Now let us insert an ideal low pass filter of bandwidth BW; then the PSD at
the filter output is
(1.38)
The inverse Fourier transform provides the autocorrelation function as
(1.39)
(1.40) RY(t)
No
2
# sin (2ptBW)
pt
RY (t) 3
BW
BW
No
2
# ej2pft dt
SY ( f ) eNo
2
BW f BW
0 Zf Z BW
WIRELESS TOPICS 39
1.5 REPRESENTING BAND PASS SIGNALS AND SUBSYSTEMS
In this section, we aim to provide tools the reader can use to analyze band pass signals and systems.
This tool will take the form of complex envelope theory and will be used extensively throughout this
book.
1.5.1 Complex Envelope Theory
We will use the complex envelope theory to model band pass signals and systems (see [45] and [46]).
We have decided to present the complex envelope theory with the hope of accomplishing the following
three goals:
1. Provide mathematical insight into the signal/system being considered.
2. Draw relationships to the actual HWbeing designed.
3. Provide manageable means of computer simulation.
In order to derive the complex envelope of a signal, we approach this in two directions: First, we
provide the mathematical derivation shown in a step-by-step procedure. Second, we simply use the
brute force approach and write down the complex envelope directly from the signal being observed.
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