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Diffstat (limited to 'chapter02')
| -rw-r--r-- | chapter02/content_ch02.tex | 13 |
1 files changed, 9 insertions, 4 deletions
diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex index 8ae492c..d596c93 100644 --- a/chapter02/content_ch02.tex +++ b/chapter02/content_ch02.tex @@ -325,11 +325,11 @@ Now, you can prove that the cosine and sine functions are orthogonal to each oth Furthermore, the sine and cosine functions with \underline{different} indices are orthogonal to each other. \begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \cos\left(p \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{N} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \cos\left(p \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{N} \backslash \{0\} \label{eq:ch02:orth_rel_cos} \end{equation} \begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \sin\left(m \omega_0 t\right) \sin\left(q \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{mq} \qquad \forall \; m, q \in \mathbb{N} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \sin\left(m \omega_0 t\right) \sin\left(q \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{mq} \qquad \forall \; m, q \in \mathbb{N} \backslash \{0\} \label{eq:ch02:orth_rel_sin} \end{equation} with the Kronecker delta @@ -362,10 +362,13 @@ The orthogonality relations are useful to extract the coefficients $a_n$ and $b_ Using the orthogonality relations, the coefficients $\tilde{a}_n$ and $\tilde{b}_n$ can be obtained by: \begin{subequations} \begin{align} - \tilde{a}_n &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_an} \\ - \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(m \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} + \tilde{a}_n &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_an} \qquad \forall \; n > 0 \\ + \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(m \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} \qquad \forall \; m > 0 \\ + \tilde{a}_0 &= \frac{\omega_0}{2 \pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \, \mathrm{d} t \\ + \tilde{b}_0 &= 0 \end{align} \end{subequations} +\textit{Remark: } $a_0$ and $b_0$ need a special treatment, because of slightly changed orthogonality relations. \begin{proof}{Parameter Extraction for $\tilde{a}_n$} Given is a periodic function $\tilde{x}_p(t)$, which can be decomposed into: @@ -441,6 +444,8 @@ It is based on the orthogonality relation: \label{eq:ch02:orth_rel_exp} \end{equation} +\vspace*{1em} + \begin{definition}{Complex-Valued Fourier series} A complex-valued, periodic signal $\underline{x_p}(t)$ can be decomposed into a series complex-valued mono-chromatic signals \eqref{eq:ch02:fourier_series_cmplx} -- the \index{Fourier series!complex-valued} \textbf{complex-valued Fourier series}. \begin{equation*} |