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

Choices:  
A: Fourier analysis  
B: Vector analysis  
C: Numerical analysis  
D: Differential analysis    

Correct Answer is  A: Fourier analysis  

Correct answer: A — Fourier analysis

Why A is correct:
- Fourier analysis decomposes any periodic waveform into a sum of sinusoids (sines and cosines) at integer multiples of the fundamental frequency.
- A 50% duty-cycle, symmetric square wave contains only odd harmonics whose amplitudes decrease as $1/n$.
- Its Fourier series is:
  $$x(t) = \frac{4V_p}{\pi}\left(\sin \omega_0 t + \frac{1}{3}\sin 3\omega_0 t + \frac{1}{5}\sin 5\omega_0 t + \cdots \right)$$
  where $V_p$ is the peak amplitude and $\omega_0=2\pi f_0$.

Why the others are incorrect:
- B: Vector analysis — concerns vectors and operations on them; not the method for decomposing waveforms into harmonics.
- C: Numerical analysis — focuses on computational methods; while FFT is computed numerically, the underlying technique is still Fourier analysis.
- D: Differential analysis — uses derivatives/differential equations; not the method for harmonic decomposition of waveforms.

Useful formulas and rules of thumb:
- General Fourier series for a periodic signal with period $T$:
  $$x(t) = a_0 + \sum_{n=1}^{\infty}\left[a_n\cos(n\omega_0 t)+b_n\sin(n\omega_0 t)\right],\quad \omega_0=\frac{2\pi}{T}$$
- Symmetry shortcuts:
  - Odd symmetry ($x(t)=-x(-t)$) → only sine terms ($b_n$).
  - Even symmetry ($x(t)=x(-t)$) → only cosine terms ($a_n$).
  - Half-wave symmetry ($x(t)=-x(t+T/2)$), as in a square wave → only odd harmonics appear.