Evaluating the accuracy of an integrator

Question

I'm designing a digital signal processing chain that includes integration of a periodic input followed by decimation/resampling. The processor is integer/fixed point only, so the primary source of error is numerical. I'm simulating the process in python in order to evaluate the accuracy of different algorithms and parameters.

I have a target maximum error of 0.3%, and I'm in conversation with the project owner about how to interpret that, but I'm curious about standard practices for evaluating accuracy in a signal processing chain.

I could take the absolute value of the difference between my signal and the target (in this case a synthesized ideal integral) and divide by the value of the target. But this doesn't work when the signal passes through 0.

If I divide instead by the peak value, I find that the primary contributor to the instantaneous error is a small phase discrepancy. If the two signals have the same shape, but one is delayed by half a sample, I somehow feel like that shouldn't count as much toward the real error.

Answer 1

The standard practice for this is to develop the algorithm in floating point first test it and verify that it meets all performance requirements. Once that's signed off, you can start doing a fixed point implementation and compare the output to that of the floating point reference. In most cases a signal to noise metric is useful, i.e.

$$SNR = 10\cdot log_{10}(\frac{\sum x_{float}^2}{\sum (x_{fixed} -x_{float})^2})$$

You have to make sure that you have a good set of test vectors that cover not only the normal use cases but also the outliers and/or most challenging signals.

Integrators are tricky since they are only marginally stable and have infinite gain at 0Hz. The maximum gain is determined by the lowest frequency in your signal, so some sort of high pass or DC blocker would be useful. You need to know this maximum gain to scale your fixed point implementation properly and avoid clipping.