Plot of the Fourier transform of the Sine Window

Question

In Wikipedia, a Fourier transform of the Sine Window \begin{align} w[n] = \sin \left(\frac{\pi n}{N} \right), \ 0\leq n \leq N \end{align}

is plotted like below. However, I am not sure how to obtain this plot.

The Fourier transform of the sine function should be \begin{align} \mathcal{F}\{\sin(2\pi A t)\} = \frac{1}{2i}[\delta(f-A)-\delta(f+A)] \end{align} where $\delta$ is a Dirac-delta function. Thus, the analytical form of the Fourier transform of the Sine Window should be the discrete version of this expression, which seems to be far away from the orange curve in this plot. However, on the other hand, if the Fourier transform is really expressed in terms of the delta function, I don't understand the practical meaning of the Sine Window either, so the plot in Wikipedia seems to be more reasonable.

Is the plot correctly describing the Fourier transform of the Sine Window? If so, what is the analytic form?

Answer 1

The main difference here is that you are either looking at a sine wave that's infinitely long or just half a period a sine wave (with the rest being zero).

Roughly speaking we have $$\int_{-\infty}^{\infty} \sin(\omega_0 t)e^{-j2\pi \omega} dt \neq \int_{0}^{T/2} \sin(\omega_0 t)e^{-j2\pi \omega}dt$$

where $T = \frac{1}{2\pi \omega_0}$

Is the plot correctly describing the Fourier transform of the Sine Window?

Yes, of course.

There are 4 different types of "Fourier transforms" depending on whether the signals are discrete/continuous or periodic/aperiodic in either domain. The plot describes the Discrete Fourier Transform DFT of sine window accurately.

If so, what is the analytic form?

Just plug it into the definition of the DFT

$$X[k]= \sum_{n=0}^{N-1} x[n] \cdot e^{-j2\pi\frac{kn}{N}} = \sum_{n=0}^{N-1} \sin(\frac{n\pi}{N}) \cdot e^{-j2\pi\frac{kn}{N}}$$

If the frequency of the sine wave is an integer multiple of the DFT frequency resolution (sample rate divided by DFT length) you will indeed get something like a delta impulse, but for any other frequency/period you get some amount of spectral leakage. The sine window is an extreme example: the multiple is 0.5 which is NOT an integer.