[signal denoising] signal denoising based on Kalman filter with matlab code

1 Content introduction

1.1  Background and significance of Kalman filter theory  

Signal is the carrier of information . Generally, it can be divided into two categories , They are deterministic signal and random signal . obey   A signal of a fixed function relationship , It's a deterministic signal , Its changes follow certain rules , have   Determined spectral characteristics , For example, sine and cosine signals 、 Step signal 、 Rectangular pulse signal with fixed pulse width . if   The value of the signal at each time is a random variable , It is called random signal , It cannot be expressed as a definite   Time function or space function . This signal even under the same initial and environmental conditions , Every letter   The implementation result of No. is also different , For example, waves 、 Gyro drift 、 Navigation error of inertial navigation system 、 The global   positioning system (Global Position System, GPS) Choice availability (Selective Availability, SA) error   etc. , Although random signals have no definite spectral characteristics , But in practice , You can use its statistical characteristics   To calculate , That is, the random signal has a certain power spectrum [1]1-5.  In the process of signal acquisition and transmission , It will inevitably be disturbed by the surrounding environment and internal noise , So I   We need to process the signal correctly , Eliminate the influence of interference , Try to get the signal you need , This is   It is necessary to filter the received signal . wave filtering , It is the process of extracting signals from noisy observation data . according to   The characteristics of signals are different , The filtering methods used are also different . For deterministic signals , Because of its special spectrum   Sex is certain , According to the different distribution of each signal in the frequency band , Set the... Corresponding to its frequency characteristic   filter , Such as high pass filter 、 Bandpass filter 、 Low pass filter and band stop filter , Make what you need useful   The signal passes through without attenuation , And suppress interference signals . This kind of filter can be realized by hardware , That is, we   Commonly referred to as analog filter , It can also be realized by software , Digital filter . For deterministic signals   The filtering we usually call it conventional filtering .  Although the spectral characteristics of random signals are uncertain , But its power spectrum characteristics are certain , The design of filter can be based on   According to the power spectrum of the required useful signal and interference signal . American scholar Wiener (N.Wiener) And so forth  Wiener wave filtering , The filter is designed by decomposing the power spectrum , Similar to conventional filtering, it is necessary to suppress the signal   Control and gating [2,3]. The signal is processed in the frequency domain Wiener Filter design , Ask for Wiener - Hope side   cheng , It takes a lot of calculation , And it also needs a lot of storage space , To some extent, it hinders Wiener Filtered   extensive use . 1960 Kalman (R.E.Kalman) For discrete data, recursive method is used to solve its linear filtering   Paper on wave problem [4]. In this article , Kalman filter is proposed (Kalman Filter), A new and gram   I'm impressed Wiener Filtering disadvantages of filtering methods . Kalman introduced the concept of state space into random estimation theory ,  Consider the signal as the output of a linear system excited by white noise , This kind of input - The output relationship passes through the state party   In the form of process , The filtering algorithm uses the system state equation in the estimation process 、 The observation equation 、 The statistical characteristics of system noise and observation noise , Because the signal information used is time domain , So not only can   Estimate one-dimensional stationary random process , You can also estimate multidimensional 、 Nonstationary stochastic process . This overcomes the problem of   In the frequency domain Wiener Various limitations encountered in filter design , The scope of application is relatively wide .Kalman The theoretical basis of filtering is the estimation problem . From the point of view of signal processing , It is estimated that the problem is as follows   How to recover the signal or approximate the signal or deduce a set of parameter values from the observation polluted by noise . In Mathematics   On , The estimation problem belongs to the category of statistics . In the estimation problem , Observation space is the only source of information , Only   Observations can provide the information needed for estimation [5]17-24. The problem is that these observations contain errors . spread   error 、 Equipment error 、 Calculation error and human error are the main sources of these errors [6]. Under evaluation   In question , Error mainly refers to random error . therefore , The estimation problem can be summarized as ： At a given observation  z  love

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2 Simulation code

<span style="color:#333333"><span style="background-color:rgba(0, 0, 0, 0.03)"><code>addpath('./filters');</code><code>addpath('./IP_raytracing');</code><code>%%  Simulate a trajectory , Then add Gaussian observation noise , As the track of observation position . Then use Kalman filter to get the filtered result .</code><code>%  The speed is the average 0.6m Standard deviation 0.05 Gaussian distribution of </code><code>%  The standard deviation of the observed noise is 2</code><code>​</code><code>%%  Draw the actual real path </code><code>roomLength = 1000;</code><code>roomWidth = 1000;</code><code>t = 500;</code><code>trace_real = get_random_trace(roomLength, roomWidth, t);</code><code>figure; </code><code>subplot(1, 3, 1); plot(trace_real(:, 1), trace_real(:, 2), '.');</code><code>title(' The actual real path ');</code><code>​</code><code>%%  Path with observation noise </code><code>noise = 2; %2m Position fluctuation noise </code><code>trace = trace_real + normrnd(0, noise, size(trace_real));</code><code>subplot(1, 3, 2); plot(trace(:, 1), trace(:, 2), '.');</code><code>title(' Path with noise ');</code><code>fprintf(' Positioning accuracy before Kalman filter ：%f m\n', accuracy(trace, trace_real));</code><code>​</code><code>%%  Kalman filter the noisy path </code><code>kf_params_record = zeros(size(trace, 1), 4);</code><code>for i = 1 : t</code><code>    if i == 1</code><code>        kf_params = kf_init(trace(i, 1), trace(i, 2), 0, 0); %  initialization </code><code>    <span style="color:#ca7d37">else</span></code><code>        kf_params.z = trace(i, 1:2)'; % Set the observation position at the current time </code><code>        kf_params = kf_update(kf_params); %  Kalman filtering </code><code>    end</code><code>    kf_params_record(i, :) = kf_params.x';</code><code>end</code><code>kf_trace = kf_params_record(:, 1:2);</code><code>subplot(1, 3, 3); plot(kf_trace(:, 1), kf_trace(:, 2), '.');</code><code>title(' The effect of Kalman filter ');</code><code>fprintf(' Positioning accuracy after Kalman filter ：%f m\n', accuracy(kf_trace, trace_real));</code><code>​</code></span></span>

3 Running results

4 reference

[1] Hou Yapei . Research on EEG signal denoising method based on Kalman filter [D]. Yanshan University .

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