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Ideal Reconstruction of Sampled Signals

Module by: Robert Nowak

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)?

Figure 1
Figure 1 (sec12_fig1.png)

Step 1

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

x s t=∑n=-∞∞xnδt-nT x s t n x n δ t n T (1)

Figure 2
Figure 2 (sec12_fig2.png)

Step 2

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

Figure 3
Figure 3 (sec12_fig3.png)

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

Figure 4
Figure 4 (no_image.png)

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=-∞∞xnδt-nT x s t n x n δ t n T
  2. x r t=∑n=-∞∞xnδt-nT* 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=-∞∞xnsincπTt-nT x r t n x n sinc T t n T

Figure 5
Figure 5 (sec12_fig6.png)

h LP t=sincπTt=sinπTtπTt 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).

Figure 6
Figure 6 (sec12_fig7.png)

Sinc Interpolator

x r t=∑n=-∞∞xnsinπTt-nTπTt-nT x r t n x n T t n T T t n T (3)

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