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Grasping the intuitive sampling theory (Part 3)

25 Oct 2013  | Michael Dunn

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In part 1, we began to make some intuitive connections between near-Nyquist sampling, the addition of close-frequency sines, and how those signals would interact with perfect LP filters. Part 2 looks at aliasing, filtering, and how they tie in to the concepts discussed in part 1.

As we discussed at the beginning of this series, sampling has nothing to do with digitisation, or quantisation. A sampled signal can remain in the analogue domain, though digitized signals are sampled by nature. So, what exactly does quantisation bring to the mix?

Let's get quantised
What are the pros and cons of turning our sampled signals into numeric form? After all, virtually anything that can be done in the digital domain can also be done in the analogue sampled domain (yes, there have been analogue storage and signal processing chips, often using CCD technology). But like most other analogue technologies turned digital, the vast improvements in performance and implementation that digital brings makes the "choice to quantise" a no-brainer.

The biggest con in going digital is the potential to introduce noise. "Quantisation noise," to be specific. When an analogue signal is sampled, its value at that instant is unlikely to fall exactly on an ADC quantisation level. The difference between the actual signal, and the theoretical value given by its digital representation, can effectively be modelled as noise added to the system.

Figure 1: I've always seen quantisation error thus represented, but it's not really correct. One can't compare the continuous signal to the digitized one. The noise is actually the difference between the signals on a sample-by-sample basis. The two signals could also be compared once the quantised one was converted back to a continuous signal. (Source: Wikipedia).

Of course, AD & DA converters are themselves imperfect, with integral and differential non-linearities. These further compound the noise, and introduce some distortion too. Still, we've learned to live with, account for, and/or mitigate these errors, because the digital advantage is just so great.

Besides, converters have gotten really good too. A 24bit, 90kHz BW chip can be had for a buck. Such parts cannot even hope to live up to their own potential! Consider – 24 bits translates to 16 million levels of quantisation – making each step 350nV for a typical audio line-level signal. The part's own noise swamps out the quantised lsb (least-significant bit) size. That's just an audio example – converters cover the gamut of bandwidths and word sizes.

What kind of SNRs (signal to noise ratios) do we get from different word widths? The simple answer is: multiply by six. Decibels, that is. An 8bit representation provides 48dB. At 16 bits, SNR is 96dB. The 24bit example has a theoretical SNR of 144dB – not easily achieved in the real world.

We've considered converters so far, but once we have the signal in our digital clutches, there are no physical limits to the precision with which we can represent it. We can use 16, 32, or 64 bits, or even floating-point. Noise tends to extreme insignificance at the 64bit level. And as DSP operations tend to increase word widths as they are applied, it's good to have lots of bits available.

But quantisation brings problems. Turn the page and discover some fixes.

Dithering about...
Now that you've acquired an intuitive and practical understanding of sampling and quantisation, it's time to consider some of the non-obvious problems you might run into, and what to do about them.

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