Question

I. In the given scenario where the Fourier series has only odd harmonics, and (x(t)) is an even function, the sketch of (x(t)) in the interval (0 < t < T) would resemble which of the following?

a. A symmetric waveform with respect to the y-axis
b. An asymmetric waveform with respect to the y-axis
c. A waveform with both even and odd harmonics
d. A flat line at the x-axis

ii. In the given scenario where the Fourier series has only odd harmonics, and (x(t)) is an odd function, the sketch of (x(t)) in the interval (0 < t < T) would resemble which of the following?

a. A symmetric waveform with respect to the y-axis
b. An asymmetric waveform with respect to the y-axis
c. A waveform with both even and odd harmonics
d. A flat line at the x-axis

Answer 1

In the context of Fourier series with only odd harmonics, an even function (x(t)) would result in a symmetric waveform about the y-axis. An odd function (x(t)) would result in an asymmetric waveform about the y-axis.

Step-by-step explanation:

In the given scenario where the Fourier series has only odd harmonics, and (x(t)) is an even function, the sketch of (x(t)) in the interval (0 < t < T) would resemble a symmetric waveform with respect to the y-axis. This is because even functions are defined by their property of symmetry about the y-axis, which means for an even function, y(x) = y(-x). In contrast, if we consider the Fourier series has only odd harmonics, and (x(t)) is an odd function, the sketch of (x(t)) will be an asymmetric waveform with respect to the y-axis. This is because odd functions are anti-symmetric with respect to the y-axis, satisfying y(x) = -y(-x).