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I am new to the signals area and have been reading through a lot and have some questions.

Suppose I have a saved audio file on my computer, .wav file. I can view the time domain of the signal by identifying the sampling rate and bit depth. The resulted domain signal is a discrete signal, since everything stored in a computer is numbers.Now, I can convert this discrete signal into the frequency domain using the Fourier Transform algorithm. Since it is discrete, I must be using DFT equation below

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Now, I can view the spectrogram of either continuous or discrete using the STFT, which computes the Fourier Transform over segmented windows of the whole signal (where we use FFT for DFT to reduce the complexity). The STFT equations of the continuous and discrete signals are:

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My questions are:

1- Is my understanding correct?

2- Can we apply Hann Windowing on discrete signals?

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Your understanding is correct and you can apply Hann windowing on discrete signals. The choice of your window depends on how easily you can visually resolve two sinusoids that are closely spaced in the spectrogram. $$STFT = X(m,\omega) = \sum_{n=-\infty}^{n=+\infty}x_nw_{n-mL}e^{-j\omega n}, $$ where $L$ is the step size of spectrogram (interval after which you place next window). and $W$ is the size of window, which decides resolution of frequency (row-wise separation of frequencies). Higher the $W$, more finer the resolution. You can see there is overlap between adjacent windows, the overlap being $W-L$.

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