[Filter Special Topic-Part 4] Evaluation indicators of filter filtering effect (signal-to-noise ratio SNR, mean square error MSE, waveform similarity parameter NCC)

The previous two articles talked about the two most basic theories of filtering algorithms, FIR and IIR . This article will talk about the evaluation indicators and usage of the filtering effect. Evaluation indicators are mainly used for quantitative evaluation of filtering effects, which are often used in papers.

1. Signal-to-noise ratio SNR

Signal-to- noise ratio (English: Signal-to-noise ratio , abbreviated as SNR or S/N ), refers to the ratio of signal power (Power of Signal) to noise power (Power of Noise), and is also the ratio of the square of amplitude (Amplitude) :

$SNR=\frac{P_{signal}}{P_{noise}}=\frac{A^{2}_{signal}}{A^{2}_{noise}}$

Its unit generally uses decibels, and its value is ten times the logarithmic signal-to-noise power ratio:

$SNR(dB)=10log_{10}(\frac{P_{signal}}{P_{noise}})=20log_{10}(\frac{A_{signal}}{A_{noise}})$

The signal-to-noise ratio is the most intuitive quantity to measure the degree of noise reduction. The larger the signal-to-noise ratio, the less noise the signal contains and the better the noise reduction effect.

Special emphasis should be placed here:

The above-mentioned signal ( $P_{signal}$ or $A_{signal}$ ) refers to the "useful signal" that is ready to calculate the signal-to-noise ratio, and the useful signal here usually refers to the "pure signal";

$P_{noise}$ The noise ( or ) mentioned above $A_{noise}$ refers to pure noise, and this "pure noise" refers to the "filtered signal" minus the "pure signal". Note: [Using the filtered signal to subtract the pre-filtered signal as noise is a big mistake! 】

This brings up a question:

In most cases, if you want to ask for the signal-to-noise ratio of a noisy signal, you need to know the "clean signal" in advance.

Therefore, in many cases, the calculation of the SNR needs to cooperate with the simulation signal. Because only the simulated signal can accurately know the pure signal.

2. Waveform similarity parameter NCC

The waveform similarity parameter (Normalized Correlation Coefficient) [1] is to be distinguished from the correlation coefficient. The waveform similarity coefficient reflects the overall similarity of the signal waveform before and after denoising, and cannot represent the details of the waveform oscillation change. Its calculation formula is:

$NCC=\frac{\sum_{n=1}^{N}{As(n)Ad(n)}}{\sqrt{(\sum_{n=1}^{N}{As^2(n)})(\sum_{n=1}^{N}{Ad^2(n)})}}$

where As refers to the "clean signal" and Ad refers to the filtered signal.

3. Mean square error MSE

The mean square error is a measure that reflects the degree of difference between the estimator and the estimated quantity.

Its physical meaning is easy to understand, which is "the difference between the filtered signal and the pure signal, and the average of the sum of squares".

Expressed by the formula is:

$MSE=\frac{\sum_{i=1}^{N}{\left| A_{signal}-A_{denoised} \right|^2}}{N}$

where $A_{signal}$ "clean signal" $A_{denoised}$ refers to the filtered signal.

4. About the use of the above indicators

Seeing this, you may have discovered that the above-mentioned SNR, NCC and MSE indicators all need to know the " pure signal " to calculate.

The perfect " pure signal " can only be obtained in simulation, that is to say, use MATLAB or other software to generate a section of pure signal $A_{signal}$ , and then add noise $A_{noise}$ to get a "polluted" signal, which is filtered after using some kind of filtering and noise reduction method As a result $A_{denoised}$ , it is enough to calculate the above-mentioned indicators at this time as long as the corresponding values are brought in.

Therefore, the practice of many papers is that in order to verify the effectiveness of a certain filtering algorithm, the simulation signal is first used for experiments, which may involve horizontal comparison of indicators obtained by several different filtering algorithms. It shows that after the current filtering algorithm is optimal, and then use this algorithm to filter a real signal, the above three indicators can no longer be calculated at this time, and the filtering effect can only be analyzed through subjective research and judgment.

Of course, for some fields, it may be possible to obtain "pure signal" approximately by controlling external noise in physical experiments. At this time, these indicators can also be used for measurement.

V. Horizontal comparison of indicators

The paper often involves the horizontal comparison of evaluation indicators. Here, we especially emphasize which scenes can be compared and which ones cannot. So as not to cause the illusion of "why my calculation is so much worse than others".

Comparable scenario 1: For the same data, different algorithms are used for processing, and the obtained filtering results can be compared with indicators to judge the pros and cons of different algorithms. This situation is the most common and easiest to understand, so I won't explain too much.

Scenario 2 that can be compared: For similar data of the same research object, different algorithms are used for processing, and limited comparison can be made. For example, if the same type of gearbox is also sampled from different individuals, the filtered indicators can be compared in order of magnitude. Unless the filtering effect is extremely different, it cannot directly explain the advantages and disadvantages of the filtering algorithm, so it is "Limited Contrast".

Scenarios that cannot be compared: Different research objects, different types of data, and filtering indicators cannot be compared horizontally. Special attention should be paid to this. Some students reproduced the filtering algorithm of a reference paper, and then applied it to their own data. They found that the evaluation index was several orders of magnitude worse (especially for MSE). They thought it was the recurrence of the algorithm. Problem - Actually, it is not. The research objects are different, especially when the data of the object of analysis is not in the same order of magnitude, the comparison of filter indicators can be said to be meaningless, so don't spend time entangled in this point!

end. About the topic of filtering

Current plans include:

-1. Theoretical explanation of FIR finite impulse response and IIR infinite impulse response digital filter algorithm, filter design method and MATLAB code implementation -2
. Evaluation index of filter filtering effect
-3. "EMD-like" method (that is, including EMD, EEMD, CEEMD, VMD and a series of methods) filter algorithm explanation and implementation
-4. "EMD-like" method combined with ICA filter algorithm explanation and implementation
-5. Wavelet threshold filtering method explanation and implementation
-6. Kalman Explanation and implementation of filtering methods
... (Other methods are added at any time)

refer to

^ A block-based adaptive complex threshold algorithm for extracting PD signals based on complex wavelet transform