# Reading — Angle of Arrival By the end of this lesson you should be able to: 1. Explain why **angle of arrival (AoA)** is the hardest field in the pulse-descriptor word (PDW) to measure well, and what it does for the receiver. 2. Describe **amplitude comparison** — squinted-beam quadrant bearing — and where its accuracy runs out. 3. Derive the **interferometer** relation $\Delta\phi = \frac{2\pi d}{\lambda}\sin\theta$ and invert it for bearing. 4. Trace the **accuracy–ambiguity trade** that sets baseline length, and explain how multi-baseline arrays buy both. ## The one field that says *where* L12 built the PDW: frequency, pulse width, time of arrival, power — and one field the receiver measures but the last lesson never explained. **Angle of arrival (AoA)**. Every other parameter in the word is a *timing* measurement made on a single antenna: how long the pulse lasted, when it arrived, how often it repeats. AoA is different in kind. It is the only **spatial** parameter in the word — it says not *what* the emitter is, but *where* it is — and it is the one field that depends on geometry rather than on a stopwatch. That makes it both the most useful field and the hardest to get right. Useful, because AoA is what the processor **deinterleaves** on: when two emitters share a band and their pulses overlap in time, frequency and pulse width may look nearly identical, but a pulse from the left and a pulse from the right arrive at different angles. Bearing is what splits the merged stream back into separate tracks. AoA is also what **cues** the aircrew — it points the cockpit symbol and steers countermeasures at the correct threat rather than the wrong one. Hard, because the receiver is bolted to a maneuvering aircraft with no scanning dish, and a single bearing is a **line**, not a fix. One receiver gives you a direction; a position needs two of them crossed (the subject of L14). :::{admonition} Key Concept :class: key-concept Every other PDW field says *what* the emitter is — its frequency, its pulse width, its repetition pattern. AoA is the only one that says *where*. It is the field the processor deinterleaves on and the field that cues the shooter, and it is the hardest to measure because it is geometry, not timing. ::: ## Two physical handles on bearing A passive receiver — one that only listens — has exactly two physical quantities it can turn into an angle: - **Amplitude** — how *loud* the signal is in antennas that point different ways. A source off to one side is louder in the beam aimed nearer to it. - **Phase** — how the *wavefront* arrives at antennas spaced apart. A wave coming in at an angle reaches one element slightly before the other, and that delay is a measurable phase shift. These define the two methods. **Amplitude comparison** reads a power ratio across overlapping beams: coarse, cheap, and instant. **Phase interferometry** reads the phase difference across a baseline: fine, but it can wrap into ambiguity. Most fielded radar warning receivers (RWRs) do amplitude comparison; precision direction-finding systems add interferometry on top. Both are passive and both work on a single pulse — there is no scanning and nothing radiated. ## Amplitude comparison: squinted beams Point several broad antennas in different directions so their patterns overlap, then compare how loud the same pulse is in each. A signal off boresight is louder in the beam aimed nearer to it, and the *ratio* of powers in adjacent beams maps to a bearing within the overlap region. The classic RWR does this with **four antennas, one squinted into each quadrant**; the relative power between the two beams that hear the pulse places it inside that quadrant. The method's virtues are practical ones. It is **single-pulse** — no waiting for a second hit, no second emitter required. It is wideband and cheap, just a few antennas and a comparator. And it degrades gracefully: as a signal weakens, the bearing gets noisier rather than dropping out. Its limit is **calibration**. The bearing is only as good as your knowledge of each antenna's gain pattern, and real patterns ripple, shift with frequency, and distort once mounted on an airframe. Typical four-quadrant bearing lands within roughly **10–30°** — enough to label a quadrant and cue a head-turn, not enough to target. When precision matters, you need the other handle. ## Phase interferometry: the geometry Place two antennas a baseline distance $d$ apart and let a plane wave arrive from an angle $\theta$ off boresight. Because the wave is tilted, it reaches the far element after traveling an extra distance — the leg of a right triangle with hypotenuse $d$: $$ \text{extra path} = d\sin\theta. $$ At boresight ($\theta = 0$) the wave hits both elements at once and the extra path is zero; the farther off boresight the source, the longer that extra leg. That extra path shows up as a **phase difference** between the two receiver channels, and phase is something we can measure precisely. ## From phase to angle Convert the extra path to phase by dividing by wavelength and scaling to radians — one full wavelength of path is $2\pi$ of phase: $$ \Delta\phi = \frac{2\pi d}{\lambda}\,\sin\theta \qquad\Longrightarrow\qquad \theta = \arcsin\!\left(\frac{\lambda\,\Delta\phi}{2\pi d}\right). $$ Measure the single number $\Delta\phi$, plug it in, and read out the bearing $\theta$. This is the centerpiece of the lesson. The payoff is in the ratio $d/\lambda$: a *fixed* phase-measurement error maps to a **smaller** angle error as the baseline grows, because the same slice of phase corresponds to a finer slice of angle. The accuracy scales like $\lambda/(2\pi d)$, so long baselines push the bearing to **sub-degree** precision — far past anything amplitude comparison can reach. :::{admonition} Key Concept :class: key-concept The interferometer turns one phase measurement into a bearing: $\Delta\phi = \frac{2\pi d}{\lambda}\sin\theta$, inverted to $\theta = \arcsin\!\big(\frac{\lambda\,\Delta\phi}{2\pi d}\big)$. Because angle accuracy scales like $\lambda/(2\pi d)$, **stretching the baseline sharpens the bearing** — a longer $d$ buys finer angle. ::: ## The catch: phase wraps There is a price for the long baseline, and it is built into how a phase meter works. A phase detector cannot tell $\Delta\phi$ from $\Delta\phi + 2\pi$ — both fold to the same reading in $[-\pi, \pi]$. Phase is known only **modulo** $2\pi$. So once the true phase difference exceeds half a turn, several different arrival angles produce the *same* measured phase, and the inversion has no way to know which one is real. Work out when this first bites. To stay unambiguous across the full $\pm 90^\circ$ field of regard, the phase must never run past half a turn — $|\Delta\phi| \le \pi$ — and setting $\sin\theta = 1$ in the relation gives the condition $$ d \le \frac{\lambda}{2}. $$ But $d \le \lambda/2$ is a *short* baseline, which puts you right back at coarse accuracy. Make $d$ large for precision and the phase wraps: you get a beautifully fine bearing that may simply be **wrong**, landing on the wrong $2\pi$ cycle. Accuracy wants a long baseline; unambiguity wants a short one. **You cannot have both from a single pair.** ## Resolving it: multi-baseline The escape is to stop using just two elements. Build the array with at least a **short** baseline and a **long** one and let each do the job it is good at: 1. The short pair ($d \le \lambda/2$) gives a **coarse but unambiguous** bearing — no wrap, just imprecise. 2. That coarse bearing predicts **which $2\pi$ cycle** the long pair's phase is sitting in. 3. Use that cycle to *unwrap* the long pair's measured phase, then invert the relation for a **fine** bearing. Said in one line: the **short pair picks the cycle; the long pair sets the precision**. Real direction-finding arrays carry several baselines of graded length and calibrate each across the operating band, so the unwrap stays reliable as frequency changes. ## Amplitude vs interferometry | | Amplitude comparison | Interferometry | |---|---|---| | Reads | Power ratio across beams | Phase across a baseline | | Accuracy | Coarse ($\sim 10$–$30^\circ$) | Fine ($<1^\circ$ possible) | | Pulses | Single pulse | Single pulse | | Cost / size | Low; a few antennas | Higher; spaced array + calibration | | Failure mode | Pattern-calibration error | Phase wrap / ambiguity | The two are complementary, not competing. Many fielded RWRs **amplitude-compare for the quadrant, then interferometer-refine** when precision is needed — the coarse bearing can even seed the interferometer's ambiguity resolution. And both are **passive** and **instantaneous**: no scan, no transmit, an answer on the very first pulse. ::::{admonition} Quick Exercise :class: quick-exercise A 10 GHz threat, so $\lambda \approx 3$ cm. Reason about each choice: 1. You set $d = 1.5$ cm. Ambiguous or not? What does that cost you? 2. You set $d = 15$ cm. What do you gain, and what breaks? 3. Both elements drift by $20^\circ$ in measured phase. Which baseline's *bearing* moves more? 4. Why does amplitude comparison never have this ambiguity problem? :::{admonition} Solution :class: dropdown 1. $d = 1.5$ cm is exactly $\lambda/2$, so it is **unambiguous** across the full $\pm 90^\circ$ — but it is a short baseline, so the bearing is **coarse**. 2. $d = 15$ cm is $5\lambda$, about **10× finer** bearing — but the phase now wraps, so the naive inversion can land on the wrong cycle. You need the short pair (or another baseline) to resolve it. 3. The same $\Delta\phi$ error maps to a *smaller* angle error on the **long** baseline, so the long pair's *bearing* moves **less**. (Accuracy scales like $\lambda/(2\pi d)$.) 4. Amplitude comparison reads a **monotone power ratio**, not a periodic phase — there is no $2\pi$ to wrap and confuse it. It pays for that immunity with poor accuracy. ::: :::: ## Wrap-Up AoA is the only spatial field in the PDW — it deinterleaves overlapping emitters, cues the aircrew, and steers countermeasures, and it is the hardest field to measure because it is geometry on a moving platform. **Amplitude comparison** reads a power ratio across squinted beams: coarse ($\sim 10$–$30^\circ$), cheap, and single-pulse, limited by pattern calibration. **Interferometry** turns one phase measurement into a fine bearing, $\Delta\phi = \frac{2\pi d}{\lambda}\sin\theta$ inverted to $\theta = \arcsin\!\big(\frac{\lambda\,\Delta\phi}{2\pi d}\big)$; a longer baseline sharpens accuracy but wraps the phase, and **multi-baseline** arrays buy both — short pair picks the cycle, long pair sets the precision. Every one of these bearings is still just a *line*. Next, **L14 — Triangulation and Fusion** takes the bearings from two receivers and crosses them into a fix.