Page 698 - The Mechatronics Handbook

0066_Frame_C23 Page 6 Wednesday, January 9, 2002 1:52 PM

It is also referred to as the unit impulse sequence or Kronecker delta function. The working properties
of the unit sample sequence are analogous to that of δ(t) and these are shown here:

∞
∑ xn()δ nm) = xm()
(
–
n=∞
–
xn()δ nm) = xm()δ nm)
(
(
–
–

(
δ an ± b) = δ n ± b 
--

a
∞
xn() ∗ δ nm) = ∑ xr()δ nr m) = xn m)
(
(
(
–
–
–
–
n=−∞
Note that the scaling property is only applicable when both a and b/a are integers. Two other basic signals
that are useful for analysis are the unit step and unit ramp signals. The unit step sequence, u(n), is defined as
 1, n ≥ 0
un() =  (23.10)
 0, n < 0
whereas the unit ramp signal, denoted as r(n), is given by
 n, n ≥ 0
rn() = 
 0, n < 0
The above three sequences are related as follows:

(
δ nk) = un k) un k 1)
(
(
–
–
–
–
–
n
(
un() = ∑ δ nm)
–
m=∞
–
rn() = un() ∗ un 1)
(
–
Figure 23.5 illustrates the above DT sequences.
Analysis of Continuous-Time Signals
Basic Operations on Signals
There are some important operations that are often performed on signals so as to understand either their
characteristics or the physical phenomena generating them. The three most common operations are
shifting, time scaling, and reflection. Examples of these operations are illustrated in Fig. 23.6, where x(t)
is expressed as
 t + 1, – ≤ t ≤ 3
1
xt() =  3, 3 < t ≤ 6


 0, otherwise

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