Some challenges of drone acoustic location: why sound is not radio
On systemic issues that get lost in the conversation about spectra and neural networks
I'm 80 years old and have been retired for a long time. In the past, I served for a long time at a military academy and was involved in applied research in the field Defense, so I'm familiar with this area professionally. Reading open publications about the fight against drones, I increasingly find myself with the same feeling: the techniques for specific issues have been sorted out, but the systemic level is lagging. The comments keep throwing up "complex numbers," "Fourier spectrum," "cepstrum," "neural networks," and "artificial intelligence," and they're far from always appropriate. I want to partially fill this gap. My starting point was The Acoustic Signature of Drones: Detection, Identification, and Countermeasure Possibilities, where technical details are present but higher-level tasks are absent. The word "some" in the title is not coquettish: this is a sketch, not a finished solution.
First questions, then spectra
Before diving into analyzing the signal structure in the time and frequency domain, it's helpful to ask and answer a few questions before doing any math.
First: what are we actually considering—a single receiving station or a distributed system? Second: what do we want to obtain—just the signal detection or an estimate of its parameters? And which ones specifically—bearing (that is, the direction of the sound source), coordinates, speed? The answer is up to the user: an air defense crew needs one thing, a warning system another.
Next comes the location method. Off the top of my head, there are three: angle-measuring (measuring directions), differential-range-measuring (measuring distance differences), and a combination of both. The choice isn't abstract: it determines how many points to place and how to count them.
And only now – about the range. According to open data, a single microphone “hears” Drone per unit of kilometers. This value fluctuates widely and depends on the type of apparatus, background levels, and atmospheric conditions. Precise values should be verified using specialized sources. If the microphone array is highly focused, a separate question arises: how to survey the space with it? Without answering this, it is impossible to estimate the size of the detection zone, and therefore the cost of covering a given defense area. Angular accuracy also comes into play. Does a single microphone provide an acceptable bearing? What about the array? Until these accuracies are known, it is simply too early to talk about pinpointing the source with the required accuracy.
Why one point is not enough and where do ghosts come from?
From a single point, the range to a target can only be estimated very roughly, based on sound volume. And if there's more than one target in the air, especially at close bearings, separating them from a single point is impossible. Therefore, at least two measuring points are needed. The target's position is then determined by the intersection of the bearings, simply and clearly.
It's simple, as long as there's only one target. As soon as there are multiple, the problem of identification arises. Let's say two points each provide N bearings to N targets. There will be N² points of intersection of these bearings, and only N of them are true. The remaining N² − N are false, geometric ghosts created when a ray from one point intersected with a foreign ray from another. They need to be filtered out, preferably cheaply, without a ton of calculations.
Three or more points significantly simplify matters: the true point is aligned with all pairs of bearings at once, while the false one is not. This is a reliable indicator for rejection. But there is a second type of ghost, independent of the number of points—multipath. Sound arrives not only directly but also reflected from the ground and local objects. Each such reflection acts as an imaginary source and generates additional false intersections. Eliminating them is a separate task, and it will not go away.
Correlation, hyperbole, and "we're there"
Let's return to the difference-ranging method. The idea is familiar: the coordinates of a source are found by the difference in signal arrival times at two points. A pair of points yields one such difference. On a plane, this difference corresponds not to a point or a line, but to a hyperbola—a line along which the difference in distances to two points is constant. To obtain a point, a second hyperbola is needed, that is, a third point. Or, instead, a sufficiently accurate bearing from one of the two points.
All that remains is to find the actual arrival time difference. A correlation function comes in handy here—in simple terms, it's a measure of how much two microphones hear the same thing, but with a time shift. We shift one recording relative to the other and see at what shift they match best. This shift is the desired arrival time difference. It seems the job is done, and we've reached our goal.
But don't rush. In the radio range, the correlation-difference-ranging method works perfectly: there, the signals are noisy, meaning weakly correlated, and the correlation peak is sharp and singular. With sound, the first difficulty arises: periodicity.
The drone's sound source is periodic: the fundamental harmonic, the engine RPM, dominates the spectrum. Since the signal is periodic, so is the correlation function. It has more than one maximum: false peaks repeat with the signal period. And when the signal period is shorter than the desired delay (the difference in arrival time), distinguishing the true maximum from the adjacent, false one becomes difficult, creating ambiguity. These false peaks have to be filtered out. One approach is to not focus on a single fundamental harmonic, but to use the broadband component of the spectrum. Why this is truly important, I'll explain later.
The Doppler Effect: Why Sound Doesn't Behave Like Radio
The second challenge is more serious, and I once encountered it firsthand. While observing an An-2 aircraft with the S-200 anti-aircraft system's target illumination radar, I saw a Doppler shift of approximately 2 kHz. Let me clarify up front: this is an empirical estimate from personal experience, an illustration of the scale, and not a universal reference result. But it illustrates the point well.
From the textbook, we take the general formula for the Doppler shift for the case where the source is moving, while the receiver and environment are stationary (no wind). This is precisely the case we need, since the sound is emitted by a drone:
dF = F V/(c−V),
Where F is the radiation frequency, V is the source's velocity relative to the receiver (positive for an approaching source), and c is the propagation velocity of the oscillations. This case should not be confused with others: if the receiver is moving and the source is stationary, the formula is different (as a first approximation, dF = F V/c), while in windy conditions, it has its own formula. The total observed frequency for an approaching source can conveniently be written as F' = F c/(c − V).
The key to this formula is that the shift is directly proportional to the frequency. It is this proportionality that creates the entire difference between radio and sound.
Let's check it on the radio range. Let F = 6 109 Hz, V = 50 m/s, c = 3·108 m/s (the speed of light). Then the shift will be dF = 6 109·50/(3·108 - 50) ≈ 2000 Hz. I observed this on the S-200 while accompanying a slow-moving An-2 aircraft.
Note the proportion: the target's speed is negligible compared to the speed of light, V ≪ c. Therefore, the Doppler shift is practically the same for all sections of the spectrum, with the relative difference at its edges vanishingly small. This difference is safely ignored, and the spectrum simply shifts as a single spot.
In the audio range, the picture is the opposite. The speed of sound is only c = 340 m/s, and a drone with a speed of V = 50 m/s is already moving at a noticeable fraction of this speed. Let's take a signal with two spectral components, 100 Hz and 1000 Hz, that is, a width of 900 Hz. We calculate the shifts: for 100 Hz, this is 100 x 50/(340 − 50) ≈ 17,2 Hz, for 1000 Hz - 1000 x 50/(340 − 50) ≈ 172 Hz. At the receiving point, the lines will be at 117,2 Hz and 1172 Hz. Now the spectral width is the distance between the outer lines: 1172 − 117,2 ≈ 1055 Hz. That is, the spectrum expanded from 900 to 1055 Hz, by 155 Hz, by approximately 17%.
Here's the crux of the matter. In radio, Doppler is additive, adding almost the same amount to all frequencies. In audio, it's multiplicative: it doesn't shift the spectrum, but rather stretches it unevenly. Low frequencies shift only slightly, while high frequencies shift significantly more. And this stretching varies at different receiving stations because the target approaches them with different radial velocities. Two stations no longer hear the same shifted signal, but two differently deformed ones, and the correlation between them disintegrates. This is a significant difference from multi-position passive radar, where this subtlety can be ignored.
Why bother with this at all? For the sake of range. From an air defense tactical perspective, the range of passive sound ranging is always insufficient, and it must be squeezed out of the entire signal energy, the entire spectrum, and not just the fundamental harmonic. But it is the periodicity of the correlation function (the same one from the previous section) and multiplicative Doppler that prevent us from working with the entire spectrum. We'll overcome them, and we'll get the range. There's also a pleasant side effect: the spectrum stretching coefficient is uniquely related to the target's radial velocity. This means that by selecting a correction that removes Doppler distortion, we simultaneously measure the radial velocity itself, the projection of the target's movement toward the target. And from three points, three radial projections form a complete three-dimensional velocity vector. Knowing this, we can account for the sound delay and calculate the target's actual location at the time of calculation, not where it was when the sound was transmitted.
And there's another fly in the ointment. The speed of sound is variable: it depends on temperature, humidity, and air composition. Wind and atmospheric temperature stratification bend sound rays, a phenomenon known as refraction. All of this impacts both range and accuracy, adding to the aforementioned multipath effect.
Why "some"
Problems similar to those discussed have likely long been solved in another field, sonar. However, for obvious reasons, the results haven't appeared in the public domain there. Interestingly, physics favored the sonar scientists in this problem, not ours. Sound in water travels more than four times faster than air, about 1500 m/s versus 340. And objects of interest to sonar—submarines and torpedoes—move several times slower than drones, not to mention other aerial targets. This means the ratio of target velocity to the speed of sound is an order of magnitude smaller, and with it, the multiplicative spectral stretching. It's safe to assume that this effect is simply ignored in sonar, and the correlation method works almost without exception. Air doesn't afford us such luxury.
These notes are just a sketch of a future, more detailed analysis. In the sequel, I hope to refine two components into practical form: an algorithm for calculating the correlation function with correction for Doppler spectral stretching and a procedure for filtering out false intersections of bearings in a spaced system. If the topic proves interesting to readers, I will try to complete the solution and publish the results.
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