6.5: Impulse functions
- Page ID
- 155604
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Unit impulse function = Dirac delta function is a generalized function with the properties:
- \(\delta(t) =0, t \not=0\)
- \(\int_{\infty}^{-\infty} \delta(t) dt =1\)
- \(\mathcal{L}(\delta(t-t_0))=e^{-s t_0}\)
Since the first item is just a definition we will prove 2. and 3.
We let \(d_k(t) = \begin{cases} \frac{1}{2k} \quad -k <t<k \\ 0 \quad t \leq -k \\ 0 \quad t \geq -k \end{cases}\)
We note that \(\lim_{k \to 0} d_k(t)=0\) if \(t \not=0\) and \(\lim_{k \to 0} \int_{-\int}^{\int} d_k(t) = \lim_{k \to 0} 1=1 = \int_{-\infty}^{\infty}\delta(t) dt.\)
\[\notag
\begin{align}
\mathcal{L}(\delta(t-t_0)) &= \lim_{k \to 0^+}\mathcal{L}\left(d_k(t-t_0)\right) \notag\\
&= \lim_{k \to 0} \int_0^{\infty}e^{-st}\left(d_k(t-t_0)\right)dt \notag \\
&= \lim _{k \to 0} \frac{1}{2 k} \int_{t_0-k}^{t_0+k} e^{-s t} d t \notag\\
&= \lim _{k \to 0} \frac{-1}{2 s k} e^{-s t} \Big|_{t_0-k} ^{t_0+k} \notag\\
&= \lim _{k \to 0} \frac{1}{2 s k} e^{-s t_0}\left(e^{s k}-e^{-s k}\right) \notag\\
& =\lim _{k \to 0} \frac{\sinh (s k)}{s k} e^{-s t_0}\notag\\
&=\lim _{k \to 0} \frac{s \cosh (s k)}{s} e^{-s t_0}=e^{-s t_0} \notag
\end{align}
\]
The following are some useful identities:
- \(\sin (t)=\frac{e^{i t}-e^{-i t}}{2 i}\)
- \(\cos (t)=\frac{e^{i t}+e^{-i t}}{2}\)
- \(\sinh (t)=\frac{e^t-e^{-t}}{2}\)
- \(\cosh (t)=\frac{e^t+e^{-t}}{2}\)
- \( [\sinh(t)]' = \cosh(t)\)
- \( [\cosh(t)]' = \sinh(t)\)
- \(\sinh (0)=\frac{e^0-e^0}{2}=0\)
- \(\cosh (0)=\frac{e^0+e^0}{2}=1\)
Intro to Group Theory
Define the \(\cdot\) product on \(R^2\) by
\[ \notag
\left(x_1, y_1\right) \cdot\left(x_2, y_2\right)=\left(x_1 x_2-y_1 y_2, x_1 y_2-y_1 x_2\right)
\]
Note \(\cdot\) is
1.) commutative:\( \left(x_1, y_1\right) \cdot\left(x_2, y_2\right)=\left(x_1 x_2-y_1 y_2, x_1 y_2-y_1 x_2\right) =\left(x_2 x_1-y_2 y_1, x_2 y_1-y_2 x_1\right)=\left(x_2, y_2\right) \cdot\left(x_1, y_1\right)\)
2.) associative: \((f \cdot g) \cdot h=f \cdot(g \cdot h)\)
3.) distributive w.r.t \(+: f \cdot\left(g_1+g_2\right)=f \cdot g_1+f \cdot g_2\)
4.) \(\left(x_1, y_1\right) \cdot(0,0)=(0,0)\)
Note: \((0,1) \cdot(0,1)=(-1,0)\)