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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
children
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 % FPSDER performs the numeric time derivative
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Daniele Nicolodi <nicolodi@science.unitn.it>
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2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Daniele Nicolodi <nicolodi@science.unitn.it>
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3 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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4 % DESCRIPTION: FPSDER (Five Points Stencil Derivative) performs the numeric
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Daniele Nicolodi <nicolodi@science.unitn.it>
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5 % time derivative using the method of five points stencil.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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6 % The function can perform first, second, third and fourth derivetive
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Daniele Nicolodi <nicolodi@science.unitn.it>
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7 % of a series of input data.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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8 % Given a discrete series of data points, the five-point-stencil method for
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Daniele Nicolodi <nicolodi@science.unitn.it>
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9 % the derivative approximation, at a given time t0, is calculated by means
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Daniele Nicolodi <nicolodi@science.unitn.it>
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10 % of finite differences between the element at t0 with its four neighbors.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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11 % The n-order derivative at a certain time can be approximated by a five
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Daniele Nicolodi <nicolodi@science.unitn.it>
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12 % point difference equation:
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13 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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14 % d^{n}y[k]
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15 % --------- = (1/T^n) * {a*y[k-2] + b*y[k-1] + c*y[k] + d*y[k+1] + e*y[k+2]}
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16 % dt^{n}
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17 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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18 % It can be demonstrated [1,2] that the five coefficients [a, b, c, d, e] can
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Daniele Nicolodi <nicolodi@science.unitn.it>
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19 % be written in terms of only one of them. In fpsder the independent
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Daniele Nicolodi <nicolodi@science.unitn.it>
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20 % coefficient is fixed to be the first and is called m. It can be input as
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Daniele Nicolodi <nicolodi@science.unitn.it>
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21 % a parameter when the function is called.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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22 %
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23 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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24 % CALL: Deriv = fpsder(data, params)
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25 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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26 % INPUTS:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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27 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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28 % - a is a vector containing the data to be differentiated.
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29 % - params is a struct with the input parameters:
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30 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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31 % - 'ORDER' set the derivative order. Its allowed options are:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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32 % - 'ZERO' perform data smoothing using the couefficients
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Daniele Nicolodi <nicolodi@science.unitn.it>
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33 % vector d0 = [m -4*m 1+6*m -4*m m].
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Daniele Nicolodi <nicolodi@science.unitn.it>
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34 % - 'FIRST' perform the first derivative using the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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35 % couefficients vector d1 = [m -(0.5+2*m) 0 (0.5+2*m) m]./T.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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36 % Recomended values of m are in the interval [-0.1, 0.1].
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Daniele Nicolodi <nicolodi@science.unitn.it>
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37 % - 'SECOND' perform the second derivative using the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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38 % coefficients vector d2 = [m 1-4*m 6*m-2 1-4*m m]./(T^2).
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Daniele Nicolodi <nicolodi@science.unitn.it>
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39 % Recomended values of m are in the interval [-0.11, 0.3].
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Daniele Nicolodi <nicolodi@science.unitn.it>
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40 % - 'THIRD' perform the third derivative using the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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41 % coefficients vector d3 = []./(T^3)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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42 % - 'FOURTH' perform the third derivative using the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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43 % coefficients vector d4 = []./(T^4)
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44 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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45 % - 'COEFF' set m coefficient values.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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46 % In case of data smoothing: m = -3/35 correspond to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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47 % parabolic fit approximation.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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48 % In case of first order derivative: m = -1/5 correspond to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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49 % parabolic fit approximation, m = 1/12 correspond to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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50 % Taylor series approximation.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 % In case of second order derivative: m = 2/7 corresponds to
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Daniele Nicolodi <nicolodi@science.unitn.it>
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52 % the parabolic fit approximation, m = -1/12 corresponds to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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53 % Taylor series approximation and m = 1/4 gives the notch
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 % feature at the Nyquist frequency
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55 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 % - 'FS' set the data sampling frequency in Hz
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57 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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58 % NOTE1: T is the sampling period
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 % NOTE2: The default option for 'ORDER' is 'SECOND'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 % NOTE3: The default option for 'COEFF' is 2/7
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Daniele Nicolodi <nicolodi@science.unitn.it>
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61 % NOTE4: The default option for 'FS' is 10
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62 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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63 % OUTPUTS:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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64 % - D is a vector containing the resulting data after the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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65 % differentiation procedure
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66 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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67 % REFERENCES:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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68 % [1] L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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Daniele Nicolodi <nicolodi@science.unitn.it>
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69 % estimation in LISA Pathfinder data reduction,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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70 % <a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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71 % href="matlab:web('http://www.iop.org/EJ/abstract/0264-9381/26/9/094013/','-browser')">Class. Quantum Grav. 26 (2009) 094013.</a>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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72 % [2] L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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73 % estimation in LISA Pathfinder data reduction,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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74 % <a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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75 % href="matlab:web('http://arxiv.org/abs/0903.0324v1','-browser')">http://arxiv.org/abs/0903.0324v1</a>
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76 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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77 % EXAMPLES:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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78 % - Performing the second order derivative of a series of data, m
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 % coefficient is fixed to 2/7 and data sampling frequency is
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Daniele Nicolodi <nicolodi@science.unitn.it>
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80 % fixed to 10 Hz.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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81 % params = struct('ORDER', 'SECOND', 'COEFF', 2/7, 'FS', 10);
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82 % Deriv = fpsder(data, params);
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83 %
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84 % VERSION: '$Id: fpsder.m,v 1.8 2010/04/09 09:59:59 mauro Exp $';
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85 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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86 % HISTORY: 18-03-2008 L Ferraioli
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Daniele Nicolodi <nicolodi@science.unitn.it>
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87 % Creation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 %
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89 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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90
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91
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Daniele Nicolodi <nicolodi@science.unitn.it>
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92 function Deriv = fpsder(a, params)
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93
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94 % Getting input parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Daniele Nicolodi <nicolodi@science.unitn.it>
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95 % Collect inputs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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96
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Daniele Nicolodi <nicolodi@science.unitn.it>
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97 % Default input struct
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98 defaultparams = struct('ORDER','SECOND',...
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99 'COEFF',2/7,...
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100 'FS',10);
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101
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102 names = {'ORDER','COEFF','FS'};
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103
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Daniele Nicolodi <nicolodi@science.unitn.it>
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104 % collecting input and default params
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Daniele Nicolodi <nicolodi@science.unitn.it>
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105 if ~isempty(params)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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106 for jj=1:length(names)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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107 if isfield(params, names(jj)) && ~isempty(params.(names{1,jj}))
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Daniele Nicolodi <nicolodi@science.unitn.it>
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108 defaultparams.(names{1,jj}) = params.(names{1,jj});
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109 end
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110 end
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111 end
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112
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113 % values for input variables
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114 order = defaultparams.ORDER;
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115 m = defaultparams.COEFF;
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116 fs = defaultparams.FS;
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117
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118 % willing to work with columns
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Daniele Nicolodi <nicolodi@science.unitn.it>
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119 if size(a,2)>1
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Daniele Nicolodi <nicolodi@science.unitn.it>
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120 a = a.';
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121 end
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122
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123 % Assigning coefficients values %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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124
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125 % Assigning coefficients values based on the input options
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Daniele Nicolodi <nicolodi@science.unitn.it>
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126 switch upper(order)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 case 'ZERO'
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128 Coeffs = [m -4*m 1+6*m -4*m m];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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129 case 'FIRST'
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130 Coeffs = [m -(0.5+2*m) 0 (0.5+2*m) -m];
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131 case 'SECOND'
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132 Coeffs = [m 1-4*m 6*m-2 1-4*m m];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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133 case 'THIRD'
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134 Coeffs = [0 0 0 0 0];
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135 disp('Not yet implemented, sorry!');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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136 case 'FOURTH'
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137 Coeffs = [0 0 0 0 0];
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138 disp('Not yet implemented, sorry!');
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139 otherwise
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140 error('### Unknown order %s', order);
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141 end
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142
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143 % Sampling period
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144 T = 1/fs;
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145
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146 % Building vectors for calculation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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147
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148 % Building the 'extended' vector for calculation
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149 % a_temp = [a(1);a(1);a(1);a(1);a;a(end);a(end);a(end);a(end)];
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150 a_temp = [2*a(1)-a((4+1):-1:2);a;2*a(end)-a((end-1):-1:end-4)];
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151
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152 % Switching between the input options differentiate
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Daniele Nicolodi <nicolodi@science.unitn.it>
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153 switch upper(order)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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154 case 'ZERO'
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diff changeset
155 Deriv = (Coeffs(1)*a_temp(1:end-4) + Coeffs(2)*a_temp(2:end-3) + Coeffs(3)*a_temp(3:end-2) + Coeffs(4)*a_temp(4:end-1) + Coeffs(5)*a_temp(5:end));
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
156 Deriv = Deriv(3:end-2);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
157 case 'FIRST'
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
158 Deriv = (1/T).*(Coeffs(1)*a_temp(1:end-4) + Coeffs(2)*a_temp(2:end-3) + Coeffs(3)*a_temp(3:end-2) + Coeffs(4)*a_temp(4:end-1) + Coeffs(5)*a_temp(5:end));
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
159 Deriv = Deriv(3:end-2);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
160 case 'SECOND'
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
161 Deriv = (1/T^2).*(Coeffs(1)*a_temp(1:end-4) + Coeffs(2)*a_temp(2:end-3) + Coeffs(3)*a_temp(3:end-2) + Coeffs(4)*a_temp(4:end-1) + Coeffs(5)*a_temp(5:end));
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
162 Deriv = Deriv(3:end-2);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
163 case 'THIRD'
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
164 Deriv = (1/T^3).*(Coeffs(1)*a_temp(1:end-4) + Coeffs(2)*a_temp(2:end-3) + Coeffs(3)*a_temp(3:end-2) + Coeffs(4)*a_temp(4:end-1) + Coeffs(5)*a_temp(5:end));
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
165 Deriv = Deriv(3:end-2);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
166 case 'FOURTH'
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
167 Deriv = (1/T^4).*(Coeffs(1)*a_temp(1:end-4) + Coeffs(2)*a_temp(2:end-3) + Coeffs(3)*a_temp(3:end-2) + Coeffs(4)*a_temp(4:end-1) + Coeffs(5)*a_temp(5:end));
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
168 Deriv = Deriv(3:end-2);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
169 otherwise
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
170 error('### Unknown order %s', order);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
171 end
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
172
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
173 end