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 bandlimited signal from its samples on the integer is to do the following steps: turn the sample sequence fs n into an impulse train fimp t lowpass filter fimp t to get the reconstruction f ~ t (cutoff freq. = π)

Reconstruction block diagram with lowpass filter (LPF). The lowpass filter's impulse response is g t . The following equations allow us to reconstruct our signal (), f ~ t . f ~ t g t fimp t g t n fs n δ t n f ~ t n fs n g t δ t n n fs n g t n

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

Zero Order Hold How does f ~ t reconstructed with a zero order hold compare to the original f t in the frequency domain? 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.

First Order Hold Second Order Hold ∞ Order Hold Nth Order Hold Examples (nth order hold is equal to an nth order B-spline)

Ultimate Reconstruction Filter What is the ultimate reconstruction filter? Recall that (see )

Our current reconstruction block diagram. Note that each of these signals has its own corresponding CTFT or DTFT. If G ω has the following shape ():

Ideal lowpass filter then f ~ t f t Therefore, an ideal lowpass filter will give us perfect reconstruction! In the time domain, impulse response g t t t f ~ t n fs n g t n n fs n t n t n f t

Amazing Conclusions If f t is bandlimited to , it can be reconstructed perfectly from its samples on the integers fs n t n f t f t n fs n t n t n 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: What does t n t n equal at integers other than n? What is the support of t n t n ?