| fftderiv - derivative of a vector using FFT |
[Dy, DY] = fftderiv(y, [n , sigma, delta, in, out]) [Dy, DY] = fftderiv(y, <named_args>) |
| y |
| vector containing a periodic window of a function to be differentiated. |
| n |
| the order of the derivative. It is 1 for 1st derivative, 2 for 2nd derivative, and so on. |
| sigma |
| the standard deviation of the gaussian kernel used to smooth the input. If sigma is zero, fftderiv will not smooth the input. (Defaults to 5) |
| delta |
| a double number. If the input is in the time domain, this is the time between samples (delta t), and defaults to 1. If the input is in the frequency domain, this is the frequency increment between samples (delta f), and defaults to 1/N, where N is the number of samples. |
| in |
| indicates if the input, x, is a function of time (no FFT has been applied) or frequency (FFT has already been applied). Can be 'time' or 'frequency'. (Defaults to 'time') |
| out |
| indicates if the output, xsm, is a function of time (inverse FFT will be applied) or frequency (inverse FFT will not be applied). Can be 'time' or 'frequency'. (Defaults to 'time') |
| <named_args> |
| This is a sequence of statements key1=value1, key2=value2,... where key1, key2,... can be any of the optional arguments above, in any order. |
| Dy |
| the derivative vector in "time" or "frequency" domain. |
| FDy |
| the derivative vector in "frequency" domain. |
| Function fftderiv performs the n-th derivative of a periodic function, stored in a vector, using FFT. The optional arguments in and out enables the user to reuse previously done FFTs. Here are some possible uses of gsm: |
| Dy = fftderiv(y) |
| n defaults to 1, sigma defaults to 5, in and out both defaults to 'time'. |
| Dy = fftderiv(y,2,3) |
| n equals 2, sigma equals 3, in and out both defaults to 'time'. |
| Dy = fftderiv(y,sigma=3, in='frequency',out='frequency') |
| n dafaults to 1, y in frequency domain (fft has already been done). Dy in frequency domain (inverse fft is NOT done by gsm) |
| In all above examples, FDy is in the frequency domain. It is the second output parameter, and thus it was discarded in the above examples. |
step = 2*%pi/100; y = sin(step:step:2*%pi); // from 2pi/100 to 2pi xbasc() plot(y); // 1st derivative, a sigma of 3 steps to the left and to the right d = fftderiv(y,1,3*step,step); xbasc() plot(d) // a cosine period |
| For a derivative without noises, the vector y must be an exact period of a continuous periodic function, i.e., its repetition has to be continuous. A direct way for checking this is to plot z = [y y] and look close in the middle. If there is not a minimal discontinuity, then fftderiv will certainly work without need for smoothing. |
| y should be smoothed before using fftderiv so the derivative is less sensitive to discontinuities and aliasing. For an estimation of the sigma parameter, please refer to the references below. |
| "Shape Analysis and Classification", L. da F. Costa and R. M Cesar Jr., CRC Press, pp. 335-347. |
| "1D and 2D Fourier-based approaches to numeric curvature estimation and their comparative performance assessment", L. F. Estrozi, L. G. R. Filho, A. G. Campos and L. da F. Costa, Digital Signal Processing, 2002, accepted paper. |
| Ricardo Fabbri <rfabbri@if.sc.usp.br> |
| The latest version of the Scilab Image Processing toolbox can be found at |
| http://siptoolbox.sourceforge.net |
| follow, gsm, fftshift, curvature |