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This blog post is the third part in a series presenting an overview of the theories and practices of analog-to-digital conversion. In the previous post, we briefly discussed how analog-to-digital converters (ADCs) perform the conversion of continuous analog signals into a sequence of digital numbers (samples). In this post, we’ll examine the process of sampling an analog signal that varies in time, and how the interpretation of the results can be significantly altered by the dynamic characteristics of that signal.

Bandwidth-Limited Analog Signals

Towards the end of the previous post in this series, we mentioned the Nyquist-Shannon sampling theorem, one of the fundamental principles that make it possible to transform an analog signal into a completely equivalent digital counterpart. The theorem states that a continuous bandwidth-limited signal (or function) can be completely defined or reconstructed from an infinite sequence of samples that describes it. This means that, in principle, the sequence of samples coming out of an ideal ADC should entirely describe the incoming analog signal and that an equivalent continuous analog signal can be reconstructed from the sequence of discrete samples. The key factor to ensuring that this will be possible is the condition in the theorem that states that the signal be bandwidth-limited. We’ll examine the meaning of this condition in the next few sections.

To get a more intuitive feel for what the sampling theorem actually implies, we’ll use a graphical metaphor that everybody will be familiar with. This metaphor might be considered a bit simplistic, but it can really help with understanding some fundamental principles that might otherwise be buried in more elaborate mathematical explanations.

Imagine doing a flipbook-type animation of a wheel slowly rotating in a clockwise direction. Each page of the book represents a sample of the position of the wheel as it slowly revolves around its center axis. This is exactly what a movie camera does: it takes 24 or 30 pictures (samples) per second to capture a series of discrete images that can be played back in sequence to recreate the original movement in the scene.

In our flipbook example, the wheel revolves clockwise at a rate of one turn per second and the drawings are updated eight times per rotation. That is, the wheel moves by 1/8 of a turn (+45°) for each page of the book. The resulting sequence of drawings might look like the following:

wheel clockwise revolution updated eight times per rotation

If the sequence were to repeat, it’s easy to see how the eye would interpret this as a wheel spinning in the clockwise direction at a rate of one revolution per second.

If we look at the rotating movement of the wheel from a different perspective, we can observe that the movement can be decomposed into separate horizontal and vertical components, as shown in the following figure. We can see that both components fluctuate at the revolving frequency of the wheel and that they follow a sine-wave pattern.

horizontal and vertical components fluctuate at the revolving frequency of the wheel and follow a sine-wave pattern

To simplify things, we’ll concentrate on the vertical motion of the wheel as it spins around its axis. The following figure plots the vertical motion of the wheel, and its sampled position is marked using a    symbol. If we connect the dots between samples, we can clearly see the underlying sine-like waveform.

 vertical motion of the wheel underlying sine-like waveform

Now imagine that instead of revolving by 1/8 of a turn for each drawing, the wheel now rotates by 7/8 of a turn (+315°). The resulting sequence of drawings in our flipbook now looks like the following:

 wheel rotation by 7/8 of a turn (+315 degree)

Even though the wheel is quickly spinning in the clockwise direction, what the eye will perceive is a slow counter-clockwise rotation! How can this be possible? The answer is easy to understand, as shown in the figure below, where the time axis has been compressed to show multiple cycles of the wheel’s fast vertical motion (shown with a dotted line):

 aliasing phenomenon - reconstructed signal is different from the original signal

As you can see, the underlying pattern connecting the sampled positions also looks like a slow varying waveform, even though the actual rotation is much faster. This phenomenon is known as aliasing because the reconstructed signal is different from the original signal.

If we were to look for a way to prevent aliasing from occurring when sampling the movement of the rotating wheel, we would observe that the phenomenon starts to appear at the exact moment when the wheel starts revolving by more than 180° between samples. As long as the wheel rotates by less than 180°, it will always be possible to exactly define and reconstruct its original movement.

This is what the bandwidth-limited condition in the Nyquist-Shannon sampling theorem actually means. It means that we must sample a signal (the rotation of the wheel in our example) at least twice per its highest cyclical component to prevent aliasing from occurring. The term highest cyclical component means the highest frequency component of the signal, which by definition is also called its bandwidth. The reciprocal of this is that, given a known sampling frequency, the bandwidth of a signal must be smaller than half this frequency to prevent aliasing from occurring (thus the bandwidth-limited condition imposed on a signal by the Nyquist-Shannon sampling theorem).

Welcome to the Nyquist Zone

Let’s take a closer look at what actually happens as the rotation frequency of the wheel is gradually increased beyond the Nyquist-Shannon (usually shortened to just Nyquist) bandwidth limit. The following table summarizes the perceived rotation speed for various increments ranging from +1/8 of a turn to +8/8 of a turn between samples:

 perceived rotation speed for various increments ranging from +1/8 of a turn to +8/8 of a turn between samples

 As expected, we see that for increasing values below +4/8 (one half turn), the motion is perceived as increasing in frequency in a clockwise direction. +4/8 is a special ambiguous case where the samples are frozen at identical values and the exact rotation can’t be resolved. Above +4/8, the motion will be perceived as moving in a counter-clockwise direction, but decreasing in frequency as the increments get closer to a full turn per sample. +8/8 is similar to +4/8, as the samples are all identical and no motion can be determined for the wheel.

The following figure presents, in a graphical manner, what’s happening when frequencies are aliased (CCW stands for counter-clockwise wheel, and CW for clockwise wheel):

 what’s happening when frequencies are aliased

Frequencies that are below half the sampling rate are perceived normally, whereas frequencies above half the sampling rate are shifted on the negative side of the axis, causing them to be negative (rotating in the opposite direction) and slowing down towards zero as they approach the sampling frequency. We refer to these aliased frequencies as inverted, mirrored, or being mirror images of their actual frequencies.

We can continue to explore what happens if the frequency is increased even beyond one cycle per sample. If, for example, the increment is set just over one complete cycle at +9/8 of a turn, we can observe that the motion will instead be interpreted as being just +1/8 of a turn. +10/8 will be interpreted as +2/8, and so on. We can generalize that adding one (or any multiple of) full turn(s) to each increment will simply be ignored and will bring the frequency back (through aliasing) to the equivalent lower frequency located below half the sampling rate. Adding full turns to the increments that are located in the second half of the circle (increments which cause frequencies to be inverted and mirrored), will also be aliased and the frequencies will be brought back to the equivalent negative frequency located below half the sampling rate.

The following figure illustrates the generalized principle that frequencies that are located in zones that are at odd multiples (1, 3, 5, and so on) of half the sampling frequency are aliased back to the origin, whereas frequencies that are located in zones that are at even multiples (2,4,6, and so on) of half the sampling frequency are mirrored back to the origin:

 frequencies at odd multiples are aliased back to the origin while frequencies at even multiples are mirrored back to the origin

The different regions shown in the figure above are known as Nyquist zones and are identified in the following figure. The sampling frequency is labeled Fs and the frequency limit above which aliasing will start to occur, located at half the sampling frequency (Fs/2), is usually referred to as the Nyquist Frequency.

 Nyquist zones

ADCs normally digitize bandwidth-limited signals for which all the constituent frequencies are located in the first Nyquist zone. This is the simplest and usually the most convenient way to digitize an analog signal. Nevertheless, it can sometimes be advantageous to digitize signals that have their bandwidth located in one of the upper Nyquist zones, and this is a subject we’ll discuss in blog posts to come.

Conclusion

In this post, we used a simple graphical metaphor to develop an intuitive understanding of the Nyquist-Shannon sampling theorem. An understanding of its underlying principles is absolutely necessary in planning for the conversion of analog signals using analog-to-digital converters. We introduced the concept of bandwidth in relation to Nyquist zones. This concept will serve as the basis for our next post in this series in which we’ll discuss the frequency contents of continuous and transient analog signals.