# Windowing a time domain OFDM signal

Windowing is a popular method of reducing the spectral sidelobes of OFDM. A popular window usually used for this purpose in the raised cosine (RC) window, because of its tapered and smooth edges. I understand the concept of windowing and why it reduces the sidelobes, however what I don't understand is this: "windowing a time domain OFDM signal is equivalent to every subcarrier being individually windowed by the window function (RC for example), basically the sinc sidelobes of every subcarrier will be reduced". This is true and you can actually see it when the window function is applied to a non-contiguous OFDM signal (some subcarriers are deactivated somewhere in the middle of the spectrum), the spectrum is not only reduced at the edges of the band, but also in the in-band. You can't get this if you apply a filter to the signal, with a filter you only see power spectrum reduction in the edges but not in the band. I'd appreciate if someone can explain why this is true, preferably with some equations.

## 1 Answer

If I understand the question properly, then it sounds like the difference between time-domain multiplication and frequency domain multiplication.

Windowing is a time-domain multiplication operation. As such, it is equivalent to convolving the spectrum with the window kernel, which will operate all frequencies the same way. On the other hand, filtering is (can be) a frequency domain multiplication, so the filter can be designed to operate specifically on chosen frequencies (or bands of frequencies if you like).

To look at it slightly differently: if you consider your signal as the sum of all your subcarriers (which you can do with linearity), then windowing your signal with an RC is the same as windowing each of your subcarriers with an RC before adding them together.