5. Pulse Radar in Practice

5.1 Pulse parameters and their tradeoffs

Pulse radars are characterized by:

• Pulse width $\tau_p$: duration of each transmitted pulse. Sets range resolution $c\tau_p/2$.
• Pulse repetition frequency PRF: how often pulses go out per second. Sets unambiguous range $c/(2\,\text{PRF})$ and unambiguous Doppler $\pm \text{PRF}/2$.
• Duty cycle $D = \tau_p \cdot \text{PRF}$: fraction of time transmitting. Typical 0.001 to 0.05.
• Peak power $P_t$: power during the pulse.
• Average power $\bar P = P_t \cdot D$: power averaged over time. This is what determines transmitter cooling and energy budget.

Modern radars almost universally apply pulse compression: transmit a long pulse with a wideband chirp or coded modulation, then matched-filter the receive to compress it. The compressed pulse has duration approximately $1/B$ where $B$ is the chirp bandwidth, recovering range resolution $c/(2B)$. Average and peak power keep their original ratio, but range resolution is decoupled from pulse duration, allowing simultaneously long range and fine resolution. Almost every modern surveillance radar uses linear FM (chirp) compression with bandwidth-time products of 100 to 1000.

5.2 Blind speeds and PRF stagger

We saw that a target's Doppler shift wraps around modulo PRF. If the target moves at exactly $f_d = n \cdot \text{PRF}$ for some integer $n$, the Doppler appears to be zero, and the radar cannot distinguish it from stationary clutter. These are blind speeds:

$v_{blind,n} = \frac{n \cdot \lambda \cdot \text{PRF}}{2}$

For an X-band radar at $\lambda = 3$ cm and PRF = 1 kHz: $v_{blind,1} = 15$ m/s, $v_{blind,2} = 30$ m/s, and so on. A target flying at exactly 15 m/s along the line of sight is invisible to MTI processing.

The fix is PRF stagger: vary PRF pulse-to-pulse, so the blind speeds at one PRF do not coincide with the blind speeds at another. Combine results across PRFs and you fill in the blind-speed gaps. This also helps disambiguate second-time-around echoes.

5.3 Pulse-Doppler radar

A pulse-Doppler radar fires a coherent train of pulses, samples returns into range bins, and applies an FFT across pulses to recover Doppler for each range bin. The output is a 2D range-Doppler map showing where targets are in both dimensions.

                 Doppler (velocity)
                 →   →   →   →   →
                ┌─────────────────────┐
                │  .          *       │  Range bin 1 (close)
                │              .      │  Range bin 2
   Range  ↓     │    *                │  Range bin 3
                │                  *  │  Range bin 4
                │ . .  .  .  .  . .  .│  Range bin 5  (clutter at zero Doppler)
                │                     │
                │       *             │  Range bin 6 (target moving)
                │                     │
                └─────────────────────┘
                 ←─ stationary ─→
                clutter        targets

Real radars almost always use this 2D representation because it makes the difference between clutter and moving targets visually obvious, and it lets later processing apply different detection thresholds at different Doppler velocities. Stationary clutter sits at zero Doppler; moving targets are off to the side; falling rain occupies a smear at moderate negative Doppler.

The price of pulse-Doppler is data rate. A radar with 1000 range bins and 256 Doppler bins computes and analyzes a 256k-cell map at every dwell, perhaps every 10 ms. This is straightforward on a modern GPU but was hard in the 1980s.

5.4 Low, medium, and high PRF: choose two of three

Pulse-Doppler radars operate in one of three PRF regimes, each getting two of three things you want.

• Low PRF: long unambiguous range, small unambiguous Doppler. Used by surveillance radars that care about range out to hundreds of km; Doppler is resolved by PRF stagger or cross-radar correlation.
• High PRF: large unambiguous Doppler (good for fast aircraft and Doppler-resolving close clutter), small unambiguous range. Used by fighter look-down radars separating a target from ground clutter spread in Doppler.
• Medium PRF: ambiguous in both. Resolved by alternating multiple PRFs and applying the Chinese Remainder Theorem.

The AN/APG-77 in the F-22 alternates among many PRFs to cover everything from helicopters at low Doppler to oncoming missiles at high Doppler.