FMCW Radar: The Mathematical Foundation How one radar waveform can reveal an object's range, velocity, and direction using frequency, phase, Fourier transforms, and antenna geometry. An FMCW radar does not directly measure range, velocity, and angle. Instead, it measures changes in frequency and phase . Those measurements are transformed into physical quantities. Dimension What changes? Information Samples within a chirp Beat frequency Range Chirp to chirp Doppler phase Velocity Receiver to receiver Spatial phase Angle 1. Start With an FMCW Chirp FMCW stands for Frequency-Modulated Continuous Wave . Instead of transmitting a single frequency, the radar continuously sweeps its frequency. A simple up-chirp can be written as: [ f(t) = f_c + St ] where: (f_c) = carrier frequency (S) = chirp slope in Hz/s (t) = time The slope is: [ S = \frac{B}{T_c} ] where (B) is bandwidth and (T_c) is chirp duration. Numerical Exam...

📡 Interactive FMCW Radar Simulator See how a chirp signal detects an object's range and velocity from the returned echo. 📡 Radar Parameters Carrier Frequency 77 GHz Chirp Bandwidth 500 MHz Chirp Duration 1 ms Chirp Slope 5.00 × 10¹¹ Hz/s 🎯 Target Parameters Target Range 50 m Target Velocity 20 m/s Interpretation ...

  📡 How Does a Chirp Signal Detect an Object? Understanding the difference between steady-frequency radar and chirp-based FMCW radar. The Basic Idea A radar can transmit a radio-frequency signal and listen for the signal reflected from an object. The important difference is whether the transmitted frequency is constant or changes with time. 📻 Steady-Frequency Signal f(t) = f c The carrier frequency remains constant. 📈 Chirp S...

Chirp Signal Simulator fs: Start Freq: End Freq: Up Chirp Down Chirp Show Spectrogram (approx) Linear Chirp Signal A chirp is a sinusoidal signal whose frequency changes continuously with time. In a linear chirp , the instantaneous frequency changes linearly. $$f(t)=f_{start}+kt,\qquad k=\frac{f_{end}-f_{start}}{T}$$ where: \(f_{start}\) = starting frequency (Hz) \(f_{end}\) = ending frequency (Hz) \(T\) = chirp duration (seconds) \(k\) = chirp rate (Hz/s) The phase of the chirp signal is obtained by integrating frequency: $$\phi(t)=2\pi\left(f_{start}t+\frac{k}{2}t^2\right)$$ $$s(t)=\sin\left(2\pi \left(f_{start}t+\frac{k}{2}t^2\right)\right)$$ For a down-chirp , frequency decreases with time: $$s(t)=\sin\left(2\pi \left(f_{end}t-\frac{k}{2}t^2\right)\right)$$ Understanding the Spectrogram A spectrogram shows how the frequency components of a signal change with time. It is obtained by ap...

Chirp Signal Simulator Starting Frequency (Hz) Ending Frequency (Hz) Amplitude phase Up-Chirp (unchecked = Down-Chirp) Generate Chirp Demodulate Return to DSP Simulations Main Page →

  for up-chirp generation fs = 1000;           % Sampling frequency (Hz) t = 0:1/fs:1;         % Time vector (1 second duration) f_start = 50;         % Starting frequency (Hz) f_end = 200;          % Ending frequency (Hz) % Generate up-chirp signal up_chirp = chirp(t, f_start, 1, f_end, 'linear'); plot(up_chirp) Output         for down-chirp generation fs = 1000;           % Sampling frequency (Hz) t = 0:1/fs:1;         % Time vector (1 second duration) f_start = 200;         % Starting frequency (Hz) f_end = 50;          % Ending frequency (Hz) % Generate up-chirp signal down_chirp = chirp(t, f_start, 1, f_end, 'line...

📘 Overview & Theory 🧮 MATLAB Code 🧮 Chirp Signal Simuator 📚 Further Reading   Chirp signals are often used to find target objects. In a chirp signal, the frequency varies with time. For up-chirp signals, frequency increases with time. Oppositely, for down-chirp signals, the frequency decreases with time. Advantages of a chirp signal over a single-toned signal Better resolution Better Security The wide bandwidth of a chirp signal allows for capturing more detailed info about the target or object In a chirp signal, pulse compression enhances resolution by concentrating the signal energy into a shorter duration of time It is less susceptible to noise  It improves signal to noise ratio Up-Chirp Signal A sinusoidal up-chirp signal is denoted as Where A is the amplitude of this signal             f0 is the starting frequency of the chirp at t=0             α is the chirp rate or the...