In “Why DAC and ADC responses droop,” we looked at the two common places where the sinc() function crops up in a data acquisition system. A DAC with zero-order hold has a frequency response that follows a sinc() response, falling as signal frequency rises. Commonly, instrumentation-grade delta-sigma ADCs have responses that are higher powers of the sinc() function. This is a Filter Wizard column, so you just know that filters are going to be involved in the remedy for this response droop. In the ADC case, a filter was the cause of it in the first place!

In a system that suffers from both types of droop when measured from input to output, we could perhaps just try to apply one fix for both. But I recommend that you treat the input side and the output side separately. That’s because your data acquisition system probably shares data with other systems and analysis computers, so you want both the ‘record’ and ‘replay’ processes to be correct. I’ll assume from now on that you do have a droop problem on one or both ends. If you don’t, it’s either because you already recognized the issue and fixed it, or you used components (for instance, audio-grade converters) that didn’t suffer from the issue in the first place. I hope the rest of the column will still be interesting and useful for you, if that’s the case!

Let’s look at the DAC side of things first, just because that’s the order implied by the title. Matters are broadly similar for the ADC case, as we’ll see. In a typical data ‘replay’ path, your data might pass through a digital filter on the way to the DAC. It might also go through an analogue ‘reconstruction’ filter after conversion, before it reaches the final output of the system. If you have neither an analogue nor a digital filter in your system (harrumph! Why?), and you’re suffering from droop, then you will need to add something to your system. If there is a filter in the signal path, the chances are that it can be altered to compensate for the droop.

I’ll start with the case that I’m not going to cover here (if that makes sense). If you only have an analogue filter, it will need to be redesigned to have a different, rising frequency response. Whether this can be done, and how to do it if it is possible, is just too complicated to go into here. In the “good old days,” I designed many reconstruction filters with a peaked response to correct for a DAC’s sinc() droop, while still suppressing the ‘images’ from the sampling process. Rather than using tables or coefficient equations, the design usually requires careful ‘fitting’ of the response with a dedicated computer program. There are certainly some neat ‘short cuts’ to get you to a good initial solution. If you need to fix a replay droop problem by altering your analogue reconstruction filter, drop me an email or a LinkedIn message.

Incidentally, on a historical note, modern active filter design was heavily driven in the 1970s by the needs of the telephone industry, which was making a transition to digital telephony at the still-standard sample rate of 8 ksps. The highest required audio frequency is 3.4 kHz, which makes the frequency of the lowest sampling image (images are the additional high frequency components that appear in the reconstructed output signal) equal to 4.6 kHz, which is only a factor of 1.35 away. Separating these two frequencies requires fairly sharp filtering, and this was typically done using active elliptic lowpass filters. Had they chosen a higher sample rate for digital telephony, the filtering requirements would have been alleviated, and much less effort would have gone into active-filter-design methodologies. The need for Filter Wizards would have been reduced, and I might have ended up doing something completely different!

OK, nostalgic waffle mode off, let’s look at what I can contribute to droop management on the replay side without tweaking the analogue filtering. The key question: is there a digital filter in your signal path? The digital filter might be implemented in dedicated hardware, such as the Digital Filter Block in Cypress’s PSoC3 and PSoC5LP devices – or your fave DSP. It might be buried in the code that creates or massages the data that you’re reproducing.

No digital filter? Then here’s the easiest way to fix things up. You can add a very simple digital filter – so simple that it can be done in software – that can give a very good correction of sinc() droop up to a useful fraction of the sample rate. It’s the simplest case of what I call the “zero-adding” method. You might recall from the discussion in “Now synthesize your filters using high-school algebra” that a ‘zero pair’ can be expressed as a second-order polynomial in the unit delay variable z^{-1}. The effect that this small filter has on a system’s frequency response depends on the coefficients. In the earlier FIR articles, I looked at using them to produce deep nulls in the stopband. Here we’ll use just one zero pair (giving a second-order polynomial) to provide a gentle *boost* to the passband frequency response, without doing significant harm anywhere else.

**Table 1** Here are the z^{-1} coefficients for the “zero-adding” method.

**Table 1** contains all you’re likely to need (all thanks to the Million Monkeys). It gives the coefficient K of the z^{-1} term in the added second order polynomial when the constant and z^{-2} terms are equal to 1. To leave the overall gain unaffected, we need to scale all the coefficients by 1/(K-2). So the added term has the form:

with positive K as given in the table. The table also shows the peak dB error to expect for input frequencies up to the column limit frequency.

To use the method, first decide the fraction of the data sampling frequency up to which you’d like to fix up the droop, then pick the column in Table 1 with the next-higher value. The row you’ll use for regular DAC droop correction is the sinc1 row. The accuracy of the correction falls as the fractional frequency range rises. For sinc1, Table 1 extends to Fs/3. You really shouldn’t be expecting an otherwise-unfiltered system to be functional at higher frequencies than that – it’s too close to the sampling frequency and you will be getting significant amounts of image frequency in your output signal.

Let’s say your sample rate is 100 ksps and you want a good flat frequency response up to 20 kHz, which is Fs/5. Choose the Fs/4 column and the sinc1 row of Table 1, then read out K = 21.04096. Calculating the coefficients in equation 1 gives the filter calculation you need to do for the ith sample, whether in hardware or in the software flow, as:

output(i) = -0.05252*input(i) + 1.10504*input(i-1) -0.05252*input(i-2) [2]

If there’s already a gain scaling operation in your system, you can roll equation 1’s constant factor of (K-2) into that calculation instead. Table 1 indicates that the worst-case approximation error in this case is ±0.045 dB up to Fs/4, so this is a pretty good fix for the droop. The frequency response of equation 2 right up to Nyquist is shown in **Figure 1**. It continues to boost at frequencies above 20 kHz, but not excessively.

**Figure 1**: This graph shows the frequency response of 0.5252 + 1.10504z^{-1} 0.5252z^{-2}.

If there is a digital filter in your system and you nevertheless still have a droop issue, then it’s finger-wagging time for someone. The most common function for such a filter is to help out with interpolating or ‘upsampling’ your replay data so that you can apply it at a higher sample rate to the physical DAC. One reason why you do this is to mitigate the droop problem in the first place! So someone should have fixed it already.

If your digital filter implementation allows you to add an extra zero pair to the transfer function, just use the method described above. The addition is straightforward when you’re using an FIR filter; it increases the tap count by two. If you know the existing coefficients, a simple spreadsheet calculation can do the multiplication needed to get the new coefficients. In the PSoC Creator Filter customizer, it can be done automatically by just adding the new three-tap term as an extra FIR filter stage.

If you’re using an IIR filter architecture and can add another biquad, you just use the polynomial created above as the numerator, then define a do-nothing denominator (with the z^{-1} and z^{-2} terms both zero); a slight waste of computing cycles but it will work fine. You can also use the zero-adding method if you can see that one of the numerators in your IIR filter only has one term, in which case you just discard it and add the new zero pair.

If you are using an IIR filter and can change the coefficients but not the order of the filter (i.e. the number of biquads), then you need a slightly different method, the “zero-shifting” method. Instead of just adding a zero pair with the right amount of boosting frequency response, the zero-shifting method removes an existing zero pair, replacing it with one that has less droop. The effect of this is to boost the frequency response without changing the filter order.

**Table 2**: This table shows the z^{-1} multiplying factors for the zero-shifting method.

The zero-shifting method could work in principle with any choice of initial zero pair. To simplify matters and permit the use of another precalculated table, the approach given here assumes you can find and remove a zero pair at Nyquist. In an IIR filter, such a pair is the result of the bilinear transform acting on a pair of s-plane zeroes at infinity (sorry for the sudden burst of filter jargon!).

It’s very unusual to find a workable IIR transfer function that doesn’t have at least one of these as a biquad numerator. For instance, this is the case for nearly all the lowpass and bandpass IIR filters produced by the filter customizer in PSoC Creator, which uses the standard bilinear transform method to create its IIR functions.

The only exceptions are first-order lowpass and second-order bandpass filters. If you study the numerators of most other lowpass or bandpass filter functions from PSoC Creator or similar filtering packages, you’ll find that at least one of them will be have the form (C + 2Cz^{-1} + Cz^{-2}) where C is a constant. The compensation method here involves changing just one of these terms to be (C + C * (pick a K from Table 2) * z^{-1} + Cz^{-2}). I could hardly have made it easier for you; you just have to change *one coefficient* in your factorized transfer function. If you can arrange it, I think it’s a good idea to start with a transfer function that has at least two zero pairs at Nyquist. This will help to preserve good stopband rejection.

Adjusting coefficients in this way increases the DC gain of the filter (it’s proportional to the sum of the numerator coefficients). To get back to the original gain, you’ll need an extra scaling factor of 2/(K+1), which you can either multiply into one of the numerators or into your overall system gain constant.

Using the same frequency example as before, let’s now assume that you initially have a nice flat passband 0.02 dB ripple eighth-order Chebychev filter in the system, cutting off at 20 kHz. The passband response is shown in **Figure 2**. The coefficients of the four biquad sections, pasted straight out of the PSoC Creator customizer, are:

0.0323762893676758

0.0647528171539307

0.0323762893676758

-0.823582649230957

0.41172194480896

0.173670530319214

0.347340822219849

0.173670530319214

-0.588811874389648

0.612842559814453

0.11561107635498

0.231222152709961

0.11561107635498

-1.01051115989685

0.286822557449341

1.40126466751099

2.80252933502197

1.40126466751099

-0.452846765518188

0.857346057891846

The first three numbers of each quintuplet are the numerator coefficients, and you can see that they all follow the (C + 2Cz^{-1} + Cz^{-2}) pattern. If we go to Table 2 and find the intersection of the sinc1 row and the Fs/4 column, it tells us to multiply one of the z^{-1} coefficients – it doesn’t matter which one – by K = 1.269685. Doing this results in the response shown in **Figure 3** – though here, the customizer has helpfully adjusted the gain to ensure that the peak magnitude doesn’t exceed 0 dB. This highlights an important issue, namely that by raising the gain of the filter at some frequencies, we run the risk of premature overload as the digital path clips early. The droop caused by the zero order hold at the output of a DAC is ‘real’ attenuation, and you need to feed a larger digital signal into the DAC in order to get a given magnitude of sinewave component out. This takes us into the realm of digital filter overload, which is a great subject for a whole Filter Wizard in itself (duly noted).

**Figure 2**: This graph shows the (flat) frequency response of the n=8 Chebychev lowpass filter.

**Figure 3**: The response of Figure 2’s filter is shown here, with zero-shifting sinc^{1}() compensation.

The stopband rejection of Figure 3’s filter is not significantly degraded. That’s because the original Chebychev filter had all four zero pairs at Nyquist, and we only shifted one of them.

Incidentally both the zero-adding and zero-shifting methods are linear phase, when implemented as described here. Changing the numerator with the zero-shifting method doesn’t change the filter’s group delay at all. The zero-adding method increases the group delay by exactly one sample time.

What if you have an FIR filter that already has the maximum number of taps that your hardware or firmware can cope with, and you really can’t stretch it by another two to use the zero-adding method? If you have access to a suitable ‘ordinary’ FIR filter design program, one approach is to design a new starting filter with a reduced tap count (by two or more) compared to your present filter. If this can still meet all the passband and stopband requirements you originally had, use this as your new starting point and apply the zero-adding method as described previously. Some filter design programs will even do the droop compensation for you, but I insist that you at least try to do it once yourself before you click that box!

The toughest gig is when you have the coefficients, but no access to the design methodology that produced them. If you need to keep the tap count the same but change the existing response slightly to rectify a droop issue, you will have to roll your sleeves up and do it the hard way. In “Analyze FIR filters with high-school algebra,” we saw how an FIR coefficient set can be treated as a polynomial whose roots can be found, and the zero pair factors calculated from them. That’s what you’d do here. Paste the filter coefficients into a root-finder program or spreadsheet (such as this) and find the roots.

If you find a pair of zeroes very close to Nyquist (a pair of roots with a value very close to 1), then you can use the zero-shifting method “as-is.” If so, replace them by the polynomial for the new, shifted zero pair (you don’t need to know the roots of that), and multiply everything back up to get a new set of coefficients. If there are no zeroes very close to Nyquist, there is a technique that can modify the filter you’ve got in order to force this. It’s all rather long-winded so I won’t give an example here. If anyone out there thinks this method might help them solve a particular problem, just let me know.

What about the ADC side of things? The main difference compared to the DAC case is that a droopy delta-sigma ADC usually has a response with a higher power of sinc(). That’s why tables 1 and 2 run right up to sinc^{5}().

Feeling active? Don’t be tempted to correct the droop of a delta-sigma ADC by using a ‘peaking’ analogue filter in front of the ADC. This will give you horrible overload problems and will further degrade the aliasing tendency of the sinc() decimator. It will be hard to design and may well spoil your system’s offset and noise performance too. I hope you understand how hard it is for me to advise you not to use an analogue filter, but there it is!

Correcting the ADC droop digitally proceeds in the same way as for the DAC compensation, and both the zero-adding and the zero-shifting methods can be employed. Now, if you have significant droop, chances are you have a sinc()-responding decimator. That will cause some aliasing, which is bad news in frequency-domain applications. If you want to use an instrumentation-grade delta-sigma ADC for audio or other frequency-domain applications, you really ought to have a better grade of decimation filter designed into your system, then run the ADC at say 2× or 4× the desired output sampling rate. For instance, the PSoC Creator filter customizer has a dedicated lowpass filter type for the Digital Filter Block, providing sinc^{4}() correction for the ADC. This is a good place to start if you’re using PSoC3, PSoC5LP, or any other delta-sigma ADC with a 3rd order modulator.

If you really can’t afford to change the system sample rate but do need that flatter frequency response, you can use the zero-adding method to build a response compensator, as shown for the DAC case. It’s really not the optimum solution, though, it’s just a Band-Aid, as my US friends would call it. For higher powers of sinc(), a single section doesn’t provide particularly accurate correction; up to Fs/4, it’s accurate to a bit more than a quarter of a dB.

The zero-adding method also works fine for ADC use. Reading along the sinc4 row of Table 2, in the Fs/4 column we find K = 3.42206. If we take the Chebychev filter example of Figure 1 and multiply one of its numerator z^{-1} coefficients by 3.42206, we get the response shown in **Figure 4**. This will correct the droop of our ADC to within 0.146 dB. Again, the PSoC Creator filter software has brought the peak gain down to 0 dB.

**Figure 4**: This graph shows the response of Figure 2’s filter with zero-shifting sinc^{4}() compensation.

With luck, this article and its prequel have given you a picture of where response droop comes from and how you can make it go away. These simple techniques can often save you from having to do battle with a full-blown filter design. They also give an insight into what the intricate moving parts of a transfer function can be made to achieve. So, filter fans, lift up your responses with a little snap of polynomial magic. Don’t let droop spoil your day. And I mean that most sinc-erely! best / Kendall

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