M2. Pulse & Modulation

1. Pulse Train Basics: PRI, PRF, Duty Cycle

A pulse radar transmits for time $\tau$ (pulse width) every PRI (pulse repetition interval). The pulse repetition frequency PRF and duty cycle are

Typical duty cycles: pulse magnetron radars 0.001, modern TWTA/SSPA pulsed 0.01-0.1, pulse-Doppler and FMCW approach unity. The PRF choice trades range ambiguity against Doppler ambiguity (Module 3).

2. Range Resolution

Two targets at ranges $R_1, R_2$ produce echoes separated by $\Delta t = 2(R_2-R_1)/c$. For the envelope detector to distinguish them, $\Delta t \ge \tau$:

The second equality uses $B\approx 1/\tau$ for a rectangular pulse; a 1 m resolution requires 150 MHz bandwidth. For a pulse-compressed chirp with bandwidth $B$ (independent of pulse duration $T$), the same formula applies using the instantaneousbandwidth. This decouples energy from resolution: we can have a long high-energy pulse with fine resolution.

3. Range Ambiguity

Echoes must return before the next pulse is transmitted; otherwise they alias. Setting$2R/c < \text{PRI}$ gives the unambiguous range:

At PRF = 1 kHz, $R_u$ = 150 km. For pulse-Doppler operation at PRF = 10 kHz,$R_u$ = 15 km — second-time-around echoes appear as false targets. Multiple staggered PRFs resolve ambiguity via the Chinese Remainder Theorem.

4. Linear Frequency Modulation (LFM)

An LFM pulse sweeps frequency linearly over duration $T$:

Applying the stationary-phase approximation, its Fourier transform is approximately a rectangle of width $B$. The autocorrelation (output of the matched filter $h(t)=s^*(-t)$) is

The mainlobe width is $\sim 1/B$, giving compressed range resolution $c/(2B)$. The compression gain is $BT$ (time-bandwidth product): a 20 us pulse with 10 MHz bandwidth gives 23 dB SNR improvement.

5. The Matched Filter

Given received signal $r(t) = s(t-t_0) + n(t)$ with white Gaussian noise of PSD$N_0/2$, the linear filter maximizing output SNR is the matched filter:

Remarkably, the output SNR depends only on the energy $E_s=\int|s(t)|^2 dt$of the signal — not its shape. Short vs. long pulses, LFM vs. Barker, all achieve the same SNR provided their energies are equal. The waveform's shape controls only the range resolution and sidelobes.

See our Signal Theory course for a proof via the Cauchy-Schwarz inequality.

6. Phase Codes: Barker and Beyond

Phase-coded waveforms divide the pulse into $N$ subpulses (chips) of duration$\tau_c$, each given a phase $\phi_i$. For a biphase (0 or π) code,$\phi_i \in \{0,\pi\}$, represented as $\pm 1$. The Barker codes have optimal peak-sidelobe-ratio (PSR) for short lengths:

No Barker code longer than 13 exists. Longer codes are constructed from pseudo-random sequences: maximal-length shift-register (m-sequences), Kasami, Gold codes, or complementary (Golay) pair codes which give zero sidelobes when two pulses are averaged.

Polyphase codes (Frank, P1-P4, Zadoff-Chu) allow more than two phase values and achieve lower sidelobes still — essential for weather radar and low-probability-of-intercept (LPI) systems.

7. Stepped Frequency Waveforms

Instead of a continuous chirp, one can transmit $N$ pulses at frequencies$f_0, f_0+\Delta f, \ldots, f_0+(N-1)\Delta f$ and coherently combine them. The synthetic bandwidth is $B = N\Delta f$ with range resolution $c/(2B)$. Advantages:

• Narrow instantaneous bandwidth — lower-cost ADCs suffice.
• Each step can be individually filtered — clutter and interference can be excised per-step.
• Compatible with inexpensive CW oscillators tuned to each frequency.

GPR (ground-penetrating radar, Module 8) commonly uses stepped-frequency continuous wave (SFCW).

8. FMCW (Frequency-Modulated Continuous Wave)

Automotive and short-range radars use FMCW waveforms where the transmitted chirp is always on and the target echo is mixed with a copy of the transmitted signal to produce a beat frequency proportional to range:

A short FFT gives the range profile at low hardware cost; a separate FFT across chirps gives Doppler. This is the basis of 77 GHz automotive radars (Module 8).

9. Polyphase Codes and Zadoff-Chu

Polyphase codes use multiple phase values. A Zadoff-Chu sequence of length $N$ and root $r$ (coprime with $N$) is

Zadoff-Chu sequences have constant amplitude and ideal cyclic autocorrelation — a δ-function in the periodic sense. They are used in LTE synchronization and modern wideband radar. Frank codes, Px codes (P1-P4), and Huffman codes are approximations to LFM on a discrete chip grid.

10. Sidelobe Suppression via Windowing

An LFM pulse compressed with its exact matched filter gives a sinc-like response with first sidelobe at -13.3 dB. For surveillance radar this is too high — sidelobes of a strong target mask nearby weak targets. Applying a window $w(t)$ before compression (a mismatched filter):

lowers sidelobes at the cost of broadened mainlobe and reduced SNR. Common choices:

11. Eclipsing and Blind Ranges

A monostatic radar cannot listen while transmitting. Echoes arriving during pulse transmission are lost (eclipsed). The fraction of pulse intervals during which a target of range $R$ is eclipsed equals the duty cycle, times any straddling factor:

Additional blind ranges arise at multiples of $c\,\text{PRI}/2$ (the range bins where targets alias into the transmit interval). Staggered PRF (changing PRI each pulse) ensures no target stays blind over a full dwell.

12. Range-Doppler Coupling in LFM

The LFM pulse suffers from a subtle problem: a Doppler shift $f_d$ translates the compressed peak in range by

This range-Doppler coupling is a feature — it allows range-Doppler coupled search — or a bug, depending on context. Costas codes, stepped-frequency, and up/down chirps can decouple the two axes. We develop this fully in Module 3.

Simulation: LFM, Pulse Compression & Barker Codes

The Python simulation generates a 20 \u03BCs LFM chirp with 10 MHz bandwidth, shows its time-domain waveform and spectrogram, performs matched-filter pulse compression on a simulated echo with two closely spaced targets, computes the Barker-13 autocorrelation, and plots range resolution vs bandwidth and unambiguous range vs PRF.

13. Non-Linear FM and Shaped Waveforms

Non-linear frequency modulation (NLFM) sweeps the instantaneous frequency along a schedule$f(t)$ designed to match a desired spectral window (Taylor, Hamming). By the stationary-phase principle,

So slowing the sweep near band edges lowers the tails of the spectrum, giving low sidelobeswithout a mismatched filter — and therefore without SNR loss. NLFM is the state-of-the-art for weather radar (Module 7) where a strong convective cell's sidelobes must not mask nearby stratiform rain.

14. Digital Implementation of Pulse Compression

Two equivalent digital methods implement the matched filter:

1. Time-domain convolution: $y[n] = \sum_k r[n-k]\,h[k]$. Cost: $O(NM)$.
2. FFT-based: $Y = R\cdot H^{*}$ (overlap-save or overlap-add). Cost: $O(N\log N)$.

For typical radar $N\sim 10^5$, FFT is orders of magnitude faster. The stretch processing technique avoids the wide-bandwidth ADC by mixing the received signal with a copy of the transmit chirp — range maps to beat frequency exactly as in FMCW. This is standard on wideband imaging radars.

Fixed-point quantization of the reference code introduces an SNR loss of roughly 0.2 dB per bit dropped below 10 bits. Modern radars use 14-16 bit ADCs at 500 MSPS+.

15. Radar Equation with Pulse Compression

With pulse compression gain $BT$, the effective SNR becomes

SNR depends only on pulse energy $E_t = P_t\tau$ — the bandwidth cancels. A long low-power pulse gives the same SNR as a short high-power pulse of equal energy, while retaining fine range resolution. This is the single most important lesson of the module.

16. Low Probability of Intercept (LPI) Waveforms

Military radars often want to detect while avoiding detection themselves. Key design levers: (a) wide bandwidth spreading the transmit power, (b) continuous transmission (FMCW) at low peak power, (c) frequency agility hopping between PRIs, (d) polyphase codes with nearly white spectra. Long-integration LPI receivers (Intrepid, SIGINT) nevertheless defeat all but the most aggressive spreading.