Lecture 2: Linear Transformations on the Independent
Variableof Signals
Tie Liu
August 30, 2011
1 Transformation of signals in CT
The are many ways of transforming a CT signal into another. For instance, we can scale
it, shift it in time, differentiate it, or perform a combination of these actions.
Later in
this course, we will introduce the idea of transforming a signal as a system. To familiarize
you with manipulating signals, well examine a particular type of transformation in this
subsection: transformation on the independent variable of signals. More formally, let us for
now restrict ourselves to transformations of the form:
x
(
t
)
→
y
(
t
) =
x
(
f
(
t
))
where
x
(
t
) is the starting signal given to us,
y
(
t
) is the signal we end up with after the
transformation, and
f
(
t
) is a function of
t
. The arrow “
→
” denotes the action and direction
of transformation. The function
f
(
t
) can be any welldefined function, of course, but for the
study of ECEN 314, we will look at the class of linear functions
f
(
t
) =
at
+
b
where
a
and
b
are arbitrary real constants. The resulting transformation of
x
(
t
) into
y
(
t
) is
hence called “linear transformations on the independent variable.”
All such transformations can be decomposed into just three fundamental types of signal
transformations on the independent variable:
time shift, time scaling, and time reversal.
They involve a change of the variable
t
into something else:
•
Time shift:
f
(
t
) =
t

t
0
for some
t
0
∈
R
.
•
Time scaling:
f
(
t
) =
at
for some
a
∈
R
+
.
•
Time reversal (or flip):
f
(
t
) =

t
.
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 Fall '09
 Signal Processing, Scale factor, Linear function

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