Signals and Systems - Electrical Engineering

0.3 Analog or Discrete? 15

FIGURE 0.7

Approximation of area under

x(t)=t,t≥ 0 , 0 otherwise, by pulses of

width 1 and heightnTs, whereTs= 1 and

n=0, 1,...

0 2 4 6 8 10

0

2

4

6

8

10

t

t, n

t

The approximation of the area usingTs=1 is very poor (see Figure 0.7). In the above, we used the
fact that the sum is not changed whether we add the numbers from 0 to 9 or backwards from 9 to 0,
and that doubling the sum and dividing by 2 would not change the final answer. The above sum can
thus be generalized to

N∑− 1

n= 0

n=

1

2

[N− 1

∑

n= 0

n+

N∑− 1

n= 0

(N− 1 −n)

]

=

1

2

N∑− 1

n= 0

(N− 1 )

=

N×(N− 1 )

2

(0.9)

a result that Gauss found out when he was a preschooler!^3

To improve the approximation of the integral we useTs= 10 −^3 , which gives a discretized signalnTs
for 0≤nTs<10 or 0≤n≤( 10 /Ts)−1. The area of the pulses isnTs^2 and the approximation to the
integral is then

(^10) ∑^4 − 1
n= 0
p[n]=
(^10) ∑^4 − 1
n= 0
n 10 −^6

=

104 ×( 104 − 1 )

106 × 2

=49.995

(^3) Carl Friedrich Gauss (1777–1855) was a German mathematician. He was seven years old when he amazed his teachers with his trick
for adding the numbers from 1 to 100 [7]. Gauss is one of the most accomplished mathematicians of all times [2]. He is in a group of
selected mathematicians and scientists whose pictures appear in the currency of a country. His picture was on the Mark, the previous
currency of Germany [6].