Concerning your preliminary note: depending on the use of windows, they are sometimes normalized, for instance toward unit energy (values divided by $\left(\sum_n |w_n|^2\right)^{1/2}$) or unit sum (values divided by $\sum_n w_n$).
Here, in appearance, as you mentioned, the product by a unit amplitude window does not seem to change anything on the signal. But you are assuming here, with point-wise product:
x*1
that x
and the window already have the same size. So for any processing that applies to the finite-sized x
, multiplying with a unit window won't change the numerical outcomes.
However, you should take into account that:
- the finite-sized
x
is already discretized, and can be trimmed from the left and the right, or set to zero outside the window, and this already affects interpretations
- the processing on this finite-sized, discrete signal can be an approximation of some continuous processing.
Hence, interpretations or features extracted can be altered by the above considerations, from the underlying continuous time, infinite support signal. What is classically done is to consider an (infinite) signal $x = \ldots,x_0,x_1,\ldots,x_{N-1},\ldots$, unknown where the dots are, multiplied by a infinite window $w = \ldots,0,1_0,1_1,\ldots,1_{N-1},0,\ldots$, which is fully known outside the support. Hence, to analyze a potentially infinite-length signal $x$, unknown to the left and the right, it is more practical to consider that it is multiplying by an infinite window (known to be zero to the left and the right), whose properties are well-established, and whose effects on the signal can be measured.
As @hotpaw2 mentioned, such a window turns in the Fourier domain into an aliased cardinal sine, or a periodic cardinal sine.