Polynomial Variance (Time Scale and Shift)

Function shifts and scales a polynomial taken over arbitrary interval [x0,xf] onto a normalized interval of [0,1], or vice versa.

Forward transform:
Let p(x) = c_0 + c_1x + c_2x^2...c_Kx^K be a polynomial over an interval [x0,xf]. We seek the polynomial p(v) = d_0 + d_1v + d_2v^2...d_Kv^K over the normalized interval [0,1]. It can be said that all functions that have the same p(v) are equivalent up to time-scale and time-shift.

For function call
[d] = shiftrescale(c,t_sh,t_sc)

Input:
c is the row vector corresponding to the input polynomial coefficients: c_0,1,...K
The normalization parameters are defined via the desired input interval
t_sh = x0
t_sc = xf-x0

Output d is the row vector corresponding to the normalized polynomial coefficients: d_0,1,...K.

Inverse transform:
For an input polynomial defined as p(v) = d_0 + d_1v + d_2v^2...d_Kv^K over the normalized interval [0,1], let an output polynomial be p(x) = c_0 + c_1x + c_2x^2...c_Kx^K over an arbitrary interval [x0,xf]. These polynomials can be said to be equivalent up to time scale and time shift.

For function call
[c] = ishiftrescale(d,t_sh,t_sc)

Input:
d is the row vector corresponding to the normalized polynomial coefficients: d_0,1,...K
The restoration parameters are defined via the desired output interval
t_sh = x0
t_sc = xf-x0

Output c is the row vector corresponding to the restored polynomial coefficients: c_0,1,...K.