Throughout this course, we shall be interested in the analog to
digital conversion of signals f(t),t∈Rf(t),t∈R. We shall always assume f∈L 2 f∈L 2 and usually assume additional properties of ff in order to get meaningful results. In
particular, we want to study two mappings: the
encoding of ff into bit streams and the decoding of
the bit streams into approximations or estimates of ff,
E:f→bitsstreams(Encoder)D:bitsstreams→f ¯(Decoder)
E:f→bitsstreams(Encoder)D:bitsstreams→f ¯(Decoder)
(1)
where
f ¯f ¯ is the approximation of
ff defined by
f ¯:=D(E(f))f ¯:=D(E(f)). In general,
f ¯≠ff ¯≠f, so we shall need
some way of quantifying how well
f ¯f ¯ approximates
ff.
Normally, the distortion between is measured by some norm
∥f-f ¯∥∥f-f ¯∥. Typical choices include:
the
L
2
norm
∥
f
∥
L
2
:
=
∫
|
f
(
t
)
|
2
d
t
1
/
2
the
L
∞
norm
∥
f
∥
L
∞
:
=
sup
t
|
f
(
t
)
|
the
L
p
norm
∥
f
∥
L
p
:
=
∫
|
f
(
t
)
|
p
d
t
1
/
p
the
L
2
norm
∥
f
∥
L
2
:
=
∫
|
f
(
t
)
|
2
d
t
1
/
2
the
L
∞
norm
∥
f
∥
L
∞
:
=
sup
t
|
f
(
t
)
|
the
L
p
norm
∥
f
∥
L
p
:
=
∫
|
f
(
t
)
|
p
d
t
1
/
p
(2)
