Re: [Discuss-gnuradio] quadrature_demod losing weak signals

[Kevin Reid]

On Nov 30, 2013, at 8:22, Tom Rondeau <address@hidden> wrote:

> On Sat, Nov 30, 2013 at 10:55 AM, Kevin Reid <address@hidden> wrote:
>>    /* don't divide by zero! */
>>    if((y_abs < 1.5E-5) && (x_abs < 1.5E-5))
>>      return 0.0;
>> 
>> Since the preceding multiply_conjugate should square the magnitude, this 
>> choice of epsilon accounts for the threshold I found empirically.
>> 
>> Given this, I'm going to file an issue requesting that this epsilon be 
>> reduced. (The only alternative that comes to mind is to place an AGC block 
>> before the demodulator, which seems unlikely to be better from a numerical 
>> perspective).
> 
> Agreed. This threshold should be reduced. Can you experiment with that
> threshold to see if there's any lower limit that might be problematic
> (my default threshold, for some reason, is 1e-20; would want to make
> sure that doesn't cause something else in the math to blow up).


I have modified a copy of fast_atan2f to have the test "(y_abs == 0.0f) && 
(x_abs == 0.0f)" and plotted the results of fast_atan2f(k*sin(angle), 
k*cos(angle)) against both angle and atan2f(k*sin(angle), k*cos(angle)) at 
various exponents for k.

At small magnitudes fast_atan2f is a _better_ approximation to regular atan2f 
than it is at magnitudes close to 1, and the error compared to the original 
angle (which should partly be attributed to representing sin and cos values) 
has the same patterning at all scales. At no point does fast_atan2f emit 
complete nonsense; it is consistent all the way down to the point at which both 
y and x are rounded to 0.0.

My conclusion is that fast_atan2f is as good as it ever is all the way down to 
the minimum representable values (~ 1e-45), and that one might as well revert 
the code to an equality test with 0.0 (or perhaps "y_abs <= 0.0" if a linter 
needs to be satisfied).

That said, I'm no expert on numerics and I might well have missed some 
particular undesirable behavior that should be avoided. Therefore, I will 
attach the tools I wrote to test the behavior. "main.cc" generates a table of 
error values (the parameters are hardcoded), and "run.sh" will build it and 
plot the values using the gnuplot script.

(Note that there is a special case in main.cc to emit a constant nonzero value 
when x and y are exactly zero after being scaled down; that value is intended 
as an out-of-band marker and is not an actual error.)

main.cc
Description: Binary data

plot.gnuplot
Description: Binary data

run.sh
Description: Binary data

-- 
Kevin Reid                                  <http://switchb.org/kpreid/>