Nyquist Theorem

A slecture by ECE student Miguel Rodrigo Castellanos

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

## Outline

1. Introduction
2. Proof
3. Example
4. Conclusion
5. References

## Introduction

The Nyquist theorem provides a sufficient condition for perfect reconstruction of a signal from its sampling. In this slecture I will give a proof of the Nyquist theorem and give an example demonstrating how perfect reconstruction is possible even when violating the Nyquist condition.

## Proof

Let the ideal sampling $x_s(t) \text{ of } x(t)$ be defined as

$x_s(t) := x(t)p_{\frac{1}{f_s}}(t)$,

where

$p_{\frac{1}{f_s}}(t) = \sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s})$.

Nyquist Theorem: A signal $x(t)$ that has the property $X(f) = 0$ for $|f| \ge f_M$ can be perfectly reconstructed from its sampling $x_s(t)$ if sampled at a rate $f_s > 2f_M$.

To prove that perfect reconstruction is possible, we must find an expression for $x(t)$ in terms of $x_s(t)$.

Given that $\mathcal{F}(x(t)) = X(f)$, we can find $X_s(f)$ using the convolution property.

\begin{align} X_s(f) &= X(f)*\mathcal{F}(p_{\frac{1}{f_s}})\\ &= X(f)*\mathcal{F}(\sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s}))\\ &= X(f)*f_s\sum_{k = -\infty}^\infty \delta(f-kf_s)\\ &= f_s\sum_{k = -\infty}^\infty X(f)*\delta(t-\frac{k}{f_s})\\ &= f_s\sum_{k = -\infty}^\infty X(f-kf_s)\\ \end{align}

Without loss of generality, we can assume that the signal $x(t)$ has the spectrum shown in the figure below. The shape of the graph of $X(f)$ does not matter because the only important feature of $X(f)$ is that $X(f) = 0$ for $|f| \ge f_M$.

We would like to determine what $X_s(f)$ looks like in order to find a way to reconstruct $x(t)$.

Since we have sampled at a rate $f_s > 2f_M$, the following inequalities hold:

$f_s > 2f_M \iff f_s - f_M > f_M \iff -f_s + f_M < f_M$.

Together, these inequalities, the graph of $X(f)$, and the expression for $X_s(f)$ in terms of $X(f)$ imply that $X_s(f)$ will have the spectrum shown in the figure below.

Notice that the spectrum of the ideal sampling of a signal is an amplitude scaled periodic repetition of the original spectrum. Since $x(t)$ is bandlimited and we have sampled at a rate $f_s > 2f_M$, the periodic repetitions of $X(f)$ do not overlap.

All the information needed to reconstruct $X(f)$ can be found in the portion of $X_s(f)$ that corresponds to $X(f)$ (shown in red). Therefore we can use a simple lowpass filter with gain $\tfrac{1}{f_s}$ and cutoff frequency $\tfrac{f_s}{2}$ to recover $X(f)$ from $X_s(f)$.

$X(f) = X_s(f)\left\{ \begin{array}{ll} \frac{1}{f_s}, & |f| \le \frac{f_s}{2}\\ 0, & \text{else} \end{array} \right.$

$\iff x(t) = x_s(t)*\text{sinc}(f_st)$

$\therefore$ We can perfectly reconstruct $x(t)$ from $x_s(t)$.

## Example

Though the Nyquist theorem states that perfect reconstruction is possible if we satisfy the Nyquist condition $(f_s > 2f_M)$, it is important to note that this condition is not necessary. The following example demonstrates how perfect reconstruction is sometimes possible even when undersampling.

Let the signal $x(t)$ have a spectrum $X(f)$ as seen in the figure below.

The Nyquist condition states that we should sample at a rate $f_s > 2(2a) = 4a$. Instead, let us sample at $f_s = 2a$.

As before, we have $x_s(t) = x(t)p_{\frac{1}{f_s}}(t)$ and $X_s(f) = f_s\sum_{k = -\infty}^\infty X(f-kf_s)$

$\implies X_s(f) = 2a\sum_{k = -\infty}^\infty X(f-2ka)$.

Therefore, $X_s(f)$ will have the spectrum shown in the figure below.

Notice that there is no aliasing in $X_s(f)$ even though $f_s < 4a$. In addition, the portion of $X_s(f)$ that corresponds to $X(f)$ (shown in red) can be recovered using a bandpass filter with gain $\tfrac{1}{2a}$ and cutoff frequencies $a \text{ and } 2a$.

## Conclusion

To summarize, the Nyquist theorem states that any bandlimited signal can be perfectly reconstructed from its sampling if sampled at a rate greater than twice its bandwidth $(f_s > 2f_M)$. However, the Nyquist condition is not necessary for perfect reconstruction as shown in the example above.

## References

[1] John G. Proakis, Dimitris G. Manolakis, "Digital Signal Processing with Principles, Algorithms, and Applications" 4th Edition,2006