[ |\chi(\tau, f_d)| = \left| \int_-\infty^\infty s(t) s^*(t+\tau) e^j2\pi f_d t dt \right| ]
After matched filtering, the output envelope is a function with first nulls at (\pm 1/B). LFM is Doppler-tolerant (slight frequency shifts cause small range shifts but minimal SNR loss). 4.2 Phase-Coded Signals The pulse is divided into sub-pulses (chips), each with 0° or 180° phase according to a binary sequence (e.g., Barker code, Gold code). Bandwidth (B \approx 1/t_chip), time-bandwidth product equals the number of chips. radar signal
Advantage: Lower range sidelobes than LFM (Barker codes give peak sidelobe 1/N). Disadvantage: More sensitive to Doppler shifts. A train of narrowband pulses at different carrier frequencies synthesizes wide total bandwidth. Enables high range resolution with lower instantaneous bandwidth hardware. 5. The Ambiguity Function The ambiguity function (\chi(\tau, f_d)) is the 2D autocorrelation of the radar signal, showing the response to a target at range delay (\tau) and Doppler shift (f_d): A train of narrowband pulses at different carrier
Resulting compressed pulse width: (\tau_comp = 1/B). Range resolution improves to (\Delta R = \fracc2B), independent of the original pulse width. 4.1 Linear Frequency Modulated (LFM) Chirp The instantaneous frequency changes linearly over time: (f(t) = f_0 + \fracB\taut). Bandwidth (B \approx 1/t_chip)
The matched filter output for a signal (s(t)) in white noise maximizes the Signal-to-Noise Ratio (SNR) and is given by the convolution with the time-reversed complex conjugate: (h(t) = s^*(-t)).
[ s(t) = A \cdot \textrect\left(\fract\tau\right) \cos(2\pi f_0 t) ]
