Linear Ramp of Velocity with Range

Suppose that a continuous-wave IF waveform has an instantaneous frequency f (t)in Hertz (cycles/sec). Consider a range bin located at time

Ʈbin within a set of pulses that are separated by Ʈs = 1/PRF. The phase measured at that bin on the nth pulse is the integral of the frequency within that pulse starting from range zero (since RVP10 is phase-locked to range 0):

Φ n = ∫ n τ s n τ s + τ b i n f ( t ) ⅆ t

If we assume that the input frequency is a linear Frequency Modulation (FM) at the rate of M cycles/sec/sec on top of a base frequency To, then:

Φ n + 1 − Φ n = ∫ ( n + 1 ) τ s ( n + 1 ) τ s + τ b i n ( T 0 + M t ) ⅆ t − ∫ n τ s n τ s + τ b i n ( T 0 + M t ) ⅆ t = ( M τ s ) τ b i n

which is independent of both To and n. Thus, a linear FM input signal produces a fixed (I,Q) phase difference from pulse-to-pulse at any given range. The magnitude of the phase difference is proportional to the range, and the slope is(M Ʈs) cycles for each second of delay in range. For example, if the test signal generator is sweeping 100 KHz every two seconds, then the velocity observed at a range of 300 km at 250 Hz PRF is:

Φ n + 1 − Φ n = ( 100 K H z 2 ⁢ sec ⁡ ) × ( 1 250 sec ) × ( 300 k m ) × ( 6.6 μ ⁢ sec ⁡ 1 k m ) = 0.40 ⁢     c y c l e s

We would thus observe a velocity of (0.8 × Vu) at 300 km, where Vu is the unambiguous Doppler velocity in meters/sec. Note that these phase difference calculations have made no assumptions about the RVP10 processing mode, and thus are valid in all major modes (PPP, FFT, DPRT, RPH), as well as in all Dual-PRF unfolding modes.

This simple FM signal generator also produces valid second trip velocities that can be seen during Random Phase processing. This follows from the above analysis because we've never assumed that Ʈbin was smaller than Ʈs, that is, it is fine for the range bin to be located in any higher-order trip.