M3. Doppler & Moving Targets

Motion shifts the frequency of the echo. Exploiting that shift lets us discriminate aircraft from mountains, missiles from chaff, cars from guardrails. The pulse-Doppler processor, MTI filter, and Woodward ambiguity function are the core mathematical tools.

1. The Doppler Shift: Classical and Relativistic

A target moving with radial velocity $v_r$ (positive = receding) has its range changing at that rate. A transmitted carrier at frequency $f_0$ experiences a round-trip phase

$$\phi(t) = -\frac{4\pi R(t)}{\lambda} = -\frac{4\pi (R_0 + v_r t)}{\lambda}.$$

The time-derivative $d\phi/dt$ gives the Doppler shift:

$$\boxed{\;f_d = -\frac{1}{2\pi}\frac{d\phi}{dt} = \frac{2 v_r}{\lambda}\;}.$$

At X-band (10 GHz, $\lambda = 3$ cm), 1 m/s produces 67 Hz; a Mach-2 fighter produces 45 kHz. The factor of 2 distinguishes radar Doppler (two-way) from optical Doppler (one-way).

The relativistic Doppler formula is

$$f_r = f_0\cdot\frac{1-v_r/c}{\sqrt{1-v^2/c^2}}\quad\longrightarrow\quad f_d = \frac{2 v_r}{\lambda}\left[1 + \mathcal{O}(v^2/c^2)\right].$$

Relativistic corrections matter only for space-based intercept of ballistic missiles and for radar astronomy of fast-spinning asteroids.

2. The Pulse-Doppler Data Cube

A coherent pulse train collected over a coherent processing interval (CPI) of $N_p$pulses samples the return on a 2-D grid:

Fast time (within one PRI): encodes range via $\tau = 2R/c$.
Slow time (across PRIs): encodes Doppler via phase change $\Delta\phi = 2\pi f_d \text{PRI}$.

Compressing along fast time (Module 2) then FFTing along slow time produces the range-Doppler map:

$$X(R, f_d) = \sum_{p=0}^{N_p-1} x_p(R)\,e^{-j 2\pi f_d p\,\text{PRI}}.$$

Doppler resolution is $\Delta f_d = 1/(N_p\,\text{PRI}) = \text{PRF}/N_p$. For$N_p=128$ at PRF=5 kHz, $\Delta f_d$=39 Hz, i.e. 0.6 m/s at X-band.

Range-Doppler Map

3. Moving Target Indication (MTI)

MTI filters suppress clutter (returns from terrain, sea, rain) by exploiting its near-zero Doppler. The simplest two-pulse canceller subtracts adjacent pulses:

$$y[n] = x[n] - x[n-1],\qquad |H(f)| = 2\,|\sin(\pi f\,\text{PRI})|.$$

This has a deep notch at $f_d=0$ (and at all multiples of PRF — the blind speeds). A three-pulse (double) canceller $y = x[n]-2x[n-1]+x[n-2]$ squares the response:

$$|H_2(f)| = 4\sin^2(\pi f\,\text{PRI}),$$

widening and deepening the clutter notch at the cost of also rejecting slow-moving targets. Staggered PRI and optimal MTI (derived from clutter covariance) combat both blind speeds and sidelobe clutter; see Hsiao (1974).

4. Blind Speeds

Because Doppler is sampled at the PRF, the velocity spectrum repeats with period PRF, producing aliased copies of the clutter notch at

$$v_{\text{blind}} = n\cdot\frac{\lambda\,\text{PRF}}{2},\qquad n=1,2,\ldots$$

At X-band with PRF = 5 kHz, first blind speed is 75 m/s — a serious problem for tracking jets. Solutions: raise PRF (trades range ambiguity), stagger PRI, or use multiple dwells at different PRFs (PRF diversity).

5. The Woodward Ambiguity Function

The ambiguity function characterizes a waveform's joint range-Doppler resolution:

$$\chi(\tau,\nu) = \int_{-\infty}^{\infty} s(t)\,s^{*}(t+\tau)\,e^{j 2\pi\nu t}\,dt.$$

The matched-filter response to a target with delay $\tau$ and Doppler $\nu$ is$|\chi(\tau,\nu)|$. Three volume theorems (Woodward, 1953) constrain waveform design:

Total volume: $\iint |\chi(\tau,\nu)|^2\,d\tau\,d\nu = E_s^2$ — a conservation law. You cannot reduce ambiguity everywhere simultaneously.
Peak: $\chi(0,0) = E_s$, maximum.
Symmetry: $\chi(-\tau,-\nu)=\chi^{*}(\tau,\nu)$.

Canonical shapes:

Single pulse: “thumbtack + ridge” pattern. Range resolution poor ($c\tau/2$).
LFM chirp: “knife edge” — tilted ridge in ($\tau,\nu$) plane (range-Doppler coupling).
Coherent train: multiple ridges at PRF intervals (blind speeds and ambiguous ranges).
Costas code: nearly optimal thumbtack — frequency jumps follow a Costas array.

6. Space-Time Adaptive Processing (STAP)

An airborne radar sees ground clutter spread along a clutter ridge in angle-Doppler space: at look direction $\theta$ the ground appears with Doppler $f_c = (2v_p/\lambda)\cos\theta$where $v_p$ is platform velocity. STAP adaptively combines $N$ antenna channels and $M$ pulses to estimate the interference covariance $\mathbf{R}$and steer nulls along the clutter ridge:

$$\mathbf{w}_{\text{opt}} = \mu\,\mathbf{R}^{-1}\mathbf{s}(\theta_t, f_t),\qquad \mathbf{R} = E\left[\mathbf{x}_{\text{int}}\mathbf{x}_{\text{int}}^{H}\right].$$

This is the multidimensional generalization of MTI. Reed-Mallett-Brennan (RMB) rule: to achieve within 3 dB of optimum, use $K \ge 2NM$ secondary samples. Reduced-dimension STAP (JDL, PRI-staggered) lowers compute cost dramatically.

Airborne Clutter Ridge

7. Pulse-Doppler vs. CW Radar

Two extreme architectures:

Feature	Pulse	Pulse-Doppler	CW / FMCW
Duty cycle	0.001	0.01-0.1	1
Range	Unambiguous	Ambiguous	Via beat freq
Doppler	Ambiguous	Unambiguous	Unambiguous
Tx/Rx isolation	Easy	Medium	Hard (100 dB+)
Typical use	Surveillance	Airborne AI/FC	Automotive, police

Pulse-Doppler radars operate in three PRF regimes: low PRF (range unambiguous, Doppler aliased), medium PRF (both ambiguous, resolved by multiple PRFs), high PRF (Doppler unambiguous, range aliased). Airborne fighter radars use medium or high PRF to combat MLC (main-lobe clutter).

8. MTI Improvement Factor

The MTI Improvement Factor $I_f$ is the ratio of output SCR to input SCR averaged over target Doppler, and is a standard figure of merit:

$$I_f = \frac{\text{SCR}_{\text{out}}}{\text{SCR}_{\text{in}}} = \frac{\langle |H|^2\rangle_v}{\langle |H|^2\rangle_{\text{clutter}}}.$$

For a 2-pulse canceller with Gaussian clutter spectrum of rms $\sigma_c$ (in Hz):$I_f \approx 2\text{PRF}^2/(2\pi\sigma_c)^2$. Ground clutter at L-band has$\sigma_c\sim 1$ Hz (leaves blowing). Weather clutter can have $\sigma_c\sim$tens of Hz. Chaff and sea clutter are worse still. Three- and four-pulse cancellers improve$I_f$ at the cost of target sensitivity.

9. Micro-Doppler Signatures

Targets are rarely rigid. Rotating propeller blades, helicopter rotors, vibrating engines, and human limbs all modulate the Doppler spectrum. Applying a short-time Fourier transform (STFT) reveals time-varying frequency signatures that classify targets: jet vs prop, helicopter vs drone, walking person vs vehicle.

For a rotor blade of length $L$ spinning at angular rate $\Omega$, the tip velocity oscillates as $v_{\text{tip}} = \Omega L \cos(\Omega t)$, giving a sinusoidally modulated Doppler line of extent $\pm 2\Omega L/\lambda$. Chen (2011) gives a full theory.

10. Doppler Processing via FFT

Across the CPI, each range bin is a slow-time signal $x_p = A\,e^{j 2\pi f_d p\,\text{PRI}}$. An $N_p$-point FFT maps pulses to Doppler bins. The window applied before FFT controls Doppler sidelobe level — typically Taylor 35 dB or Hamming.

Equivalent-time operations: zero-padding to $M > N_p$ interpolates Doppler; CZT (chirp-Z transform) zooms into a band; filter-bank architectures provide overlapping Doppler filters for bin-centered energy collection.

Non-coherent integration across CPIs (after envelope detection) further improves SNR at the cost of losing Doppler resolution.

11. Track-while-Scan and Doppler Memory

Raw range-Doppler detections feed a tracker (Module 5). Doppler adds an instantaneous velocity measurement to the kinematic state, dramatically reducing velocity uncertainty over range-only tracking. For a Kalman filter, Doppler reduces the 1-step velocity variance by roughly$\sigma_v^2/\sigma_{\dot r}^2$.

Simulation: Range-Doppler Processing and MTI

The code below builds a pulse-Doppler data cube with three targets (approaching fighter, receding car, zero-Doppler clutter), performs range compression and slow-time FFT to produce the range-Doppler map, plots the LFM ambiguity function, the MTI filter transfer function, and enumerates the blind speeds.

Python

script.py124 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ---- Range-Doppler simulation ----
fc = 10e9             # X band
c = 3e8
lam = c/fc
PRF = 5000            # pulse repetition frequency
PRI = 1/PRF
Np  = 128             # number of pulses in a CPI
fs  = 20e6
Tp  = 10e-6
B   = 8e6

# fast-time chirp
t_fast = np.arange(0, Tp, 1/fs)
tx = np.exp(1j*(2*np.pi*(-B/2)*t_fast + np.pi*(B/Tp)*t_fast**2))
Ns = len(t_fast)

# Build data cube for 3 targets: (range, velocity, RCS)
targets = [
    (3000,   -80, 1.0),   # approaching
    (5000,    30, 0.7),   # receding
    (7500,     0, 0.5),   # stationary (clutter)
]

range_bins = int(PRI*fs)
data = np.zeros((Np, range_bins), dtype=complex)
for tgt_R, tgt_v, tgt_A in targets:
    tau = 2*tgt_R/c
    fD  = 2*tgt_v/lam
    n0  = int(tau*fs)
    for p in range(Np):
        phase = np.exp(1j*2*np.pi*fD*(p*PRI))
        if n0 + Ns < range_bins:
            data[p, n0:n0+Ns] += tgt_A * phase * tx

data += (np.random.randn(*data.shape) + 1j*np.random.randn(*data.shape))*0.15

# Range compression
h = np.conj(tx[::-1])
rc = np.zeros_like(data)
for p in range(Np):
    rc[p] = np.convolve(data[p], h, mode='same')/Ns

# Doppler FFT across slow-time
doppler = np.fft.fftshift(np.fft.fft(rc, axis=0), axes=0)
vel_bins = np.fft.fftshift(np.fft.fftfreq(Np, PRI)) * lam/2
rng_bins = np.arange(range_bins) * c/(2*fs)

# ---- Ambiguity function of LFM pulse ----
def ambiguity_lfm(tau, fd, T, B):
    mu = B/T
    arg = np.pi*T*(fd + mu*tau)
    # avoid div by 0
    eps = 1e-12
    chi = np.where(np.abs(tau) < T,
                   np.abs((1 - np.abs(tau)/T) * np.sinc(T*(fd + mu*tau)*(1-np.abs(tau)/T))),
                   0)
    return chi

Tau = np.linspace(-Tp*1.1, Tp*1.1, 200)
Fd  = np.linspace(-3e5, 3e5, 200)
TAU, FD = np.meshgrid(Tau, Fd)
chi = ambiguity_lfm(TAU, FD, Tp, B)

# ---- Plot ----
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for sp in ax.spines.values(): sp.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.3)

# Range-Doppler map
im0 = axes[0,0].pcolormesh(rng_bins/1000, vel_bins, 20*np.log10(np.abs(doppler)+1e-6),
                           cmap='magma', vmin=-30, vmax=20)
axes[0,0].set_xlim(0, 12); axes[0,0].set_ylim(-120, 120)
axes[0,0].set_xlabel('Range [km]', color='#e2e8f0'); axes[0,0].set_ylabel('Velocity [m/s]', color='#e2e8f0')
axes[0,0].set_title('Range-Doppler Map (128 pulse CPI)', color='#fecaca', fontsize=13)
plt.colorbar(im0, ax=axes[0,0], label='dB')

# Ambiguity function surface
im1 = axes[0,1].pcolormesh(Tau*1e6, Fd/1e3, 20*np.log10(chi+1e-3), cmap='magma', vmin=-40, vmax=0)
axes[0,1].set_xlabel('Delay $\tau$ [μs]', color='#e2e8f0'); axes[0,1].set_ylabel('Doppler $f_d$ [kHz]', color='#e2e8f0')
axes[0,1].set_title('LFM Ambiguity Function (knife-edge)', color='#fecaca', fontsize=13)
plt.colorbar(im1, ax=axes[0,1], label='dB')

# MTI filter response
f = np.linspace(-PRF/2, PRF/2, 1000)
# single-pulse canceller H(f) = 1 - exp(-j2pi f/PRF)
H1 = 2*np.abs(np.sin(np.pi*f/PRF))
# double canceller
H2 = H1**2
axes[1,0].plot(f, 20*np.log10(H1/2+1e-3), color='#f87171', linewidth=1.5, label='Single (2-pulse)')
axes[1,0].plot(f, 20*np.log10(H2/4+1e-3), color='#fbbf24', linewidth=1.5, label='Double (3-pulse)')
axes[1,0].set_xlabel('Doppler [Hz]', color='#e2e8f0'); axes[1,0].set_ylabel('Response [dB]', color='#e2e8f0')
axes[1,0].set_title('MTI Filter Transfer Function (PRF = 5 kHz)', color='#fecaca', fontsize=13)
axes[1,0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
axes[1,0].set_ylim(-50, 10)
axes[1,0].axhline(-30, color='#60a5fa', linestyle='--', alpha=0.4)

# Blind speeds
n_blind = np.arange(0, 6)
v_blind = n_blind * PRF * lam/2
axes[1,1].bar(n_blind, v_blind, color='#a78bfa', edgecolor='#c4b5fd')
for i, v in zip(n_blind, v_blind):
    axes[1,1].text(i, v+5, f"{v:.0f} m/s", ha='center', color='#e2e8f0', fontsize=9)
axes[1,1].set_xlabel('Blind-speed harmonic n', color='#e2e8f0')
axes[1,1].set_ylabel('Blind speed [m/s]', color='#e2e8f0')
axes[1,1].set_title(f'Blind speeds at PRF={PRF/1000:.1f} kHz, λ={lam*100:.1f} cm', color='#fecaca', fontsize=13)

plt.tight_layout()
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0f172a')

print(f"Wavelength: {lam*100:.2f} cm")
print(f"Doppler resolution (Np={Np}): {1/(Np*PRI):.1f} Hz -> {(1/(Np*PRI))*lam/2:.2f} m/s")
print(f"Unambiguous velocity: ±{PRF*lam/4:.1f} m/s")
print(f"First blind speed: {PRF*lam/2:.1f} m/s")
print(f"Unambiguous range: {c/(2*PRF)/1000:.1f} km")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

12. Clutter Spectral Models

Clutter is a stochastic process, not a single tone. Common spectral models:

$$G(f) \propto \begin{cases} \exp(-f^2/2\sigma_c^2) & \text{Gaussian (foliage)} \\[3pt] 1/(1+(f/f_c)^n) & \text{power-law (sea)} \end{cases}.$$

Typical $\sigma_c$ values at L-band: leaves 0.1 Hz, small branches 1 Hz, trees in wind 5 Hz, rain 30 Hz, sea state 4 ~50 Hz. These drive the MTI filter design and STAP covariance estimation.

13. Velocity Ambiguity and the Doppler-Range Tradeoff

Unambiguous Doppler is $\pm\text{PRF}/2$, giving unambiguous velocity$|v| \le \text{PRF}\cdot\lambda/4$. Combined with $R_u = c/(2\,\text{PRF})$, the product

$$R_u\cdot v_u = \frac{c\lambda}{8}.$$

is fixed by wavelength alone. At X-band, $R_u v_u \approx 1.1\times 10^{6}\,\text{m}^2/\text{s}$; one can unambiguously measure 100 km with 11 m/s or 10 km with 110 m/s, but not both. Multi-PRF resolution via the Chinese Remainder Theorem breaks this deadlock by using staggered dwells.

14. Coherent Integration Gain in Doppler Processing

Coherent integration across $N_p$ pulses boosts SNR by exactly $N_p$(for a constant-Doppler target in the matched Doppler bin). This integration gain is equivalently the FFT's processing gain: concentrating energy from $N_p$ pulses into a single Doppler cell divides noise variance by $N_p$. The matched filter principle from Module 2 extends naturally into slow time.

15. Displaced Phase Center Antenna (DPCA)

An airborne radar with two phase centers separated by $d$ along the velocity vector can synthesize a stationary antenna by transmitting alternately from each. The returns from stationary clutter are then identical and subtract to zero — the airborne analog of ground MTI. The DPCA condition is

$$d = 2 v_p\,\text{PRI},$$

a strict mechanical constraint. STAP generalizes DPCA to arbitrary geometry and adapts to non-ideal conditions.

References

Woodward, P.M. — Probability and Information Theory, with Applications to Radar, Pergamon (1953).
Morris, G.V. & Harkness, L. — Airborne Pulse Doppler Radar, Artech House (1996).
Klemm, R. — Principles of Space-Time Adaptive Processing, IEE/IET (2002).
Ward, J. — “Space-Time Adaptive Processing for Airborne Radar”, MIT Lincoln Lab TR-1015 (1994).
Chen, V.C. — The Micro-Doppler Effect in Radar, Artech House (2011).
Stimson, G.W. — Introduction to Airborne Radar, 3rd ed., chs. 15-22.
Reed, I.S., Mallett, J.D., Brennan, L.E. — “Rapid Convergence Rate in Adaptive Arrays”, IEEE Trans. AES, 10, 853 (1974).

Cross-References

DFT & FFT Digital Filters M2: Pulse Modulation

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