噪声正弦时间序列中的实时峰值检测 [英] Real-time peak detection in noisy sinusoidal time-series

问题描述

我一直在尝试实时检测正弦时间序列数据中的峰值，但是到目前为止我还没有成功.我似乎无法找到一种实时算法，可以以合理的准确度检测正弦信号中的峰值.我要么没有检测到峰值，要么沿着正弦波得到无数个点被检测为峰值.

对于类似正弦波且可能包含一些随机噪声的输入信号，什么是好的实时算法?

<小时>

作为一个简单的测试用例，考虑一个频率和幅度始终相同的平稳正弦波.(确切的频率和幅度无关紧要；我任意选择了 60 Hz 的频率，幅度为 +/-1 单位，采样率为 8 KS/s.)以下 MATLAB 代码将生成这样的正弦信号:

dt = 1/8000;t = (0:dt:(1-dt)/4)';x = sin(2*pi*60*t);

使用

按照

尽管 Eli Billauer 的算法要简单得多，并且确实倾向于可靠地产生我想要的结果，但它并不适合实时应用程序.

<小时>

作为我想应用这种算法的信号的另一个示例，考虑 Eli Billauer 为他自己的算法给出的测试用例:

t = 0:0.001:10;x = 0.3*sin(t) + sin(1.3*t) + 0.9*sin(4.2*t) + 0.02*randn(1, 10001);

这是一个更不寻常(不太均匀/规则)的信号，具有不同的频率和幅度，但通常仍为正弦波.绘制时，峰值很明显，但很难通过算法识别.

<小时>

什么是正确识别正弦输入信号中的峰值的好的实时算法?在信号处理方面我并不是真正的专家，所以获得一些会有所帮助考虑正弦输入的经验法则.或者，也许我需要修改例如Jean-Paul 的算法本身是为了在正弦信号上正常工作.如果是这种情况，需要进行哪些修改，我将如何进行这些修改?

解决方案

案例1:正弦波没有噪声

如果您的正弦曲线不包含任何噪声，您可以使用非常经典的信号处理技术:取一阶导数并检测它何时等于零.

例如:

函数信号=派生信号(d)% 识别信号信号 = 零(大小(d))；对于 i=2:长度(d)如果 d(i-1) >0&&d(i) <= 0信号(i) = +1;% 检测到的峰否则 d(i-1) <0&&d(i) >= 0信号(i) = -1;% 低谷检测到结尾结尾结尾

使用您的示例数据:

% 生成数据dt = 1/8000；t = (0:dt:(1-dt)/4)';y = sin(2*pi*60*t);% 添加一些趋势y(1:1000) = y(1:1000) + 0.001*(1:1000)';y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';% 近似一阶导数 (delta y/delta x)d = [0;差异(y)]；% 识别信号信号=派生信号(d)；% 绘图结果图1);clf;设置(gcf，'位置'，[0 0 677 600])子图(4,1,1)；坚持，稍等;标题(数据")；情节(t，y)；子图(4,1,2)；坚持，稍等;title('一阶导数');(d)区；ylim([-0.05, 0.05]);子图(4,1,3)；坚持，稍等;title('信号(-1 表示波谷，+1 表示波峰)');绘图(t，信号)；ylim([-1.5 1.5]);子图(4,1,4)；坚持，稍等;title('数据上标记的信号');标记=绝对(信号)>0;情节(t，y)；scatter(t(markers),y(markers),30,'or','MarkerFaceColor','red');

这会产生:

这种方法对任何类型的正弦曲线都非常有效，唯一的要求是输入信号不包含噪声.

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案例2:正弦有噪声

一旦您的输入信号包含噪声，导数方法就会失败.例如:

% 生成数据dt = 1/8000；t = (0:dt:(1-dt)/4)';y = sin(2*pi*60*t);% 添加一些趋势y(1:1000) = y(1:1000) + 0.001*(1:1000)';y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';% 添加一些噪音y = y + 0.2.*randn(2000,1);

现在将生成此结果，因为

现在有很多方法可以处理噪声，最标准的方法是应用

这种方法的主要优点是:

• 您拥有数据的实际模型，因此您可以在信号发生之前预测未来的信号！(例如修复模型并通过输入未来时间段来计算结果)
• 你不需要每个周期估计模型(见代码中的参数interval)

缺点是需要选择回溯window，但是任何用于实时检测的方法都会有这个问题.

视频演示

Data 是输入数据，Model fit 是拟合到数据的正弦波(见代码)，Signal 表示峰值和波谷和 数据上标记的信号 给人以算法准确度的印象.注意:观察模型拟合调整到图表中间的趋势！

这应该让你开始.还有很多关于信号检测理论的优秀书籍(只需 google 那个术语)，它们将更深入地介绍这些类型的技术.祝你好运！

I have been attempting to detect peaks in sinusoidal time-series data in real time, however I've had no success thus far. I cannot seem to find a real-time algorithm that works to detect peaks in sinusoidal signals with a reasonable level of accuracy. I either get no peaks detected, or I get a zillion points along the sine wave being detected as peaks.

What is a good real-time algorithm for input signals that resemble a sine wave, and may contain some random noise?

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As a simple test case, consider a stationary, sine wave that is always the same frequency and amplitude. (The exact frequency and amplitude don't matter; I have arbitrarily chosen a frequency of 60 Hz, an amplitude of +/− 1 unit, at a sampling rate of 8 KS/s.) The following MATLAB code will generate such a sinusoidal signal:

dt = 1/8000;
t  = (0:dt:(1-dt)/4)';
x  = sin(2*pi*60*t);

Using the algorithm developed and published by Jean-Paul, I either get no peaks detected (left) or a zillion "peaks" detected (right):

I've tried just about every combination of values for these 3 parameters that I could think of, following the "rules of thumb" that Jean-Paul gives, but I have so far been unable to get my expected result.

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I found an alternative algorithm, developed and published by Eli Billauer, that does give me the results that I want—e.g.:

Even though Eli Billauer's algorithm is much simpler and does tend to reliably produce the results that I want, it is not suitable for real-time applications.

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As another example of a signal that I'd like to apply such an algorithm to, consider the test case given by Eli Billauer for his own algorithm:

t = 0:0.001:10;
x = 0.3*sin(t) + sin(1.3*t) + 0.9*sin(4.2*t) + 0.02*randn(1, 10001);

This is a more unusual (less uniform/regular) signal, with a varying frequency and amplitude, but still generally sinusoidal. The peaks are plainly obvious to the eye when plotted, but hard to identify with an algorithm.

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What is a good real-time algorithm to correctly identify the peaks in a sinusoidal input signal? I am not really an expert when it comes to signal processing, so it would be helpful to get some rules of thumb that consider sinusoidal inputs. Or, perhaps I need to modify e.g. Jean-Paul's algorithm itself in order to work properly on sinusoidal signals. If that's the case, what modifications would be required, and how would I go about making these?

解决方案

Case 1: sinusoid without noise

If your sinusoid does not contain any noise, you can use a very classic signal processing technique: taking the first derivative and detecting when it is equal to zero.

For example:

function signal = derivesignal( d )

% Identify signal
signal = zeros(size(d));
for i=2:length(d)
    if d(i-1) > 0 && d(i) <= 0
        signal(i) = +1;     % peak detected
    elseif d(i-1) < 0 && d(i) >= 0
        signal(i) = -1;     % trough detected
    end
end

end

Using your example data:

% Generate data
dt = 1/8000;
t  = (0:dt:(1-dt)/4)';
y  = sin(2*pi*60*t);

% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';

% Approximate first derivative (delta y / delta x)
d = [0; diff(y)];

% Identify signal
signal = derivesignal(d);

% Plot result
figure(1); clf; set(gcf,'Position',[0 0 677 600])
subplot(4,1,1); hold on;
title('Data');
plot(t,y);
subplot(4,1,2); hold on;
title('First derivative');
area(d);
ylim([-0.05, 0.05]);
subplot(4,1,3); hold on;
title('Signal (-1 for trough, +1 for peak)');
plot(t,signal); ylim([-1.5 1.5]);
subplot(4,1,4); hold on;
title('Signals marked on data');
markers = abs(signal) > 0;
plot(t,y); scatter(t(markers),y(markers),30,'or','MarkerFaceColor','red');

This yields:

This method will work extremely well for any type of sinusoid, with the only requirement that the input signal contains no noise.

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Case 2: sinusoid with noise

As soon as your input signal contains noise, the derivative method will fail. For example:

% Generate data
dt = 1/8000;
t  = (0:dt:(1-dt)/4)';
y  = sin(2*pi*60*t);

% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';

% Add some noise
y = y + 0.2.*randn(2000,1);

Will now generate this result because first differences amplify noise:

Now there are many ways to deal with noise, and the most standard way is to apply a moving average filter. One disadvantage of moving averages is that they are slow to adapt to new information, such that signals may be identified after they have occurred (moving averages have a lag).

Another very typical approach is to use Fourier Analysis to identify all the frequencies in your input data, disregard all low-amplitude and high-frequency sinusoids, and use the remaining sinusoid as a filter. The remaining sinusoid will be (largely) cleansed from the noise and you can then use first-differencing again to determine the peaks and troughs (or for a single sine wave you know the peaks and troughs happen at 1/4 and 3/4 pi of the phase). I suggest you pick up any signal processing theory book to learn more about this technique. Matlab also has some educational material about this.

If you want to use this algorithm in hardware, I would suggest you also take a look at WFLC (Weighted Fourier Linear Combiner) with e.g. 1 oscillator or PLL (Phase-Locked Loop) that can estimate the phase of a noisy wave without doing a full Fast Fourier Transform. You can find a Matlab algorithm for a phase-locked loop on Wikipedia.

I will suggest a slightly more sophisticated approach here that will identify the peaks and troughs in real-time: fitting a sine wave function to your data using moving least squares minimization with initial estimates from Fourier analysis.

Here is my function to do that:

function [result, peaks, troughs] = fitsine(y, t, eps)

% Fast fourier-transform
f = fft(y);
l = length(y);
p2 = abs(f/l);
p1 = p2(1:ceil(l/2+1));
p1(2:end-1) = 2*p1(2:end-1);
freq = (1/mean(diff(t)))*(0:ceil(l/2))/l;

% Find maximum amplitude and frequency
maxPeak = p1 == max(p1(2:end)); % disregard 0 frequency!
maxAmplitude = p1(maxPeak);     % find maximum amplitude
maxFrequency = freq(maxPeak);   % find maximum frequency

% Initialize guesses
p = [];
p(1) = mean(y);         % vertical shift
p(2) = maxAmplitude;    % amplitude estimate
p(3) = maxFrequency;    % phase estimate
p(4) = 0;               % phase shift (no guess)
p(5) = 0;               % trend (no guess)

% Create model
f = @(p) p(1) + p(2)*sin( p(3)*2*pi*t+p(4) ) + p(5)*t;
ferror = @(p) sum((f(p) - y).^2);
% Nonlinear least squares
% If you have the Optimization toolbox, use [lsqcurvefit] instead!
options = optimset('MaxFunEvals',50000,'MaxIter',50000,'TolFun',1e-25);
[param,fval,exitflag,output] = fminsearch(ferror,p,options);

% Calculate result
result = f(param);

% Find peaks
peaks = abs(sin(param(3)*2*pi*t+param(4)) - 1) < eps;

% Find troughs
troughs = abs(sin(param(3)*2*pi*t+param(4)) + 1) < eps;

end

As you can see, I first perform a Fourier transform to find initial estimates of the amplitude and frequency of the data. I then fit a sinusoid to the data using the model a + b sin(ct + d) + et. The fitted values represent a sine wave of which I know that +1 and -1 are the peaks and troughs, respectively. I can therefore identify these values as the signals.

This works very well for sinusoids with (slowly changing) trends and general (white) noise:

% Generate data
dt = 1/8000;
t  = (0:dt:(1-dt)/4)';
y  = sin(2*pi*60*t);

% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';

% Add some noise
y = y + 0.2.*randn(2000,1);

% Loop through data (moving window) and fit sine wave
window = 250;   % How many data points to consider
interval = 10;  % How often to estimate
result = nan(size(y));
signal = zeros(size(y));
for i = window+1:interval:length(y)
    data = y(i-window:i);   % Get data window
    period = t(i-window:i); % Get time window
    [output, peaks, troughs] = fitsine(data,period,0.01);
    
    result(i-interval:i) = output(end-interval:end);
    signal(i-interval:i) = peaks(end-interval:end) - troughs(end-interval:end);
end

% Plot result
figure(1); clf; set(gcf,'Position',[0 0 677 600])
subplot(4,1,1); hold on;
title('Data');
plot(t,y); xlim([0 max(t)]); ylim([-4 4]);
subplot(4,1,2); hold on;
title('Model fit');
plot(t,result,'-k'); xlim([0 max(t)]); ylim([-4 4]);
subplot(4,1,3); hold on;
title('Signal (-1 for trough, +1 for peak)');
plot(t,signal,'r','LineWidth',2); ylim([-1.5 1.5]);
subplot(4,1,4); hold on;
title('Signals marked on data');
markers = abs(signal) > 0;
plot(t,y,'-','Color',[0.1 0.1 0.1]);
scatter(t(markers),result(markers),30,'or','MarkerFaceColor','red');
xlim([0 max(t)]); ylim([-4 4]);

Main advantages of this approach are:

• You have an actual model of your data, so you can predict signals in the future before they happen! (e.g. fix the model and calculate the result by inputting future time periods)
• You don't need to estimate the model every period (see parameter interval in the code)

The disadvantage is that you need to select a lookback window, but you will have this problem with any method that you use for real-time detection.

Video demonstration

Data is the input data, Model fit is the fitted sine wave to the data (see code), Signal indicates the peaks and troughs and Signals marked on data gives an impression of how accurate the algorithm is. Note: watch the model fit adjust itself to the trend in the middle of the graph!

That should get you started. There are also a lot of excellent books on signal detection theory (just google that term), which will go much further into these types of techniques. Good luck!

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