7. The Radar Receiver: Pulling Echoes from Noise // Radar Systems // bhaswanth

7.1 The superheterodyne front-end

The job of the receiver is to take a fantastically weak echo, sometimes $-130$ dBm, and turn it into a digitized number ready for signal processing. Almost universally, this is done with the superheterodyne architecture from Chapter 7.

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   Antenna → LNA → Mixer → IF amp → Matched filter → Envelope/IQ → ADC → DSP
                    ↑
                LO (local oscillator)

The LNA sets the system noise figure (Friis formula: first stage dominates). The mixer multiplies the RF echo by the LO, producing sum and difference frequencies; we keep the difference, called the intermediate frequency (IF), typically tens of MHz. The IF is much easier to amplify and filter than the original RF, especially at radar frequencies of 10 GHz and up. Modern radars often use direct-IQ sampling, where the RF is mixed down to baseband I and Q components and sampled directly, but the principle is the same.

7.2 The matched filter: the optimum receiver for known signals

This is one of the most powerful results in signal processing, and we already met it in Chapter 3. Here it is again, in radar-specific language.

You transmit a known signal $s(t)$ . The echo is a delayed and possibly Doppler-shifted copy, embedded in noise: $r(t) = A s(t - \tau) + n(t)$ . You want to detect the presence of the echo and estimate $\tau$ .

The optimal linear filter, in the sense of maximizing output SNR at the moment of maximum signal, has impulse response equal to a time-reversed copy of the transmitted signal: $h(t) = s(T - t)$ for some constant $T$ . The output of this filter is the cross-correlation of $r(t)$ with $s(t)$ :

$y(t) = \int r(\tau)\, s(\tau - t + T)\, d\tau$

The output peaks at $t = T + \tau_{target}$ , with peak SNR

$\text{SNR}_{out} = \frac{2E}{N_0}$

where $E = \int |s(t)|^2 dt$ is the signal energy and $N_0$ is the noise spectral density. The peak SNR depends only on the signal energy, not on its shape.

This last point is the enabler of pulse compression. You can transmit a long, low-peak-power pulse with a wideband chirp inside it. The matched filter recovers the same SNR at its output as a short, high-peak-power, wideband pulse would have, but the temporal sharpness of the output peak is set by the chirp's bandwidth, not the pulse's duration. Long pulse duration $\to$ low peak power $\to$ high average power (good for transmitters), and the matched filter's output is short $\to$ fine range resolution.

A common implementation uses an LFM (linear-frequency-modulated) pulse and a matched-filter implemented digitally as an FFT-multiply-IFFT. The compressed pulse output looks like a sinc function with width $\sim 1/B$ .

python

import numpy as np
import matplotlib.pyplot as plt
 
# Synthesize a chirp pulse and matched-filter its echo
fs = 100e6           # 100 MS/s sampling
T_pulse = 10e-6      # 10 us pulse
B = 50e6             # 50 MHz chirp bandwidth (compressed res = 6 ns ≈ 1 m)
t = np.arange(0, T_pulse, 1/fs)
s = np.exp(1j * np.pi * (B/T_pulse) * t**2)  # LFM up-chirp
 
# Place a target at "range bin" R, add noise
N = 4096
echo = np.zeros(N, dtype=complex)
target_offset = 1500          # samples
echo[target_offset:target_offset+len(s)] = 1.0 * s
echo += 0.3 * (np.random.randn(N) + 1j*np.random.randn(N))
 
# Matched filter = correlate with conjugate-reversed pulse
h = np.conj(s[::-1])
y = np.convolve(echo, h, mode='same')
 
plt.subplot(2,1,1); plt.plot(np.real(echo)); plt.title("Raw echo (real part)")
plt.subplot(2,1,2); plt.plot(np.abs(y)); plt.title("After matched filter")
plt.tight_layout(); plt.show()

Run this and you see something striking. The raw echo looks like noise. The matched filter output has a sharp peak at the target's range bin, towering above the noise floor. That peak height is what radar signal processing is built on.

7.3 Constant false-alarm rate (CFAR) detection

After the matched filter, the radar must decide which peaks are targets. A fixed threshold fails because noise level varies with range, terrain, weather, and gain settings.

CFAR sets the threshold adaptively from local noise estimates. The simplest form, cell-averaging CFAR, averages the cells around (but not including) the cell under test and sets the threshold as a constant $\alpha$ times this average:

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   range bins:  ...  c_-3  c_-2  c_-1  [CUT]  c_+1  c_+2  c_+3  ...
                     ←─── reference cells ───→
                    average → multiply by α → threshold
                    if magnitude(CUT) > threshold → declare detection

$\alpha$ is set by the desired false-alarm probability. Modern radars use variants (greatest-of CFAR, ordered-statistics CFAR) robust against multi-target scenarios where a strong target in the reference cells would inflate the threshold and mask another target.

7.4 The detector

After the matched filter and CFAR, the receiver decides what to send to the display. Square-law detection (magnitude squared of the complex IF signal) is the standard for simple amplitude detection. For Doppler-aware processing, I/Q (in-phase and quadrature) sampling preserves the complex signal, allowing later FFT-based Doppler processing. Most modern radars are I/Q to the bone.