Matched Filter and Pulse Compression

The received power from a target at range $R$ is:

$$P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4}$$

where $P_t$ is transmit power, $G$ is antenna gain, $\sigma$ is radar cross section, and $\lambda$ is wavelength.

A chirp signal sweeps frequency linearly across bandwidth $B$ during pulse duration $T_p$:

$$s(t) = \exp\!\left(j\pi \frac{B}{T_p} t^2\right), \quad 0 \le t \le T_p$$

The time-bandwidth product $BT_p \gg 1$ enables fine range resolution with long pulses (high energy).

def chirp_signal(B, Tp, fs):
    t = np.arange(0, Tp, 1/fs)
    return np.exp(1j * np.pi * B/Tp * t**2)

The matched filter for a chirp is its time-reversed conjugate. The output is the compressed pulse with width $\approx 1/B$:

$$y(t) = s(t) * s^*(-t) = BT_p \cdot \text{sinc}(Bt) \cdot e^{j\pi B t}$$

Range resolution: $\Delta R = c/(2B)$.

def pulse_compress(rx_signal, tx_chirp):
    """Matched filter via FFT cross-correlation."""
    N = len(rx_signal) + len(tx_chirp) - 1
    Y = np.fft.fft(rx_signal, N) * np.conj(np.fft.fft(tx_chirp, N))
    return np.fft.ifft(Y)

The range resolution is the minimum distance between two distinguishable targets:

$$\Delta R = \frac{c}{2B}$$

For $B = 100$ MHz, $\Delta R = 1.5$ m.

The ambiguity function characterizes the matched filter response in both delay and Doppler:

$$\chi(\tau, f_d) = \int s(t)\,s^*(t-\tau)\,e^{j2\pi f_d t}\,dt$$

The zero-delay cut gives the Doppler response; the zero-Doppler cut gives the range response (compressed pulse shape).