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Nyquist Theorem

Module by: Justin Romberg. E-mail the author

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Summary: This module introduces the Nyquist theorem.

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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.

Theorem 1: Nyquist Theorem ("Fundamental Theorem of DSP")

If ft f t is bandlimited to -ΩBΩB ΩB ΩB , we can reconstruct it perfectly from its samples fsn=fnT fs n f n T for Ωs=2πT>2ΩB Ωs 2 T 2 ΩB

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
Figure 1 (nyq_f1.png)

Example 1: Examples:

  • Human ear hears frequencies up to 20 kHz → CD sample rate is 44.1 kHz.
  • Phone line passes frequencies up to 4 kHz → phone company samples at 8 kHz.

Reconstruction

The reconstruction formula in the time domain looks like ft=∑n=-∞∞fsnsinπTt−nTπTt−nT f t n fs n T t n T T t n T We can conclude, just as before, that ∀n,n∈ℤ:sinπTt−nTπTt−nT n n T t n T T t n T is a basis for the space of -ΩBΩB ΩB ΩB bandlimited functions, ΩB=πT ΩB T . The expansion coefficient for this basis are calculated by sampling ft f t at rate 2πT=2ΩB 2 T 2 ΩB .

Note:

The basis is also orthogonal. To make it orthonormal, we need a normalization factor of T T .

The Big Question

Exercise 1

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

Solution

Go through the steps: (see Figure 2)

Figure 2
Figure 2 (nyq_f2.png)

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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