I thought that DSP would be done by using FFT of portions of a signal, modify the samples that result from FFT (since they represent the spectrum of our signal + noise) and remove any unwanted signals and than do an inverse FFT to get a time domain representation of the filtered signal (noise has been removed now). This however is not done, instead we do all the work in time domain using window functions. Why?
If we multiply window function in time domain than we are convolving the window function's frequency response with spectrum of our signal in frequency domain, how does that work out? I mean if we just did all the work in frequency domain by multiplying our signal with the frequency response of the filter, that would be like filtering right? But here we do all the stuff in time domain instead using window.
->Lets see where my confusion comes from. For analogue filters e.g low pass filter, we have this pulse like frequency response. When we filter a signal, we are effectively multiplying our signal's spectrum with the pulse like frequency response of the filter. This will reduce all frequencies in our signal above a cut-off to 0. This is how a low pass filter works in essence. Why not do the same with digital filters as well?