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# spline() - Signal Processing

### Syntax

### Example

### Output / Return Value

### Limitations

### Alternatives / See Also

### Reference

yy = spline(x,Y,xx) uses a cubic spline interpolation to find yy, the values of the underlying function Y at the values of the interpolant xx. For the interpolation, the independent variable is assumed to be the final dimension of Y with the breakpoints defined by x. The values in x must be distinct. The sizes of xx and yy are related as follows:If Y is a scalar or vector, yy has the same size as xx. If Y is an array that is not a vector,If xx is a scalar or vector, size(yy) equals [d1, d2, ..., dk, length(xx)].If xx is an array of size [m1,m2,...,mj], size(yy) equals [d1,d2,...,dk,m1,m2,...,mj].pp = spline(x,Y) returns the piecewise polynomial form of the cubic spline interpolant for later use with ppval and the spline utility unmkpp. x must be a vector with distinct values. Y can be a scalar, a vector, or an array of any dimension, subject to the following conditions:If x and Y are vectors of the same size, the not-a-knot end conditions are used.If x or Y is a scalar, it is expanded to have the same length as the other and the not-a-knot end conditions are used. (See Exceptions (1) below).If Y is a vector that contains two more values than x has entries, the first and last value in Y are used as the endslopes for the cubic spline. (See Exceptions (2) below.)ExceptionsIf Y is a vector that contains two more values than x has entries, the first and last value in Y are used as the endslopes for the cubic spline. If Y is a vector, this meansf(x) = Y(2:end-1)df(min(x)) = Y(1)df(max(x)) = Y(end)If Y is a matrix or an N-dimensional array with size(Y,N) equal to length(x)+2, the following hold:f(x(j)) matches the value Y(:,...,:,j+1) for j=1:length(x)Df(min(x)) matches Y(:,:,...:,1)Df(max(x)) matches Y(:,:,...:,end)Note You can also perform spline interpolation using the interp1 function with the command interp1(x,y,xx,'spline'). Note that while spline performs interpolation on rows of an input matrix, interp1 performs interpolation on columns of an input matrix.

yy = spline(x,Y,xx)pp = spline(x,Y)

Spline Interpolation of Sine DataOpen This ExampleThis generates a sine curve, then samples the spline over a finer mesh.x = 0:10; y = sin(x); xx = 0:.25:10; yy = spline(x,y,xx); plot(x,y,'o',xx,yy) Spline Interpolation of Distribution and Specify Endpoint SlopesOpen This ExampleThis illustrates the use of clamped or complete spline interpolation where end slopes are prescribed. Zero slopes at the ends of an interpolant to the values of a certain distribution are enforced.x = -4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0]; cs = spline(x,[0 y 0]); xx = linspace(-4,4,101); plot(x,y,'o',xx,ppval(cs,xx),'-'); Extrapolation Using Cubic SplineOpen This Example The two vectorst = 1900:10:1990; p = [ 75.995 91.972 105.711 123.203 131.669 ... 150.697 179.323 203.212 226.505 249.633 ]; represent the census years from 1900 to 1990 and the corresponding United States population in millions of people. The expressionspline(t,p,2000) uses the cubic spline to extrapolate and predict the population in the year 2000. The result is ans = 270.6060 Spline Interpolation of Angular DataOpen This Example The statementsx = pi*[0:.5:2]; y = [0 1 0 -1 0 1 0; 1 0 1 0 -1 0 1]; pp = spline(x,y); yy = ppval(pp, linspace(0,2*pi,101)); plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or'), axis equal generate the plot of a circle, with the five data points y(:,2),...,y(:,6) marked with o's. Note that this y contains two more values (i.e., two more columns) than does x, hence y(:,1) and y(:,end) are used as endslopes. Spline Interpolation of Sine and Cosine DataOpen This ExampleThe following code generates sine and cosine curves, then samples the splines over a finer mesh.x = 0:.25:1; Y = [sin(x); cos(x)]; xx = 0:.1:1; YY = spline(x,Y,xx); plot(x,Y(1,:),'o',xx,YY(1,:),'-') hold on plot(x,Y(2,:),'o',xx,YY(2,:),':') hold off