Signal Distortion and Noise Reduction
It is inevitable that the signal will be distorted in the convolution process. From property 4 above, when data which which has a peak is smoothed the peak height will be reduced and the half-width will be increased. Both the extent of the distortion and S/N (signal-to-noise ratio) improvement:
- decrease as the degree of the polynomial increases
- increase as the width, m of the convolution function increases
For example, If the noise in all data points is uncorrelated and has a constant standard deviation, σ, the standard deviation on the noise will be decreased by convolution with an m-point smoothing function to
- linear polynomial:
- quadratic polynomial: .
Thus, with a 9-point linear function (moving average) two thirds of the noise is removed and with a 9-point quadratic smoothing function only about half the noise is removed. To remove two thirds of the noise with a quadratic smoothing function 21 points will be needed.
Although the moving average (polynomial order 0 or 1) gives the best noise reduction it is unsuitable for smoothing data which has curvature over m points. A quadratic filter is unsuitable for getting a derivative of a data curve with an inflection point because a quadratic polynomial does not have one. The optimal choice of polynomial order and number of convolution coefficients will be a compromise between noise reduction and distortion.
Read more about this topic: Numerical Smoothing And Differentiation
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