Digital Signal Processing Reference
In-Depth Information
1,
t 7 0
b
u(t) =
0,
t 6 0.
We will assume that the initial condition
y
(0) is zero; that is,
y(0) = 0.
Sampling a unit step
function yields
x(nH) = 1
for
n G 0,
and thus
x[n] = 1
for
n G 0.
(We have assumed that
x[0] = 1.
) From (9.5), the difference equation to be solved is
y[n] = y[n - 1] + Hx[n - 1].
This equation is solved iteratively, beginning with
n = 1:
y[1] = y[0] + Hx[0] = 0 + H = H;
y[2] = y[1] + Hx[1] = H + H = 2H;
y[3] = y[2] + Hx[2] = 2H + H = 3H;
o
y[n] = y[n - 1] + Hx[n - 1] = (n - 1)H + H = nH.
Thus,
y(nH) = nH.
The exact integral of the unit step function gives the result
t
t
t
y(t) =
L
u(t)dt =
L
dt = t
= t, t 7 0,
0
0
0
and
y(t)
evaluated at is equal to
nH
. Hence, Euler's rule gives the exact value for the
integral of the unit step function. In general, the Euler rule is not exact. The reader may wish
to consider why the results are exact, by constructing a figure of the form of Figure 9.2 for the
unit step function.
t = nH
■
From Figure 9.2, we see that the integral of a general function
x(t)
by Euler's
rule yields the summation of
x
[
k
] multiplied by the constant
H
:
n- 1
Á
y[n] = Hx[0] + Hx[1] + Hx[2] +
+ Hx[n - 1] = H
a
x[k].
(9.8)
k= 0
Hence, we see that in this case, there is a relation between integration in continu-
ous time and summation in discrete time. This relation carries over to many other
situations.
We begin the study of discrete-time signals by defining two signals. First, the
discrete-time unit step function u
[
n
] is defined by
1,
n G 0
b
u[n] =
(9.9)
0,
n 6 0.
