Question

Which of the following cannot be the Fourier series expansion of a periodic signal?

(a) x1(t) = 2 cos  + 3 cos 3

(b) x2(t) = 2 cos  + 7 cos

(c) x3(t) = cos  + 0.5

(d) x4(t) = 2 cos 1.5 + sin 3.5t

I had been asked this question in class test.

Question is from Common Fourier Transforms in chapter Fourier Transform of Signals and Systems

Answer 1

Right choice is (b) x2(t) = 2 cos  + 7 cos

Easy explanation: x1(t) = 2 cost  + 3 cost  is periodic signal with fundamental frequency w0 = 1.

x2(t) = 2 cos πt + 7 cos t The frequency of first term w1 =  π frequency of 2nd term is w2 = 1. Since, \(\frac{ω_1}{ω_2}\) = π, which is not rational. So, x(t) is not periodic.

x3(t) = cos t + 0.5 is a periodic function with w0 = 1

x4(t) = 2 cos(1.5)t + sin(3.5)t first term has frequency w1 = 1.5π and 2nd term has frequency w2 = 3.5π. Since, \(\frac{ω_1}{ω_2}  = \frac{3}{7}\), which is rational.

Since x2(t) is not periodic, so it cannot be expanded in Fourier series.