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2.40 The fundamental period of a periodic sequence with an angular frequency
0
satisfies
Eq. (2.53a) with the smallest value of and .
(a) For this problem,
0
= 0.3 , so the equation reduces to 0.3 = 2 , which is satisfied
with = 20, = 3.
(b) For this problem,
0
= 0.48 , so the equation reduces to 0.48 = 2 , which is
satisfied with = 25, = 6.
(c) For this problem,
0
= 0.525 , so the equation reduces to 0.525 = 2 , which is
satisfied with = 80, = 21.
(d) For this problem,
0
= 0.7 , so the equation reduces to 0.7 = 2 , which is satisfied
with N = 20, r = 7.
(e) For this problem,
0
= 0.75 , so the equation reduces to 0.75 = 2 , which is
satisfied with = 8, = 3.
2.41 Here
0
= 0.06 , so that Eq. (2.53a) reduces to 0.06 = 2 which is satisfied with =
100 and = 3. Other sequences
with the same fundamental period will
satisfy
Choosing = 1 and = 2, two possible sequences with the same
fundamental period would have
2
= 0.02 and
3
= 0.01 , respectively, resulting in:
= and =
2.42 In each of the following parts, is the fundamental period and is a positive integer. Eqs.
(2.53a) and (2.39) are used the compute the fundamental period and the average power of
the periodic sequences.
(a) For
=
and must satisfy the relation ( /3) = 2 , which is
satisfied by = 6 and = 1, which are the smallest values of and . The average
power can be calculated as follows:
=
=
=
=
=
=
=
(b) For
=
and must satisfy the relation (2 /5) = 2 , which is
satisfied by = 5 and = 1, which are the smallest values of and . The average
power can be calculated as follows:
=
=
=
=
=
=
=
.
(c) For
=
and must satisfy the relation (2 /7) = 2 , which is
satisfied by = 7 and = 1, which are the smallest values of and . The average
power can be calculated as follows: