Suppose PSD of a signal $x(n)$ is $ PSD_{xx}(w)=1$.
Now if a window function w(n) of length $ 0\le n \le N-1$ is applied to $x(n)$. what will be the new PSD?
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When you "window" a signal in the time domain you are multiplying a block of signal data, element by element, with the window's elements. This corresponds to convolution in the frequency domain. Windows are usually "pseudo-gaussian", in the sense that they look more-or-less like a bell curve (since they have finite length, of curse, the tails only go so far). Like true gaussians, their Fourier transforms also look like gaussians, so in the frequency domain you will get some smearing. The smearing effect is reduced as the window gets larger. EDIT: As Dilip and Hilmar pointed out, the correct formulation is $S_{out}(\omega) = S_{in}(\omega)\mid H(\omega)\mid^2$. |
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Windowing is multiplication in the time domain which corresponds to convolution in the frequency domain, so the spectrum of the windowed signal is the spectrum of the original signal convolved with the spectrum of the window. The PSD is simply the magnitude squared of the spectrum of the windowed signal, i.e. $$P(\omega)=\left \| X(\omega)\ast W(\omega) \right \|^{^{2}}$$ where $\ast$ is the convolution operator. Hence the desire to have the spectrum of the window look like an impulse as that has the least impact on the PSD. |
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