Sampling theory, the best explanation you’ve ever heard—Part 1

I’ll start by giving away secrets first:

1. Individual digital samples are impulses. Not bandlimited impulses, ideal ones.
2. We know what lies between samples—virtual zero samples.
3. Audio samples don’t represent the source audio. They represent a modulated version of the audio. We modulated the audio to ensure points #1 and #2.

Well, not secrets, but many smart people—people who’ve done DSP programming for years—don’t know these points. They have other beliefs that have served them well, but have left gaps.

Let’s see why

Analog audio, to digital for processing and storage, and back to analog

Component details—first the analog-to-digital converter (ADC)

The digital-to-analog converter (DAC)

Analog to digital conversion, and conversion back to analog are symmetrical processes—not surprising.

But we can make another important observation: We know that the bandlimiting lowpass filter of the ADC is there as a precaution, to ensure that the source signal is limited to frequencies below half the sample rate. But we have an identical filter at the output of the DAC—why do we need that, after eliminating the higher frequencies at the beginning of the ADC? The answer is that conversion to discrete time adds high frequency components not in the original signal.

Stop and think about this—it’s key to understanding digital audio. It means that the digital audio samples do not represent the spectrum of the bandlimited analog signal—the samples represent the spectrum of the bandlimited analog signal and additional higher frequencies.

To understand the nature of the higher frequencies added in the sampling process, it helps to look at the origins of sampling.

Next: We explore the origins of sampling in Part 2

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4 Responses to Sampling theory, the best explanation you’ve ever heard—Part 1

1. How did you know this would be the best explanation I’ve ever read? I think it might be…

• Nigel Redmon says:

That’s great to hear, Brian! I took a little heat for naming the series that, but explained that my hope was that, for some people at least, it would be the best explanation they’ve ever heard. Nice to hear that it may be that for you.

2. Noisey says:

I understand that the incoming continuous information is sampled say 44100 times per second (sample rate) and then the amplitudes mapped onto a kind of grid (bit depth). How does it preserve the frequency information?

• Nigel Redmon says:

You’ve probably read the other parts to the series, and I hope that has answered your question.