Filter Evolution

We're working on the FPGA signal processing for The Wayback Machine. This is a box that stores user-supplied waveform files and plays them back at some programmable rate.

If we use a DDS clock to determine the playback sample rate, and dump the file data into a DAC, we get the obvious stairsteps in the data, at the sample rate SR. A sine wave at frequency Fs, which was sampled at, say, SR=64Fs, will look pretty stair-steppy on a scope, but not too bad. But a sinewave sampled at something like SR=3Fs will look awful, a jumble not recognizable as a sine wave at all. Spice has a sample/hold, so you could play with this if you're interested.

Mr Shannon said that a lowpass filter will fix this.

My first inclination was to take the last two data values and linear interpolate at some higher rate than F. It can be done like this:

This removes the steps and looks pretty good at high ratios of SR/Fs. But it can't output a value that's bigger than either of the inputs, so we get this problem as the signal frequency approaches Nyquist:

We need a bigger red peak than the interpolator can make. The result is squirmy waveforms and residual AM.

The classic analog transversal filter is...

which is simple if you think about it right: the delayed signal is multiplied by a list tap gains that literally draw the desired filter impulse response.

The digital version is the classic Finite Impulse Response filter

where the Z-1 blocks are just delays, namely a clocked data latch. This can approximate any filter response (as long as a decent ratio SR/Fs is maintained) but the output rate equals the input rate, so it won't help my stairstep situation much, without squashing the system bandwidth.

So we sort of combined the interpolator with the FIR filter:

This is a digital FIR filter, but we change the tap gains Kn much faster than the data sample rate... probably at 64 MHz. This upsamples the data rate and smooths things out, an interpolator on steroids.

This turns out to be a known trick for DSP filtering.

The cool thing is that if we use K0 and K1 and set the rest of the Ks to 0, I get my linear interpolation, at a much higher rate than the customer's SR. K0=1 alone is "filtering off". If we use just the first four Ks, I can approximate a cubic spline curve fit onto four samples, which gives me the red curve slingshot effect that I need.

Adding more K terms makes the filter better, nicer waveforms and less AM as we approach Nyquist, at the cost of more time delay.

The math is an FPGA MAC (multiply/accumulate) operation. The Kn coefficients will be stored in a block RAM lookup table. To keep the table size down, we can break up the M bits into two chunks and do some linear interpolation.