So, what’s the problem sequential sampling? After all, if you’ve got, let us say, four channels that you need to sample at some rate F_{S}, and an ADC that’ll sample fully at 4F_{S} and that can be driven through a 4‑way multiplexer, then isn’t that a nice economical way of getting the job done?

Here’s the catch. If you need to do any sample-by-sample processing that involves data from more than one channel, you’re going to have to ‘pretend’ that the readings you took from different channels at different times actually happened at the **same** time, so that you can combine them in a calculation.

Say you want to calculate the instantaneous energy being absorbed in an AC power circuit. You sample the voltage and the current and multiply those readings together. The result, when multiplied by the length of time you assumed the samples were valid over—essentially, the sample period—gives you the energy. Integrate these readings over time and you accumulate the total energy. Hey Presto! It’s an electricity meter!

But! If you use a multiplexed ADC, *you didn’t take the voltage and the current readings at the same time*. By pretending that you did, you’ve introduced a time delay between the voltage and the current sequences. This is equivalent to a small phase shift, which really messes up any calculation that’s sensitive to the phase difference between voltage and current (like, um, power).

The error doesn’t come from the sampling process itself, but from our choice of how to interpret and manipulate the sample values. Let’s set up an example with four channels (much of this material is taken from a real project I architected for a super-precise electricity meter built with Cypress Semiconductor’s PSoC 3).

Let’s refer to the four channels by the letters W through Z, and with sample number i written as W(i), and so on. With our ADC multiplexing around the four channels in order, it’s pretty clear that the sequence of data words straight out of that ADC goes like **sequence {1}** and arrives at rate 4F_{S}:

Sequence {1} Samples straight from ADC

Our problems come when we take the simplistic approach of breaking this stream up into four separate streams in which we assume that samples with the same index can be treated as simultaneous, as in the set of **sequence {2},** each now at rate F_{S}:

Sequence {2} ADC samples broken into four streams

The standard way of breaking up sequence {1} is to ‘stuff’ zeros into the locations where other channels are being sampled, to get four streams, **sequence {3}.** This is an example of interpolation:

Sequence {3} Streams “stuffed” to preserve timing relationships

These streams are all running at 4F_{S}, and they clearly don’t ‘line up’ anymore. It’s meaningless to do any arithmetic on values that are vertically aligned when written out as in {3}, i.e., happening “at the same time.” One of any pair is guaranteed to be zero, and therefore so is their product.

What’s more, by taking the readings from each channel at F_{S} and producing an output stream that repeats at 4F_{S}, we’ve created ‘images’ of the frequency spectrum of our signals. The images are centered on multiples of F_{S}, as shown in **Figure 1**, taken from the very useful National Instruments website. This high frequency information wasn’t present on our W~Z inputs originally. Haven’t we made things even worse now?

**Figure 1** Interpolation results in frequency spectrum “images.” (Image source: National Instruments).

* *All is not lost. There’s a method for getting rid of the high frequency rubbish and getting our sample rate back down to F_{S}. It’s called decimation, and to do it we use a decimation filter.

The term ‘decimation’ has drifted in meaning from its origin as a translation of a dreadful punishment exacted by the Romans on disobedient or underperforming army units. In the rather less blood-thirsty discipline of signal processing, it refers to the selection of every N^{th} sample (“decimation by N”) from a stream of data. It’s sometimes called down-sampling, especially in the audio world. In our example here, N=4, of course.

Decimation involves reducing the sample rate, and we know that we run the risk of aliasing, unless we take steps to remove those components from the signal whose frequency is higher than 0.5F_{S}. In other words, a decimation filter is really just a digital antialiasing filter.

There are several types of filter topology we could choose to do the job in our case. Here I’ll use a finite impulse response (FIR) lowpass filter, and the benefits of that choice will become obvious later. We’re going to filter the sequences in {3} in such a way that we can then decimate them from 4F_{S} down to F_{S}. So, we need a lowpass filter with a good stopband that begins at 0.5 F_{S} when it’s operating on 4F_{S} data.

The frequency response of a suitable 128‑tap FIR filter is shown in **Figure 2**. It’s a linear phase filter designed with the ever-so-easy “windowed sinc” method, using nothing more complicated than a spreadsheet. The stopband is never poorer than 75 dB down, which limits potential aliasing errors to a reasonable level for this application. You might spot that the passband gain is 12 dB, i.e. 4x. The reason for that choice of scaling factor will also become clear later.

**Figure 2** The magnitude and phase response of the initial 128‑tap FIR filter.

So, let’s take four of these 128‑tap filters, one for each of the sequences in {3}. Samples go into, and come out of, each filter at 4F_{S}, to make new sequences shown in {4}. Now, none of the samples are zero.

Sequence {4} Output of FIR filters

Because we’ve filtered off the high frequency stuff, we can now decimate each stream by a factor of four, meaning that we only take every fourth output sample, for example to give the sequences in {5} at F_{S}.

Sequence {5} Decimated FIR output streams

So far, we could have used any form of digital filter, but using an FIR offers a great simplification. Look at the sequences in {3} again. Three-quarters of the data values are zero. Each output value from the 128‑tap FIR filter used to create the sequences in {4} is the result of adding together 128 individual products of input data value and coefficient value. But there’s no point doing the multiply inside the filter structure when we know in advance that the data value is going to be zero. There’s a large potential saving here; how do we exploit this?

The solution is to partition the FIR filter’s impulse response into four subsets that match the positions of the non-zero samples in the sequences in {3}. This turns our 128‑tap filter into four distinct 32‑tap sub-filters. Then, we use the ‘misaligned’ input sequences in {2} as the inputs to these filters, each just running at F_{S}. **Figure 3** shows how we split up the impulse response into the four ‘phases’ that get assembled into the four different filters. We still get the sequences in {4} as the output; it’s just that we’ve eliminated all the redundant multiplies by zero.

**Figure 3** Four 32-tap sub-filters with four different phases yields the same impulse response as the 128-tap FIR filter.

Now, instead of implementing four (identical) 128‑tap filters running at 4F_{S}, we just have to implement four (different) 32‑tap filters each running at F_{S}. That’s a pretty significant savings. We never actually need to create and decimate the sequences in {3}. The new filters have the original frequency response, but each just hits its stopband right at Nyquist.

This is an example of **polyphase decimation**. There’s a delay difference between each of these sub-filters, and it’s **exactly equal** to the time error caused by our original misalignment of the sample data in creating the sequences in {2}, which was one-quarter of the sample period of the output data between channels. This delay difference therefore ‘realigns’ the data in the time domain, eliminating the error. How cool is that?

Because we divided our impulse response into four sets, each ‘carries’ one-quarter of the signal. Figure 2 showed a gain of 4x to compensate this right from the start. You could also start with a unity gain filter and just multiply all the final coefficients by four at the end, with the same result.

**Figure 4** The response curves of the four sub-filters show overlapping magnitude and staggered phase.

**Figure 4** shows the frequency and phase responses of the four sub-filters; they now have unity gain. The amplitude responses are identical, but the phase responses are skewed apart as you’d expect from four filters that have slightly different delay. **Figure 5** shows the group delay values of the four filters directly, along with an expanded passband plot. The simulations were taken for F_{S} of 3000 Hz, which was the sample rate we used in the metering work.

**Figure 5** This close-up of the passband gain and the group delay (dotted) responses for the sub-filters shows the phase skew.

If you’ll pardon the cliché, the proof of the pudding is in the eating. **Figures 6** and **7 **are “before and after” plots for a segment of data from a four-channel .wav file acquired from a multiplexed-ADC project built on a development board. The source signal consists of equal amounts of 50 Hz and its first 21 harmonics, phased randomly. It was generated by playing out a synthesized .wav file through a PC’s line output and applied to the four multiplexed inputs (through carefully matched input AC coupling networks). We took a lot of care to ensure that everything on one channel settled really well before switching over to the next multiplexer channel (otherwise you get crosstalk, which can cause significant accuracy problems in a meter; perhaps a future article will quantify this).

**Figure 6** Sequentially acquired data from a single input signal, plotted together, shows how the timing mismatch yields different-looking data curves.

Because the four channels aren’t sampled at the same time, the data points for the four streams plotted together in Figure 6 aren’t identical. This extreme waveform was chosen to make a visual point, even though it would be a rather unusual signal to apply to an actual electricity meter.

**Figure 7** shows the outputs from the four sub-filters, all fed with the samples of Figure 6, implemented using the ‘Digital Filter Block’ in Cypress Semiconductor’s PSoC 3. Remember that the four filters have identical magnitude responses and differ only in that their group delays are spread apart by units of a quarter of a sample time. Passage through these filters rather magically reshapes the applied waveforms so that the output signals are all identical, to within very close tolerances – close enough for precision electricity metering, as we showed.

**Figure 7** The four channels after passage through their ‘realignment filters’ yield matching data curves.

Are there downsides to this approach? Well, we’ve put each channel’s data through a 32‑tap filter, and the average group delay experienced by the channels is that of a 32‑tap linear phase filter running at F_{S}, and that’s 16/F_{S}. You might want that to be rather lower, especially at lower sample rates or in a closed-loop system (it’s not at all critical for the metering application). If you don’t need such a wide band of flat frequency response, you can reduce the number of taps in the initial filter to reduce the delay.

If you really do need the bandwidth, another approach is to use a **minimum-phase** FIR filter as your starting point. In sacrificing linear phase response, the low frequency group delay can be greatly reduced. A method of creating minimum-phase FIR filters from a linear phase prototype will have to wait for a future article…

This work was done to support our electricity metering analysis, but it has quite a few other applications. If you’re analyzing structures, for instance, the cross-correlation terms in a vibration analysis will be useless unless you can rely on simultaneity in the data sets. One intriguing application I also examined was a gunshot detection system that sampled eight wideband microphones at 60 ksps each. The Digital Filter Block implemented eight 12‑tap sub-filters derived from a minimum phase prototype, at an aggregate rate of 480 kHz. You can’t do that in your average microcontroller – even a fancy 32-bit one!

So, remember, you don’t need multiple ADCs just because you want data samples that behave as though they were taken simultaneously across all channels. One single high-quality ADC with a good input mux and some digital Filter Wizardry, and you’re all set. Hope we’re all aligned on that!

**Editor’s note**: A version of this article originally appeared on EE Times-Europe.

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