The sum of two or more sinusoids may or may not be periodic depending on the relationship
Question:
The sum of two or more sinusoids may or may not be periodic depending on the relationship of their separate frequencies. For the sum of two sinusoids, let the frequencies of the individual terms be f1 and f2, respectively. For the sum to be periodic, f1 and f2 must be commensurable; i.e., there must be a number f0 contained in each an integral number of times. Thus, if f0 is the largest such number,
f1 = n1f0 and f2 = n2f0
where n1 and n2 are integers; f0 is the fundamental frequency. Which of the signals given below are peridic?
Find the periods of those that are periodic.
(a) x1 (t) = 2 cos (2t) + 4 sin(6πt)
(b) x2 (t) = cos (6πt) 7 cos(30πt)
(c) x3 (t) = cos (4πt) + 9 sin(21πt)
(d) x4 (t) = 3 sin (4πt) + 5 cos(7πt) + 6 sin (11πt)
(e) x5 (t) = cos (17πt) + 5cos(18πt)
(f) x6 (t) = cos (2πt) + 7 sin(3πt)
(g) x7 (t) = 4 cos (7πt) + 5 cos(11πt)
(h) x8 (t) = cos (120πt) + 3 cos(377t)
(i) x9 (t) = cos (19πt) + 2 sin (21πt)
(j) x10 (t) = 5 cos (6πt) + 6 sin(7πt)
Step by Step Answer:
Principles of Communications Systems, Modulation and Noise
ISBN: 978-8126556793
7th edition
Authors: Rodger E. Ziemer, William H. Tranter