Introduction

Earlier you should have been exposed to the concepts behind sampling and the sampling theorem. While learning about these ideas, you should have begun to notice that if we sample at too low of a rate, there is a chance that our original signal will not be uniquely defined by our sampled signal. If this happens, then there is no guarantee that we can correctly reconstruct the signal. As a result of this, the Nyquist Theorem was created. Below, we will discuss just what exactly this theorem tells us.

Nyquist Theorem

We will let TT equal our sampling period (distance between samples). Then let Ωs=2πT Ωs 2 T (sampling frequency in radians/sec). We have seen that if ft f t is bandlimited to -ΩBΩB ΩB ΩB and we sample with period T<πΩb⇒2πΩs<πΩB⇒Ωs>2ΩB T Ωb 2 Ωs ΩB Ωs 2 ΩB then we can reconstruct ft f t from its samples.

ΩN=2ΩB ΩN 2 ΩB is called the "Nyquist frequency" for ft f t . For perfect reconstruction to be possible Ωs≥2ΩB Ωs 2 ΩB where Ωs Ωs is the sampling frequency and ΩB ΩB is the highest frequency in the signal.

Figure 1: Illustration of Nyquist Frequency

The Big Question

Exercise 1

What if Ωs<2ΩB Ωs 2 ΩB ? What happens when we sample below the Nyquist rate?

Solution 1

Go through the steps: (see Figure 2)

Figure 2:

Finally, what will happen to F s ⅇⅈω F s ω now? To answer this final question, we will now need to look into the concept of aliasing.

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