Continuous-Time Unit Impulse: Ilya Pollak ECE 301 Signals and Systems Section 2, Fall 2010 Purdue University
Continuous-Time Unit Impulse: Ilya Pollak ECE 301 Signals and Systems Section 2, Fall 2010 Purdue University
Continuous-Time Unit Impulse: Ilya Pollak ECE 301 Signals and Systems Section 2, Fall 2010 Purdue University
Functionals 1
1 Functionals
As we have seen before, a signal operates on numbers and produces numbers. In
other words, any signal x is a mapping from a set of numbers (called the domain
of x) to another set of numbers (called the range of x), as shown in Fig. 1(a). For
example, a continuous-time complex-valued signal x is a mapping from the set of
real numbers to the set of complex numbers: given any real number t, the value
x(t) is a complex number—see Fig. 2(a).
T T T T
T
Domain of x, Range of x,T Domain of v,T Range of v,T
D, a set of T signal x R, a set of T D, a set of T functional v R, a set of T
T numbers T numbers T signals T numbers
T T T T
T T T T
(a) (b)
Figure 1. (a) A signal is a mapping between two sets of numbers. (b) A functional is a
mapping between a set of signals and a set of numbers.
(a) (b)
Figure 2. (a) A generic block diagram for a signal. (b) A generic block diagram for a functional.
2
The domain of such a functional can only contain signals x for which xs is inte-
grable. For example, the functional v2 defined above is induced by the unit step.
As another example, consider a short pulse δ∆ of duration ∆, illustrated in Fig. 3:
0, t≤0
δ∆ (t) = 1/∆, 0 < t ≤ ∆ (2)
0, t>∆
δ∆ (t) C
1
∆
X
∆ t
lim δ∆ = δ. (4)
∆→0
This is correct; however, the limiting operation here is not the usual pointwise
limit.
As a matter of fact, the pointwise limit of δ∆ (t) for any t is zero. To see this,
consider the cases t ≤ 0 and t > 0 separately. For t ≤ 0, we have from Eq. (2) that
δ∆ (t) = 0 regardless of ∆. Therefore, for any t ≤ 0, the limit δ∆ (t) as ∆ → 0 is
zero. For t ≥ 0, as soon as ∆ is smaller than t, we also have δ∆ (t) = 0, according
to Eq. (2). Thus, no matter how small t is, as long as t is positive, the limiting
value of δ∆ (t) is zero as ∆ → 0. For all t we therefore have:
lim δ∆ (t) = 0.
∆→0
This shows that Eq. (4) cannot be interpreted pointwise. The correct interpre-
tation of Eq. (4) is in the sense of functionals: as ∆ → 0, what the functional
induced by δ∆ does to any signal approaches what δ does to the same signal. To
see this, we look at the output value of the functional induced by δ∆ for some
4
5 Notation
It is possible to demonstrate that there does not exist a function s such that
Z ∞
s(t)x(t)dt = x(0)
−∞
for every signal x which is continuous at zero. Therefore, the CT unit impulse is
a functional that is not induced by any function. Despite this, the notation δ(t)
is used in much engineering literature, including the textbook. We will therefore
have to use it throughout the course. Another piece of really bad, confusing
notation is: Z ∞
δ(t)x(t)dt
−∞
This notation simply means x(0), nothing more! Unfortunately, this notation is
also used throughout the text, and we will therefore have to use it class.
If t0 is a fixed real number, the unit impulse shifted by t0 is denoted δt0 and
is defined as the following functional:
v(x) = x(t0 ).
Two more pieces of bad but widely used notation:
δt0 (t) = δ(t − t0 )
Z ∞
δ(t − t0 )x(t)dt = x(t0 )
−∞