what is the the resultant, or, "final" frequency of the FFT of a chirped signal Let's say we have a time-transient with a chirped signal,
$$V(t) = \cos\left( 2 \pi f(t) t \right)$$
where $f(t)$ is the frequency of the signal and is time dependant. My question is how would one describe the final frequency, as produced by an FFT spectrum? My intuition is that it would be the time average such that
$$f^{\rm{FFT}}_{\rm{final}} = \frac{1}{T}\int^{T}_{0} f(t) \ {\rm{d}}t$$
where $T$ is the length of the time-transient signal used in the FFT. Strictly speaking I think would use a Gabor transform for non-stationary signals, but seeing as the spectrum analyser I am working with uses an FFT algorithm.
Depending on the chirp, if you seek a closed form solution, there may not be one. For empirical estimation, one can use non-stationary methods.
Take exponential with frequency ranging from 0.005 to .125 of sampling frequency. Applying synchrosqueezed CWT and plotting energy norm per row vs cumulative sum of rows' associated frequency:

The red line marks where energy contributions grow insignificant (and thus max freq is likely reached), which can be found with a flatline detection algorithm. It's heuristics and engineering.
Code below, can be improved.
import numpy as np
from ssqueezepy import ssq_cwt, TestSignals
from ssqueezepy.visuals import plot, imshow

def flatline_after_max(x, count=10, rel_dist=.1, max_frac=.99):
    # pick slightly less than max
    max_idx = np.where(x &gt; max_frac*x.max())[0][0]
    ctr = 0
    for i in range(max_idx, len(x)):
        if abs(x[i] - x[i - 1]) / x[i] &lt; rel_dist:
            ctr += 1
        else:
            ctr = 0
        if ctr == count:
            return i - (count - 1)
    
# define test signal ########################################################
N = 4096  # sampling rate (or number of samples)
fmin = .005
fmax = .125
x = TestSignals().echirp(N=4096, fmin=fmin*N, fmax=fmax*N)[0]

# synchrosqueeze &amp; visualize
Tx, _, ssq_freqs, *_ = ssq_cwt(x, ('gmw', {'beta': 4}))
imshow(Tx, abs=1, yticks=ssq_freqs[::-1], ylabel=&quot;Fraction of fs&quot;,
       title=&quot;abs(SSQ_CWT)&quot;)

# compute row energies and find 'stopping point' #############################
Tx_energies = np.sum(np.abs(Tx[::-1])**2, axis=1)
Tx_energies_cs = np.cumsum(Tx_energies)
max_idx = flatline_after_max(Tx_energies_cs)
est_freq = ssq_freqs[max_idx]

# plot #######################################################################
plot(ssq_freqs, Tx_energies_cs, xlabel=&quot;Fraction of fs&quot;,
     title=&quot;cumsum(energy(Tx, axis=1)) | est_freq=%.3f&quot; % est_freq,
     vlines=(est_freq, {'color': 'tab:red'}))