ECE 3250
CONTINUOUSTIME CONCEPTS I
Fall 2008
Continuoustime Signals and Convoluton
1. Continuoustime signals
We think of the real numbers
R
as a mathematical model for “continuous time.”
Real
number
t
corresponds to “time
t
.” Real number 0 corresponds to “time 0.” If
s > t
, then
“time
s
is later than time
t
.” Having used
R
to model continuous time, we can think of
a
continuoustime signal over
F
as a function with domain
R
that takes values in
F
. As
usual,
F
is either
R
or
C
. We denote the set of all continuoustime signals by
F
R
.
Working with continuoustime signals is a lot touchier than working with discrete
time signals. We have all kinds of analytical things to worry about, e.g. continuity and
differentiability. Many continuoustime signals are quite nasty. Fortunately, those signals
play a limited role in applications, and we will end up essentially wishing them out of the
picture by restricting our attention to what I’ll be calling decent signals.
Definition 1:
A
decent signal
is a signal
x
∈
F
R
that has the following properties:
(1)
x
is continuous except possibly for jump discontinuities. Furthermore,
x
has at
most finitely many jumps in any bounded interval [
t
1
, t
2
]
⊂
R
.
(2)
x
is bounded on any bounded interval [
t
1
, t
2
]
⊂
R
.
I.e., for any such interval,
there exists
R >
0 such that

x
(
t
)
 ≤
R
for every
t
∈
[
t
1
, t
2
].
Requirement 1 eliminates a number of signals we’d rather not deal with. Consider, for
example, the signal
x
1
∈
R
R
with specification
x
1
(
t
) =
0
if
t
is rational
1
if
t
is irrational
.
The signal
x
1
has no points of continuity. In any bounded interval,
x
1
has uncountably
many “jumps,” if you even want to call them jumps.
Requirement 1 also rules out the
slightly less pathological signal
x
2
with specification
x
2
(
t
) =
8
>
>
<
>
>
:
0
if
t <
0
0
if 1
/
(
n
+ 1)
< t
≤
1
/n
and
n
is even
1
if 1
/
(
n
+ 1)
< t
≤
1
/n
and
n
is odd
1
if
t
≥
1
.
x
2
has countably infinitely many jumps in the interval [0
,
1], but is flat between the jumps.
Note that a decent signal can have at most countably infinitely many jumps because of
Requirement 1. Can you see why?
Requirement 2 in the definition of decent signals eliminates signals that blow up some
where other than as
t
→ ±∞
. Consider, for example, the signal
x
3
with specification
x
3
(
t
) =
1
t

1
if
t
= 1
0
if
t
= 1
.
x
3
(
t
) blows up as
t
→
1 from either side, so
x
is unbounded on any interval that contains
time
t
= 1.
Every continuous signal is a decent signal. Many discontinuous signals are decent as
well, including some that we’ll be encountering frequently. The
continuoustime unit step
is the signal
u
that has specification
u
(
t
) =
1
if
t
≥
0
0
if
t <
0
.
1
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I hope you don’t object to my using the same notation
u
for both the discrete and
continuoustime unit steps.
u
is not continuous, but its only discontinuity is a jump at
t
= 0. One comment on the unit step: I’ve defined it so
u
(0) = 1. As it happens, in all the
manipulations we do that involve decent signals, it will
not matter at all
how we define
the signals’ values at jumpdiscontinuity points. I could just as well have set
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 '04
 LIPSON
 Signal Processing, Continuous signal

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