Sample multiple channels 'simultaneously' with a single ADC: Page 2 of 7

September 16, 2011 //By Kendall Castor-Perry
Sample multiple channels 'simultaneously' with a single ADC
Kendall Castor-Perry (aka The Filter Wizard) explains that despite its title, this is indeed an article about a filtering technique. Using it, you can sample a number of analog input channels sequentially with a single ADC in such a way that the data appears to have been acquired at the same instant in time on every channel. It sounds like magic – but it's just Wizardry!
that the sequence of data words straight out of that ADC goes like sequence {1} and arrives at rate 4Fs:

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 sequences {2}, each now at rate Fs:

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, sequences {3}. This is an example of interpolation:

These streams are all running at 4Fs, and they clearly don’t ‘line up’ any more. 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 Fs and producing an output stream that repeats at 4Fs, we’ve created ‘images’ of the frequency spectrum of our signals, centred on multiples of Fs; see Figure 1, taken from the 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:
The generation of images through interpolation (from 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 Fs. 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 Nth sample (“decimation by N”) from a stream of data. It’s sometimes called downsampling, especially in

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