ECE 325
CONTINUOUSTIME CONCEPTS I
Fall 2007
Continuoustime Signals, Convolution,
and LTI Systems
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
) =
1
if
t
is rational
0
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 odd
1
if 1
/
(
n
+ 1)
< t
≤
1
/n
and
n
is even
1
if
t
≥
1
.
x
2
has countably inﬁnitely many jumps in the interval [0
,
1], but is ﬂat between the jumps.
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
, which has speciﬁcation
x
3
(
t
) =
1
t

1
if
t
6
= 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