Understanding the Signal u(sin(t))
The signal u(sin(t)) is a function that is periodic with a fundamental frequency. To analyze its Fourier series representation, we must consider its symmetry and harmonic content.
Fourier Series Representation
- The Fourier series of a periodic function decomposes it into a sum of sine and cosine functions with different frequencies (harmonics).
- A key aspect of the Fourier series is that it can include both even and odd harmonics, as well as a DC component, depending on the signal's symmetry.
Symmetry Analysis
- The function u(sin(t)) is an odd function, meaning it is symmetric about the origin. This characteristic leads to the following implications:
- It contains only sine terms (which are odd functions) in its Fourier series expansion.
- There is no DC component since a DC term would be represented by a cosine term.
Harmonics Present
- The Fourier series will consist of the following:
- Odd Harmonics of Sine: The periodic nature of sin(t) leads to the presence of odd harmonics (3rd, 5th, etc.) in the expansion.
- DC Component: Since the signal is odd, the average value over one period is zero, leading to no DC term.
Conclusion
- Thus, the Fourier series of u(sin(t)) includes:
- Both odd harmonics of sine.
- Zero DC component.
- Therefore, the correct answer is option 'C': DC and odd harmonics of sine.
This understanding highlights the significance of symmetry in determining the harmonic content of periodic signals in the context of Fourier analysis.