Part 8 - Signal reconstruction

Introduction

In this part we’ll cover how we can recover a signal from a sampled signal, given that it follows Nyquist-Shannons theorem.

Signal reconstruction

If we want to reconstruct a signal from a sampled version, assuming it follows Nyquist-Shannos theorem.

We have two options. Let’s start with the first option.

Filter

Given the spectrum of the signal, we can isolate the replica and perform the inverse Fourier transform on this.

How do we isolate the replica?

Quite simple, we multiply our spectrum with a gate function with a width of $2B$.

Meaning: $$ \bar{F}(\omega) H(\omega) = F(\omega) $$

Then: $$ \mathcal{F}^{-1}(F(\omega)) = f(t) $$

Remember that we could also do this in the time domain: $$ \bar{f}(t) * h(t) = f(t) $$

As we said, $H(\omega) = T\ rect\left(\dfrac{\omega}{4\pi B}\right)$, therefore $h(t) = sinc(2\pi Bt)$. This is from the Fourier table.

Interpolation

We can also interpolate using the sinc function. $$ \sum_k f(kT)\ sinc(2\pi Bt - k\pi) $$