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(c) Argue that if for some
k
,
s
3
(
n
)=
s
1
(
n
+
k
)
for every
n
∈
Z
, then the recon-
struction
f
3
of
s
3
is just
f
3
(
x
)=
f
1
(
x
+
k
)
. We say that reconstruction is
translation
equivariant.
(d) Explain why the reconstruction of the signal
s
with
s
(
0
)=
1, and
s
(
n
)=
0for
all other
n
,isjust
f
(
x
)=
sinc
(
x
)
.
(e) From your answers to parts (a) through (d), conclude that reconstruction for
any
discrete signal comes from convolution with sinc.
Exercise 19.6:
Suppose we band-limit
f
(
x
)=
sinc
(
x
)
to
v
0
=
4
. Describe
the resultant signal precisely.
Exercise 19.7:
We suggested that one bad way to downsample a 300
×
300
image to a 150
150 image was to simply take pixels from every second row and
every second column. Suppose the source image is a checkerboard: Pixel
(
i
,
j
)
is
white if
i
+
j
is even, and it's black if it's odd. At a distance, this image looks
uniformly gray.
(a) Show that if we choose pixels from rows and columns with odd indices, the
resultant subsampled image is all white.
(b) Show that if we use odd row indices, but even column indices, the resultant
subsampled image is all black.
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