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Countdown to S-to-Z

See the updated 2019 version of this article here.

 

This column is a “Part 2” and it’ll make much more sense if you read “an Excelent Fit, Sir!” first. Well, I hope it will, anyway. In that column, we used Excel’s ‘solver’ functionality to bend a filter transfer function to our will. Or, rather, our customer’s will, since he is much more important than we are.

Recap: we computed a filter for use in flattening out the frequency response of a sensor with a frequency response looking like figure 1:

We ‘turned the response upside down’; extended it (with orders to ‘come back downwards’ again); guessed some filter coefficients, and fine-tuned them with Excel’s solver. The light blue curve in figure 2 shows what we got:

The ‘accuracy’ of our fit is shown in figure 3. It’s rather wiggly; that just reflects the rather approximate way in which we eyeballed the required frequency response off figure 1 in the first place.

Now, our customer doesn’t want to buy big, expensive, high quality capacitors to build an active filter with this response (good, all the more of these rare beasts for me, buwahahah). So, having established the principle of optimising in the analog domain, let’s look at whether need to make any changes in order to use it to create a digital filter.

Well, the first thing to do is to choose a sample rate. The noise bandwidth of our system will be dominated by this equalizing filter, whose response falls at 6dB per octave above about 30Hz. This means that the output noise will be essentially independent of the amount of decimation that we dial into PSoC3’s highly configurable low-noise delta-sigma ADC. If we set the intrinsic sample rate too high, we’ll use power unnecessarily, and may have to watch for arithmetic issues in the digital filter. If we set the sample rate too low, then the inherent response of the decimation filter may have to be accounted for. This response gives both a slight response fall-off in the passband, and can also result in some residual aliasing; this is typical of all instrumentation-grade delta-sigma ADCs.

Well, we need not worry; if we pick a sample rate of 1ksps, the droop in the passband is completely negligible, and the falling response of the sensor itself ensures that aliasing isn’t going to be an issue. Another more advanced digital filter design issue called ‘frequency warping’ can also be completely ignored (something for a future column, methinks).

Even-order harmonics of the AC line frequency will cause a modulation of the incident light level (dang! I’ve given the game away, yes, it is a PIR sensor). We can address this potential interferer with some extra filtering. So, just because we can, I’ve put in another lowpass filter section designed to be flat up to 10Hz (our equalization limit), and I’ve also sneakily dropped in a notch (or ‘zero’, as we Filter Wizards tend to call them) at 100Hz, which is where most of the trouble is going to be, in Europe anyway. I used a Butterworth section, which has a dissipation of sqrt(2); positioned the -3dB point out of the way at 20Hz, and instead of leaving the a2 coefficient at 0. I made it equal to 1/25, which gives a zero at sqrt(25) times the 20Hz normalization frequency, i.e. 100Hz. After a final ‘solve’, Excel gives:

We can’t put it off any longer; let’s make that move from the s-domain to the z-domain. A spreadsheet is certainly a nice warm place for the s-to-z transform to hide. If you feel like doing the algebra yourself, just substitute the bilinear z transform expression for s:

into equation [1] in the first installment of this article, and do the manipulation. Or, you can go and look it up in Lyons, time-poor engineer-style (Understanding Digital Signal Processing, 2nd edition, pp266-267). This gives the coefficients of powers of z in H(z), which translate more or less directly to the actual multiplication factors used in the digital filter sections, depending on the particular topology you use. The Digital Filter Block in PSoC3 can be programmed to implement many different forms of digital filter; we’ll save that detail for another day. What now remains in the last hundred words is to work out the transfer function, and then run it through a simulator to show that it does indeed do what it “says on the tin”.

So that you can check your own work, here are the coefficients in standard equation form that I ended up with for the three filter sections in table 1 (your optimisations may be slightly different):

To see a bigger version of this graphic click here

I’ll wheel out trusty LTSpice to analyse the filter. The block I use is a z-domain biquad model that is actually built using transmission lines as delay elements. It runs cleanly and quickly in both time and frequency domains. I embed the whole circuit in ASCII schematic form in another worksheet in the Excel file; it picks up the design values directly from the s-to-z transform sheet. Just paste the worksheet back to a text file with .asc extension and it opens straight up with LTSpice (figure 7)

To see a bigger version of this graphic click here

And, finally the suspense is over: here’s the response of the final filter (figure 8). The notch at 100Hz can clearly be seen, as can the fall to another notch at 500Hz – half the sample rate – which is characteristic of a digital filter designed with the bilinear transform method:

Space limitations mean this could only be a whistle-stop tour through this design approach. I’ll enthuse more about uses for my new friend the PSoC3 from time to time. What filter problems do you wish you could solve in a single chip? – Kendall.

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