LANCZOSFILTER Filters a time series via Lanczos method (cosine filter).
[Y,coef,window,Cx,Ff] = LANCZOSFILTER(X,dT,Cf,M,pass) Filters the time
series via the Lanczos filter in the frequency space (FFT), where
INPUTS:
X - Time series
dT - Sampling interval (default: 1)
Cf - Cut-off frequency (default: half Nyquist)
M - Number of coefficients (default: 10)
pass - Low or high-pass filter (default: 'low')
OUTPUTS:
Y - Filtered time series
coef - Coefficients of the time window (cosine)
window - Frequency window (aprox. ones for Ff lower(greater) than Fc
if low(high)-pass filter and ceros otherwise)
Cx - Complex Fourier Transform of X for Ff frequencies
Ff - Fourier frequencies, from 0 to the Nyquist frequency.
The program removes from the time series the frequencies greater than
the cut off frequency if "pass" is 'low', i.e., low-pass filter .
Otherwise, if pass is 'high', frequencies from zero to Cf are removed,
i.e., a high-pass filter. Units of the cut-off frequency, [Cf], must be
[dT]^{-1}.
In consequence, when used as a low-pass the time series is smoothed
like a cosine filter in time space with M coefficients where greater is
better (see the reference).
If any option is empty, defaults are used.
Note: NaN's elements are replaced by the mean of the time series and
ignored. If you have a better idea, just let me know.
Reference:
Emery, W. J. and R. E. Thomson. "Data Analysis Methods in Physical
Oceanography". Elsevier, 2d ed., 2004. On pages 533-539.
Example:
dT = 30; % min
N = 7*24*60/dT;
t = (0:N-1)*dT; % data for 7 days
pnoise = 0.30;
T1 = 12.4*60;
T2 = 24*60;
T3 = 15*24*60;
Tc = 10*60; % min
xn = 5 + 3*cos(2*pi*t/T1) + 2*cos(2*pi*t/T2) + 1*cos(2*pi*t/T3);
xn = xn + pnoise*max(xn-mean(xn))*(0.5 - rand(size(xn)));
[xs,c,h,Cx,f] = lanczosfilter(xn,dT,1/Tc,[],'low');
subplot(211), plot(t,xn,t,xs), legend('noisy','smooth'), axis tight
subplot(212), plot(f,h,f,abs(Cx)/max(abs(Cx)),...
[1 1]/Tc,[min(h) max(h)],'-.',...
[1/T1 1/T2 1/T3],([1/T1 1/T2 1/T3]<=1/Tc),'o'), axis tight
See also FILTER, FFT, IFFT0001 function [y,coef,window,Cx,Ff] = lanczosfilter(x,varargin) 0002 %LANCZOSFILTER Filters a time series via Lanczos method (cosine filter). 0003 % [Y,coef,window,Cx,Ff] = LANCZOSFILTER(X,dT,Cf,M,pass) Filters the time 0004 % series via the Lanczos filter in the frequency space (FFT), where 0005 % 0006 % INPUTS: 0007 % X - Time series 0008 % dT - Sampling interval (default: 1) 0009 % Cf - Cut-off frequency (default: half Nyquist) 0010 % M - Number of coefficients (default: 10) 0011 % pass - Low or high-pass filter (default: 'low') 0012 % 0013 % OUTPUTS: 0014 % Y - Filtered time series 0015 % coef - Coefficients of the time window (cosine) 0016 % window - Frequency window (aprox. ones for Ff lower(greater) than Fc 0017 % if low(high)-pass filter and ceros otherwise) 0018 % Cx - Complex Fourier Transform of X for Ff frequencies 0019 % Ff - Fourier frequencies, from 0 to the Nyquist frequency. 0020 % 0021 % The program removes from the time series the frequencies greater than 0022 % the cut off frequency if "pass" is 'low', i.e., low-pass filter . 0023 % Otherwise, if pass is 'high', frequencies from zero to Cf are removed, 0024 % i.e., a high-pass filter. Units of the cut-off frequency, [Cf], must be 0025 % [dT]^{-1}. 0026 % 0027 % In consequence, when used as a low-pass the time series is smoothed 0028 % like a cosine filter in time space with M coefficients where greater is 0029 % better (see the reference). 0030 % 0031 % If any option is empty, defaults are used. 0032 % 0033 % Note: NaN's elements are replaced by the mean of the time series and 0034 % ignored. If you have a better idea, just let me know. 0035 % 0036 % Reference: 0037 % Emery, W. J. and R. E. Thomson. "Data Analysis Methods in Physical 0038 % Oceanography". Elsevier, 2d ed., 2004. On pages 533-539. 0039 % 0040 % Example: 0041 % dT = 30; % min 0042 % N = 7*24*60/dT; 0043 % t = (0:N-1)*dT; % data for 7 days 0044 % pnoise = 0.30; 0045 % T1 = 12.4*60; 0046 % T2 = 24*60; 0047 % T3 = 15*24*60; 0048 % Tc = 10*60; % min 0049 % xn = 5 + 3*cos(2*pi*t/T1) + 2*cos(2*pi*t/T2) + 1*cos(2*pi*t/T3); 0050 % xn = xn + pnoise*max(xn-mean(xn))*(0.5 - rand(size(xn))); 0051 % [xs,c,h,Cx,f] = lanczosfilter(xn,dT,1/Tc,[],'low'); 0052 % subplot(211), plot(t,xn,t,xs), legend('noisy','smooth'), axis tight 0053 % subplot(212), plot(f,h,f,abs(Cx)/max(abs(Cx)),... 0054 % [1 1]/Tc,[min(h) max(h)],'-.',... 0055 % [1/T1 1/T2 1/T3],([1/T1 1/T2 1/T3]<=1/Tc),'o'), axis tight 0056 % 0057 % See also FILTER, FFT, IFFT 0058 0059 % Written by 0060 % Lic. on Physics Carlos Adrián Vargas Aguilera 0061 % Physical Oceanography MS candidate 0062 % UNIVERSIDAD DE GUADALAJARA 0063 % Mexico, 2004 0064 % 0065 % nubeobscura@hotmail.com 0066 0067 % Check arguments: 0068 if nargin<1 || nargin>5 0069 error('Lanczosfilter:ArgumentNumber','Incorrect number of arguments.') 0070 elseif ~isvector(x) || ~isreal(x) 0071 error('Lanczosfilter:ArgumentType','Incorrect time series.') 0072 end 0073 if nargin<2 || isempty(varargin{1}) 0074 dT = 1; 0075 elseif ~(numel(varargin{1})==1) || ~isreal(varargin{1}) 0076 error('Lanczosfilter:ArgumentType','Incorrect time interval.') 0077 else 0078 dT = varargin{1}; 0079 end 0080 Nf = 1/(2*dT); % Nyquist frequency 0081 if nargin<3 || isempty(varargin{2}) 0082 Cf = Nf/2; 0083 elseif ~(numel(varargin{2})==1) || ~isreal(varargin{2}) || varargin{2}<=0 || varargin{2}>Nf 0084 error('Lanczosfilter:ArgumentType','Incorrect cut-off frequency.') 0085 else 0086 Cf = varargin{2}; 0087 end 0088 if nargin<4 || isempty(varargin{3}) 0089 M = 100; 0090 elseif ~(numel(varargin{3})==1) || ~isreal(varargin{3}) || ~(varargin{3}==round(varargin{3})) 0091 error('Lanczosfilter:ArgumentType','Incorrect Number of coeffients.') 0092 else 0093 M = varargin{3}; 0094 end 0095 if nargin<5 || isempty(varargin{4}) 0096 LoH = 'l'; 0097 elseif ~ischar(varargin{4}) || isempty(strfind('lh',lower(varargin{4}(1)))) 0098 error('Lanczosfilter:ArgumentType','Incorrect filter pass type.') 0099 else 0100 LoH = varargin{4}; 0101 end 0102 if strcmpi(LoH(1),'h') 0103 LoH = 2; 0104 else 0105 LoH = 1; 0106 end 0107 0108 % Normalize the cut off frequency with the Nyquist frequency: 0109 Cf = Cf/Nf; 0110 0111 % Lanczos cosine coeficients: 0112 coef = lanczos_filter_coef(Cf,M); 0113 coef = coef(:,LoH); 0114 0115 % Filter in frequency space: 0116 [window,Ff] = spectral_window(coef,length(x)); 0117 Ff = Ff*Nf; 0118 0119 % Replace NaN's with the mean (ideas?): 0120 inan = isnan(x); 0121 xmean = mean(x(~inan)); 0122 x(inan) = xmean; 0123 0124 % Filtering: 0125 [y,Cx] = spectral_filtering(x,window); 0126 0127 function coef = lanczos_filter_coef(Cf,M) 0128 % Positive coeficients of Lanczos [low high]-pass. 0129 hkcs = lowpass_cosine_filter_coef(Cf,M); 0130 sigma = [1 sin(pi * (1:M) / M) ./ (pi * (1:M) / M)]; 0131 hkB = hkcs.*sigma; 0132 hkA = -hkB; hkA(1) = hkA(1)+1; 0133 coef = [hkB(:) hkA(:)]; 0134 0135 function coef = lowpass_cosine_filter_coef(Cf,M) 0136 % Positive coeficients of cosine filter low-pass. 0137 coef = Cf*[1 sin(pi*(1:M)*Cf)./(pi*(1:M)*Cf)]; 0138 0139 function [window,Ff] = spectral_window(coef,N) 0140 % Window of cosine filter in frequency space. 0141 Ff = 0:2/N:1; 0142 window = zeros(length(Ff),1); 0143 for i = 1:length(Ff) 0144 window(i) = coef(1) + 2*sum(coef(2:end).*cos((1:length(coef)-1)'*pi*Ff(i))); 0145 end 0146 0147 function [y,Cx] = spectral_filtering(x,window) 0148 % Filtering in frequency space is multiplication, (convolution in time 0149 % space). 0150 Nx = length(x); 0151 Cx = fft(x(:)); 0152 Cx = Cx(1:floor(Nx/2)+1); 0153 CxH = Cx.*window(:); 0154 CxH(length(CxH)+1:Nx) = conj(flipud(CxH(2:end-1))); 0155 y = real(ifft(CxH)); 0156 0157 0158 % Carlos Adrián Vargas Aguilera. nubeobscura@hotmail.com