Power spectrum when the wave number is not an integer

[arcTomato]

Homework Statement

I would like to know the about the power spectrum of a sin wave when the wave number is not integer.

Relevant Equations

Fourier transform

Hi all.

I made a program of DFT, so I made the power spectrum of a sin wave.
This is the sin wave I used.
All data number $N=100$ and the frequency of sine wave is 4.5Hz.

And the power spectrum is this.

The wave number is not integer so the spectrum has the side lobe.
But I think this is not right.
I think the power spectrum shape should be the product of the data's power spectrum and the window function's power spectrum(in my case the window is Square wave, and in the under image is also square wave too).
like this.

　　　　　　

I would like to know why the shape of the spectrum I made is like this.

Thank you.

 

[BvU]

Keep in mind that the FT assumes periodic continuation. Your 4 1/2 periods don't transform to a single spike that way.

 

[arcTomato]

Hi @BvU again!
yes!
because there is the discontinuity point, right??
but I would like to understand why in another way(the spectrum of window).
I think the spectrum of the data which wave number is 9 is single spike.
If I multiply the data (9 wave number) with square wave(it becomes 4.5 wave number ), I think the spectrum should look like the third image.
But the spectrum I made is not look like.