Inputs are calculated from Outputs & Transfer Functions in three different ways,
Input Time Waveforms or Fourier spectra from Output Time Waveforms or Fourier spectra & Transfer Functions
Input Auto spectra or PSDs from (Input-Output) Cross spectra & Transfer Functions
Input Auto spectra or PSDs from Output Auto spectra or PSDs & Transfer Functions.
Input Fourier spectra are calculated from Output Fourier spectra & Transfer Functions using the formula;
{F(w)} = [T(w)] {X(w)}
where:
[T(w)] = [[H(w)]t [H(w)]]-1[H(w)]t matrix (m by n)
[H(w)] - Transfer Function matrix (n by m)
{F(w)} - Input Fourier spectra (m - vector)
{X(w)} - Output Fourier spectra (n - vector)
m - number of Inputs
n - number of Outputs
w - frequency variable (radians per second)
t - denotes transposed conjugate
-1- denotes matrix inverse
If Output Time Waveforms are provided, they are transformed to Fourier spectra before using the above equation.
The calculated Input Fourier spectra are then transformed to Input Time Waveforms.
Input Auto spectra are calculated from Cross spectra & Transfer Functions using the formula;
[{F(w)}{F(w)}t ] = [T(w)] {X(w)}{F(w)}t
where:
[{F(w)}{F(w)}t ] = Input Auto spectrum matrix (m by m)
[T(w)] = [[H(w)]t [H(w)]]-1[H(w)]t matrix (m by n)
[H(w)] = Transfer Function matrix (n by m)
[{X(w)}{F(w)}t ] = Cross spectrum matrix (n by m)
m - number of Inputs
n - number of Outputs
w - frequency variable (radians per second)
t - denotes transposed conjugate
-1- denotes matrix inverse
Input Auto spectra or PSDs are calculated from Output Auto spectra or PSDs & Transfer Functions using the formula;
[{F(w)}{F(w)}t ] = [T(w)] [{X(w)}{X(w)}t ] [T(w)]t
where:
[{F(w)}{F(w)}t ] = Input Auto spectrum matrix (m by m)
[T(w)] = [[H(w)]t [H(w)]]-1[H(w)]t matrix (m by n)
[H(w)] = Transfer Function matrix (n by m)
[{X(w)}{X(w)}t ] = Output Auto spectrum matrix (n by n)
m - number of Inputs
n - number of Outputs
w - frequency variable (radians per second)
t - denotes transposed conjugate
-1- denotes matrix inverse
Only the diagonal elements of the Output Auto spectrum matrix are used for this calculation.