Lesson 27 found that noise power in one cell is exponentially distributed with mean \(P_n\), so noise alone crosses a threshold \(T\) with probability

The threshold for \(P_{fa} = 10^{-6}\) is \(T = 13.8\ P_n\), 11.4 dB above the mean noise. If the noise rises by a factor \(g\) while \(T\) stays put, the probability becomes \(P_{fa}^{1/g}\), and a 3 dB rise turns \(10^{-6}\) into \(10^{-3}\). At the half a million decisions a second of the L29 burst, that is 500 false targets a second.

The noise on a real map does move. Stationary clutter fills the cells near zero velocity, strong reflections fill the cells nearest the radar, and the receiver gain sets the level of everything. A fixed threshold high enough for the worst of these misses targets everywhere else. Lesson 27’s last figure showed the alternative: estimate the noise power from the cells around the one being tested, and set the threshold from that estimate.

The detector tests one cell at a time, the cell under test. On each side of it, it skips a few guard cells and then averages a block of reference cells. The average is the estimate of the noise power at this cell, and the threshold is that estimate multiplied by a bias. If the cell under test exceeds the threshold, the detector declares a target there. Then the window moves one cell and the detector repeats the test, so the detector recomputes the threshold at every cell from that cell’s own neighbors.

The figure draws 2 guard and 4 reference cells a side so that the cells are readable. The running example in this lesson uses 2 guard and 16 reference cells a side, \(N = 32\) reference cells in all, which is the local-average threshold in Lesson 27’s last figure. ADI’s CFAR lab lists the same steps for the PHASER: pick the cell, skip the guard cells, average the reference cells, multiply by the bias, and compare.

With \(z_i\) the power in reference cell \(i\), the cell-averaging (CA) CFAR estimate of the noise power is the sample mean \(\hat{P}_n\), and the threshold is \(T = \alpha\hat{P}_n\). The bias \(\alpha\) is the number we choose.

For a given estimate, the cell under test crosses the threshold with probability \(e^{-T/P_n}\), the Lesson 27 result. The estimate is itself random, so the probability of false alarm is the average of that over the reference cells. Each \(z_i\) is independent and exponential with mean \(P_n\), and for one such cell the average of \(e^{-cz_i/P_n}\) is \(1/(1 + c)\). The exponential of a sum is a product, so

where the angle brackets mean the average over the noise. \(P_n\) has dropped out. Wherever the noise sits, whether 0 dB or 15 dB, the probability that noise alone crosses the threshold is the same, which is what constant false-alarm rate means. Two assumptions came with it: the reference cells hold only noise, and their noise has the same mean as the noise in the cell under test. The rest of the lesson is about what happens when either one fails.

Inverting the false-alarm expression gives the bias for a chosen \(P_{fa}\):

For \(P_{fa} = 10^{-6}\) and \(N = 32\), \(\alpha = 32(10^{6/32} - 1) = 17.3\), or 12.4 dB. With 16 reference cells it is 21.9, 13.4 dB; with 144 it is 14.5, 11.6 dB. As \(N\) grows,

which is the 11.4 dB threshold of Lesson 27, because an average over infinitely many cells equals the noise power.

The bias is larger with fewer cells because the estimate scatters. Sometimes it comes out low, the threshold drops with it, and noise crosses more often than it would with the true \(P_n\). Raising \(\alpha\) compensates for those low estimates. In the code later in this lesson, bias_db is \(10\log_{10}\alpha\).

A target that a known-noise detector finds with \(P_d = 0.9\) at \(P_{fa} = 10^{-6}\) needs 13.2 dB of SNR (Albersheim’s fit in Lesson 27 gave 13.1 dB). The CA-CFAR detector needs more, because its threshold is higher on average and scatters with the estimate. The difference is the CFAR loss. The figure computes it by averaging the Lesson 27 detection probability over the scatter of \(\hat{P}_n\): 2.1 dB with 16 reference cells, 1.0 dB with 32, and 0.2 dB with 144.

The figure favors many reference cells, but a wider window is more likely to reach noise of a different level. The estimate assumes that every reference cell has the same mean noise as the cell under test. A window 32 cells wide in range spans 9.6 m at the 0.30 m cells of the L29 burst, and the clutter near the radar does not stay level over 9.6 m. The number of reference cells trades the CFAR loss against how local the estimate is.

A target does not occupy exactly one cell. The window applied before the FFT, Hann in ADI’s scripts and in the L29 range-Doppler map, widens the target’s peak so that each neighboring cell holds a quarter of its power, 6 dB down. A person is also larger than one 0.30 m range cell, and moving arms and legs spread the echo over several Doppler cells.

Without guard cells, that spill lands in the reference cells. Take a 20 dB target with no guard cells and 32 reference cells. Its two neighbors hold 14 dB each, 25 times the noise power, and they raise the estimate from \(P_n\) to about \((30 + 2 \times 26)/32 = 2.6\ P_n\), which lifts the threshold by 4.1 dB. The target still crosses, but its margin shrinks from 7.6 dB to 3.5 dB, and a weaker target would not. One or two guard cells a side remove the problem. More guard cells than the target’s extent do no harm to the target, but they push the reference cells farther out, where the noise is less likely to match.

Cell averaging assumes that every reference cell holds noise of the same mean as the cell under test. The figure shows the two common ways that fails.

On the left, an 18 dB target sits eight cells from a 24 dB target. When the weaker target is the cell under test, the stronger one and its two neighbors are in its reference window, and they hold about \(251 + 2 \times 63\) times the noise power. Spread over 32 cells, that raises the estimate by a factor of about 12.7, 11 dB, so the threshold rises to 23.5 dB and the detector misses the 18 dB target. This is masking, and it happens wherever targets are close together: two people walking side by side, or a hand in front of a body.

On the right, the noise rises 15 dB at cell 60, the edge of a clutter region. A cell in the clear just before the edge has half its reference cells in the clutter, so its threshold rises by about 12 dB and a target there is masked. A cell in the clutter just past the edge has half its reference cells in the clear, so its estimate is half what it should be, the threshold sits 3 dB low, and its \(P_{fa}\) rises from \(10^{-6}\) to \((1 + \alpha/32)^{-16} = 10^{-3}\). The clutter edge produces false alarms on one side and misses on the other.

Both variants average the two sides of the window separately, \(\hat{P}_\text{near}\) over the reference cells nearer the radar and \(\hat{P}_\text{far}\) over those farther away, and then keep one of the two.

Greatest-of (GO) keeps the larger. At a clutter edge, the side in the clutter wins, so GO compares a cell just inside the clutter with the clutter level, and its false alarms stay near the design value; the dashed red line in the right panel of the last figure does that. GO misses targets in the clear cells just before the edge, which it also compares with the clutter, and it makes masking worse.

Smallest-of (SO) keeps the smaller. When a second target sits on one side, the clean side wins and the weaker target is found, as the dashed orange line in the left panel shows. SO fails at clutter edges instead, where it takes the quiet side and lets the clutter through.

Neither max nor min of two averages is an average, so each has its own relation between \(\alpha\) and \(P_{fa}\). With 16 cells a side and \(P_{fa} = 10^{-6}\), GO needs a bias of 12.0 dB on its larger half and SO 13.7 dB on its smaller half, against 12.4 dB for CA. For \(P_d = 0.9\), GO needs 0.2 dB more SNR than CA and SO 0.7 dB more. ADI’s cfar() takes the three methods as 'average', 'greatest', and 'smallest'; the widget switches among them.

The window runs out at the ends of the data. In range, a cell within \(g + n\) of the first or last cell has fewer reference cells on one side, or none. There are three choices: skip those cells and never declare a detection there; average whatever reference cells exist, which is what this lesson’s code does; or raise the bias for the cells with fewer neighbors. With the second choice, 16 reference cells instead of 32, and the bias left at 12.4 dB, \(P_{fa}\) rises to \((1 + 17.3/16)^{-16} = 8 \times 10^{-6}\), a small increase that keeps those cells in the test. The one choice to avoid is padding with zeros and counting them, which drags the estimate down and the false alarms up. ADI’s cfar() does not test the end cells against their own neighbors: it gives them the lowest threshold found anywhere in the profile, so a strong return near either end is reported whatever the noise around it. The cells nearest 0 m on the PHASER hold the kit’s own transmit-to-receive leakage, so they are usually excluded in any case.

Doppler is different. The Doppler FFT is periodic: the cell above the highest velocity, +15 m/s for the L29 burst, is the lowest, −15 m/s. A reference window that runs off one end of the Doppler axis continues at the other, so it never runs out, and the Doppler edge needs no special case.

The range-Doppler map has two axes, so the window does too. The guard cells become a box around the cell under test, and the reference cells are the ring between that box and a larger one. With 2 guard and 4 reference cells each way from the cell under test, in both range and Doppler, the outer box is 13 by 13 cells and the guard box 5 by 5, so \(N = 169 - 25 = 144\). The bias for \(P_{fa} = 10^{-6}\) is 11.6 dB and the CFAR loss is 0.2 dB, far better than any one-dimensional window can do while staying within a few cells of the cell under test.

The two directions need not match. At the L29 burst’s 0.30 m range cells and 0.12 m/s velocity cells, a walking person spans a cell or two in range but many cells in Doppler, because the arms and legs move at different speeds from the body. A wider guard band in Doppler than in range keeps that spread out of the estimate.

Without MTI, every stationary object puts its echo at zero velocity, so the map has a ridge of clutter along the zero-Doppler column. To a CFAR window, that ridge is a clutter edge in the Doppler direction: cells on the ridge are compared with an average that is mostly noise, and they are declared targets. MTI from Lesson 29 removes most of the ridge before CFAR runs.

The L29 burst’s map of \(269 \times 256\) cells, with 144 reference cells each, is 10 million additions. A Python loop over every cell is slow, but a convolution of the map with the window does the same sums in a fraction of a second, and the next frame’s code works that way.

The map is synthetic, built with the L29 burst’s cells: a walker at 6.0 m and +1.2 m/s, a hand at 2.1 m and −0.6 m/s, and the clutter a 2-pulse canceller leaves at zero velocity near the radar. CFAR with 2 guard and 4 reference cells each way and an 11.6 dB bias declares both targets and the strongest clutter cell. Each target produces a cluster of detections, because its peak spreads into its neighbors and those neighbors are well above the noise too. A detection list that feeds a classifier or a tracker needs one entry per object, so we reduce each cluster to its strongest cell.

The excerpt is the cell-averaging path of the provided cfar_2d in radar_lab.py, which the L30 Lab notebook calls; the provided version also takes method="ca", "go", or "so" for the variants above. It takes the map in dB, as Lesson 29 plots it, and converts it back to power, because the CFAR statistics of this lesson are for power. guard and ref are (range, Doppler) pairs of half-widths. The two convolutions sum each cell’s reference window and count how many cells it held, so an end cell in range averages only the cells that exist; the code wraps the Doppler axis. Pass the map’s axes, and the detection list comes back in meters and meters per second.

import numpy as np
from scipy import signal
def cfar_2d(rd_map, guard, ref, bias_db, axes=None):
    P = 10 ** (rd_map / 10)                                  # dB back to power
    (gr, gv), (rr, rv) = guard, ref                          # (range, Doppler) half-widths
    win = np.ones((2 * (gr + rr) + 1, 2 * (gv + rv) + 1))
    win[rr:rr + 2 * gr + 1, rv:rv + 2 * gv + 1] = 0          # guard cells and the cell under test
    def pad(A): return np.pad(np.pad(A, ((0, 0), (gv + rv,) * 2), "wrap"), ((gr + rr,) * 2, (0, 0)))
    total = signal.fftconvolve(pad(P), win, mode="valid")
    count = signal.fftconvolve(pad(np.ones_like(P)), win, mode="valid")
    snr_db = 10 * np.log10(P / (total / count))              # each cell over its local noise
    det = snr_db > bias_db
    r_ax, v_ax = axes or (np.arange(P.shape[0]), np.arange(P.shape[1]))
    hits = [(r_ax[i], v_ax[j], snr_db[i, j]) for i, j in zip(*np.nonzero(det))]
    return det, hits

The SNR in each entry is the cell’s power over its own local noise estimate, the same quantity the threshold tests, so it is the number to compare with Lesson 27.

We set the bias from a design \(P_{fa}\) and check it by counting. On maps with no target, or in the cells far from any target, every detection is a false alarm, and the measured probability is the number of false detections over the number of cells tested.

Counting false alarms at the design rate takes many maps: the L29 burst’s map of \(269 \times 256\) cells is 68 864 decisions, so at \(10^{-6}\) it averages 0.07 false alarms, one every 15 maps. Counting ten of them to get the rate within about a third takes 145 maps. Instead, we lower the bias to a design \(P_{fa}\) of \(10^{-3}\) or \(10^{-4}\), where a single map has 7 to 69 false alarms, and check the measured rate against \((1 + \alpha/N)^{-N}\) at a few biases. If the counts follow the curve, the extrapolation to \(10^{-6}\) is sound; if they come out high, the noise in the reference cells does not match the noise in the cells under test.

We measure the probability of detection the same way, against a target you placed: the fraction of maps in which CFAR declares it. Compare it with Lesson 27’s curves at the SNR the detection list reports, with the CFAR loss added. For a hand or a person, which fluctuate from map to map, expect the Swerling numbers of Lesson 27 rather than the steady ones.

The L30 Lab runs this lesson’s detector twice. It starts in the Phaser app, on its Range (CFAR) preset: one chirp of 500 MHz in 0.5 ms at 10 GHz each display update, with a one-dimensional CFAR threshold drawn over the live range profile. You place the threshold by eye, apply it, walk, and count what crosses it on an empty scene. The rest runs in the lab’s notebook, which calls the provided cfar_2d on range-Doppler bursts of the Range + Velocity preset, the L29 burst: you set the bias for a false-alarm rate and count the false alarms, switch among cell averaging, greatest-of, and smallest-of, measure the probability of detection against Lesson 27, and make a detection list. You work the lab on your own, outside class, and turn in one packet in three parts: Part 1, Pre-Lab, the predictions; Part 2, Measurements, from the app and the notebook; and Part 3, Writeup. I check the packet and return it with feedback. It is scaffolding for your final report, not the report.

The bench part follows ADI’s CFAR lab, which tunes the bias, guard cells, and reference cells of its script’s threshold by eye against a live range spectrum. The app’s Range (CFAR) preset is that lab’s waveform, so its range cells are 0.30 m, 6.67 kHz of beat per meter. The notebook’s bursts are the app’s Range + Velocity preset: 256 chirps of 300 µs, 0.5 ms apart, 500 MHz at 9.9 GHz, so the map’s cells are 0.30 m by 0.118 m/s.

The table is the core of Part 1. Each bias is \(10\log_{10}\alpha\) with \(\alpha = N(P_{fa}^{-1/N} - 1)\), from the bias frame of this lesson. The first row is a short window in the app, 4 reference cells on each side of the cell under test; the second is the window the lesson used for one range profile, 2 guard and 16 reference cells a side; the third is the 2D window of the notebook, 2 guard and 4 reference cells each way.

The app’s Ref Cells and Guard Cells count one side of the cell under test, and the app averages the reference cells as power, as this lesson does, so Ref Cells = 4 is \(N = 8\) and this lesson’s bias formula applies to it. Its profile runs from 0 to 20 m in 63 cells beyond 1 m. At a 13 dB bias, \(P_{fa} = (1 + 20/8)^{-8} = 4.5 \times 10^{-5}\), and 30 empty updates should hold a crossing in about 0.1 of them; at 10 dB, about 3; at 7 dB, about 22. The notebook’s empty map tests 64 106 cells, so its design counts are 64 at \(10^{-3}\), 6 at \(10^{-4}\), and 0.06 at \(10^{-6}\), each with a scatter of about its own square root. That is why the notebook checks the design at the lower biases, as the measuring frame of this lesson describes.

Part 1 also predicts how far you could walk before the threshold loses you. Your echo falls as \(R^{-4}\), so a peak that stands \(M\) dB over the threshold at 3 m stays over it out to \(3 \times 10^{M/40}\) m, if nothing else enters the window, and a bias 3 dB higher shortens that by \(10^{-3/40} = 0.84\).

Every cadet has a kit, so every beam in the room crosses someone. Point yours along a clear line, at least 4 m long if you can, and agree with your neighbors on who walks when. For the empty scene, step behind the kit or well to one side, and check the range profile before you count: a peak that moves is a person, not a false alarm.

ADI’s lab asks you to move slowly while the threshold is applied, because each update computes a threshold for every cell, and a fast walk smears your echo across the updates. ADI also asks you to leave the display’s limits alone while you set the threshold: making the clutter fade from the plot hides it from you, not from the detector. No step needs a corner reflector.

Keep L30_detections.csv with your captures, radar_lab.py, and this lab’s notebook, in the same folder as the L28 and L29 work. The L31 Lab runs the same chain, the canceller, the map, and cfar_2d, on its own records, so the file is your record of what the chain found, not an input to L31. For the final report, keep three results from this lab: the bias you chose and the false-alarm rate you measured with it, with your explanation of why the two differ; what greatest-of and smallest-of did to your scene, and which one you would run on it; and the probability of detection you measured against the Lesson 27 prediction. The report asks you to explain how your radar decides that a target is present, in your own words. The packet’s tables are the evidence for it, not the explanation.