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:
“Change begets change. Nothing propagates so fast. If a man habituated to a narrow circle of cares and pleasures, out of which he seldom travels, step beyond it, though for never so brief a space, his departure from the monotonous scene on which he has been an actor of importance would seem to be the signal for instant confusion.... The mine which Time has slowly dug beneath familiar objects is sprung in an instant; and what was rock before, becomes but sand and dust.”
—Charles Dickens (1812–1870)
“This is our fate: eight hundred years’ disaster,
crazily tangled like the Book of Kells:
the dream’s distortion and the land’s division,
the midnight raiders and the prison cells.”
—John Hewitt (b. 1907)
“Soon silence will have passed into legend. Man has turned his back on silence. Day after day he invents machines and devices that increase noise and distract humanity from the essence of life, contemplation, meditation.... Tooting, howling, screeching, booming, crashing, whistling, grinding, and trilling bolster his ego. His anxiety subsides. His inhuman void spreads monstrously like a gray vegetation.”
—Jean Arp (1887–1948)
“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.)