LFM signal denoising, time-frequency analysis, filtering

Preface

LFM （Linear Frequency Modulation,LFM） The signal has a large time bandwidth product , A large pulse compression ratio can be obtained , It is a signal form widely used in radar system and sonar system .
LFM The mathematical expression of the signal is ：

among ,A Is the signal amplitude ,t For time ,T Is the pulse duration （ Pulse width ）,fc Is the carrier frequency ,K Is the linear frequency modulation rate of the signal ,rect() Rectangular window function , The mathematical expression is as follows ：

Assume signal bandwidth B=10MHz, Pulse width T=100us, Carrier frequency fc=100MHz, Sampling frequency is fs=2B=20MHz. Linear frequency modulation rate K=B/T.

The design requirements are as follows ：
1. use Matlab For this parameter LFM Time frequency analysis of the signal , Come to the conclusion ;
2. Interference to signal noise （ Single frequency noise or Gaussian white noise ）, Time frequency analysis of the noisy signal ;
3. Design a suitable filter , For noisy LFM The signal is filtered , Compare the effect before and after filtering .

1.LFM Time frequency analysis of signal

LFM （LFM） Signal refers to the signal whose instantaneous frequency changes linearly with time , Also known as Chirp The signal . Expression han’you With time t Item pair of t We can derive K*t, Instantaneous frequency . stay Matlab in , Generated 2000 A little long LFM The signal , Draw the real part 、 Imaginary part 、 phase 、 The instantaneous frequency is shown in the figure below ：

LFM The time and frequency domain waveform of the signal is as follows ：

2. Add noise interference

In order to compare the influence of different kinds of noise on the signal , The addition frequency is 6MHz And Gaussian white noise （ After noise SNR=10dB）.
The time domain diagram and spectrum diagram of adding single frequency noise are as follows ：

Time domain diagram with Gaussian white noise 、 The spectrum diagram is as follows ：

3. Noisy LFM Signal filtering

3.1 Filter out single frequency noise

utilize Matlab Inside Filter Design hold-all , To design a IIRLPF, Its low-pass frequency is 5500Hz, The cut-off frequency is 5700Hz, The stopband attenuation is 40dB.

The signal containing a single frequency is passed through the filter , Observe the output time domain diagram and frequency domain diagram .6000Hz Frequency noise is obviously filtered out , But the signal is partially distorted .

3.2 Filter the Gaussian white noise in the signal

Because the frequency of Gaussian white noise is LFM Any frequency of the signal covers , So you can't filter by filter . The commonly used method is adaptive signal filtering , for example LMS wave filtering 、RLS Filtering, etc . The principle is not repeated here . In this paper RLS Filtering method .
stay Matlab Compiling RLS The filtering procedure is as follows ：

function [e,w]=my_rls(lambda,M,u,d,delta)
% recursive least squares,rls.
% Call:
% [e,w]=rls(lambda,M,u,d,delta)
%
% Input arguments:
% lambda = constant, (0,1]
% M = filter length, dim 1x1
% u = input signal, dim Nx1
% d = desired signal, dim Nx1
% delta = constant for initializaton, suggest 1e-7.
%
% Output arguments:
% e = estimation error, dim Nx1
% w = final filter coefficients, dim Mx1
% Step1:initialize
w=zeros(M,1);
P=eye(M)/delta;
u=u(:);
d=d(:);
% input signal length
N=length(u);
% error vector
e=d.';
% Step2: Loop, RLS
for n=M:N
    uvec=u(n:-1:n-M+1);
    e(n)=d(n)-w'*uvec;
    k=lambda^(-1)*P*uvec/(1+lambda^(-1)*uvec'*P*uvec);
    P=lambda^(-1)*P-lambda^(-1)*k*uvec'*P;
    w=w+k*conj(e(n));
end

Before filtering 、 After LFM The time domain diagram of the signal is as follows ：

Filtered LFM The signal spectrum is as follows ：

（ The simulation analysis is left to everyone ！）

Code

clear;
clc;
close all;
%%  Parameter setting 
B=10e6;    % Signal bandwidth 10MHz
tao=10e-5;   % Pulse width 100us
fs=2*B;    % sampling frequency 
Ts=1/fs;   % Sampling period 
K=B/tao;     % Linear frequency modulation rate 
fc =1e8;   % Carrier frequency 
%%  produce LFM The signal 
t=-tao/2:1/fs:tao/2-1/fs;
LFM=exp(j*2*pi*(fc*t+0.5*K*t.^2));  % Send a signal 
theta =  pi*K*t.^2;      % Signal radian 
ft = K*t;                 % signal frequency 

figure();
subplot(2,2,1);
plot(real(LFM));
xlabel(' Signal points n');ylabel(' Range ');
title('LFM Signal real part ');
subplot(2,2,2);
plot(imag(LFM));
xlabel(' Signal points n');ylabel(' Range ');
title('LFM Imaginary part of signal ');
subplot(2,2,3);
plot(theta);
xlabel(' Signal points n');
title(' Signal phase ');
subplot(2,2,4);
plot(ft);
xlabel(' Signal points n');
title(' signal frequency  Hz');

X=fftshift(fft(LFM));
f=linspace(0,fs,length(t))-fs/2;        % Set the frequency variable 
figure();
subplot(211);
plot(t,real(LFM));
xlabel(' Time /t');ylabel(' Range ');
title('LFM Signal time domain ');
subplot(212);
plot(f,abs(X));
xlabel(' frequency /f');ylabel(' Range ');
title('LFM Signal spectrum ');

%%  Add a single frequency noise 
fn1=6e6;        % Noise frequency 6MHz
n1=0.1*cos(2*pi*fn1*t);
S1=LFM+n1;
figure();
subplot(211);
plot(t,S1);
xlabel(' Time /t');ylabel(' Range ');
title(' Time domain of noisy signal ');
subplot(212);
plot(f,abs(fftshift(fft(S1))));
xlabel(' frequency /f');ylabel(' Range ');
title(' Signal spectrum after noise ');

%%  Design bandpass filter 
y=filter(IIRBPF2,S1);       % use Matlab Self contained filter design hold-all , Design low-pass filter , The code is shown IIRBPF2.m
[b,a]=tf(IIRBPF2);        % Put the low-pass filter IIRBPF2 Into a transfer function , The coefficient is b,a
figure();
freqz(b,a);     % do IIRBPF2  Amplitude frequency characteristic curve 

figure();
subplot(211);
plot(t,y);
xlabel(' Time /t');ylabel(' Range ');
title(' Time domain diagram of denoised signal ');
subplot(212)
plot(f,abs(fftshift(fft(y))));  % Denoised spectrum 
xlabel(' frequency /f');ylabel(' Range ');
title(' Denoised signal spectrum ');
%%  Add Gaussian white noise 
S2=awgn(LFM,10,'measured');    % Add Gaussian white noise , The SNR is 10dB
sn=S2-LFM;                     % noise 

figure();
subplot(211);
plot(t,S2);
xlabel(' Time /t');ylabel(' Range ');
title('LFM+ Time domain waveform of Gaussian white noise ');
subplot(212);
plot(f,abs(fftshift(fft(S2))));
xlabel(' Time /t');ylabel(' Range ');
title('LFM+ Gaussian white noise spectrum ');

%% RLS Algorithm filtering 
mu2=0.002;
M=2;
espon=1e-5;
delta=1e-7;
lambda=0.99;
[en,w2]=my_rls(lambda,M,sn,S2,delta);%RLS Algorithm subroutine 
er=en-LFM;            %er For error signal , Filter output - Input 
figure();
subplot(311);
plot(t,S2);
xlabel(' Time /t');ylabel(' Range ');
title('LFM+ Time domain waveform of Gaussian white noise ');
subplot(312);
plot(t,en);
xlabel(' Time /t');ylabel(' Range ');
title(' Time domain waveform of filtered signal ');
subplot(313);
plot(t,er);
xlabel(' Time /t');ylabel(' Range ');
title(' Error signal ');

figure;
plot(f,abs(fftshift(fft(en))));
xlabel(' frequency /f');ylabel(' Range ');
title(' Signal spectrum after filtering Gaussian white noise ');