DFT and windows

Question

I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following FT (Hann window), will suppress the side lobes found in a signal FT (second picture) :

But I dont understand how are the values |F($\omega$)| related to suppressing the signals side lobes ... e.g. Tukey window plotted as |F($\omega$)|

How is the width and the pace of decreasing sidelobes (of the above plot) related to getting rid of sidelobes of signals FT ? Is there any intuitive way to explain ?

Answer 1

the DFT (of size $N$, with a window $h_n$) of $$x_n = \sum_{m=1}^M C_m e^{2 i \pi f_m n} \qquad \qquad n \in \{0,\ldots,N-1\}$$ (a mixture of (complex) sines) is $$X_k = \sum_{n=0}^{N-1} h_n x_n e^{-2 i \pi k n /N} = \sum_{m=1}^M C_m H(k/N-f_m)$$ where $$H(\xi) = \int_{-\infty}^\infty \sum_{n=0}^{N-1} h_n \delta(x-n) e^{-2 i \pi \xi x} dx$$ is the Fourier transform of $\sum_{n=0}^{N-1} h_n \delta(x-n)$ i.e. of the (discretized) window.

Proving it is just a matter of the frequency shift theorem for the Fourier transform (of distributions). with $$G(\xi) = FT[g(x)](\xi) = \int_{-\infty}^\infty g(x) e^{-2 i \pi \xi x} dx$$ then the Fourier transform of $g(x) e^{2 i \pi f x}$ is $$FT[g(x) e^{2 i \pi f x}] (\xi) = \int_{-\infty}^\infty g(x) e^{2 i \pi f x} e^{-2 i \pi \xi x} dx = G(\xi-f)$$

Answer 2

Alright, here we come. Even though the above answer is great and actually explains quite a lot, here is an answer using crayons.

compare the following, left side Rectangular window, right side Tukey window with ${\alpha} = 1$ (generated by Matlab).

enter image description here

My question was, how does the lower pair of pictures tells how the window works ? Well, it cant be stated directly from those pictures, or at least it is not as obvious as from the upper pair, where we can see the real part going to positive and negative values.

As mentioned in my question, $$Time Domain(multiplication) = Frequency Domain(Convolution)$$

For Rectangular Window:

No change on the signal DFT. The discrete window have DFT values $c_0 = 1$ and the rest of the $c_k = 0$

For Tukey Window:

The Tukey DFT has 2 spikes. 1 positive and 1 negative. Therefore when it passes the signal DFT during convolution, these 2 spikes flatten everything except big spikes (the side lobes or "skirts" of the signal DFT are flattened, while the main spikes of signal DFT remain)