# Demo — Pulse Timeline

The [PRF Explorer](L04_PRFExplorer_Demo.md) gives you the numbers the tradeoff produces. This one shows you where those numbers come from. A radar has exactly one pulse train, and every limit in this lesson falls out of it: how long each pulse is sets $\Delta R$, how long the radar waits between pulses sets $R_u$, and how fast it revisits a target sets $v_u$. Watch the pulses physically travel out, bounce, and come back — and watch the A-scope underneath report what the radar *believes* it saw.

## Interactive demo

<a class="demo-fullscreen" href="../_static/demos/PulseTimeline.html" target="_blank" rel="noopener">Open in full screen</a>

<div class="demo-wrap">
<iframe src="../_static/demos/PulseTimeline.html"
        title="Interactive pulse timeline showing range resolution, range ambiguity and velocity ambiguity"
        width="100%"
        loading="lazy">
</iframe>
</div>

## Walkthrough

1. **Drag B toward A until the echoes merge.** Each returning echo is a band of length $c\tau$; two of them stay separable only while the targets are more than $\Delta R = c\tau/2$ apart. At the default $\tau = 10$ µs, $\Delta R = 1.5$ km — so A at 12.0 km and B at 12.9 km already arrive as **one lump**, on the wire and on the A-scope. Drag B past 13.5 km and the lump splits in two. That threshold *is* range resolution: closer than $\Delta R$, the radar reports one target where there are two.
2. **On the Medium preset, follow C's drop-line.** C sits at 18 km, past $R_u = 15$ km, so its echo arrives after the next pulse has already left — the radar times it from the wrong launch and reports it at **3.0 km**. The dashed drop-line elbows left by exactly $R_u$ to show where the truth lands. The blip itself is indistinguishable from a real 3 km target. Switch to **Low 1 kHz** and the fold disappears: $R_u$ jumps to 150 km, off the end of the wire, and every target drops straight down.
3. **Sweep target A's speed.** The dial steps once per echo, by $\Delta\phi = 4\pi v T/\lambda$. At **50 m/s** the step is 120° and the radar reads the speed true. At **100 m/s** the step is 240°, which is indistinguishable from −120°, so the radar reports **−50 m/s** — an inbound fighter read as outbound. At **150 m/s** the step is a full 360°: the needle freezes, the target looks stationary, and MTI throws it away with the clutter. That is the first blind speed, and it is L5's opening move.

## Key observations

- **All three limits are one pulse train.** $\Delta R$ is the pulse's length, $R_u$ is the wait between pulses, $v_u$ is the revisit rate. There is nothing else in the picture.
- **The radar cannot tell a fold from the truth.** Only the drop-lines — which the radar does not have — reveal which blip is lying.
- **Velocity ambiguity is aliasing, literally.** The dial is a wagon wheel filmed at the PRF: past $v_u$ it turns backward, and at one full turn per pulse it stands still.