders I know, the 45
degrees quadrants approach I was mentioning is used. By this method, you only get numbers
between -1 and +1 (-π/4 ~ π/4 on the other axis), which is very handy when using fixed point
arithmetic or even integers.

Take a look at the original set of data.

That is incorrect.

Just see the plot of tan(x) vs x:

View attachment 156682

(we cannot plot ±∞ and the graph is cropped).

It is not easy to use a LUT for tan(x).

I disagree, what I said is perfectly correct. You may consider reading my post
before claiming I'm wrong.
I was talking of tan, taken in shifted quadrants (from -π/4 to π/4, see my post
above), in which case you have only tan values between -1 and 1. You can then use
the symmetry to reduce further to 1/8 of a circle (0, π/4).
In this case, it's extremely easy to use a LUT, which gives you values that can be
used in fixed point or integer.
The symmetry around π/4 allows to use tan((π/2) -x) to find the angle in the
interval [π/4, π/2], and then the other symmetries for x and y are just a matter of sign.

Now, for the easy / not easy issue, that's really a matter of taste / ability.
What I meant is that this lookup table is not more difficult to make than any other LUT.

Dora.