Question

Which of the following signal(s) has (have) Fourier series representation?

(i) cos(t)+sin(2t)
(ii) cos(πt)+sin(2πt)
(iii) cos(π2t)+sin(2π2t)


(a) (i) only
(b) (ii) only
(c) (iii) only
(d) (i) and (ii) and (iii)
(e) none of the above

Answer 1

The signal cos(t) + sin(2t) and the signal cos(πt) + sin(2πt) have Fourier series representations, while the signal cos(π2t) + sin(2π2t) does not.

Step-by-step explanation:

The Fourier series representation can be used to represent periodic functions as a sum of sines and cosines. A function has a Fourier series representation if it is periodic and has a finite number of discontinuities. Let's analyze each signal:

(i) cos(t) + sin(2t): This signal has a Fourier series representation because it is a sum of a cosine function and a sine function, both with frequencies that are integer multiples of the fundamental frequency.

(ii) cos(πt) + sin(2πt): This signal also has a Fourier series representation because it is a sum of a cosine function and a sine function, just like in the previous signal. The fact that the frequencies are multiples of π does not affect the existence of a Fourier series representation.

(iii) cos(π2t) + sin(2π2t): This signal does not have a Fourier series representation because the frequencies are not integer multiples of the fundamental frequency. The frequency of the cosine function is π^2 and the frequency of the sine function is 4π, which means they cannot be represented as a sum of sines and cosines with integer frequencies.

Therefore, the answer is (b) (ii) only.

Answer 2

Signals (i), (ii), and (iii) have Fourier series representations.

Step-by-step explanation:

In order for a signal to have a Fourier series representation, it must be periodic. A signal is periodic if it repeats itself after a certain interval. We can determine if a signal has a Fourier series representation by examining its frequency and phase.

(i) cos(t)+sin(2t) has a frequency of 1 and a phase of 0 which makes it periodic and suitable for a Fourier series representation.

(ii) cos(πt)+sin(2πt) has a frequency of π and a phase of 0 which makes it periodic and suitable for a Fourier series representation.

(iii) cos(π2t)+sin(2π2t) has a frequency of 2π and a phase of 0 which makes it periodic and suitable for a Fourier series representation.

Therefore, the correct answer is (d) (i) and (ii) and (iii).