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# Ideal Reconstruction of Sampled Signals

Module by: Robert Nowak. E-mail the author

Summary: Examines ideas of ideal reconstruction of signals for a sampled, noisy original.

## Reconstruction of Sampled Signals

How do we go from xn x n to CT (Figure 1)?

### Step 1

Place xn x n into CT on an impulse train st s t (Figure 2).

x s t= n =xnδtnT x s t n x n δ t n T
(1)

### Step 2

Pass x s t x s t through an idea lowpass filter H LP Ω H LP Ω (Figure 3).

If we had no aliasing then x r t= x c t x r t x c t , where xn= x c nT x n x c n T .

## Ideal Reconstruction System

### In Frequency Domain:

1. X s Ω=XΩT X s Ω X Ω T where XΩT X Ω T is the DTFT of xn x n at digital frequency ω=ΩT ω Ω T .
2. X r Ω= H LP Ω X s Ω X r Ω H LP Ω X s Ω

#### Result:

X r Ω= H LP ΩXΩT X r Ω H LP Ω X Ω T

### In Time Domain:

1. x s t= n =xnδtnT x s t n x n δ t n T
2. x r t= n =xnδtnT* h LP t x r t n x n δ t n T h LP t h LP t=sincπTt h LP t sinc T t

### result:

x r t= n =xnsincπT(tnT) x r t n x n sinc T t n T

h LP t=sincπTt=sinπTt(πTt) h LP t sinc T t T t T t
(2)
h LP t h LP t "interpolates" the values of xn x n to generate x r t x r t (Figure 6).

### Sinc Interpolator

x r t= n =xnsinπT(tnT)πT(tnT) x r t n x n T t n T T t n T
(3)

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