How to create a linearly interpolated symbolic function from data?

I have measurements that can be described with unknown (and probably complicated) functions. I need symbolic functions that return the linear interpolated numbers between any two measurement points for any given time so I can define the Laplace-transforms of these functions. How can this be done?

[Kuifeng]

Maybe elaborate more about your question with an example?

[Tibor Guba]

Here is the working example of the program in Mathcad. The red dots are measured, the blue line is symbolic. For the Laplace transform I have to use the function represented by the blue line, but its explicit formula is unknown and irrelevant. It is enough to have the linear interpolated values between the red dots. My goal is to obtain this symbolic function.

Answer 1

[Walter Roberson]

Let x be the timepoints and y be the associated data values. Then for any time t between x(J) and x(J+1)

heaviside(t - x(J)) * heaviside(x(J+1) - t) * (y(J) + (t-x(J))/(x(J+1)-x(J)) * (y(J+1)-y(J))

so sum those formula over J = 1 to length(x)-1

I might have messed up the code slightly, as I simply wrote it out. It is the standard "origin plus a ratio of the change in value" calculation, together with a pair of heaviside that will only multiply to non-zero when the value is within the range.

Caution: you need to adjust this to take into account the value associated with heaviside(0), which there is no standard value for. Sometimes you want heaviside(0) to be 0, sometimes you want it to be 1, and it is not uncommon to see it defined as 1/2 . If it is defined as 0, then you need to use a dirac delta to add in the value at every boundary. If it is 1/2 then you need to use a dirac delta to add in half of the value at each boundary.

If you have R2015a or later see sympref() for adjusting the value of heaviside at 0.

[Tibor Guba]

I will check this out, thank you!

[Tibor Guba]

I checked the code out. The only issue with the syntax in the beginning was, that it lacked a parenthesis. I rewrote it to my variable names and attached everything for you to try out:

DATA = csvread('l1=1.0mm_l2=2.7mm SIMA.csv');

%everything in millimetres

l1 = 1; l2=2.7; V0 = 150000; r = 45;

A = pi*r^2;

Skala = 1e6 * A * l2 / V0; % scaling from ppm to number

tm = DATA(:,1); % time measured
qm = DATA(:,2) / Skala; % quantity measured

kar = 0.006;  
w0 = 0; w1=0.02;  
% N=101;  
M = length(qm);  
% dw = (w1 - w0) / N;

w = linspace(w0,w1,N);

pm = kar + 1i*w;

%sympref();

syms t s;

syms interpolalt(t);

J = linspace (1, M-1, M-1)';

interpolalt (t) =  heaviside( t - tm(J) ) *    heaviside(tm(J+1) - t) * ( qm(J) + ( t - tm(J) ) *  (qm (J+1) - qm(J) ) / ( tm(J+1) - tm(J) ) );

I see, how your line works, but I am not sure how to do the summing symbolically. Could you please help me with it?