fc=sgsdf(x,n,dn,x0,flag)

function fc=sgsdf(x,n,dn,x0,flag)
%Function:
%       Savitzky-Golay Smoothing and Differentiation Filter
%       The Savitzky-Golay smoothing/differentiation filter (i.e., the polynomial smoothing/differentiation filter, 
%       or  the least-squares smoothing/differentiation filters) optimally fit a set of data points to polynomials
%       of different degrees. 
%       See for details in Matlab Documents (help sgolay).  The sgolay function in Matlab can deal with only 
%       symmetrical and uniformly spaced data of even number.
%       This function presented here is a general implement of the sgolay function in Matlab. The Savitzky-Golay filter
%       coefficients for even number, nonsymmetrical and nonuniformly spaced data can be obtained. And the filter coefficients
%       for the initial point or the end point can be obtained too.In addition, either numerical results or symbolical 
%       results can be obtained.
%Usage:
%       fc   = sgsdf(x,n,dn,x0,flag)
%       x    = the original data point, e.g., -5:5 
%       n    = polynomial order,                    default=1
%       dn   = differentation order (0=smoothing),  default=0
%       x0   = estimation point,                    default=0
%       flag = numerical(0) or symbolical(1),       default=0
%       fc   = filter coefficients obtained.
%Notes:
%1.     x  can be arbitrary, e.g., odd number or even number,symmetrical or nonsymmetrical, uniformly spaced 
%          or nonuniformly spaced, etc.       
%2.     x0 can be arbitrary, e.g., the initial point,the end point,etc.
%3.     Either numerical results or symbolical results can be obtained.
%Example:
%       sgsdf([-3:3],2,0,0,0)
%       sgsdf([-3:3],2,0,0,1)
%       sgsdf([-3:3],2,0,-3,1)
%       sgsdf([-3:3],2,1,2,1)
%       sgsdf([-2:3],2,1,1/2,1)
%       sgsdf([-5:2:5],2,1,0,1)     
%       sgsdf([-1:1 2:2:8],2,0,0,1)
%Author:
%       Jianwen Luo <[e-mail], [e-mail]> 2003-10-05
%       Department of Biomedical Engineering, Department of Electrical Engineering
%       Tsinghua University, Beijing 100084, P. R. China  
%Reference
%[1]A. Savitzky and M. J. E. Golay, "Smoothing and Differentiation of Data by Simplified Least Squares Procedures," 
%   Analytical Chemistry, vol. 36, pp. 1627-1639, 1964.
%[2]J. Steinier, Y. Termonia, and J. Deltour, "Comments on Smoothing and Differentiation of Data by Simplified Least Square Procedures,"
%   Analytical Chemistry, vol. 44, pp. 1906-1909, 1972.
%[3]H. H. Madden, "Comments on Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data," 
%   Analytical Chemistry, vol. 50, pp. 1383-1386, 1978.
%[4]R. A. Leach, C. A. Carter, and J. M. Harris, "Least-Squares Polynomial Filters for Initial Point and Slope Estimation," 
%   Analytical Chemistry, vol. 56, pp. 2304-2307, 1984.
%[5]P. A. Baedecker, "Comments on Least-Square Polynomial Filters for Initial Point and Slope Estimation," 
%   Analytical Chemistry, vol. 57, pp. 1477-1479, 1985.
%[6]P. A. Gorry, "General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method," 
%   Analytical Chemistry, vol. 62, pp. 570-573, 1990.
%See also:
%       sglay, sgsdf_2d2

if nargin<5
    flag=0;
end
if nargin<4
    x0=0;
end
if nargin<3
    dn=0;
end
if nargin<2
    n=1;
end


if(n>length(x)-1)
    error('Polynomial Order too Large');    
end
if(dn>n)
    error('Differentiation Order too Large!');    
end

if flag
    x=sym(x);
    x0=sym(x0);
end

x=x(:);
A=ones(length(x),1);
for k=1:n
    A=[A x.^k];
end
h=inv(A'*A)*A';

if flag
    hx=sym([(zeros(1,dn)) prod(1:dn)]);  %order=0:dn-1,& dn,respectively
    for k=1:n-dn                        %order=dn+1:n=dn+k
        hx=[hx sym(x0^k*prod(dn+k:-1:k+1))];
    end    
else
    hx=[(zeros(1,dn)) prod(1:dn)];  %order=0:dn-1,& dn,respectively
    for k=1:n-dn                        %order=dn+1:n=dn+k
        hx=[hx x0^k*prod(dn+k:-1:k+1)];
    end    
end
% h
% hx
hx=repmat(hx',[1,size(h,2)]);
h=h.*hx;
fc=sum(h);