An FMCW radar transmits continuously, and its frequency climbs linearly across a bandwidth \(B\) during each chirp of length \(T_s\). On the PHASER, in the Phaser app’s Range (Basic) preset, the chirp starts at 12.145 GHz and sweeps \(B = 500\) MHz in \(T_s = 1\) ms. The echo from a target at range \(R\) is a copy of the transmitted chirp delayed by the round trip,

which is 6.7 ns for every meter. The receiver compares the echo with the chirp that is still going out. Because both ramps climb at the same rate, the echo is always below the transmitted chirp by the same frequency, and that difference is the beat frequency \(f_b\). The figure exaggerates the delay so that it is visible: on the PHASER, a target at 3 m delays the echo by 20 ns, 0.002% of a 1 ms chirp.

The transmitted frequency rises at the chirp slope \(B/T_s\), which for the PHASER is \(500\ \text{MHz}/1\ \text{ms} = 0.5\ \text{MHz}\) per microsecond. In the delay \(\tau\) the chirp climbs \(\tau B/T_s\), so the beat frequency is

The beat is proportional to range, so the radar reads range by measuring a frequency. A target at 1 m gives 3.33 kHz, and one at 3 m gives 10.0 kHz. These frequencies are low enough for the Pluto to sample directly, even though the chirp itself spans 500 MHz: comparing the echo with the outgoing chirp removes the sweep and leaves only the small difference that carries the range. ADI’s FMCW lab calls this stretch processing.

The kHz per meter depends on both waveform settings. Halving the bandwidth to 250 MHz halves it, to 1.67 kHz per meter, and halving the chirp length to 0.5 ms doubles it, to 6.67 kHz per meter. As the next frames show, only the bandwidth changes the resolution.

The radar sees each target’s beat for one chirp, \(T_s\), and an FFT over a record of length \(T_s\) separates two tones only if they differ by at least \(1/T_s\). Over one chirp, that is a difference of one cycle. Converting that frequency step to range with the beat per meter gives

which is the result of Lesson 26. At 500 MHz it is 0.30 m. A longer chirp makes the beat per meter smaller but lets the FFT resolve finer frequency steps in the same proportion, so \(T_s\) cancels. Halving the bandwidth to 250 MHz doubles the resolution cell to 0.60 m; halving the chirp length to 0.5 ms leaves it at 0.30 m.

The number of beat cycles in one chirp is \(f_bT_s = 2RB/c\), which does not depend on \(T_s\) either. Two plates 0.30 m apart differ by exactly one cycle a chirp at 500 MHz; closer than that, their beats stay within one cycle of each other for the whole chirp and the FFT returns one peak.

The PHASER’s Pluto samples the beat at \(f_s = 0.6\) MSPS, so one 1 ms chirp yields 600 complex samples. The app discards the first 0.1 ms of each chirp, while the frequency ramp settles, and keeps \(N = 540\). The FFT of those samples is the range profile: each bin is a frequency, and the beat-per-meter relation converts it to a range,

where \(f_0\) is the 0 m frequency of the next frame. The bins are \(f_s/N = 1.11\) kHz apart, which is 0.33 m, one resolution cell. That is a little more than the 0.30 m of the resolution frame because the 0.9 ms of each chirp the app keeps sweeps 450 MHz, not 500, and \(c/(2 \cdot 450\ \text{MHz}) = 0.33\) m. Padding the record with zeros before the FFT, to 4096 points or more, draws the same spectrum on a finer grid and makes the peaks easier to read, but it does not resolve anything new.

The figure is one chirp from the lab’s recorded data: a 10 cm corner reflector at 2.0 m, the transmit antenna’s direct path into the array at 0 m, and the far wall at 5.5 m. The top axis shows the same profile as beat frequency above \(f_0\), at 3.33 kHz per meter.

The FFT also integrates. The 540 samples of a steady tone add in phase while the noise adds in power, so the tone rises \(10\log_{10}540 = 27.3\) dB above the noise per sample, as Lesson 27 described. Seen another way, each bin collects noise from only 1.11 kHz. That is the narrowing Lesson 25 described: the corner at 3 m, 31.1 dB above the noise in 1 MHz, stands

above it in one bin.

The PHASER does not put a 0 m echo at zero frequency. The Pluto transmits its tone offset by signal_freq in ADI’s scripts, about 100 kHz, from the frequency the receiver compares the echo with, so every beat lands about 100 kHz above zero. That keeps the targets clear of zero frequency, where the receiver’s own offset and low-frequency noise sit. The offset is not exactly 100 kHz on every kit, and the cables and the board add a little delay of their own, so each kit has its own 0 m frequency \(f_0\), and the range equation measures beats from it.

The calibration is simple. Hold a flat metal plate against the array face, read the frequency of the peak, and record it as \(f_0\); ADI’s lab expects a value such as 99 or 103 kHz. With the plate gone, a peak stays at \(f_0\): it is the transmit antenna’s direct path into the array, the leakage, and it marks 0 m on every capture you take. A kit whose \(f_0\) is off by 1 kHz reports every range 0.30 m long or short.

The Pluto samples complex data at \(f_s\), so it can represent beat frequencies up to \(f_s/2 = 300\) kHz. A target far enough away to put its beat above that limit aliases: it reappears at a wrong frequency, and the range profile shows it at a wrong range. With the 0 m frequency at 100 kHz, the beats above it have 200 kHz of room, so the farthest range the profile reports is

This limit comes from the sample rate, not from the transmit power; Lesson 25 put the detection range of a 10 cm corner near 8.5 m in a 1 MHz bandwidth, and farther out in the 1.11 kHz of one bin. A slower slope reaches farther: halving \(B\) halves the beat per meter and doubles \(R_\text{max}\) to 120 m, but it also doubles the resolution cell. A steeper slope reaches less far at the same sample rate, so the app’s Range (Fast) preset, 500 MHz in 0.5 ms, also samples at 5 MSPS: its 6.67 kHz per meter then reaches \((2.5\ \text{MHz} - 100\ \text{kHz})/(6.67\ \text{kHz/m}) = 360\) m. The lab measures both changes.

An FFT of one chirp treats the record as if it started and stopped abruptly, which gives each tone a \(\sin x/x\) response with sidelobes 13 dB below the peak, falling slowly. A strong target’s sidelobes then cover the bins around it, and a weak target nearby disappears under them. In the figure the second target, 30 dB weaker and 2.0 m beyond the first, is invisible with the rectangular window.

Multiplying the samples by a window that tapers smoothly to zero at both ends of the chirp lowers the sidelobes. The Hann window,

puts the first sidelobe at −31 dB and the rest falling fast, and the weak target appears. This is the amplitude taper of Lesson 22 applied in time instead of across the array, with the same consequences: the main lobe widens, from a 3 dB width of 0.89 bins to 1.44 bins, so two equal targets one cell apart may merge, and the window discards some signal energy, a 1.8 dB loss in SNR. We use Hann by default; the rectangular window is the choice only when two equal targets must be split at the edge of the resolution.

One range profile is a snapshot. Stacking the profiles of successive chirps, one row per chirp and darker for a stronger echo, gives the range waterfall: range across, time up. A fixed object, the leakage at 0 m or the wall at 5.5 m, keeps the same range in every row and draws a vertical line. A person walking from 1 m out to 4 m and back draws a zigzag, and the slope of each leg is the walking speed: a 3 m leg in 3 s gives 1 m/s.

The waterfall records range against time and nothing else. Its rows here are 50 ms apart, one profile each time the display updates, and over 50 ms a walker’s carrier phase changes by many cycles, so successive rows say nothing about the walker’s velocity except through the range they show. Lesson 29 takes chirps a fixed 0.5 ms apart and reads the velocity from that phase.

The four labs that follow share one Python module, radar_lab.py, and one capture format. The course provides the module complete: you run it from a notebook, one for each lab, and each lab’s tasks change a parameter or a line or two, never a whole function. The excerpt is the function this lesson describes: it windows every chirp, takes the FFT along each one, and divides by the sum of the window, so that a tone of amplitude 1 reads 0 dBFS whatever the window. n_fft larger than the samples in a chirp zero-pads it. The result has one row per chirp and one column per range bin.

A capture is an .npz file holding iq, the complex beat samples of the summed array with one chirp a row, shape n_chirps × n_samples; optionally iq_ch, the same samples from the array’s two halves, shape 2 × n_chirps × n_samples, which Lesson 31 uses to measure angle; the sample rate fs; the carrier fc; the bandwidth B; the chirp length T_ramp, the lesson’s \(T_s\); the time PRI from one chirp to the next; and f0, the 0 m frequency from your calibration, or the 100 kHz IF tone in a capture saved before it. The number of chirps is not fixed: 32 in an L28 range capture, 256 in the L29 burst, and thousands in a record of several seconds. The range of each bin comes from range_axis, the range equation of the range FFT frame with k the meters per hertz of beat:

def range_axis(cap, n_fft=None):
    n = n_fft or cap["iq"].shape[1]
    f = np.fft.fftfreq(n, 1 / cap["fs"])
    k = C * cap["T_ramp"] / (2 * cap["B"])
    return (f - cap["f0"]) * k

The bins come out in the order np.fft.fft returns them, and the bins below \(f_0\) come out as negative ranges. Two more L28 functions build on range_fft: range_profile(cap, chirp=0, window="hann") returns one chirp’s ranges at or beyond 0 m and its power in dB, (r, p_db), and waterfall(cap) returns (r, t, w_db), one row per chirp. Lesson 29 uses range_doppler and mti, Lesson 30 uses cfar_2d, and Lesson 31 uses the spectrogram, a classifier, and a tracker, all in the same module. The L28 notebook is L28_range.ipynb.

The L28 Lab measures everything this lesson derived, on your own kit: the 0 m frequency, the beat per meter, how the chirp length and the bandwidth change it, and the range waterfall. The Phaser app does the live work: the 0 m calibration, the range profile, the waterfall, and saving captures. The course’s notebook does the rest. It reads the frequency of each peak from the captures you saved, which the app does not display, and it runs the provided range FFT on them, with one line changed for each of three tasks. You work the lab on your own, outside class, 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 own kit; and Part 3, Writeup, after it. I check the packet and return it with feedback. It is the first piece of scaffolding for your final report, and it is not the report.

The procedure is ADI’s FMCW lab run on the app’s Range (Basic) preset: a 12.145 GHz carrier, 500 MHz in 1 ms, 0.6 MSPS, and \(c = 3.00\times 10^8\) m/s. The radar setup page describes the transmit path.

The table is the core of Part 1. Every number in it follows from the beat-frequency, resolution, and reach frames, with \(f_0 = 100\) kHz, \(f_s = 0.6\) MSPS for 1 ms chirps, and 5 MSPS for Range (Fast). The app has no bandwidth control, so the 250 MHz row comes from the course’s recorded capture. The bench checks each column. The beat per meter is the slope of the line through your 1, 2, and 3 m readings. The resolution shows as the width of the peak: at 250 MHz the peak from the same plate is twice as wide in range, and at 0.5 ms it is as wide as at 1 ms. The reach is not something the lab can walk out to, but the range axis shows where it ends.

The 0.5 ms chirp also changes the height of the peak. A bin’s noise bandwidth is the inverse of the record length, so halving the chirp doubles the noise in each bin while the echo’s power stays the same, and the plate’s peak stands about 3 dB lower above the noise floor.

A static measurement requires that nothing else in the beam moves. Your own body is a larger reflector than a small plate, so step behind the kit or well to one side while you read a static range; otherwise your echo adds a moving peak near the target’s and the profile you save is not the one you meant to save. Keep the beam’s path clear of walls and furniture for at least 4 m if you can, and expect the far wall to show up as a peak of its own.

Any flat metal plate about 20 cm on a side works, and so does aluminum foil glued smoothly to cardboard. A plate reflects strongly only when it faces the array squarely, so turn it slowly until its peak is highest before you save. A trihedral folded from three foil-covered cardboard squares is more forgiving of its angle. I have only one or two commercial corner reflectors; no step needs one, and if one is free, Part 2 has an optional reading for it. Measure range from the array face to the plate’s front with a tape measure.

Keep your captures, the notebook, and radar_lab.py together in your Module 4 folder. For the final report, keep three results from this lab: the beat per meter you measured against the 3.33 kHz the lesson predicts, the resolution you saw at 500 and 250 MHz, and a waterfall you can explain line by line. The report asks you to explain how a chirp becomes a range measurement and the resolution you measured, in your own words; the packet’s tables are the evidence for it, not the explanation.

This lesson used one chirp at a time and discarded its phase, which Lesson 29 keeps. A burst of 256 chirps, each 300 µs long and repeated every 0.5 ms, gives each target a phase that advances from chirp to chirp at its Doppler frequency, and a second FFT across the chirps reads that frequency as a velocity, with the resolution of Lesson 26,

The first FFT of that map is the range FFT you ran here, applied to every chirp of the burst. Moving target indication, in the same lesson, removes the fixed lines of the waterfall, the leakage and the wall, so that only what moves remains.