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Lecture 1Sampling of Signals

by

Graham C. GoodwinUniversity of Newcastle

Australia

Lecture 1

Presented at the Zaborszky Distinguished Lecture Series

December 3rd, 4th and 5th, 2007

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Recall Basic Idea of Sampling

and Quantization

Quantization

Sampling

t1 t3t2 t4t

0

1

2

3

4

5

6

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In this lecture we will ignore quantizationissues and focus on the impact of differentsampling patterns for scalar and

multidimensional signals

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Outline

1. One Dimensional Sampling

2. Multidimensional Sampling

3. Sampling and Reciprocal Lattices

4. Undersampled Signals5. Filter Banks

6. Generalized Sampling Expansion (GSE)

7. Recurrent Sampling

8. Application: Video Compression at Source9. Conclusions

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Sampling: Assume amplitude quantizationsufficiently fine to be negligible.

Question: Say we are given

Under what conditions can we recover

from the samples?

( );f t t

( );if t i Z

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A Well Known Result (Shannons

Reconstruction Theorem for Uniform

Sampling)

Consider a scalar signal f(t) consisting of

frequency components in the range . If

this signal is sampled at period , then thesignal can be perfectly reconstructed from the

samples using:

[ ]

( )

( )

sin2

( )

2

s

sk

t k

y t y k

t k

w

w

= -

- D = - D

,2 2

s sw w-

2

s

pwD