In Lesson 28 one chirp gave one range profile. The echo, compared with the chirp still going out, is a tone whose frequency is proportional to range, and the range FFT sorts the echoes into range cells. The burst the L29–L31 Labs use sweeps 500 MHz in 300 µs, so the beat rises by 11.1 kHz for every meter of range, and the 0.30 m range cells of \(\Delta R = c/2B\) are 3.33 kHz apart.

A moving target also shifts the echo by its Doppler frequency, \(f_D = 2v/\lambda\), which is 66.7 Hz for every meter per second at the 3.0 cm wavelength of a 10 GHz carrier. Within a single chirp that shift is invisible. A hand moving at 1 m/s shifts the beat by 66.7 Hz, a fiftieth of one range cell, and moves only 0.3 mm while the chirp lasts. By the rule of Lesson 26, an observation 300 µs long resolves frequencies 3.33 kHz apart, which is 50 m/s of velocity.

To measure velocity we have to watch the target for longer, and an FMCW radar does it by repeating the chirp. The burst sends \(M = 256\) chirps of 300 µs at a pulse repetition interval (PRI) of 0.5 ms, so it watches every range cell for 128 ms. The rest of the lesson is how the radar turns those 256 range profiles into a velocity, following ADI’s Range Doppler Radar Processing lab.

The range FFT of Lesson 28 needs only the frequency of the beat, so it does not care when the chirp started. The velocity measurement does. It compares the phase of the beat in one chirp with its phase in the next, and a moving target changes that phase by only a small, steady amount from chirp to chirp. A hand closing at 1 m/s moves 0.5 mm in one PRI, which shortens the round trip by 1.0 mm, a thirtieth of a wavelength, and advances the phase by 12°.

The beat phase also depends on when the samples start. A hand at 1.5 m has a beat of 117 kHz, the 0 m frequency of about 100 kHz that the L28 calibration measures plus 11.1 kHz per meter, so starting the samples 1 µs late changes the phase by \(360° \times 0.117 = 42°\). If the start time wanders by a microsecond from chirp to chirp, the error is larger than the signal we are trying to measure.

By default the board’s receiver simply fills a buffer with samples, with no relation between the start of the buffer and the start of any chirp. ADI’s Synchronizing Radar Chirps lab describes how the PHASER fixes this: a timing engine in the radio starts the capture and then fires each chirp at a programmed delay after it, one every PRI, so every chirp’s samples begin at the same point of its ramp. The capture format records that PRI as PRI, and every velocity on this page depends on it being exact.

The figure shows the difference in the hand’s range cell. With the chirps synchronized, the Doppler FFT of the next frames collects the hand into one sharp peak at +1 m/s. With each chirp starting at a random time within its PRI, the phase from chirp to chirp is random, and the same energy spreads evenly across every velocity: the hand is still there in range, but it has no velocity, and it sits 20 dB lower, close to the noise.

Stack the 256 chirps of a burst as the rows of a matrix. Along a row the samples are 0.25 µs apart, the 4 MSPS sample rate, and a row covers the 300 µs ramp in 1200 samples. That direction is fast time. Down a column the samples are one PRI apart, 0.5 ms, the same instant of each chirp; that direction is slow time. ADI draws the matrix the other way, chirps as columns; the capture format of the L28–L31 Labs stores it as the iq array, one row per chirp, n_chirps × n_samples, so the code on this page follows that layout.

The figure is the real part of that matrix for a single hand at 1.5 m, closing at 1 m/s, for the first 96 chirps and 160 samples. Across a row the beat oscillates at 117 kHz, one cycle every 34 samples; that frequency sets the range. Down a column the same sample oscillates once every 30 chirps, because the phase advances 12° a chirp; that rate measures the velocity. The stripes are diagonal because both are happening at once. A stationary target would draw vertical stripes, the same in every row.

Take the range FFT of every row. The target now sits in the same range cell of every chirp, and that cell holds a complex number whose phase is the phase of the echo, \(-4\pi R_m/\lambda\), set by the round trip \(2R_m\) in wavelengths. Between chirps the target moves. With a closing speed \(v\), the range at chirp \(m\) is \(R_m = R_0 - v\ m\ T_\text{PRI}\), and substituting it gives the chain on the slide. The first term is a constant. The second grows linearly with \(m\), which is a tone in slow time at

the Doppler shift of Lesson 26, with the same sign: a closing target has a positive frequency. At \(\lambda = 3.0\) cm, 1 m/s gives 66.7 Hz, and in one PRI of 0.5 ms the phase advances \(360° \times 66.7 \times 0.5 \times 10^{-3} = 12°\).

Two effects that the chain leaves out are small for this burst. The Doppler shift also moves the beat in fast time, but at the ±15 m/s limit of the next frames it is 1.0 kHz against 11.1 kHz per meter, a range error of 0.09 m, under a third of a range cell. And the target moves during the burst: at 1 m/s it covers 0.128 m in 128 ms, less than one 0.30 m cell, which is the dwell limit of Lesson 27. A much faster target moves across several cells during the burst and smears in range; the widget later in the lesson shows it.

The second FFT, down each column of the range-transformed matrix, finds the slow-time tone in every range cell. The result is the range-Doppler map: range on one axis, radial velocity on the other, and the power of each cell. Each FFT is the coherent integration of Lesson 27, so a target’s energy from all 1200 × 256 samples collects into one cell.

The figure is a map computed with the course’s burst numbers for a scene like the one in the L29 Lab. The kit’s own transmit leakage appears at 0 m and a wall at 3.0 m; neither moves, so both sit in the 0 m/s column, and they are the strongest returns on the map. A hand at 1.5 m closing at 1.0 m/s sits at +1.0 m/s. A fidget spinner at 2.1 m, three arms 3.5 cm long at 10 revolutions a second, has arm tips moving at \(2\pi \times 0.035 \times 10 = 2.2\) m/s; the arms move toward and away from the radar as they turn, so the spinner covers a band of velocities from −2.2 to +2.2 m/s at a single range. That spread is micro-Doppler, which Lesson 31 uses to classify targets.

Both FFTs use a Hann window, as the range FFT did in L28. The window matters more in Doppler than in range: the wall is 18 dB stronger than the hand here, and the first sidelobe of an unwindowed FFT is only 13 dB down, so without the window the wall’s sidelobes would spread across every velocity at its range. The Hann window puts the first sidelobe 31 dB down.

The provided radar_lab.py does this, and the L29 notebook runs it on your captures. The range FFT on each row is the one from L28.

def range_doppler(cap, window="hann"):
    """Range-Doppler map: (rd_db, range_m, vel_mps), rd_db indexed [range, velocity]."""
    rp = range_fft(cap, window)                 # n_chirps x n_bins, from L28
    r = range_axis(cap)
    keep = r >= 0
    rp = rp[:, keep]
    m = rp.shape[0]
    wd = _window(window, m)
    rd = np.fft.fftshift(np.fft.fft(rp * wd[:, None], axis=0), axes=0) / wd.sum()
    lam = C / cap["fc"]
    vel = np.fft.fftshift(np.fft.fftfreq(m, cap["PRI"])) * lam / 2
    rd_db = 20 * np.log10(np.abs(rd.T) + 1e-12)
    return rd_db, r[keep], vel

The velocity axis converts each Doppler bin to velocity with \(v = f_D\lambda/2\). The module follows the conventions of this page: positive velocity is closing, toward the kit, as in Lesson 26; the map is indexed [range, velocity], which is the order Lesson 30’s cfar_2d expects; only ranges at or beyond 0 m are kept; and range_doppler divides each FFT by the sum of its window, so a tone of amplitude 1 reads 0 dBFS whatever the window. A hand moving toward the kit therefore lands at positive velocity, and checking that on your own map is the first test of the velocity axis.

The Doppler FFT of \(M\) slow-time samples, \(T_\text{PRI}\) apart, has bins \(1/(M\ T_\text{PRI})\) apart in frequency, which is the rule of Lesson 26: watching for \(M\ T_\text{PRI}\) resolves its reciprocal. Converting with \(v = f_D\lambda/2\) gives the velocity resolution

The slow-time samples come once per PRI, so the Doppler FFT sees the tone sampled at the PRF, 2 kHz, and it can only tell apart frequencies within ±1 kHz. That is the maximum unambiguous velocity,

A faster target aliases. At +22 m/s the phase advances 264° a chirp, which the FFT cannot tell from −96°, so the map reports \(22 - 30 = -8\) m/s. These are the same two formulas ADI’s Range_Doppler_Processing.py prints as its velocity resolution and maximum velocity.

Each parameter of the burst controls one axis of the map. The number of chirps \(M\) sets the velocity resolution and nothing else: 32 chirps watch for 16 ms and resolve 0.94 m/s, 128 chirps resolve 0.23 m/s, and 256 chirps resolve 0.12 m/s. The chirp bandwidth sets the range resolution and nothing else: dropping from 500 MHz to 200 MHz widens every target from 0.30 m to 0.75 m in range. The PRI sets the velocity limit, and a longer chirp needs a longer PRI to fit in, so a 500 µs ramp, which cannot fit in a 0.5 ms PRI with time to reset, forces a slower PRF and a smaller \(v_\text{max}\). These are the experiments ADI’s range-Doppler lab suggests, and the L29 Lab runs the first two on your own kit.

More chirps also have two drawbacks. The integration gain of Lesson 27 is \(10\log_{10}M\): 24.1 dB for 256 chirps and 15.1 dB for 32, so a short burst has a higher noise floor on the map. A long burst refreshes the map less often, 128 ms for 256 chirps, and lets a fast target cross range cells while the burst lasts.

Everything that does not move returns the same echo on every chirp: the leakage from the transmit antenna straight into the receiver, the walls, the bench, the stand the kit sits on. Radar engineers call these returns clutter. On the map they all sit at 0 m/s, and in a lab room they are by far the strongest returns. In the scene of the earlier figure the leakage is 30 dB above the hand and the wall is 18 dB above it.

The Doppler FFT already separates clutter from a moving target, but removing the clutter still helps, for three reasons. A strong return has sidelobes in Doppler even with a window, and anything that changes the clutter slightly from chirp to chirp, a vibrating stand, a fan’s air moving a cable, the small timing error that chirp sync cannot remove, spreads its energy across nearby velocities, where a slow target competes with it. The CFAR detector of Lesson 30 estimates the noise from the cells around each test cell, and a clutter ridge in those cells raises the threshold for the targets beside it. And the spectrogram of one range bin, which the L29 Lab keeps for Lesson 31, has far less Doppler resolution than the full burst, so a strong clutter line there covers a wide band. Removing clutter before the Doppler FFT addresses all three, and that is moving target indication (MTI).

The simplest MTI filter subtracts each chirp from the one before it, sample by sample. ADI’s MTI Radar Processing lab calls it the two-pulse canceller. A return that is identical in two consecutive chirps, which is what a stationary return is when the chirps are synchronized, subtracts to zero. A moving target’s phase has advanced between the chirps, so the difference is not zero, and the target survives.

Applying the same subtraction to the output a second time gives

the three-pulse canceller. Its notch at 0 m/s is deeper and wider, as the next frame shows.

Both filters work across chirps, so we can apply them to the raw beat samples before the range FFT, as ADI’s script does, or to the range-transformed rows after it; the FFT is linear, so the two give the same map. Each output has one or two fewer chirps than the input. The provided radar_lab.py does this too, and the L29 notebook runs it on your captures. It returns a capture, so range_doppler(mti(cap, "2pulse")) works without any other change:

def mti(cap, kind="2pulse"):
    """A pulse canceller across chirps; returns a capture, one or two chirps shorter."""
    def cancel(x):                              # chirps on the second-to-last axis
        if kind == "2pulse":
            return x[..., 1:, :] - x[..., :-1, :]
        if kind == "3pulse":
            return x[..., 2:, :] - 2 * x[..., 1:-1, :] + x[..., :-2, :]
        if kind == "none":
            return x
        raise ValueError(f"kind must be '2pulse', '3pulse', or 'none', not {kind!r}")
    out = {**cap, "iq": cancel(cap["iq"])}
    if "iq_ch" in cap:
        out["iq_ch"] = cancel(cap["iq_ch"])
    return out

To find what the canceller does to a target of any speed, feed it the slow-time tone of the earlier frame,

The two-pulse output is

so the tone comes out with its power multiplied by

The three-pulse canceller is two of these in a row, so its power gain is the square,

Both gains are zero at \(f_D = 0\), the clutter. Both peak at \(f_D = \pm 1/(2T_\text{PRI})\), the edge of the unambiguous window at ±15 m/s, at 4 (+6.0 dB) and 16 (+12.0 dB).

The gain is also zero wherever \(f_D T_\text{PRI}\) is a whole number \(k\): the target’s phase then advances a whole number of turns from one chirp to the next, and the canceller cannot tell it from a stationary return. Those are the blind speeds,

every 30 m/s for this burst. The first blind speed is twice \(v_\text{max}\), which follows from the aliasing of the velocity-axis frame seen from the other side: a target at 30 m/s aliases to 0 m/s on the map, and the canceller removes it with the clutter. Nothing in the L29 Lab moves that fast, but a car or a drone can, and a radar that must see them changes its PRI from burst to burst so that a target blind at one PRI is visible at the next.

The figure runs the scene of the range-Doppler frame through each canceller and draws all three maps on one scale. Both cancellers remove the leakage and the wall completely; what is left at 0 m/s is the residue of the small chirp-to-chirp timing error in the simulation, which is why chirp sync sets how deep the cancellation can go.

The cancellers also weaken moving targets, and they weaken the slow ones most. A hand at 1.0 m/s has \(f_D T_\text{PRI} = 66.7 \times 0.5 \times 10^{-3} = 0.033\), deep in the notch: the two-pulse canceller multiplies its power by \(4\sin^2(0.105) = 0.044\), −13.6 dB, and the three-pulse by −27.2 dB. The noise goes through the same filter. Each output adds independent noise samples, so the total noise power rises by the sum of the squared taps: \(1^2 + 1^2 = 2\), +3.0 dB, for the two-pulse canceller and \(1^2 + 2^2 + 1^2 = 6\), +7.8 dB, for the three-pulse. Against that total the hand’s SNR falls by 16.6 dB and 35.0 dB, and every target slower than about 7.5 m/s (two-pulse) or 8.6 m/s (three-pulse) loses SNR; that is the loss a detector would see if it looked at the canceled chirps without a Doppler FFT. The map does not compare the hand with the total. The Doppler FFT gives each velocity its own cell, and the noise in the hand’s cell passed through the same notch as the hand, so it drops by the same 13.6 or 27.2 dB, and the hand stands as high above its own floor as it did without MTI. The floor rises only toward ±15 m/s, by up to 6.0 and 12.0 dB, so the canceller shapes the floor the way it shapes the targets, and the L29 Lab’s notebook measures both. MTI improves the ratio of a target to the clutter, not to the noise, so it helps where clutter, not noise, limits what the radar can see.

Two more consequences follow from the notch. The canceller removes a target that stops moving, and also one moving across the beam, whose radial velocity is zero. The spinner shows both effects at once: the arms moving fastest along the line of sight survive, and the canceller removes the parts of its spectrum near 0 m/s with the clutter. For a hovering drone, ADI’s MTI lab points out, the body disappears but the blades’ micro-Doppler remains, which is what Lesson 31 classifies.

The choice of canceller is therefore a choice about the scene. With strong clutter and fast targets the three-pulse canceller gives the cleanest map; for a slow hand the two-pulse canceller, or none, keeps more of the target. The L29 Lab measures both on the same captures.

The L29 Lab runs this lesson on your own kit, GUI first. The Phaser app does the live work: its Range + Velocity preset sends the burst, its Range-Doppler view draws the map, its MTI switch and MTI Mode run the cancellers, and Save Capture writes each burst to your laptop. In that view you check the velocity axis with your hand, closing and then opening, and watch a fidget spinner spread into a band of velocities at one range. You read the cells at 256 chirps on the kit and at 128, 64, and 200 MHz from the notebook and a recorded burst, and you switch the canceller between off, 2-pulse, and 3-pulse with a wall and your hand in the beam. The lab ends in the course’s notebook, where you run the provided range_doppler and mti on the bursts you saved and change a line or two for each of three tasks. You work on your own and turn in one packet in three parts: Part 1, Pre-Lab, the predictions, before you go to the bench; Part 2, Measurements, from your kit and the notebook; and Part 3, Writeup, after it. I check the packet and return it with feedback. It is the second piece of scaffolding for your final report.

The procedure follows ADI’s range-Doppler and MTI labs, which wave a hand and a fidget spinner at the same burst. The lab’s numbers are the app’s preset, not the round values above: a 9.9 GHz carrier (\(\lambda\) = 3.03 cm), 500 MHz in 300 µs, 256 chirps every 0.5 ms, 2 MSPS (600 samples a chirp), and \(c = 3.00\times 10^8\) m/s. The carrier moves the lesson’s velocity numbers by 1%: a 0.118 m/s cell, ±15.2 m/s, and 66.0 Hz and 11.9° a chirp at 1 m/s. The sample rate changes nothing on the map, because the range cell is still \(c/2B\) = 0.30 m. Their chirp synchronization page describes how the kit starts every chirp at a known time.

The table is the core of Part 1, and every entry follows from the frame Choosing the Burst, at the preset’s 9.9 GHz. The velocity cell is \(\lambda/(2MT_\text{PRI})\), so halving the chirp count doubles it, and the range cell is \(c/2B\), so 200 MHz makes it 2.5 times taller. The floor column is the integration gain of Lesson 27: half the chirps add 3 dB less, while a target’s level stays where it was, because range_doppler divides each FFT by the sum of its window. A Hann window makes a target’s spot one to one and a half cells wide 3 dB down, so a spot’s width on the screen is the cell you read. Part 1 also predicts the hand’s Doppler and the cancellers’ gain at its speed, from the frame Canceller Response and Blind Speeds: at 1.0 m/s, 66.0 Hz and 11.9° a chirp, −13.7 dB through the 2-pulse canceller and −27.4 dB through the 3-pulse.

A fidget spinner held edge-on, its plane of rotation along the line of sight, moves each of its arms toward the kit for half a turn and away for the other half. An arm on a circle of radius \(a\) turning at \(f_\text{rev}\) revolutions a second has a radial velocity of

so over a burst its echo covers every velocity between \(-v_\text{tip}\) and \(+v_\text{tip}\). The arm spends the longest near the extremes, where its radial velocity turns around, so the band is brightest at its edges. For arms 3 cm from the hub at 15 revolutions a second, \(v_\text{tip} = 2\pi(0.03)(15) = 2.83\) m/s, a band of ±187 Hz at 9.9 GHz. The hub, and the hand that holds it, do not move, and sit at 0 m/s.

The band is not smooth. Three identical arms make a pattern that repeats every third of a turn, and a signal that repeats every \(1/(3f_\text{rev})\) seconds has its spectrum in lines \(3f_\text{rev}\) apart: 45 Hz, or \(\lambda(45\ \text{Hz})/2 = 0.68\) m/s, so the 0.118 m/s cells of the 256-chirp map resolve them. The spacing gives the spin rate without a ruler, and the band’s edge gives it with one; the lab measures both. Held face-on, the spinner moves its arms across the beam, with no radial velocity, and the band collapses toward the 0 m/s column: a hand never holds it exactly face-on, so a line or two either side of 0 m/s remains. This is the micro-Doppler of Lesson 26, the parts of a target moving relative to its body, and Lesson 31 uses it to tell one kind of target from another.

The burst sees everything that moves in the beam, and you are the largest mover in the room. While you read the wall or the cancellers on a static scene, step behind the kit or well to one side, or your breathing and sway appear as a smear of slow velocities across your range. When you are the target, keep it simple: a flat hand, 0.5 to 1 m out, pushed straight toward the array and pulled straight back at a steady speed. Moving the hand across the beam gives it almost no radial velocity, and the map shows it near 0 m/s.

Use a wall, or the L28 plate standing on its own, about 2.5 m in front of the kit as the static target the cancellers remove. Hold the spinner by its hub at arm’s length with its plane of rotation horizontal or vertical and pointed at the kit, so the arms move toward and away from it. A spinner with metal weights in its arms reflects more than an all-plastic one. The transmit antenna stays on OUT2, connected before you power the kit, as in the L28 Lab. No step needs a corner reflector.

The L30 Lab runs CFAR on range-Doppler maps of this burst, with and without MTI, so keep your bursts, the notebook, and radar_lab.py together in your Module 4 folder, with the file names written in your packet. For the final report, keep three results: the velocity cell you read at three chirp counts against \(\lambda/(2MT_\text{PRI})\), the spinner’s spin rate from its band and from its lines, and what each canceller removed and what it kept. The report asks you to explain how a burst of chirps becomes a velocity, and what MTI does to a scene, in your own words; the packet’s tables are the evidence, not the explanation.