Deriving the Resolution Equation of an FMCW RADAR
Definitions:


$\tau$: round-trip delay.  $\Delta \tau$ = delay resolution.
$r$: range.  $r=c\tau/2$.  Range resolution $\Delta r = c \Delta \tau / 2$.
$B$: chirp bandwidth
$T$: chirp sweep time
$R$: chirp rate.  $R=B/T$


Basic steps:


We estimate the delay by estimating the frequency shift between the received and transmitted signal.  Typically the resolution of a frequency estimate is the inverse of the observation time, which in this case is $1/T$.  Why?  Think of taking the Fourier Transform of a tone with a rectangular window T.  The window will have nulls at $f-1/T$ and $f+1/T$, where $f$ is the tone's frequency.
Because the chirp rate is $R=B/T$, the time resolution corresponding to a frequency resolution of $\frac{1}{T}$ is $\frac{1/T}{B/T} = \frac{1}{B}$.
Plugging this into the range resolution equation, we have $\Delta r = c/(2B)$.


Intuitively, we improve range resolution by (1) improving resolution of the frequency shift estimate (i.e., increasing sweep time), and (2) decreasing the time delay corresponding to a given frequency shift (i.e., increasing chirp rate).  To do one without sacrificing the other, we need to increase the bandwidth.  If we double the chirp rate but cut the sweep time in half, thus keeping the bandwidth the same, the time delay per frequency shift decreases but the frequency resolution gets worse.  Likewise, if we cut the chirp rate in half but double the sweep time, the time delay per frequency shift increases while the frequency resolution improves.  In either case, the range resolution does not change.