Reading — Doppler & MTI

Reading — Doppler & MTI#

By the end of this lesson you should be able to:

  1. Compute the Doppler shift from a target’s radial velocity.

  2. Explain how a single delay-line MTI rejects ground clutter.

  3. Identify blind speeds and how PRF moves them around.

  4. Describe pulse-Doppler processing as an FFT across pulses and read a range-Doppler map.

The second observable#

Range tells you where a target is. Doppler tells you what it is doing. A target moving radially at velocity \(v_r\) shifts the carrier frequency by

\[ \boxed{\,f_d = \frac{2 v_r}{\lambda}\,} \]

The factor of 2 is, once again, the two-way path. At X-band (\(\lambda = 3\) cm), a target closing at 300 m/s produces a 20 kHz shift; at 30 m/s, 2 kHz. Each pulse in the train carries a phase, and across pulses that phase advances in proportion to \(f_d\) — so a coherent radar reads velocity straight off the pulse-to-pulse phase.

Clutter versus targets#

A radar looking down sees everything: terrain, buildings, weather, sea, foliage. Ground clutter has an enormous RCS and sits at essentially zero Doppler because it is not moving. The aircraft return is far weaker and buried in clutter on amplitude alone.

The escape is Doppler. The clutter sits at \(f_d \approx 0\); the mover sits somewhere else. Separate them in frequency and the weak target pops out of the strong clutter.

Key Concept

Clutter is a confidently stationary signal. The radar cannot reject it by amplitude — clutter is far stronger than the target — but it can reject it by Doppler, because clutter lives at zero Doppler and movers do not.

Single delay-line MTI#

Moving-target indication (MTI) exploits exactly that. The simplest canceller subtracts each pulse return from the previous one:

\[ y[n] = x[n] - x[n-1]. \]

Anything that did not change between pulses (stationary clutter) cancels to zero; anything that changed (a mover, whose phase advanced) survives. The frequency response of this one-tap filter, with \(T = \text{PRI}\), is

\[ |H(f)| = 2\,|\sin(\pi f T)|. \]

This has notches at \(f = 0, \text{PRF}, 2\,\text{PRF}, \dots\) — it kills zero-Doppler clutter and its PRF-spaced repeats, while passing Doppler shifts in between. It costs one PRI of memory and a subtractor. Higher-order cancellers and adaptive MTI deepen and widen the notch, but the idea is this.

Blind speeds#

The notches are also a vulnerability. A target whose Doppler shift lands on a notch looks stationary and is cancelled along with the clutter. These blind speeds occur at

\[ \boxed{\,v_{\text{blind},n} = \frac{n\,\lambda\,\text{PRF}}{2}\,} \]

At X-band with PRF = 1 kHz, the first blind speeds are 15, 30, 45 m/s, … — squarely in the range of real aircraft and ground vehicles. Raising the PRF to 10 kHz pushes the first blind speed to 150 m/s, usually outside the common clutter band. This is a major reason pulse-Doppler radars favor medium-to-high PRF — and it ties straight back to Lesson 4’s ambiguity tradeoff.

Key Concept

MTI notches reject clutter at \(f = 0, \text{PRF}, 2\,\text{PRF}, \dots\), but a target on a notch is also rejected. Raising PRF moves the blind speeds higher and out of the way — at the cost of range ambiguity. The PRF choice is always a compromise.

Pulse-Doppler processing and the range-Doppler map#

Rather than a single subtraction, modern radars collect \(N\) pulses in a coherent dwell and take an \(N\)-point FFT across the pulses at each range bin. The output is a Doppler spectrum per range bin, with bin width

\[ \Delta f_d = \frac{\text{PRF}}{N}. \]

More pulses give finer Doppler resolution and, because the integration is coherent, an SNR gain proportional to \(N\). Each target lands in its own range-Doppler (R-D) cell.

Stacking the range bins against the Doppler bins gives the range-Doppler map — a 2D image where:

  • Clutter forms a vertical ridge at zero Doppler (all ranges, one velocity).

  • Movers land off the ridge, easy to threshold and track.

The EW implication is sobering for the B-21: once you are moving relative to the radar, your Doppler signature is hard to hide. Stealth aims to push the amplitude return down into the clutter band; standoff jammers raise the noise floor across all Doppler bins. Both are attacks on this very picture, and both are Block 2 material.

Quick Exercise

X-band radar, \(\lambda = 3\) cm.

  1. Compute \(f_d\) for a B-21 closing at 250 m/s.

  2. At PRF = 1 kHz, list the first two blind speeds. Are these realistic aircraft speeds?

  3. The radar switches to PRF = 10 kHz. Where do the first two blind speeds move?

  4. A jammer wants the B-21 to appear stationary on the R-D map. What property would it try to spoof?

Solution

  1. \(f_d = 2(250)/0.03 = 16.67\) kHz.

  2. \(v_{\text{blind},1} = 15\) m/s, \(v_{\text{blind},2} = 30\) m/s — yes, both are realistic.

  3. They move to 150 m/s and 300 m/s — well outside the common ground-clutter band, much safer against low/slow threats.

  4. Match the platform’s own ground-clutter Doppler line, so the false return hides on the zero-Doppler clutter ridge and gets cancelled with it. (EP counters include geometry checks and inverse-cancellation logic.)

Wrap-Up#

Doppler, \(f_d = 2 v_r/\lambda\), is the second observable after range. Single delay-line MTI, \(y[n] = x[n] - x[n-1]\), notches out zero-Doppler clutter via \(|H(f)| = 2|\sin(\pi f T)|\), but creates blind speeds at \(v_{\text{blind},n} = n\lambda\,\text{PRF}/2\) that the PRF choice must manage. Pulse-Doppler processing — an FFT across pulses — produces the range-Doppler map, with clutter as a vertical ridge and movers in their own cells.

That closes the threat model. We now understand how a radar detects, ranges, resolves, and separates a moving target from clutter — the full Detect-and-Track problem the B-21 must defeat. Block 2 turns to the other side: the electronic-attack techniques that break these links.

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