Time Domain Representation of Continuous and Discrete Signals
11
R.1.20 A right-shifted unit step, by t 0 units, denoted by u(t − t 0 ) is illustrated in Figure 1.11.
The shifted step function u(t − t 0 ) is defi ned analytically by
u t t
t t
t t
(
)
Ϫ
Ն
Ͻ
0
0
0
1
0
ϭ
for
for



R.1.21 A unit step sequence or the discrete unit step u(n) is illustrated in Figure 1.12.
The unit discrete step u(n) is defi ned analytically by
u n
n
n
( ) ϭ
1
0
0
0
for
for
Ն
Ͻ



R.1.22 A unit discrete step sequence u(n) can be constructed by a sequence of impulses
indicated as follows:
u n
n k
k
( )
(
)
ϭ
ϭ
Ϫ
ϱ
0
∑
Observe that δ(n) = u(n) − u(n − 1).
R.1.23 A shifted and amplitude-scaled step sequence, A u(n − m) is illustrated in Figure 1.13.
The sequence A u(n − m) is defi ned analytically by
Au n m
A
n m
n m
(
)
Ϫ
Ն
Ͻ
ϭ
for
for
0



sw closes at t = 0
1 V
v(t) = u(t)
FIGURE 1.10
Circuit implementation of u(t).
FIGURE 1.11
Plot of u(t − t 0 ).
u(t − t 0 )
0
t 0
t
1
u (n)
−2
−1
1
2
3
0
n
FIGURE 1.12
Plot of u(n).
CRC_47760_CH001.indd 11
CRC_47760_CH001.indd 11
7/25/2008 4:16:12 PM
7/25/2008 4:16:12 PM