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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 x⁢n x n to CT (Figure 1)?

Figure 1
Figure 1 (sec12_fig1.png)

Step 1

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

x s ⁢t=∑ n =−∞∞x⁢n⁢δ⁢t−n⁢T 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 x⁢n= x c ⁢n⁢T x n x c n T .

Ideal Reconstruction System

Figure 4
Figure 4 (no_image.png)

In Frequency Domain:

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

In Time Domain:

  1. x s ⁢t=∑ n =−∞∞x⁢n⁢δ⁢t−n⁢T x s t n x n δ t n T
  2. x r ⁢t=∑ n =−∞∞x⁢n⁢δ⁢t−n⁢T* h LP ⁢t x r t n x n δ t n T h LP t h LP ⁢t=sinc⁢πT⁢t h LP t sinc T t

result:

x r ⁢t=∑ n =−∞∞x⁢n⁢sinc⁢πT⁢(t−n⁢T) x r t n x n sinc T t n T

Figure 5
Figure 5 (sec12_fig6.png)

h LP ⁢t=sinc⁢πT⁢t=sinπT⁢t(πT⁢t) h LP t sinc T t T t T t
(2)
h LP ⁢t h LP t "interpolates" the values of x⁢n 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 =−∞∞x⁢n⁢sinπT⁢(t−n⁢T)πT⁢(t−n⁢T) x r t n x n T t n T T t n T
(3)

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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Definition of a lens

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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