I want to make a program which can determine the coefficients of a cubic spline. I have the (n+1) points and I want to calculate the value of the n polynomials of spline cubic interpolations
MATLAB: How to make a program which can determine the coefficients
cubic splineSpline Toolbox
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Yes. Just in case this is homework, I will give an example to get you started.
Let's say you have the case below (I will start with the case where we do not have derivatives at a location).
We are trying to solve for A, B, and C
y(x) = A*x^2 + B*x + CKnowns:
x1 = 0; y1 = 1;x2 = 3; y2 = 6;x3 = 7; y3 = 7;So we can construct an inverse problem of the form G * m = d, where:
G = [x1^2,x1,1; x2^2,x2,1; x3^2,x3,1];d = [y1; y2; y3];We determine m (equal to [A; B; C] in our example) by:
m = G\d;And we get:
plot(-1:10,polyval(m,-1:10),'r-', [x1,x2,x3],[y1,y2,y3],'k+')
That depends. The filter coefficients are the coefficients of the difference equation. If your filter is an FIR filter, then the filter coefficients are the values of the impulse response.
If you have an IIR filter, then the filter coefficients are not the same as the impulse response. Remember in that case the impulse response is infinite.
For example:
b = fir1(10,0.2);stem(b)h = impz(b);stem(h)isequal(b',h)but
[b,a] = butter(10,0.2); h = impz(b,a);But the ratio of Z-transforms of the numerator coefficients to denominator coefficiens is equal to the Z-transform of the impulse response.
For example - consider the IIR system with the following difference equation
y(n)-0.8*y(n-1) = x(n)
So the filter coefficients are:
A = [1 -0.8]; B =1;The impulse response is:
h(n) = 0.8^n*u(n) where u(n) is the unit step. But compare:
h1 = impz(B,A); subplot(211) stem(h1,'color',[1 0 0]); subplot(212) n = 0:length(h1)-1; h = 0.8.^n; hold on; stem(h,'color',[0 0 1]);
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