Fourier Series Visualization

Square Wave: Odd and half-wave symmetric (only odd harmonics $k=1,3,5...$). Features the Gibbs phenomenon (overshoot at discontinuities). Coefficients decay as $1/k$. Note how a negative index $k$ in the exponential sum naturally implements the complex conjugate $c_{-k} = c_k^*$.

Trigonometric Form

x_{N} (t) = \sum_{\begin{matrix} k = 1 \\ k odd \end{matrix}}^{N} \frac{4}{π k} \sin (k t)

Compact Form

x_{N} (t) = \sum_{\begin{matrix} k = 1 \\ k odd \end{matrix}}^{N} \frac{4}{π k} \cos (k t - \frac{π}{2})

Exponential Form

x_{N} (t) = \sum_{\begin{matrix} k = - N \\ k odd \end{matrix}}^{N} \frac{- j 2}{π k} e^{j k t}

Fourier Series Multi-Form Visualization

ECE 210: Analog Signal Processing — University of Illinois Urbana-Champaign

Trigonometric Form

Compact Form

Exponential Form