How to calculate STFT of a function for a rectangular window How to calculate the STFT (by hand) of $$u(n)\cos(0.2\pi n)$$ for a rectangular window of a length 20, positioned at $n = 5$.

I know that to use STFT I need to divide longer signal to a shorter parts and than calculate Fourier Transform on each part. Also doing it from the definition is very long and I assume that there is a more efficient way to do it on paper.

I don't know how to start with the question so any materials on this topic are appreciated.

I came up with this idea:

First picture is just a rectangular window of length 20 and position 5.
Second picture is how I see the STFT of it. Now to calculate STFT should I provide some coefficients like mainlobe width and highest sidelobe? How I can calculate them from my data? 

Are my pictures good?
I don't understand what you mean by "positioned at $n=5$" with length $20$. Indeed:


is "positioned" the beginning  or the center?
is length an half-length around a central position, or a full length?
finally, at a given location, this is not an STFT anymore, but a mere windowed Fourier transform.


If I interpret your question in its most obvious sense, the window starts from $n=5$ to $n=20+5-1=24$, on an interval where $u[n]=1$, so you'd just have to compute a simple DFT of a cosine. The result can be obtained via Euler/De Moivre formulae, with two finite sums of geometric series. 

If this is not the case, please provide more information, close forms are at hand