Sampling and aliasing

If we want to take samples of some analog waveform, as in doing analog to digital conversions at some particular conversion rate, there is an absolute lower limit to the rate of doing conversions versus the highest frequency component of the analog signal. That limit must not be violated if the sampling process is to yield valid results. We do not want to encounter the phenomenon called “aliasing”.

The term “aliasing” as we use it here has nothing to do with spy thrillers or crime novels. Aliasing is an unwanted effect that can arise when some analog waveform is being sampled for its instantaneous values at regular time intervals that are longer than half the reciprocal of a sampling frequency. If we were to sample some waveform once every microsecond, the sampling interval is half of that one microsecond for which we would have a sampling frequency limit of 2 MHz or faster.

Aliasing will occur if the sampled waveform has frequency component(s) that are greater in frequency than 50% of the sampling frequency. To turn that statement around, aliasing will occur if the sampling frequency is too low. Aliasing will occur at any sampling rate that is lower than twice the highest frequency component of the waveform being sampled.

The next question is: Why?

The late comedian Professor Irwin Corey once posed a similar question: “Why is the sky blue?” His answer was something like “This is a question which must be taken in two parts. The first part is ‘Why?’ ‘Why’ is a question Man has asked since the beginning of time. Why? I don’t know. The second part is ‘Is the sky blue?’ The answer is ‘Yes!'”

Fortunately, we can do a little better than that as follows.

The sampling process can be thought of as multiplying the waveform being sampled by a very narrow duty cycle pulse waveform of zero value for most of the time and of unity value for the very narrow sampling time interval. That sampling waveform will be rich in harmonics. There will be a spectral line at the sampling frequency itself plus spectral lines at each of the sampling frequency’s harmonics as well. Each spectral line will have sidebands as shown in Figure 1 which will extend from those sampling frequency spectral lines up and down the frequency spectrum in keeping with the sampled waveform’s bandwidth.

Figure 1 Sampling versus aliasing where spectral line will have sidebands that will extend from those sampling frequency spectral lines up and down the frequency spectrum in keeping with the sampled waveform’s bandwidth.

The sampling waveform is amplitude modulated by the sampled waveform and so I’ve chosen to call that sampled waveform’s highest frequency component, Fmod. Each bandwidth is 2 * Fmod.

If the sampling frequency is high enough as with Fs1, the illustrated sidebands do not overlap. There is a respectable guard band between them, and no aliasing occurs.

If the sampling frequency starts getting lower as with Fs2, the sidebands start getting closer together and there is a less comfortable, if I may use that word, guard band.

If the sampling frequency gets too low as with Fs3 which is less than twice Fmod, the sidebands overlap, and we have aliasing. Sampling integrity is lost. The sampled waveform cannot be reconstructed from the undersampled output of this now unsatisfactory system.

Consider this an homage to Claude Shannon (April 30, 1916 – February 24, 2001) and his sampling theory.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

Related Content

• Digital oscilloscopes: When things go wrong
• The alias theorems: practical undersampling for expert engineers
• Control the sample rate of digitized signals
• Basics of ADCs and DACs, part 1
• Claude Shannon, ‘father of information theory,’ is born, April 30, 1916

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