Matlab simulation of radar imaging 1 - LFM signal and its spectrum

1 Theoretical basis

1.1 Chirp The signal

LFM (LFM) Signal is commonly used in radar pulse compression , Also called chirp The signal , The time domain expression is （ Plural form ）：

$$s(t)=rect(\frac{t}{T})\times e^{j2\pi (f_c+\frac{K}{2}{t}^2)}$$
among ,$f_c$ Carrier frequency ,$rect(\frac{t}{T})$ The width is $T$ Rectangular wave signal ,$K=\frac{B}{T}$ It's the FM slope slope

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2. Simulation

In the program , Set up $f_c=0$,$T=10^{-6}s$, Signal bandwidth $B=30MHz$

First, you need to set the sampling frequency $F_s$, At least of the signal bandwidth 2 Multiple , Further sampling interval $Ts=\frac{1}{F_s}$, The number of sampling points is $N=\frac{T}{T_s}$

T = 10e-6;      % Pulse duration 10us
B = 30e6;       % Bandwidth 30MHz
K = B / T;      % chirp slope
Fs = 2 * B;     % sampling frequency
Ts = 1 / Fs;    % sampling spacing
N = T / Ts;     % Number of samples

Then generate a signal , Use linspace Function generating time series

linspace(start, end, N) For from start Start , To end end ( contain end) Generate evenly spaced N A little bit

%% signals
t = linspace(-T/2, T/2, N);
St = exp(1i*pi*K*t.^2);         % s(t)

The signal FFT （ Click to view the principle ）

freq = linspace(-Fs/2, Fs/2, N);
f = fftshift(abs(fft(St));

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Complete procedures and results

clear; clc;
set(0,'defaultfigurecolor', 'w')

%% parameters
T = 10e-6;      % Pulse duration 10us
B = 30e6;       % Bandwidth 30MHz
K = B / T;      % chirp slope
Fs = 2 * B;     % sampling frequency
Ts = 1 / Fs;    % sampling spacing
N = T / Ts;     % Number of samples

%% signals
t = linspace(-T/2, T/2, N);
St = exp(1i*pi*K*t.^2);         % s(t)

%% plot LFM signal
figure(1)
subplot(2, 1, 1)
plot(t*1e6, real(St), 'k', 'LineWidth', 1.5);
xlabel('Time in u sec');
title('Real part of chirp signal');
grid on; axis tight;

%% plot chirp FFT
subplot(2, 2, 3)
freq = linspace(-Fs/2, Fs/2, N);
plot(freq*1e-6, fftshift(abs(fft(St))), 'k', 'LineWidth', 1.5);
xlabel('Frequency in MHz');
title('Magnitude spectrum of chirp signal');
grid on; axis tight;

subplot(2, 2, 4)
freq = linspace(-Fs/2, Fs/2, N);
plot(freq*1e-6, -pi/K*freq.^2 + pi/4, 'k', 'LineWidth', 1.5);
xlabel('Frequency in MHz');
title('Magnitude spectrum of chirp signal');
grid on; axis tight;

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There is a small problem , adopt

phase = angle(sFFT); 
angle(1+1.732i)/3.14*180	% 60

The obtained phase spectrum is similar to noise , The reason is that the phase of the signal may exceed $2\pi$, But the function angle It calculates the phase according to the real part and the imaginary part , Only return $[-\pi ,+\pi ]$ Internal value , Cause phase blur , So you will get a phase spectrum similar to noise .

Related content ： Radar imaging