2. Spectrum - Interspectrum - Interspectral Matrix#
2.1. Signal processing - Conventions retained#
2.1.1. Introduction#
A signal can have two representations: a temporal representation of the form \(x=f(t)\) or a frequency representation of the form \(X=F(f)\). 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 \(x(t)\) (dimensional representation or not, factor 1/2 or not for the Fourier Transformation).
However, while the various definitions of DSP (cf. [§2.2.2] and [Annexe1]) based on the Fourier Transformation of the signal do not change anything in the calculation performed by CALC_INTE_SPEC [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 [Annexe1] for each type of signal. We are only giving the general formulas again here.
2.1.2. Fourier transformation#
For the Fourier transformation in frequency \((f)\) of a signal (of unit u) , expressed in U/Hz we adopt the following definition: \(X(f)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}x(t){e}^{-2i\pi \text{ft}}\text{dt}\)
The reverse transformation is then expressed by: \(x(t)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}X(f){e}^{+2i\pi \text{ft}}\text{df}\)
We can also express the Pulsating Fourier Transformation \((\omega =2\pi f)\), by the following definition:
\({X}^{p}(\omega )=\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}x(t){e}^{-i\omega t}\text{dt}\)
The reverse transformation is expressed by: \(x(t)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{X}^{p}(\omega ){e}^{+i\omega t}d\omega\)
Which leads to equivalence: \({X}^{p}(\omega )={X}^{p}(2\pi f)=\frac{1}{2\pi }X(f)\)
2.2. Power concept - Power Spectral Density#
2.2.1. 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:
\(p(t)=x(t)\text{.}\text{x*}(t)\)
where \(\text{x*}(t)\) is the conjugated complex quantity of \(x(t)\).
frequency power is commonly called power spectral density or spectrum:
\(\underline{{S}_{\text{xx}}(f)=X(f)\text{.}\text{X*}(f)={\mid X(f)\mid }^{2}}\)
This definition is only possible when the Fourier transform of the signal exists.
We can then express the total energy of the signal as \(E=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{S}_{\text{xx}}(f)\text{df}=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{\mid X(f)\mid }^{2}\text{df}\)
The expression for this DSP for the various types of signals is given in [Annexe1]. We will see later [§3.3] another definition - equivalent according to the Wiener—Kinchine theorem - but more general, of power spectral density based on the statistical approach.
2.2.2. Interaction power - Spectral density of interaction of two signals - Interspectrum#
We also define the**instantaneous interaction power of two signals* \(x(t)\text{et}y(t)\):
\(\begin{array}{}{p}_{\text{xy}}(t)=x(t)\text{.}\text{y*}(t)\text{et}{p}_{\text{yx}}(t)=\text{x*}(t)\text{.}y(t)\\ \text{reliées par}{p}_{\text{xy}}(t)={p}_{\text{yx}}\ast (t)\end{array}\)
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 \({p}_{\text{xy}}(t)={p}_{\text{yx}}(t)=x(t)\text{.}y(t)\) is real. But there is no reason why \({S}_{\text{XY}}(f)\) should be so real; in contrast \({S}_{\text{XY}}(f)\) 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. *
2.2.3. Interspectral matrix#
An interspectral matrix of order \(N\) is a complex \(N\times N\) matrix, each term of which depends on the frequency in the form of a function of \(f\) . 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 \(\frac{N(N+1)}{2}\) terms of the upper triangular (or lower) are sufficient to completely define the interspectral matrix.
2.3. Implementation in Code_Aster#
The interspectral matrices manipulated by the operator POST_DYNA_ALEA consist of complex frequency functions: \({S}_{\text{XY}}(f)\).
These matrices are stored in interspectrum concept tables.