Introduction

The reconstruction process begins by taking a sampled signal, which will be in discrete time, and performing a few operations in order to convert them into continuous-time and, with any luck, into an exact copy of the original signal. A basic method used to reconstruct a -ππ bandlimited signal from its samples on the integer is to do the following steps:

Figure 1: Reconstruction block diagram with lowpass filter (LPF).

The lowpass filter's impulse response is gt g t . The following equations allow us to reconstruct our signal (Figure 2), f ~ t f ~ t .

Figure 2:

Examples of Filters g

Example 1: Zero Order Hold

This type "filter" is one of the most basic types of reconstruction filters. It simply holds the value that is in fsn fs n for ττ seconds. This creates a block or step like function where each value of the pulse in fsn fs n is simply dragged over to the next pulse. The equations and illustrations below depict how this reconstruction filter works with the following gg: gt= 1if0<t<τ0otherwise g t 1 0 t τ 0

fsn=∑n=-∞∞fsngt-n fs n n fs n g t n (2)

Figure 3: Zero Order Hold

Subfigure 3.1 Subfigure 3.2

question:

How does f ~ t f ~ t reconstructed with a zero order hold compare to the original ft f t in the frequency domain?

Example 2: Nth Order Hold

Here we will look at a few quick examples of variances to the Zero Order Hold filter discussed in the previous example.

Figure 4: Nth Order Hold Examples (nth order hold is equal to an nth order B-spline)

Subfigure 4.1: First Order Hold Subfigure 4.2: Second Order Hold Subfigure 4.3: ∞ Order Hold

Ultimate Reconstruction Filter

Recall that (see Figure 5)

Figure 5: Our current reconstruction block diagram. Note that each of these signals has its own corresponding CTFT or DTFT.

If Gⅈω G ω has the following shape (Figure 6):

Figure 6: Ideal lowpass filter

then f ~ t=ft f ~ t f t Therefore, an ideal lowpass filter will give us perfect reconstruction!

In the time domain, impulse response

Amazing Conclusions

If ft f t is bandlimited to -ππ , it can be reconstructed perfectly from its samples on the integers fsn=ft|t=n fs n t n f t

The above equation for perfect reconstruction deserves a closer look, which you should continue to read in the following section to get a better understanding of reconstruction. Here are a few things to think about for now: