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
Famous quotes containing the words signal, distortion, noise and/or reduction:
“The experience of a sense of guilt for wrong-doing is necessary for the development of self-control. The guilt feelings will later serve as a warning signal which the child can produce himself when an impulse to repeat the naughty act comes over him. When the child can produce his on warning signals, independent of the actual presence of the adult, he is on the way to developing a conscience.”
—Selma H. Fraiberg (20th century)
“A man’s memory is bound to be a distortion of his past in accordance with his present interests, and the most faithful autobiography is likely to mirror less what a man was than what he has become.”
—Fawn M. Brodie (1915–1981)
“Soon, with a noise like tambourines,
Came her attendant Byzantines.”
—Wallace Stevens (1879–1955)
“The reduction of nuclear arsenals and the removal of the threat of worldwide nuclear destruction is a measure, in my judgment, of the power and strength of a great nation.”
—Jimmy Carter (James Earl Carter, Jr.)