Signals and Systems - Electrical Engineering

Problems 57

whereNTs= 1 (i.e., we segment the interval [0,1] intoNintervals of widthTs). Graphically show that

the above equation makes sense by showing the right and left bounds as approximations for the area

underx(t).

(b) LetTs=0.001. Use the symbolic functionsymsumto compute the left and right bounds for the above

integral. Find the average of these results and compare it with the actual value of the integral.

(c) Verify the symbolic results by finding the sums on the left and the right of the above inequality using

the summation given at the beginning of the problem. You need to change the dummy variables.

(d) Write a similar MATLAB script to compute the area under the signaly(t)=t^2 from 0 ≤t≤ 1. Let

Ts=0.001. Compare the average of the lower and upper bounds to the value of the integral.

0.8. Integrals and sums—MATLAB
Although sums behave like integrals, because of the discrete nature of sums one needs to be careful with
the upper and lower limits more than in the integral case. To illustrate this, consider the separation of an
integral into two integrals and compare them with the separation of a sum into two sums. For the integral
we have that
∫^1

0

tdt=

∫0.5

0

tdt+

∫^1

0.5

tdt

Show that this is true by computing the three integrals. Then consider the sum

S=

(^100) ∑
n= 0
n
Find this sum and determine which of the following is equal to this sum:
S 1 =
∑^50
n= 0
n+
(^100) ∑
n= 50
n
S 2 =
∑^50
n= 0
n+
∑^100
n= 51
n
Use symbolic MATLAB functionsymsumto verify your answers.
0.9. Sum of geometric series
The geometric series
S=
N∑− 1
n= 0
αn
will be used quite frequently in the next chapters, so let us look at some of its properties:
(a) Supposeα= 1 ; what isSequal to?
(b) Supposeα6= 1 ; show that
S=
1 −αN
1 −α
This can be done by showing that( 1 −α)S=( 1 −αN). Why do you need the constraint thatα6= 1?
Would this sum exist ifα > 1? Explain.