[Radar signal analysis] Time domain analysis of linear frequency modulation signals based on LFM with Matlab code

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Content introduction

In wireless communications, LFM (Linear Frequency Modulation) linear frequency modulation signal is a common modulation method. Its characteristic is that the frequency changes linearly with time and can be used in radar, communications and other fields. This article will introduce the time domain analysis method of LFM chirp signal, as well as some problems and solutions in its practical application.

1. Time domain analysis method of LFM linear frequency modulation signal

The mathematical expression of LFM chirp signal is:

s(t) = exp(jπkt^2)

Among them, k is the frequency modulation slope and t is time. It can be seen that s(t) is a quadratic function about t, and its frequency changes linearly with time. In order to better understand the time domain characteristics of the LFM signal, we can perform Fourier transform on it to obtain its frequency domain representation.

First, we perform Fourier transform on s(t) to obtain its frequency domain expression:

S(f) = ∫exp(-j2πft)exp(jπkt^2)dt

By substitution method, the above formula can be transformed into:

S(f) = 1/2∫exp(-jπ(f-k/2t)^2/(k/2))exp(jπk/4)df

It can be seen that S(f) is a Gaussian function about f, with a center frequency of k/2 and a bandwidth of 1/k. Therefore, the spectrum of the LFM signal is a signal with a very narrow bandwidth and a center frequency that changes linearly with time. In practical applications, we usually use the autocorrelation function and cross-correlation function of the LFM signal for time domain analysis.

2. Problems and solutions of LFM linear frequency modulation signals in practical applications

Correlation functions have high computational complexity

The calculation complexity of the autocorrelation function and cross-correlation function of the LFM signal is very high, especially in high-speed signal processing, the calculation amount will be very large. To solve this problem, we can use the Fast Fourier Transform (FFT) algorithm. Through the FFT algorithm, the calculation complexity of the relevant function can be reduced from O(N^2) to O(NlogN), which greatly improves the calculation efficiency.

The peak position of the correlation function is unstable

The peak positions of the autocorrelation function and cross-correlation function of the LFM signal in the time domain change as the signal parameters change, which will affect the accuracy of signal detection. To solve this problem, we can use matched filters. A matched filter is a specific filter that matches a signal to its template to pinpoint the peak position of the signal in the time domain. In practical applications, matched filters are often used in radar signal detection and symbol timing synchronization in communication systems.

The impact of multipath effects on signal detection

In practical applications, LFM signals are often affected by multipath effects, causing changes in signal delay and phase. This will bring certain difficulties in signal detection and positioning. To solve this problem, we can use multi-channel signal processing technology. Multi-channel signal processing technology can use multiple receivers to receive the same signal, thereby eliminating the impact of multipath effects and improving the accuracy of signal detection and positioning.

Summarize:

LFM linear frequency modulation signal is a common modulation method and is widely used in radar, communications and other fields. The time domain analysis method of LFM signal mainly includes the calculation of autocorrelation function and cross-correlation function, and the application of matched filter. In practical applications, we need to pay attention to the computational complexity of the correlation function, the stability of the peak position, and the impact of multipath effects on signal detection. By using methods such as FFT algorithms, matched filters and multi-channel signal processing technology, these problems can be effectively solved and the accuracy of signal detection and positioning improved.

Part of the code

function [naf, tau, xi]=ambifunb (x, tau, N, trace)
% if (nargin == 0)
% error('At least one parameter required');
% end

[xrow,xcol] = size(x);
% if (xcol==0)|(xcol>2)
% % error('X must have one or two columns');
% end
if (nargin == 1)
    if rem(xrow,2)==0
        tau=(-xrow/2 + 1):(xrow/2-1);
    else
        tau=(-(xrow-1)/2):((xrow + 1)/2-1);
    end
    N=xrow;
    trace=0;
elseif (nargin == 2)
    N=xrow;
    trace=0;
elseif (nargin == 3)
    trace=0;
end
[taurow,taucol] = size(tau);
if (taurow~=1)
    error('TAU must only have one row');
elseif(N<0)
    error('N must be greater than zero');
end
naf=zeros (N,taucol);
if trace
    disp('Harrow-band ambiguity function')
end
for ico1=1:taucol
    if trace
        disprog (icol, taucol, 10)
    end
    taui=tau(ico1);
    t=(1 + abs(taui)):(xrow-abs(taui));
    naf(t,ico1)=x(t + taui,1).* conj(x(t-taui,xcol));
end
naf=fft(naf);
naf=naf([(N + rem(N,2))/2 + 1:N 1:(N + rem(N,2))/2],:);
xi=(-(N-rem(N,2))/2:(N + rem(N,2))/2-1)/N;
if (nargout==0)
    contour(2*tau,xi,abs(naf).^2);
% surf(2*tau,xi,abs(naf).^2,16)
    grid on
    xlabel('Delay');
    ylabel('Doppler');
    shading interp
    title('Narrow-band ambiguity function');
end

Run results

[Radar signal analysis] Based on LFM chirp signal time domain analysis with Matlab code_path planning

[Radar signal analysis] Based on LFM linear frequency modulation signal time domain analysis with Matlab code_UAV _02

References

[1] Duan Yu. Research on chirp signal detection and parameter estimation methods under low signal-to-noise ratio [D]. National University of Defense Technology [2023-10-08]. DOI: 10.7666/d.D675816.

[2] Ding Zhiquan. Design and FPGA implementation of pulse compression system for linear frequency modulation signals [D]. University of Electronic Science and Technology of China, 2008. DOI: CNKI: CDMD: 2.2007.050799.

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