# Calculating Inputs From Outputs

Inputs are calculated from Outputs & Transfer Functions in three different ways,

1. Input Time Waveforms or Fourier spectra from Output Time Waveforms or Fourier spectra & Transfer Functions

2. Input Auto spectra or PSDs from (Input-Output) Cross spectra & Transfer Functions

3. Input Auto spectra or PSDs from Output Auto spectra or PSDs & Transfer Functions.

## Input Time Waveforms or Fourier spectra

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 From Cross spectra

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 from Output Auto spectra

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.