# Comparability of Analytic Signal with (Discrete) Fourier Analysis

I am working with a finite signal response from an experiment. Basically, I feed in a uniform amplitude sine wave which ramps from 20Hz to 20kHz over the course of 50 seconds, and I read the output, which has the same frequency but with varying amplitudes. The ramp is logarithmic, so I know exactly the conversion which tells me the frequency at a given time in my sweep.

I have two ways which I can think of to analyze this response signal. The first is a discrete Fourier transform, and the second is a Hilbert transform to find the analytic signal of the response, which I can use to get an envelope of my response signal.

My assumption with the second is that it gives me the most accurate results, since I can take the envelope, which is basically a measure of amplitude as a function of time, and convert the time coordinates to frequency to get amplitude as a measure of frequency.

I don't believe the Fourier transform is a good representation, since the discrete transform of the input signal is not a straight line, but rather grows with frequency. I imagine this is because it takes larger high-frequency signals to cancel out low frequency waves in the later part of the signal, but honestly I don't know if I properly understand the Fourier transform well enough to know if this is true.

My question is, essentially, which of these methods is more suited to understanding the frequency response? I believe the wave envelope seems to make the most sense, but some of the people I'm working with say the Fourier transform should be sufficient. I think the Fourier transform is giving a recipe to generate the signal with additive sine waves rather than the actual amplitude of a given frequency. In other words, if I fed in single frequencies and plotted them, they won't line up with a Fourier transform. Is this a correct way to understand this? If I am right, is there any way to prove it analytically?

P.S. I don't post much on this board, and I realize I haven't included any equations, so if I need to add something to my question to make it more understandable, please let me know. If I'm wrong, why am I wrong, and why do these methods give different results?