frequency components of discrete Gaussian-windowed signal When we are do a discrete Gabor transform on a 1D signal, we apply a Gaussian window on a segment of the signal. However, due to the tapering ends of the Gaussian window, the magnitude of the data points will be reduced more towards the ends, This implies that the the frequency components of the segment of the signal will be suppressed when we apply FFT to it. After performing some experiments, I found that at least 2 periods of the signals is required, so that no misjudgement of the behaviour of the signal will happen.

What is the range of frequencies in which the magnitude of the FFT is still representative of the frequencies of windowed signal?
No window will give you zero processing (coherent) gain. Check out the table in fred harris's paper.

The table says there is a processing gain of $0.37$ to $0.51$ depending on your $\alpha$ parameter: