Spectrum - Interspectrum - Interspectral Matrix
===============================================




Signal processing - Conventions retained
--------------------------------------------




Introduction
~~~~~~~~~~~~

A signal can have two representations: a temporal representation of the form :math:`x=f(t)` or a frequency representation of the form :math:`X=F(f)`. These two representations are linked together by the **Fourier transformation.**

In the numerical field and in the experimental field, there are various ways of calculating the spectral quantities relating to a time signal :math:`x(t)` (dimensional representation or not, factor 1/2 or not for the Fourier Transformation).

However, while the various definitions of DSP (cf. [:ref:`§2.2.2 <§2.2.2>`] and [:ref:`Annexe1 <Annexe1>`]) based on the Fourier Transformation of the signal do not change anything in the calculation performed by CALC_INTE_SPEC [:external:ref:`U4.56.03 <U4.56.03>`], on the other hand, in the calculations carried out by the post-processing operator POST_DYNA_ALEA, it is important that the data be consistent so that the results produced by this operator are at physical dimension of the starting signal.

It is also necessary to know, for a quantitative comparison between calculation and experiment, what conventions are adopted for calculating spectral quantities. All of these conventions are recalled in [:ref:`Annexe1 <Annexe1>`] for each type of signal. We are only giving the general formulas again here.




Fourier transformation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the **Fourier transformation** **in frequency** :math:`(f)` **of a signal** (of unit u) *,* expressed in U/Hz we adopt the following definition: :math:`X(f)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}x(t){e}^{-2i\pi \text{ft}}\text{dt}`

The reverse transformation is then expressed by: :math:`x(t)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}X(f){e}^{+2i\pi \text{ft}}\text{df}`

We can also express the **Pulsating Fourier Transformation** :math:`(\omega =2\pi f)`, by the following definition:

:math:`{X}^{p}(\omega )=\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}x(t){e}^{-i\omega t}\text{dt}`

The reverse transformation is expressed by: :math:`x(t)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{X}^{p}(\omega ){e}^{+i\omega t}d\omega`

Which leads to equivalence: :math:`{X}^{p}(\omega )={X}^{p}(2\pi f)=\frac{1}{2\pi }X(f)`




Power concept - Power Spectral Density
----------------------------------------------------




Signal strength - Signal strength spectrum
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Like the signal itself, signal strength can be expressed as a function of time or frequency:


* instantaneous time power is simply called power:

:math:`p(t)=x(t)\text{.}\text{x*}(t)`

where :math:`\text{x*}(t)` is the conjugated complex quantity of :math:`x(t)`.


* frequency power is commonly called power spectral density or spectrum:

:math:`\underline{{S}_{\text{xx}}(f)=X(f)\text{.}\text{X*}(f)={\mid X(f)\mid }^{2}}`

*This definition is only possible when the Fourier transform of the signal exists.*

We can then express the total energy of the signal as :math:`E=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{S}_{\text{xx}}(f)\text{df}=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{\mid X(f)\mid }^{2}\text{df}`

The expression for this DSP for the various types of signals is given in [:ref:`Annexe1 <Annexe1>`]. We will see later [:ref:`§3.3 <§3.3>`] another definition - equivalent according to the Wiener—Kinchine theorem - but more general, of power spectral density based on the statistical approach.




Interaction power - Spectral density of interaction of two signals - Interspectrum
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


*We also define the**instantaneous interaction power of two signals** :math:`x(t)\text{et}y(t)`:

:math:`\begin{array}{}{p}_{\text{xy}}(t)=x(t)\text{.}\text{y*}(t)\text{et}{p}_{\text{yx}}(t)=\text{x*}(t)\text{.}y(t)\\ \text{reliées par}{p}_{\text{xy}}(t)={p}_{\text{yx}}\ast (t)\end{array}`


*If both signals admit a Fourier transform :math:`X(f)\text{et}Y(f)`, you can express the**interaction frequency power**or**interspect**by :math:`{S}_{\text{XY}}(f)=X(f)\text{.}\text{Y*}(f)`


*If the**two signals are real** then the power of interaction :math:`{p}_{\text{xy}}(t)={p}_{\text{yx}}(t)=x(t)\text{.}y(t)` is real. But there is no reason why :math:`{S}_{\text{XY}}(f)` should be so real; in contrast :math:`{S}_{\text{XY}}(f)` is complex with Hermitian symmetry, namely:

Even real part and odd imaginary part or even module and odd phase


*If :math:`X(f)=Y(f)`, then we are talking about**autospectrum. **




Interspectral matrix
~~~~~~~~~~~~~~~~~~~~~~~~~

An interspectral matrix of order :math:`N` is a complex :math:`N\times N` matrix, each term of which depends on the frequency in the form of a function of :math:`f` *.* The diagonal terms are the autospectra, the extra-diagonal terms are the interspectra between the points considered (each row or column representing a point in physical mesh or a mode in modal calculation). Since the interspectra manipulated in practice are Hermitian, only the :math:`\frac{N(N+1)}{2}` terms of the upper triangular (or lower) are sufficient to completely define the interspectral matrix.




Implementation in Code_Aster
----------------------------

The interspectral matrices manipulated by the operator POST_DYNA_ALEA consist of complex frequency functions: :math:`{S}_{\text{XY}}(f)`.

These matrices are stored in interspectrum concept tables.