Square and symmetric triangle waves

Figure 10.6: Symmetric triangle wave, obtained by superposing parabolic waves with $(B, c)$ pairs equal to $(0, 4/N)$ and $(N/2, -4/N)$.

To demonstrate how to obtain Fourier series for classical waveforms in general, consider first the square wave,

$$x[n] = s[n] - s[n-{N \over 2}]$$

equal to $1/2$ for the first half cycle ($0 <= n < N/2$) and $-1/2$ for the rest. We get the Fourier series by plugging in the Fourier series for $s[n]$ twice:

$$x[n] \approx {1 \over \pi} \left [ {\sin ( \omega n )} + ... ...\over 2} + {{\sin ( 3 \omega n)} \over 3} + \cdots \right .$$

$$\left . -{\sin ( \omega n )} + {{\sin ( 2 \omega n)} \over 2} - {{\sin ( 3 \omega n)} \over 3} \pm \cdots \right ]$$

$$= {2 \over \pi} \left [ {\sin ( \omega n )} + {{\sin ( 3 ... ...\over 3} + {{\sin ( 5 \omega n)} \over 5} + \cdots \right ]$$

The symmetric triangle wave (Figure 10.6) given by

$$x[n] = {{4} \over {N}} p[n] - {{4} \over {N}} p[n-{N \over 2}]$$

similarly comes to

$$x[n] \approx {4 \over {{\pi^2}}} \left [ {\cos ( \omega n ... ...over 9} + {{\cos ( 5 \omega n)} \over 25} + \cdots \right ]$$