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Aliasing

Module by: Justin Romberg, Don Johnson

Summary: This module introduces the idea of aliasing and gives examples of it in sampling and reconstruction problems.

Introduction

When considering the reconstruction of a signal, you should already be familiar with the idea of the Nyquist rate. This concept allows us to find the sampling rate that will provide for perfect reconstruction of our signal. If we sample at too low of a rate (below the Nyquist rate), then problems will arise that will make perfect reconstruction impossible - this problem is known as aliasing. Aliasing occurs when there is an overlap in the shifted, perioidic copies of our original signal's FT, i.e. spectrum.
In the frequency domain, one will notice that part of the signal will overlap with the periodic signals next to it. In this overlap the values of the frequency will be added together and the shape of the signals spectrum will be unwantingly altered. This overlapping, or aliasing, makes it impossible to correctly determine the correct strength of that frequency. Figure 1 provides a visual example of this phenomenon:
alias_eg.png
Figure 1: The spectrum of some bandlimited (to W Hz) signal is shown in the top plot. If the sampling interval T s T s is chosen too large relative to the bandwidth W W, aliasing will occur. In the bottom plot, the sampling interval is chosen sufficiently small to avoid aliasing. Note that if the signal were not bandlimited, the component spectra would always overlap.

Aliasing and Sampling

If we sample too slowly, i.e., ∀,T>πΩB:Ωs<2ΩB T ΩB Ωs 2 ΩB We cannot recover the signal from its samples due to aliasing.
Example 1 
Let f1t f1 t have CTFT.
alia_f1.png
Figure 2: In this figure, note the following equation: ΩB-Ωs2=a ΩB Ωs 2 a
Let f2t f2 t have CTFT.
alia_f2.png
Figure 3: The horizontal portions of the signal result from overlap with shifted replicas - showing visual proof of aliasing.
Try to sketch and answer the following questions on your own:
  • What does the DTFT of f 1 , s n=f1nT f 1 , s n f1 n T look like?
  • What does the DTFT of f 2 , s n=f2nT f 2 , s n f2 n T look like?
  • Do any other signals have the same DTFT as f 1 , s n f 1 , s n and f 2 , s n f 2 , s n ?
CONCLUSION: If we sample below the Nyquist frequency, there are many signals that could have produced that given sample sequence.
alia_f3.png
Figure 4: These are all equal!
Why the term "aliasing"? Because the same sample sequence can represent different CT signals (as opposed to when we sample above the Nyquist frequency, then the sample sequence represents a unique CT signal).
alia_f4.png
Figure 5: These two signals contain the same four samples, yet are very different signals.
Example 2 
ft=cos2πt f t 2 t
cos.png
Figure 6: The cosine function, ft=cos2πt f t 2 t , and its CTFT.
Case 1: Sample Ωs=8πradsec Ωs 8 rad sec ⇒ T=14sec T 1 4 sec .
note: Ωs>2ΩB Ωs 2 ΩB
Case 2: Sample wΩs=83πradsec w Ωs 8 3 rad sec ⇒ T=34sec T 3 4 sec .
note: Ωs<2ΩB Ωs 2 ΩB
When we run the DTFT from Case #2 through the reconstruction steps, we realize that we end up with the following cosine: f ~ t=cosπ2t f ~ t 2 t This is a "stretched" out version of our original. Clearly, our sampling rate was not high enough to ensure correct reconstruction from the samples.
You may have seen some effects of aliasing such as a wagon wheel turning backwards in a western movie. Aliasing in images can result in Moire Patterns. Here is an example of an image that has Moire artifacts as a result of scanning at too low a frequency.

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