# Deriving the IF phase equation of FMCW

My understanding of the FMCW was limited to one chirp and was mostly focused on frequency analysis. (Mostly built upon this answer)
I wanted to expand my understanding to multiple chirps and was introduced to the equation below $$\phi_0-2\pi\frac{2v}{c}f_cnT_c-2\pi\left[\frac{2v}{c}(f_c+nB)+\frac{B}{T_c}t_d\right]t$$ Where,
$$v$$: Velocity of target
$$n$$: Chirp number
$$T_c$$: Chirp duration
$$B$$: Bandwidth of signal
$$t_d$$: round-trip time delay
The equation represents the phase of the received beat signal for the nth chirp. For stationary targets, the Rx signal will just be a time-delayed version of the Tx signal. So the IF signal can be represented as $$A\cos(\phi_T(t-t_d)-\phi_T(t))$$ For a sawtooth LFM chirp $$\cos(2\pi f_ct+\frac{B}{T_c}t^2)$$, the phase of the IF signal is \begin{align} \phi_{IF}&=2\pi f_c(t-t_d)+\pi\frac{B}{T_c}(t-t_d)^2-2\pi f_ct-\pi\frac{B}{T_c}t^2 \\ &=\underbrace{-2\pi f_ct_d-\pi\frac{B}{T_c}t_d^2}_{\phi_0}+2\pi\left(\frac{B}{T_c}t_d\right)t \end{align} So, I understand what $$\phi_0$$ represents and where the rightmost term came from. However, I can't figure out how to get the two terms that is related to velocity. Some help on deriving this will be very helpful.

Also, in this equation and every document I come across does not deal with time-varying velocity. Why is this? How does commercial radars deal with varying velocity?

• Perhaps not your question but I wanted to check if you understand how the “row FFT” using the FFT result from multiple chirps gives you the velocity? (As Doppler bins since it is velocity plus the Doppler offset)? Dec 24, 2022 at 15:48
• The phase change in time between two successive FFT bins (the same bin for two successive chirps in time assuming we take a FFT over a complete chirp ) is proportional to the velocity since that indicates a change in distance, assuming time in the FFT was properly reset to be the same for each chirp. So we need two or more chirps and from that we can get the time rate of phase (which itself is a frequency).. if we proceed enough chirps, we can get that easily with another FFT and hence the Range Doppler Map with a 2D FFT that is commonly done. Dec 24, 2022 at 15:54
• @DanBoschen I understand that the velocity is derived by multiple chirps because velocity makes the discrete angular frequency to change. However, I was a bit clinged to the derivative equation you gave me and was wondering if an in-chirp velocity calculation was feasible. Dec 24, 2022 at 21:45
• Only if you take multiple FFTs offset in time over the chirp duration (since the single FFT gives you the average frequency - and since frequency is distance then the average distance… ) if we only have the average distance for an object we cannot from that know it’s velocity- make sense? (You can indeed take multiple FFT’s over a single chirp, each offset by a sample - it just adds more processing but certainly feasible) Dec 24, 2022 at 22:00
• @DanBoschen Oh I never thought about offsetting the FFT. So, if the sampled IF has $N$ samples, the FFT will be done for 0~N, 1~N...N-1~N? But wouldn't this make the resolution of the FFT drop? Dec 24, 2022 at 22:13