Chirp signal won't generate continuous frequency band

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I am trying to apply an AC signal with a continuous "band" of frequencies, I generate the waveform in MATLAB and use the welch spectrum analysis to get a nice continuous band of frequencies from f1 to f2. When I actually generate the wave with a function generator, the signal reads back as a bunch of discrete frequencies between f1 and f2. I'm assuming its a problem of sampling in discrete time but I am unsure how to fix it. The end goal is to apply the AC voltages to something and find/track the resonance frequency within the band.

More specifically I am looking at 300-400 kHz and my sampling rates on the generator can range from 1MHz to 2.4GHz, I'm using a linear frequency change with respect to time (the one with sin(At^2 + f0t)).

Any idea on how I can get a more continuous signal?

I don't think its a measurement issue (not sure) because it is independent of window functions, measuring the signal applied or directly from the generator, sampling rates, etc.). I'm not sure how to check that definitively so if you know how to confirm that issue that would also be helpful.

Here's the formula for the sine wave.

y=sin(2pi((f_width/(2*tmax))t.^2+f1t));

t is a 1xn vector (solved for so that the time matches the sampling rate of the generator) and everything else is constant.

I have also used the built-in function in the generator (zurich instruments) and gotten similiar issues.

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