**Topics Covered:**

- The two definitions of aliasing
- The need for an anti-aliasing filter
- Undersampling and oversampling

So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality – Einstein

An alias is a duplicate. It’s like a signal has a split personality. Here’s how:

If we sample at equal to or below the Nyquist rate, we are inviting trouble. In the interpolation process, which signal in the above graph is more likely to be reproduced? From the dots we have, both are valid solutions. Two waves hide in the same bunch of samples! This situation is called aliasing.

In this simplified example we see two potential signals (or solutions to interpolation algorithms) that can result from the same set of sampled points. Automated systems that are preprogrammed and are expected to work under certain limitations wouldn’t know the difference between the two, and this can result in a duplicate signal (or more!), and both given equal prominence. Rest assured the result isn’t pretty. The resulting ugliness, whatever form it might take, is also called *aliasing* (to make matters confusing). But that’s what happens when non-engineers begin to use engineering terms. For better or worse, the term has stuck.

By keeping our discrete signal above the Nyquist rate, we have a much better chance of eliminating aliasing. The important thing to realize is that since filters cannot completely eliminate unwanted frequencies, aliasing will *always* be there. When a system is designed well, however, this aliasing is usually small enough to pass by undetected, but it’s there. A system that samples below the Nyquist rate is said to be undersampling.

One strategy engineers use to keep a step ahead of aliasing is to sample at much higher than the Nyquist rate. This is called oversampling. After all, the higher the sampling rate the better the chance of reproducing the original signal. Another strategy is to use a filter (what did you expect?).

One such low-pass filter is the anti-aliasing filter. In its most basic form, it tries to eliminate all possible aliasing frequencies from the sampled signal. The funny thing is, unless one knows a lot about the original signal, the sampling and interpolation system and the aliasing characteristics of each device in the system, it is pretty much impossible to design a good anti-aliasing filter. It’s sort of like the chicken and egg problem. But anti-aliasing techniques introduce its own sets of problems. You see, errors are always multiplied, by which I mean, if we have n devices in a system, and each device has its own error e_{1}, e_{2}, e_{3}…e_{n}, then the total error of the system E = e_{1}*e_{2}*e_{3}…e_{n}.

What one must learn to live with is the fact that since interpolation is everywhere, aliasing isn’t far behind. It’s practically impossible to create an alias-free sampling system when dealing with analog signals. If you can learn to live with that, you’ll be much better off. Keep things simple:

We consider it a good principle to explain the phenomena by the simplest hypothesis possible – Ptolemy

Entities must not be multiplied beyond necessity – Occam’s razor

**Takeaways:**

- Aliasing is the effect produced when different signals become indistinguishable (or aliases of one another) when sampled.
- Unfortunately, aliasing also refers to the distortion or artifacts produced when the signal reconstructed is different from the original signal.
- An anti-aliasing filter is a specially designed filter to ensure a system works within the constraints of the sampling theorem.
- Undersampling is sampling below the Nyquist rate (or even equal to the Nyquist rate). Oversampling is sampling at a frequency much higher than the Nyquist rate.

**Links for further study:**

- http://en.wikipedia.org/wiki/Aliasing
- http://en.wikipedia.org/wiki/Anti-aliasing
- http://en.wikipedia.org/wiki/Anti-aliasing_filter
- http://en.wikipedia.org/wiki/Oversampling
- http://en.wikipedia.org/wiki/Undersampling

Next: The Challenge

Previous: Professor Sampler’s Notes: Interpolation

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