Radar Signal Processing

Radar (Radio Detection And Ranging) systems are among the most demanding real-time DSP applications. A radar transmits an electromagnetic pulse, receives the echo reflected by a target, and uses signal processing to determine the target's range, velocity, and sometimes angle. DSP plays a central role in pulse compression, target detection, and clutter rejection, making radar a critical topic in ECE DSP applications.

Core Concept: Radar Range and Resolution

A radar system transmits a pulse of duration tau and measures the round-trip time delay of the echo to determine target range. The range to the target is R = c * t_delay / 2, where c is the speed of light (3 x 10^8 m/s) and t_delay is the measured round-trip delay. The factor of 2 accounts for the two-way propagation.

Range resolution delta_R is the minimum separation between two targets that allows them to be distinguished as two distinct echoes. For a simple rectangular pulse of duration tau, delta_R = c * tau / 2. To achieve fine range resolution, a short pulse is needed. However, a shorter pulse has less energy, which reduces the signal-to-noise ratio (SNR) and limits detection range. This is the fundamental range-energy trade-off in radar design.

Pulse Compression: Breaking the Range-Energy Trade-off

Pulse compression allows a radar to transmit a long, high-energy pulse while achieving the range resolution of a much shorter pulse. This is done by modulating the transmitted pulse (using frequency or phase coding) and processing the received echo through a matched filter. The matched filter is a correlation receiver whose impulse response is the time-reversed complex conjugate of the transmitted signal.

The most commonly used pulse compression waveform is the Linear Frequency Modulated (LFM) chirp. In an LFM signal, the instantaneous frequency increases (or decreases) linearly across the pulse duration: f(t) = f_c + (B/2T) * t, for -T/2 to T/2. The signal occupies a bandwidth B over a pulse of duration T, so the time-bandwidth product B*T is much greater than 1. After matched filtering, the output is compressed to an effective pulse width of 1/B, giving range resolution delta_R = c/(2B).

The pulse compression ratio (PCR) is defined as PCR = B * tau. This equals the SNR gain achieved by the matched filter compared to an unmodulated pulse of the same duration. For example, a chirp with B = 10 MHz and tau = 10 microseconds gives PCR = 100 (20 dB SNR gain) while achieving range resolution of c/(2B) = 15 m.

Mathematical Expression

The matched filter for a signal s(t) has impulse response h(t) = s*(T-t), where * denotes complex conjugate and T is a delay constant. In the frequency domain, the matched filter transfer function is H(f) = S*(f). The output SNR at the matched filter output is: SNR_out = 2E/N_0, where E is the signal energy and N_0/2 is the noise power spectral density. This is the maximum achievable SNR regardless of signal shape. For an LFM chirp, the output envelope is approximately a sinc function with first null-to-null width of 2/B.

The Doppler shift of a moving target is: f_d = 2 * v * f_c / c, where v is the radial velocity of the target, f_c is the carrier frequency, and c is the speed of light. Doppler processing is performed across multiple pulses (coherent processing interval) using FFT-based analysis. The frequency resolution in Doppler is 1 / (N * PRI), where N is the number of pulses and PRI is the pulse repetition interval.

Practical Understanding: CFAR Detection

After matched filtering, a threshold is applied to the output to decide whether a target is present or absent. A fixed threshold performs poorly because the noise and clutter level varies across range cells. Constant False Alarm Rate (CFAR) detection adaptively estimates the local noise/clutter level from surrounding range cells and sets the threshold proportionally. This maintains a constant probability of false alarm P_fa regardless of the background noise level.

Cell-Averaging CFAR (CA-CFAR) is the most common variant. It uses N reference cells on each side of the cell under test (CUT), averages their power, multiplies by a threshold scaling factor alpha (chosen to achieve target P_fa), and compares to the CUT power. The threshold is T = alpha * (sum of reference cell powers / N). This makes CFAR robust to spatially varying clutter such as ground return or weather echoes.

Given:
LFM Chirp pulse: tau = 10 microseconds, Bandwidth B = 5 MHz
Carrier frequency: f_c = 10 GHz
Target radial velocity: v = 300 m/s

Why this formula applies:
Pulse compression improves range resolution from c*tau/2 to c/(2B).
Doppler shift: f_d = 2*v*f_c/c

Formula:
Range resolution (uncompressed): delta_R1 = c * tau / 2
Range resolution (compressed): delta_R2 = c / (2 * B)
Pulse Compression Ratio: PCR = B * tau
Doppler shift: f_d = 2 * v * f_c / c

Substitution:
delta_R1 = (3e8 * 10e-6) / 2
delta_R2 = (3e8) / (2 * 5e6)
PCR = 5e6 * 10e-6
f_d = 2 * 300 * 10e9 / 3e8

Calculation:
delta_R1 = 3000 / 2 = 1500 m
delta_R2 = 3e8 / 1e7 = 30 m
PCR = 50 (17 dB SNR gain)
f_d = 2 * 300 * 10e9 / 3e8 = 6e12 / 3e8 = 20000 Hz = 20 kHz

Final Answer:
Uncompressed resolution: 1500 m
Compressed resolution: 30 m
Compression ratio: 50 (50x improvement)
Doppler shift: 20 kHz for v = 300 m/s at 10 GHz

Exam Tip: Range resolution = c*tau/2 (simple pulse) or c/(2B) (LFM, after matched filter). Pulse compression ratio = B*tau = SNR gain in dB = 10*log10(B*tau). Doppler shift f_d = 2*v*f_c/c. Max unambiguous range R_max = c*PRI/2.

Mechanism: Pulse Compression Process

Mechanism Summary

• Radar range R = c * t_delay / 2. Range resolution: c*tau/2 (simple pulse) or c/(2B) (LFM after matched filter).
• LFM chirp sweeps frequency linearly across pulse duration T. Matched filter compresses the received echo to a sinc-like peak of width 1/B, achieving fine range resolution while keeping high transmitted energy.
• Pulse compression ratio = B*tau. This equals the matched filter SNR gain. Higher B*tau means better range resolution AND better SNR simultaneously.
• Doppler shift f_d = 2*v*f_c/c. Moving target causes frequency shift in the received echo. Coherent processing across N pulses using FFT resolves Doppler with resolution 1/(N*PRI).
• CFAR detection adaptively sets threshold from surrounding reference cells, maintaining constant P_fa regardless of spatially varying clutter. CA-CFAR is the standard variant.

Quick Revision

• Range: R = c*t/2. Simple pulse resolution: delta_R = c*tau/2. LFM compressed: delta_R = c/(2B).
• LFM pulse: frequency sweeps B Hz over T seconds. Matched filter H(f) = S*(f). Output: sinc-envelope of width 1/B.
• Pulse compression ratio = B*tau = SNR gain (linear). In dB: 10*log10(B*tau).
• Doppler shift: f_d = 2*v*f_c/c. For 10 GHz radar and v=300 m/s, f_d = 20 kHz.
• Max unambiguous range: R_max = c*PRI/2. Increasing PRI improves range ambiguity but reduces max unambiguous Doppler velocity.
• Exam trap: matched filter maximizes SNR, NOT minimizes sidelobes. Sidelobe reduction requires weighting (windowing) at cost of some range resolution loss.
• CFAR maintains constant false alarm rate by adaptive thresholding. CA-CFAR performs poorly in clutter edges; OS-CFAR (ordered statistics) is used for non-homogeneous clutter.