Parabolic wave

The same analysis, with some differences in sign and normalization, works for parabolic waves. First we compute the difference:

$$p[n] - p[n-1] = { { {{({n\over N} - {1\over 2})}^2} - {{({{n-1}\over N} - {1\over 2})}^2} } \over { 2 }}$$

$$= { { {{({n\over N} - {N\over {2N}})}^2} - {{({{n}\over N} - {{N - 2}\over {2N}})}^2} } \over { 2 }}$$

$$= { { {{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}} } \over { 2 }}$$

$$\approx - s[n] / N .$$

So (again for $k \neq 0$, small compared to $N$) we get:

$${\cal FT}\{ p[n] \} (k) \approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)$$

$$\approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {{-iN} \over {2 \pi k}}$$

$$= {N \over {4 {\pi ^2} {k^2}}}$$

and as before we get the Fourier series:

$$p[n] \approx {1 \over {2 {\pi^2}}} \left [ {\sin ( \omega ... ...\over 4} + {{\sin ( 3 \omega n)} \over 9} + \cdots \right ]$$