ECE 562
Fall 2011
September 7, 2011
Signal Space Concepts
In order to proceed with the design and analysis of digital communication systems (in complex
baseband) it is important for us to understand some properties of the space in which the complex
message bearing signal
s
(
t
) lies.
Inner Product and Norm
•
Let
x
(
t
) and
y
(
t
) be complex valued signals with
t
∈
[
a, b
]. If
a
and
b
are not speciﬁed, it is
assumed that
t
∈
(
∞
,
∞
).
Deﬁnition 1.
(Inner Product)
h
x, y
i
Δ
=
Z
b
a
x
(
t
)
y
*
(
t
)
du .
(1)
The inner product satisﬁes the necessary axioms:
±
h
x, y
i
=
h
y, x
i
*
²
h
x
+
y, z
i
=
h
x, z
i
+
h
y, z
i
³
h
αx, y
i
=
α
h
x, y
i
, for any complex number
α
.
´
h
x, x
i ≥
0, and
h
x, x
i
= 0 iﬀ
x
(
t
) = 0 for all
t
.
•
Signals
x
(
t
) and
y
(
t
) are said to be orthogonal if
h
x, y
i
= 0. The orthogonality of
x
(
t
) and
y
(
t
) is
sometimes denoted by
x
⊥
y
.
Deﬁnition 2.
(Norm)
The inner product deﬁned above induces the following norm:
k
x
k
=
p
h
x, x
i
.
(2)
It is easy to show that the above quantity is a valid norm in that it satisﬁes the required axioms.
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 Fall '09
 Linear Algebra, Fourier Series, Hilbert space, inner product, Inner product space

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