Pulse-Doppler Radar Range and Velocity Processing
Pulse-Doppler radar separates target range from radial velocity by transmitting a train of phase-related pulses, then processing each echo in two dimensions: echo delay gives range, while phase change from pulse to pulse gives velocity.
The key is coherence: the radar preserves a known phase relationship between transmitted pulses and compares the phase of corresponding echoes. Without that relationship, a single pulse can reveal where an object is, but not reliably how its distance is changing.
The problem: delay and motion are different measurements
A radar estimates range from the time an emitted pulse takes to return. If the round-trip delay is τ, the target range is approximately:
R = cτ / 2
where c is the speed of light. The factor of two accounts for the path out to the target and back.
A target moving toward or away from the radar also shifts the echo's carrier frequency. This is the Doppler shift, the frequency change caused by relative motion. For a monostatic radar—one antenna used for both transmission and reception—the Doppler frequency is approximately:
f_D = 2v / λ
Here, v is radial velocity, meaning the component of velocity along the radar-target line, and λ is the transmitted wavelength.
A single echo does not provide a useful measurement of that frequency shift. The radar needs observations separated in time. Pulse-Doppler processing supplies them by transmitting many pulses and comparing the matching samples from each return.
Two time axes in the received data
Suppose the radar transmits pulses at a fixed pulse repetition frequency (PRF). The interval between pulses is the pulse repetition interval (PRI):
T_PRI = 1 / PRF
For every transmitted pulse, the receiver records a stretch of time long enough to contain expected echoes. This produces a matrix of samples:
- Fast time: sample position within one pulse interval. It measures echo delay, and therefore range.
- Slow time: pulse number across the pulse train. It measures how the echo phase changes from pulse to pulse, and therefore Doppler frequency.
The receiver usually correlates each received pulse with the known transmitted waveform. This operation, often called matched filtering, concentrates a return into a peak at its delay and improves detection in noise. The peak's position along fast time becomes a range bin.
The radar then takes the same range bin across many pulses. For a stationary object, those samples have nearly constant phase. For a moving object, the phase advances or retards regularly. The phase change per PRI is approximately:
Δφ = 2π f_D T_PRI = 4πv T_PRI / λ
A frequency analysis across slow time—commonly a discrete Fourier transform, often implemented with an FFT—groups those phase progressions into Doppler bins. The result is a range-Doppler map: one axis is range and the other is radial velocity.
Range comes from where the echo appears within each pulse interval. Velocity comes from how its phase evolves across intervals.
This is also why coherence matters. The receiver must retain phase information relative to a stable reference oscillator. If it only measures echo power and discards phase, the pulse-to-pulse pattern that carries Doppler information is lost.
Why sampling creates blind speeds
Sampling the echo once per PRI means slow-time Doppler is sampled at the PRF. Frequencies separated by integer multiples of the PRF produce the same phase sequence at those sample times:
e^(j2π(f_D + k·PRF)nT_PRI) = e^(j2πf_D nT_PRI)
for any integer k. The radar therefore cannot distinguish those Doppler frequencies without additional information. This is Doppler aliasing.
For ordinary complex sampling, the unambiguous Doppler interval is usually written as:
-PRF/2 ≤ f_D < PRF/2
Using f_D = 2v/λ, the corresponding unambiguous velocity interval is:
-λ·PRF/4 ≤ v < λ·PRF/4
A faster target can therefore appear to have a slower velocity, possibly with the opposite sign, after its Doppler frequency wraps into that interval.
A related issue is the blind speed. If f_D is exactly an integer multiple of PRF, the phase shift between sampled pulses is an integer number of cycles. The sampled echo then looks phase-stationary, just like clutter or a stationary reflector. In velocity terms, these speeds are approximately:
v_blind = k λ·PRF / 2
where k is a nonzero integer. Moving-target-indicator filters—filters designed to reject stationary or slowly changing clutter—can suppress such a target because its sampled signature lands in the clutter-like response. Changing PRF, or combining measurements made at different PRFs, moves or resolves these blind speeds.
Why range is ambiguous too
The radar cannot listen forever between pulses. If an echo arrives after the next pulse has been transmitted, the receiver may associate it with the wrong transmission. With a PRI of T_PRI, the maximum unambiguous range is approximately:
R_unamb = c T_PRI / 2 = c / (2·PRF)
A return from a target at R, R + cT_PRI/2, or R + 2cT_PRI/2 can produce the same delay within the repeating pulse schedule. These are range ambiguities.
The two sampling limits coexist. A detected object is fundamentally identified only by range modulo the unambiguous range and velocity modulo the unambiguous velocity interval. Consequently, a distant, fast target can appear in a nearer range bin with an aliased velocity. This is a range-Doppler ambiguity: more than one physical target state maps to the same measured cell.
Designers manage the trade-off with choices such as multiple or staggered PRFs. A higher PRF improves unambiguous velocity but reduces unambiguous range; a lower PRF does the opposite. Measurements from several PRFs can be combined so that only one range-velocity pair is consistent across all of them. Longer coherent pulse trains improve Doppler resolution, while waveform bandwidth and pulse compression control range resolution.
Thus, when a radar log mentions PRF staggering, blind-speed avoidance, Doppler folding, or ambiguous plots, it is describing the consequences of sampling the two-dimensional echo record—not a failure to measure range and velocity separately.