*Question E8A01:
What technique shows that a square wave is made up of a sine wave and its odd harmonics?*  

*Correct Answer is  A: Fourier analysis*  

### AI Response: (may contain errors)  

Fourier analysis is the correct answer because it is the mathematical technique used to decompose complex waveforms, like a square wave, into a series of simpler sine waves. Specifically, a square wave can be represented as a sum of its fundamental frequency and its odd harmonics (e.g., 3rd, 5th, 7th, etc.).

### Explanation:

1. **Why the correct answer is correct:**
   - **Fourier Analysis**: This technique allows us to express any periodic waveform as a sum of sine and cosine functions. For a square wave, Fourier analysis shows that it is composed of a fundamental sine wave and its odd harmonics. The formula for a square wave is:
     $$ 
     f(t) = \frac{4}{\pi} \left( \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \cdots \right)
     $$
   - Here, $\omega$ is the angular frequency of the fundamental sine wave.

2. **Why the other choices are incorrect:**
   - **Vector Analysis**: This is used in physics and engineering to study quantities that have both magnitude and direction, such as force or velocity. It is not used for decomposing waveforms into sine waves.
   - **Numerical Analysis**: This involves algorithms for solving mathematical problems numerically. It is not specifically used for breaking down waveforms into their harmonic components.
   - **Differential Analysis**: This involves the use of calculus to study rates of change. It is not used for analyzing the frequency components of waveforms.

### Key Point:
- **Fourier Analysis** is essential for understanding how complex signals can be broken down into simpler sinusoidal components, which is fundamental in signal processing and communications.