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Diffstat (limited to 'tex')
| -rw-r--r-- | tex/filters.tex | 6 | ||||
| -rw-r--r-- | tex/lti.tex | 6 | ||||
| -rw-r--r-- | tex/state-space.tex | 1 |
3 files changed, 7 insertions, 6 deletions
diff --git a/tex/filters.tex b/tex/filters.tex index f1ec820..913551c 100644 --- a/tex/filters.tex +++ b/tex/filters.tex @@ -94,13 +94,13 @@ To find the order of the filter given two parameters the formula is \[ A(\Omega) = 10 \log\left(1 + e^2C_n^2(\Omega) \right), \] -where \(C_n = \cos(n\arccos(\Omega))\) for \(|\Omega| \leq 1\) (in the passband), and when \(|\Omega| > 1\) (in the stopband) \(C_n = \cosh(n\arccosh(\Omega))\), is a so called Chebyshev polynomial of \(n\)-th order. \(e\) is a parameter, \emph{not the natural number (2.71\ldots)}. Chebyshev polynomials can be computed recursively with the formula +where \(C_n = \cos(n\arccos(\Omega))\) for \(|\Omega| \leq 1\) (in the passband), and when \(|\Omega| > 1\) (in the stopband) \(C_n = \cosh(n\arccosh(\Omega))\), is a so called Chebyshev polynomial of \(n\)-th order. The ripple factor \(e\) is a parameter, \emph{not the natural number (2.71\ldots)}. Chebyshev polynomials can be computed recursively with the formula \[ C_n = 2\Omega C_{n-1} - C_{n-2}, \] and knowing that \(C_1 = \Omega\) and \(C_2 = 2\Omega^2 - 1\). -The idea is that in the passband the attenuation is periodic and stays more or less constant, and in the stopband the function is no longer periodic and damps the frequencies. To find the parameter \(e\) given an \(A_\text{max}\) +The idea is that in the passband the attenuation is periodic and stays more or less constant, and in the stopband the function is no longer periodic and damps the frequencies. To find the ripple factor \(e\) given an \(A_\text{max}\) \[ e = \sqrt{10^{A_\text{max}/10} - 1}, \] @@ -115,7 +115,7 @@ and to find the order given two parameters } \right\rceil. \] -\paragraph{Chebyshev II} Also known as \emph{inverse} Chebyshev. Let \(K(\Omega^2) = 1/e^2 C_n^2(1/\Omega)\). +\paragraph{Chebyshev II} Also known as \emph{inverse} Chebyshev because \(K(\Omega^2) = 1/e^2 C_n^2(1/\Omega)\). \paragraph{Cauer} diff --git a/tex/lti.tex b/tex/lti.tex index 812c1d4..2e1819a 100644 --- a/tex/lti.tex +++ b/tex/lti.tex @@ -67,17 +67,17 @@ The spectrum of a sinusoidal signal of frequency \(\omega_1\) is only one weight To measure the distortion of a signal in the English literature there is the \emph{total harmonic distortion} (THD) defined as \[ - \text{THD} = \frac{1}{d_1}\sqrt{\sum_{i=1}^n d_i^2}. + \text{THD} = \frac{1}{d_1}\sqrt{\sum_{i=2}^n d_i^2}. \] In the German literature there is the distortion factor (\emph{Klirrfaktor}, always between 0 and 1) \[ - k = \sqrt{\frac{d_2 + d_3 + \cdots + d_n}{d_1 + d_2 + \cdots + d_n}}. + k = \sqrt{\frac{d^2_2 + d^2_3 + \cdots + d^2_n}{d^2_1 + d^2_2 + \cdots + d^2_n}}. \] Both are usually given in percent (\%) and are related with \[ (\text{THD})^2 = \frac{k^2}{1-k^2}, \] -thus THD \(\leq k\). +thus THD \(\geq k\). \subsection{Stochastic inputs} diff --git a/tex/state-space.tex b/tex/state-space.tex index 3923b9c..e706d61 100644 --- a/tex/state-space.tex +++ b/tex/state-space.tex @@ -16,6 +16,7 @@ A system described by a system of linear differential equations of \(n\)-th orde If the system is time \emph{variant} the matrices are functions of time. \begin{table} + \centering \begin{tabular}{ >{\(}c<{\)} >{\(}c<{\)} l } \toprule \text{\bfseries Symbol} & \text{\bfseries Size} & \bfseries Name \\ |