I am struggling to understand how linear phase shift can be determined with a transform whose main purpose is to determine frequencies rather than rate of change of a function.
In FMCW radar, received signal corresponding to a chirp is passed through a mixer and the resulting beat frequency is sampled by means of ADC. The samples are stored in a row of a matrix. Similarly, samples from other chirps are stacked up in corresponding rows. Different objects at different locations produce different frequencies (thanks to Doppler). So a FFT of each row yields dominant frequency bins. The frequency bins directly relate to object range. Absolutely fine so far.
Assuming that these objects move (and slowly), they must continue to appear in the same frequency bin (column)for all of the chirps. But since they are moving, the phase of the frequency component must change. For simplicity, we assume that the magnitude of all frequency components in a column is constant.
Thanks to Shift Theorem, a signal delayed/advanced in time results in a phase shift of the frequency components of the time domain signal.
Question-1: Am I right in assuming that a constant velocity results in a LINEAR phase shift of the frequency component across chirps?
Question-2: If the phase shift is linear, how can FFT along a column determine the same? FT/DFT essentially performs decompositions in terms of basis sinusoids. But in our case, we are interested in calculating the slope of the line (or rate of change or derivative) representing the phase-shift change across chirps. Derivative is not same as frequency. I think we need a Non-sinusoidal decomposition hich can yield various slopes. But second-stage FFT along the column seems to work.
What have I misunderstood?