7.3
Signal Space Concepts
We have seen in the previous section that the statistical relation between the hypotheses
{
H
i
}
and the observation
Y
are expressed in terms of the conditional densities
p
(
y

i
). We are now
interested in applying this framework for derive optimal decision rules (and the receiver structures
required to implement them) for the problem of
M
ary signaling in AWGN. In the language of
hypothesis testing, the observation here is the received signal
y
(
t
) modeled as follows:
H
i
:
y
(
t
) =
s
i
(
t
) +
n
(
t
)
,
i
= 0
,
1
, ..., M

1
(7.32)
where
s
i
(
t
) is the transmitted signal corresponding to hypothesis
H
i
, and
n
(
t
) is WGN with PSD
σ
2
=
N
0
/
2. Before we can apply the framework of the previous section, however, we must figure
out how to define conditional densities when the observation is a continuoustime signal. Here
is how we do it:
•
We first observe that, while the signals
s
i
(
t
) live in an infinitedimensional, continuoustime
space, if we are only interested in the
M
signals that could be transmitted under each of the
M
hypotheses, then we can limit attention to a finitedimensional subspace of dimension at most
M
. We call this the
signal space.
We can then express the signals as vectors corresponding to
an expansion with respect to an orthonormal basis for the subspace.
•
The projection of WGN onto the signal space gives us a noise vector whose components are
i.i.d. Gaussian. Furthermore, we show that the component of the received signal orthogonal to
the signal space is
irrelevant:
that is, we can throw it away without compromising performance.
•
We can therefore restrict attention to projection of the received signal onto the signal space
without loss of performance.
This projection can be expressed as a finitedimensional vector
which is modeled as a discrete time analogue of (7.32). We can now apply the hypothesis testing
framework of Section 7.2 to infer the optimal (ML and MPE) decision rules.
•
We then translate the optimal decision rules back to continuous time to infer the structure of
the optimal receiver.
7.3.1
Representing signals as vectors
Let us begin with an example illustrating how continuoustime signals can be represented as
finitedimensional vectors by projecting onto the signal space.
Example 7.3.1 (Signal space for twodimensional modulation):
Consider a single complex
valued symbol
b
=
b
c
+
jb
s
(assume that there is no intersymbol interference) sent using two
dimensional passband linear modulation. The set of possible transmitted signals are given by
s
b
c
,b
s
(
t
) =
b
c
p
(
t
) cos 2
π
f
c
t

b
s
p
(
t
) sin 2
π
f
c
t
where (
b
c
, b
s
) takes
M
possible values for an
M
ary constellation (e.g.,
M
= 4 for QPSK,
M
= 16
for 16QAM), and where
p
(
t
) is a baseband pulse of bandwidth smaller than the carrier frequency
f
c
. Setting
φ
c
(
t
) =
p
(
t
) cos 2
π
f
c
t
and
φ
s
(
t
) =

p
(
t
) sin 2
π
f
c
t
, we see that we can write the set of
transmitted signals as a linear combination of these signals as follows:
s
b
c
,b
s
(
t
) =
b
c
φ
c
(
t
) +
b
s
φ
s
(
t
)
so that the signal space has dimension at most 2.
From Chapter 2, we know that
φ
c
and
φ
s
are orthogonal (IQ orthogonality), and hence linearly independent. Thus, the signal space has
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Linear Algebra, Euclidean space, signal space, Standard basis, WGN

Click to edit the document details