M5. Signal Processing

1. Hypothesis Testing in Radar Detection

In each resolution cell we test two hypotheses:

The Neyman-Pearson optimum detector computes the likelihood ratio $\Lambda = p(r|\mathcal{H}_1)/p(r|\mathcal{H}_0)$and compares to a threshold chosen for a fixed $P_{fa}$. For Gaussian noise and known signal, this reduces to the matched filter (Module 2). For unknown random signal, averaging gives the Square-Law detector: decide $\mathcal{H}_1$ if $|r|^2 > T$.

Under $\mathcal{H}_0$, $|r|^2$ is exponentially distributed; setting$P_{fa} = e^{-T/N_0}$ gives $T = -N_0\ln P_{fa}$. For $P_{fa}=10^{-6}$,$T/N_0 \approx 13.8$ (11.4 dB above noise).

2. Probability of Detection vs. SNR

For a non-fluctuating target (Swerling 0 / 5) with known amplitude, $|r|^2/N_0$ is non-central chi-squared of 2 degrees of freedom, and

with $Q_M$ the Marcum Q-function. Albersheim's useful closed-form approximation:

At $P_d=0.9, P_{fa}=10^{-6}$, the required single-pulse SNR is 13.2 dB (Swerling 0). The often-quoted “13 dB” design value comes from this case.

3. Swerling Target Fluctuation Models

Real targets fluctuate because many scattering centers interfere. Peter Swerling (1957) introduced five canonical models:

Swerling I (Rayleigh amplitude) models a complex target like an airliner; Swerling III models a single dominant scatterer plus residuals (e.g. the engine nacelle). At $P_d=0.9$, Rayleigh targets need ~8 dB more SNR than non-fluctuating — the Swerling penalty.

Pulse integration (N pulses) partly defeats fluctuation: more pulses sample more of the amplitude distribution, bringing effective performance back toward Swerling 0.

4. Constant False Alarm Rate (CFAR)

Noise power varies across range (atmospheric absorption), azimuth (clutter regions), and time (temperature, jamming). A fixed threshold would produce either too many false alarms or missed targets. CFAR adaptively estimates local noise from reference cells surrounding the cell-under-test (CUT), then sets a threshold to maintain prescribed $P_{fa}$.

The Cell-Averaging (CA) CFAR averages $N$ reference cells and scales by $\alpha$:

Guard cells between CUT and references prevent target self-masking. Typical N=16-32 gives CFAR loss of 1-2 dB relative to a fixed-threshold detector. The inherent trade: larger $N$reduces CFAR loss but increases ground-clutter and target-edge issues.

Variants: GO-CFAR (greater-of) biases toward edge robustness; SO-CFAR (smaller-of) helps with multiple targets; OS-CFAR (order statistics) uses the k-th ordered reference sample and is robust to outliers.

5. OS-CFAR (Ordered Statistics)

Rohling (1983) proposed using the $k$-th smallest of the $N$ reference samples as the noise estimate, making CFAR robust to interfering targets and clutter edges:

OS-CFAR tolerates a few interferers in the reference window at the cost of about 1 dB additional CFAR loss. Modern systems use adaptive CFAR that switches among CA / OS / GO depending on local statistics.

6. ROC Curves

The Receiver Operating Characteristic plots $P_d$ vs $P_{fa}$ for a given SNR. Each waveform / integrator combination produces a distinct ROC curve; any detector's performance must lie on or below the likelihood-ratio test's ROC (the Neyman-Pearson bound).

A useful metric: area under ROC curve (AUC). Large AUC means easy detection; AUC = 0.5 is random. In radar we typically require $P_d > 0.9$ at $P_{fa} < 10^{-6}$ per cell. With $10^6$ range-Doppler cells per dwell, this yields ~1 false alarm per dwell.

7. Kalman Filtering for Tracking

Given a linear Gaussian state-space model $\mathbf{x}_{k+1}=\mathbf{F}\mathbf{x}_k+\mathbf{w}_k$,$\mathbf{z}_k=\mathbf{H}\mathbf{x}_k+\mathbf{v}_k$, the Kalman filter propagates the posterior mean and covariance:

Typical radar tracking uses the Constant Velocity (CV) or Constant Acceleration (CA) model. For maneuvering targets, the Interacting Multiple Model (IMM) filter runs parallel Kalman filters for different dynamics models and combines their outputs via weighted mixing.

8. Data Association: JPDA and MHT

In multi-target scenarios, which detection belongs to which track? Key algorithms:

• GNN (Global Nearest Neighbor): Hungarian assignment of detections to tracks.
• JPDA (Joint Probabilistic Data Association): weighted combination of all feasible associations.
• MHT (Multiple Hypothesis Tracking): maintains a tree of association hypotheses, prunes low-probability branches.
• PHD filter (Probability Hypothesis Density): handles unknown target number via finite-set statistics.

Modern air traffic control and air defense use IMM-JPDA as a practical compromise; MHT for high-value targets in dense environments.

9. Clutter Amplitude Statistics

For many-scatterer clutter (rain, foliage), the central limit theorem gives Gaussian in-phase and quadrature, hence Rayleigh amplitude and exponential power. For high-resolution or low-grazing-angle sea clutter, heavier-tailed distributions dominate:

The K-distribution models spiky sea clutter with shape parameter $\nu$ typically 0.1-1. Log-normal and compound-Gaussian models are also used. CFAR designs must be aware of distribution; exponential-assuming CFAR over-detects on heavy-tailed clutter.

10. Censored and Adaptive CFAR Variants

Real reference windows contain targets, clutter edges, and outliers. Censored CFAR removes the largest few samples before averaging; Trimmed Mean CFAR (TMCFAR) removes from both ends. Adaptive CFAR chooses its variant online based on an Anderson-Darling or Kolmogorov-Smirnov test of the reference distribution.

Modern AI/ML approaches learn detection thresholds from labeled data, often outperforming parametric CFAR at the cost of generalization concerns.

11. Binary Integration and M-of-N

A simple but effective multi-pulse scheme: declare detection if at least $M$ out of$N$ pulses independently detect. The overall probability is

Optimal M typically $M^* \approx 0.62N$ (Fehlner 1962). Binary integration is suboptimal by ~2 dB compared to ideal square-law integration but is robust and easy to implement in pipelined hardware.

12. EKF, UKF, and Particle Filters

Most radar measurements (range, bearing, Doppler) are non-linear in Cartesian state. The Extended Kalman Filter (EKF) linearizes the measurement function $h(\mathbf{x})$ about the current estimate. The Unscented Kalman Filter (UKF) propagates a set of deterministic sigma-points through the nonlinear map, yielding better mean/covariance accuracy at the cost of more computation.

For non-Gaussian multimodal posteriors (e.g. target appearance at unknown range, multi-target association), particle filters (SIR, sequential importance resampling) sample the posterior and propagate weighted particles. They are the state of the art for difficult tracking but have dimensionality and efficiency limitations.

13. Interacting Multiple Model (IMM) Filter

A single dynamics model fits steady-state but diverges during maneuvers. The IMM maintains a bank of filters (e.g. constant-velocity, constant-acceleration, coordinated-turn) and computes mode probabilities $\mu_k^{(i)}$ via a Markov transition matrix. The output is the weighted mixture:

IMM is the workhorse for fighter and missile tracking, handling 9-g pull-ups gracefully.

Simulation: CFAR, Kalman, Swerling, ROC

The simulation below runs a complete detection chain: a noisy range line with four targets is processed with CA-CFAR (Pfa = 10\u207B\u2074), detections are shown overlaid on the threshold. A 2-D Kalman filter tracks a constant-velocity target from noisy measurements, and Swerling Pd-vs-SNR curves plus a ROC curve illustrate detection theory limits.

14. Track Initiation and Confirmation

A tentative track is formed when $M$ detections appear in a moving gate over$N$ scans; it is promoted to a confirmed track after $M'$ of the next$N'$ scans produce associated detections. Typical logic: 2-of-2 initiation, 3-of-5 confirmation, 3-of-10 maintenance. The gate size is tied to $P_d$ and expected maneuver acceleration.

False tracks from clutter scale as $P_{fa}^M$; thus M=3 at $P_{fa}=10^{-4}$gives 10\u207B\u00B9\u00B2 false-track rate — essentially eliminating them. This is the primary mechanism by which radars reject residual CFAR false alarms.

15. Albersheim and Shnidman Approximations

For radar engineering calculations, closed-form detection formulas are invaluable. Albersheim (1981) gives single-pulse SNR for Swerling 0, valid for $10^{-7}\le P_{fa}\le 10^{-3}$and $0.1\le P_d\le 0.9$. Shnidman (1989) extends to Swerling cases and coherent integration of N pulses:

These approximations are accurate to ~0.2 dB and are the mainstay of radar link-budget spreadsheets.

16. Cramér-Rao Lower Bounds on Estimation

Any unbiased estimator's variance is bounded by the inverse Fisher information. For radar range, Doppler, and angle:

Here $\beta$ is the rms signal bandwidth. A 10 MHz-rms chirp at SNR = 20 dB gives range precision $\sigma_R \approx 0.5$ m — 6x better than the resolution cell. Matched filter and discriminator outputs asymptotically achieve these bounds.

17. Track-Before-Detect (TBD)

When target SNR is below the CFAR threshold, Track-Before-Detect integrates coherently across range-Doppler cells along hypothesized motion trajectories. The Hough transform and dynamic programming both compute

summing intensity along a candidate track. TBD recovers ~5 dB of SNR for dim targets at the cost of heavy computation; it is used in missile-defense radars to extend range.