Frequency Analysis#
| Continious | Discrete | |
|---|---|---|
| Real | Fourier | Fourier |
| Complex | Laplace | Z-Transform |
| Properties | ||
|---|---|---|
| Linearity | \(\alpha f(t) + \beta g(t)\) | \(\ftsymbol\ \alpha F(s) + \beta G(s)\) |
| Shift Time | \(x(t - \tau)\) | \(\ftsymbol\ e^{-s \tau} X(x)\) |
| Shift Frequency | \(e^{\tau t}\) | \(\ftsymbol\ X(s - \tau)\) |
| Stauchung | \(f(ct)\) | \(\ftsymbol\ \frac{1}{\abs{c}} F\bigl(\frac{s}{c}\bigr)\) |
| Ableitung | \(f^{(n)}(t)\) | \(\ftsymbol\ s^n F(s)\) |
| Integral | \(\int_{-\infty}^t \tau \diff \tau\) | \(\ftsymbol\ \frac{1}{s} X(s)\) |
| Faltung | \((f * g)(t)\) | \(\ftsymbol\ F(s) \cdot G(s)\) |
Info:
timefunctions \(f,g\), frequency functions \(X,F,G\), complex number \(\cx s\), time \(t\), time shift \(\tau\), constant \(c\)
Example Spectrums
| Time \(f(t)\) | Frequency \(X(f)\) |
|---|---|
Fourierreihe#
Approximation einer periodischen Funktion \(f(t)\) durch Überlagerung gewichteter Sinus und Cosinus einer Grundfrequenz \(\omega_0 = \frac{1}{T}\) und deren Oberschwingungen \(2\omega_0, 3\omega_0, ...\)
Important:
\[F(x) = \sum\limits_{k=-\infty}^\infty c_k e^{\i k \omega_0 x} \qquad c_k = \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} f(x) e^{\i k \omega_0 x} \diff x\]
Fouriertransformation (FT)#
Important:
\[\displaystyle \underset{\text{Zeitbereich}}{ f(t)} \FT \underset{\text{Frequenzspektrum}}{ F(\omega)} := \int\limits_{-\infty}^\infty f(t) \exp(-\i \omega t) \diff t\]
Note:
Es gibt unterschiedliche Normungen (\(1, \frac{1}{\sqrt{2\pi}}\))
Laplaceransformation (LT) \(\cx s = α + ω\i\)#
LT ist Verallgemeinerung der FT mit Dämpfungsfaktor \(α\)
\[\displaystyle \underset{\text{Zeitbereich}}{ f(t)} \LT \underset{\text{Frequenzspektrum}}{ F(\cx s) = \mathcal L\left(f(t)\right)} := \int\limits_{0}^\infty f(t) \exp(- \cx s t) \diff t\]