I am studying spectrum analyzers and a doubt came to me about real time spectrum analyzers.
Suppose you have a signal in the time domain: in order to calculate the Fourier Transform, you need to know that signal at every time instant. In other words: the computation of the Fourier Transform is something which can be done only after a signal has somehow propagated, because by definition you integrate over all time instants.
Question: how does a spectrum analyzer perform a real time frequency domain analysis of a signal if this signal still has to exist? I mean, if you have a signal in the time interval [t1, t2], how can you perform the Fourier Transform of the signal in real time? If you're recording for example a sound, this sound will gradually arrive to the spectrum analyzer, thus you ignore the future values of the signal, because the generic time instant t* at which you're receiving the signal is smaller than t2, and then you ignore all the values the signal assumes in the time interval [t*, t2]. But by definition of Fourier Transform, you must integrate all over the time interval [t1, t2], meaning that you must know a priori the shape of the function at every time instant.