Data smoothing Raymond Cuijpers

Index The moving average Convolution The difference operator Fourier transforms Gaussian smoothing Butterworth filters

The moving average Let (S i ) be a data set S i,smooth = 1/3 (S i-1 + S i + S i+1 ) =

Convolution Definition: The moving average is the same as convoluting the signal with a block function !

Convolution f * g = g * f f * (g + h) = f * g + f * h (f * g) * h = f * (g * h) f * 0 = 0 But 1 * g ≠ g

The difference operator The velocity is the derivative of the displace- ment, for discrete signals this becomes the difference operator. Discrete differentiation = convolution with difference operator Velocity estimated at i+1/2

The difference operator The difference operator amplifies noise Smoothing helps but at the cost of accuracy It becomes worse for higher order derivatives * =

Differentiation and convolution Noisy = BAD Exact = GOOD Differentiation by convolution with derivative

Fourier Transforms Definition: In the Fourier domain: convolution becomes multiplication differentiation becomes multiplication with iw

Fourier Transforms Calculating convolutions using Fourier transforms is much faster for large data sets than direct computation: There are many other transforms/expansions –Sine and Cosine transforms –Laplace transform –Legendre polynomials –Hermite polynomials (=Gaussian) –Bessel Functions –…

Gaussian smoothing Let S(t) be a signal then the blurred signal is Where is the Gaussian kernel The derivative of a noisy S(t) is ill-posed, but

Gaussian smoothing The n-th order proper derivative of scale s is So in the discrete case we get

Gaussian Smoothing Gaussian filters are the only 'natural' filters Together they form a linear Scale selective space of operators * =

Butterworth Filters Noise is usually high frequent and not the signal Butterworth filters work by throwing away high frequencies in the Fourier transform A good choice of cut-off frequency is paramount

Butterworth filters Advantage: Easy to implement in electronic circuit Disadvantage: Introduces a phase shift. Solution is to apply it twice in opposite directions 'Ringing': jumps in the signal produces oscillations Depends strongly on the nature of the noise and the choice of cut-off frequency