Sampling
Sampling
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Sampling
Introduction
Sampling
Continuous
f(t)
t
Discrete
f(t)
t
Sampling
Introduction
Sampling
Sampling a continuous function
f
to produce discrete function
ˆ
f
ˆ
f
[
n
] =
f
(
n
Δ
t
)
is just multiplying it by a comb:
ˆ
f
=
f
comb
h
where
h
= Δ
t
Sampling
Sampling In The Time/Spatial Domain
Sampling In The Time/Spatial Domain 
Graphical Example
Continuous
f(t)
t
Sampling Comb
t
f(t)
Discrete
f(t)
t
Sampling
Sampling In The Frequency Domain
Sampling In The Frequency Domain
Sampling is multiplication by a comb with spacing
h
:
ˆ
f
=
f
comb
h
What is happening in the frequency domain?
The Fourier Transform of a comb with spacing
h
is a comb with
spacing 1
/
h
:
ˆ
F
=
F
*
comb
1
/
h
Convolution of a function and a comb causes a copy of the
function to “stick” to each tooth of the comb,
and all of them add
together
.
Sampling
Sampling In The Frequency Domain
Sampling In The Frequency Domain 
Graphical Example
Signal
u
F(u)
Comb
u
F(u)
Discrete Signal
u
F(u)
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Sampling
Sampling In The Frequency Domain
Reconstruction
In theory, we can reconstruct the original
continuous
function by
removing all of the extraneous copies of its spectrum created
by the sampling process:
F
(
u
) =
ˆ
F
(
u
)
rect
1
/
h
(
u
)
In other words, keep everything in the frequency domain
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 Fall '09
 Image processing, Digital Signal Processing, Signal Processing, sampling rate

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