I’ve been following a conversation at another site for the past week or so. I’m content standing on the sidelines while a couple of authorities beat up on each other. What they’re arguing about comes down to theory vs. practice. The respective posts stake out contrary positions on the merits of the Shannon-Nyquist Theorem AKA The Sampling Theorem. The practical engineer (and individual with plenty of experience and knowledge about building high-end audio circuits) contends that the Nyquist Theorem is broken. The other expert addressed the issue from a theoretical position and is emphatic that the Nyquist Theorem is not broken. So who’s actually right? Is the notion that a continuous analog waveform can be properly sampled “only if it does not contain frequency components above one-half of the sampling rate” true or false?
Let’s start with the definition of proper sampling. It’s actually pretty straightforward. If we sample a continuous signal in some fashion and then can precisely reconstruct the analog signal from the samples we took, then we must have done the sampling properly. Regardless of whether the input signal is a pure sine wave or a complex tone, if we can reproduce the source analog signal from the samples, then we’ve properly sampled the original source.
That’s the theory and it’s still true no matter what the engineers and designers of the world say to the contrary. But there are some real world considerations that must be take into account. So I thought it might be beneficial to explore the Nyquist Theorem once again.
Here are some cases that should help to clarify the basic concepts of The Sampling Theorem. This is the theory part of the discussion. I’ve placed four sine waves that illustrate the relationship between the source analog signal and the digitized version in the figures below.
Figure 1 – An analog signal that is a constant DC value, a cosine wave of zero frequency.
This is a properly sampled continuous signal, although one that has no frequency. It is a simple steady state DC offset that can be described by a series of straight lines between each of the samples. Thus all of the information needed to reconstruct the analog signal is contained in the digital data. Clearly, this unique signal is represented 100% by the samples.
Now take a look at a periodic waveform.
Figure 2 – This illustration shows a sine wave with a frequency that is .09 or 9% of the sampling rate.
The sine wave shown in Figure 2 has a frequency that is 9% of the sampling rate. This might represent, for example, a 900 cycles/second sine wave being sampled at 10,000 Hz. In other words, there are 11.1 samples taken over a single cycle of the wave. This situation is more complex than the previous case. A straight line doesn’t exist between the sample values. So the question is, do the samples accurately represent the original source sine wave? The answer is yes, because no other sine wave, or combination of sine waves, will produce this pattern of samples. These samples correspond to only one analog signal and therefore the analog signal can be exactly reconstructed. This is proper sampling once again.
It’s pretty clear that the sine wave is identified clearly in the samples of Figure 2. But what happens when the frequency of the source is substantially higher?
Come back tomorrow and we’ll take a look.
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