ភាពខុសគ្នារវាងកំណែនានារបស់ "បំលែងឡាប្លាស"

គ្មាន​ចំណារពន្យល់ការកែប្រែ
| <math> \int_0^t f(\tau)\, d\tau = u(t) * f(t)</math>
| <math> {1 \over s} F(s) </math>
| <math> u(t) \,</math> ជាអនុគមន៍កាំជណ្តើរ Heavisideជាអនុគមន៍កាំជណ្តើរហេវីសាយ (Heaviside step function)។
|-
! Scaling
| <math> f(t - a) u(t - a) \ </math>
| <math> e^{-as} F(s) \ </math>
| <math> u(t) \,</math> ជាអនុគមន៍កាំជណ្តើរ Heaviside (Heaviside step function)ជាអនុគមន៍កាំជណ្តើរហេវីសាយ
|-
!
ដូចនេះ <math>\mathcal{L}\{ f \}={1 \over 1 - e^{-Tp}} \int_0^T e^{-pt} f(t)\,dt</math>
 
== តារាងរូបមន្តសង្ខេបបំលែងឡាប្លាស ==
តារាងខាងក្រោមផ្តល់នូវរូបមន្តបំលែងឡាប្លាសទូទៅនៃអថេរមួយ។ ចំពោះនិយមន័យនិងសេចក្តីពន្យល់សូមមើល'''សំគាល់ផ្នែកខាងចុងនៃតារាង'''។
* បំលែងឡាប្លាសនៃផលបូកគឺជាផលបូកនៃបំលែងឡាប្លាសនៃតួរនិមួយៗ។
 
::<math>\mathcal{L}\left\{f(t) + g(t) \right\} = \mathcal{L}\left\{f(t)\right\} + \mathcal{L}\left\{ g(t) \right\} </math>
 
* បំលែងឡាប្លាសច្រើនដងនៃអនុគមន៍មួយគឺ​មានបំលែងឡាប្លាសជាចំនួនច្រើនដងនៃអនុគមន៍នោះ។
::<math>\mathcal{L}\left\{a f(t)\right\} = a \mathcal{L}\left\{ f(t)\right\}</math>
 
បំលែងឡាប្លាសតែឯងគឺពិតជាត្រឹមត្រូវនៅពែល t ជាចំនួនមិនអវិជ្ជមាន ដែលគ្រប់អនុគមន៍ដើមក្នុងតារាងគឺជាអនុគមន៍កាំជណ្តើរហេវីសាយជាច្រើន u(''t'') ។
 
{| class="wikitable"
|-
! ID || ឈ្មោះអនុគមន៍ || អនុគមន៍ដើម <br> <math>x(t) = \mathcal{L}^{-1} \left\{ X(s) \right\}</math> || បំលែងឡាប្លាស <br> <math>X(s) = \mathcal{L}\left\{ x(t) \right\}</math> || Region of convergence <br> ''for causal system|causal systems''
|- align="center"
| 1 || ideal delay || <math> \delta(t-\tau) \ </math> || <math> e^{-\tau s} \ </math> ||
|- align="center"
| 1a || [[Dirac delta function|unit impulse]] || <math> \delta(t) \ </math> || <math> 1 \ </math> || <math> \mathrm{all} \ s \,</math>
|- align="center"
| 2 || delayed ''n''th power <br /> with frequency shift || <math>\frac{(t-\tau)^n}{n!} e^{-\alpha (t-\tau)} \cdot u(t-\tau) </math> || <math> \frac{e^{-\tau s}}{(s+\alpha)^{n+1}} </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 2a || ស្វ័យគុណទី ''n'' <br /> ( ចំពោះចំនួនគត់ ''n'' ) || <math>{ t^n \over n! } \cdot u(t) </math> || <math> { 1 \over s^{n+1} } </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
|- align="center"
| 2a.1 || ស្វ័យគុណទី ''q'' <br /> (ចំនួនពិត ''q'' ) || <math>{ t^q \over \Gamma(q+1) } \cdot u(t) </math> || <math> { 1 \over s^{q+1} } </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
 
| 2a.2 || អនុគមន៍កាំជណ្តើរហេវីសាយ || <math> u(t) \ </math> || <math> { 1 \over s } </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 2b || delayed unit step || <math> u(t-\tau) \ </math> || <math> { e^{-\tau s} \over s } </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 2c || ramp || <math> t \cdot u(t)\ </math> || <math>\frac{1}{s^2}</math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 2d || ''n''th power with frequency shift || <math>\frac{t^{n}}{n!}e^{-\alpha t} \cdot u(t) </math> || <math>\frac{1}{(s+\alpha)^{n+1}}</math> || <math> \textrm{Re} \{ s \} > - \alpha \, </math>
|- align="center"
| 2d.1 || exponential decay || <math> e^{-\alpha t} \cdot u(t) \ </math> || <math> { 1 \over s+\alpha } </math> || <math> \textrm{Re} \{ s \} > - \alpha \ </math>
|- align="center"
| 3 || exponential approach || <math>( 1-e^{-\alpha t}) \cdot u(t) \ </math> || <math>\frac{\alpha}{s(s+\alpha)} </math> || <math> \textrm{Re} \{ s \} > 0\ </math>
|- align="center"
| 4 || [[ស៊ីនុស]] || <math> \sin(\omega t) \cdot u(t) \ </math> || <math> { \omega \over s^2 + \omega^2 } </math> || <math> \textrm{Re} \{ s \} > 0 \ </math>
|- align="center"
| 5 || [[កូស៊ីនុស]] || <math> \cos(\omega t) \cdot u(t) \ </math> || <math> { s \over s^2 + \omega^2 } </math> || <math> \textrm{Re} \{ s \} > 0 \ </math>
|- align="center"
| 6 || [[ស៊ីនុសអ៊ីពែបូលីក]] || <math> \sinh(\alpha t) \cdot u(t) \ </math> || <math> { \alpha \over s^2 - \alpha^2 } </math> || <math> \textrm{Re} \{ s \} > | \alpha | \ </math>
|- align="center"
| 7 || [[កូស៊ីនុសអ៊ីពែលីក]] || <math> \cosh(\alpha t) \cdot u(t) \ </math> || <math> { s \over s^2 - \alpha^2 } </math> || <math> \textrm{Re} \{ s \} > | \alpha | \ </math>
|- align="center"
| 8 || Exponentially-decaying <br /> sine wave || <math>e^{\alpha t} \sin(\omega t) \cdot u(t) \ </math> || <math> { \omega \over (s-\alpha )^2 + \omega^2 } </math> || <math> \textrm{Re} \{ s \} > \alpha \ </math>
|- align="center"
| 9 || Exponentially-decaying <br /> cosine wave || <math>e^{\alpha t} \cos(\omega t) \cdot u(t) \ </math> || <math> { s-\alpha \over (s-\alpha )^2 + \omega^2 } </math> || <math> \textrm{Re} \{ s \} > \alpha \ </math>
|- align="center"
| 10 || រឺសទី''n'' || <math> \sqrt[n]{t} \cdot u(t) </math> || <math> s^{-(n+1)/n} \cdot \Gamma\left(1+\frac{1}{n}\right)</math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 11 || [[លោការីតធម្មជាតិ]] || <math> \ln \left ( { t \over t_0 } \right ) \cdot u(t) </math> || <math> - { t_0 \over s} \ [ \ \ln(t_0 s)+\gamma \ ] </math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 12 || Bessel function <br> of the first kind, <br /> of order ''n'' || <math> J_n( \omega t) \cdot u(t)</math> || <math>\frac{ \omega^n \left(s+\sqrt{s^2+ \omega^2}\right)^{-n}}{\sqrt{s^2 + \omega^2}}</math> || <math> \textrm{Re} \{ s \} > 0 \, </math> <br /> <math> (n > -1) \, </math>
|- align="center"
| 13 || Modified Bessel function <br /> of the first kind, <br /> of order ''n'' || <math>I_n(\omega t) \cdot u(t)</math> || <math> \frac{ \omega^n \left(s+\sqrt{s^2-\omega^2}\right)^{-n}}{\sqrt{s^2-\omega^2}} </math> || <math> \textrm{Re} \{ s \} > | \omega | \, </math>
|- align="center"
| 14 || Bessel function <br /> of the second kind, <br /> of order 0 || <math> Y_0(\alpha t) \cdot u(t)</math> || <math>-{2 \sinh^{-1}(s/\alpha) \over \pi \sqrt{s^2+\alpha^2}}</math> || <math>\textrm{Re} \{ s \} > 0 \, </math>
|- align="center"
| 15 || Modified Bessel function <br /> of the second kind, <br /> of order 0 || <math> K_0(\alpha t) \cdot u(t)</math> || &nbsp; || &nbsp;
|- align="center"
| 16 || Error function || <math> \mathrm{erf}(t) \cdot u(t) </math> || <math> {e^{s^2/4} \left(1 - \operatorname{erf} \left(s/2\right)\right) \over s}</math> || <math> \textrm{Re} \{ s \} > 0 \, </math>
|-
|colspan=5|'''សំគាល់:'''
{{col-begin}}
{{col-break}}
* <math> u(t) \, </math> តំណាងអោយអនុគមន៍កាំជណ្តើរហេវីសាយ
* <math> \delta(t) \, </math> តំណាងអោយ Dirac delta function
* <math> \Gamma (z) \, </math> តំណាងអោយហ្គាំម៉ា
* <math> \gamma \, </math> ជាថេរអឺលែរម៉ាសឆេរ៉ូនី
{{col-break}}
* <math>t \, </math> ជាចំនួនពិតតំណាងអោយពេល (''time'')
* <math>s \, </math> ជា[[ចំនួនកុំផ្លិច angular frequency និង<math>\textrm{Re} \{ s \}</math> ជា[[ផ្នែកពិត]].
* <math> \alpha \,</math>, <math> \beta \,</math>, <math> \tau \, </math>, និង <math>\omega \,</math> ចំនួនពិត
* <math>n \, </math> ជាចំនួនគត់
{{col-end}}
|}
 
[[Category:សមីការឌីផេរ៉ង់ស្យែល]]
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