(CT Fourier Transform Properties)
 
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(Look at the syntax of the geometric series below for an example.) This will allow other people to refer to your formula later on (by refering to the corresponding page) while still being able to view all formulas on this page.
 
(Look at the syntax of the geometric series below for an example.) This will allow other people to refer to your formula later on (by refering to the corresponding page) while still being able to view all formulas on this page.
  
==General Purpose Formulas==
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{|
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|-
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! colspan="2" style="background: #bbb; font-size: 110%;" | General Purpose Formulas
 +
|-
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! colspan="2" style="background: #eee;" | Series
 +
|-
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| align="right" style="padding-right: 1em;" | [[Finite Geometric Series Formula_ECE301Fall2008mboutin]] || {{:Finite Geometric Series Formula_ECE301Fall2008mboutin}}
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|-
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| align="right" style="padding-right: 1em;" | [[Infinite Geometric Series Formula_ECE301Fall2008mboutin]] || {{:Infinite Geometric Series Formula_ECE301Fall2008mboutin}}
 +
|-
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! colspan="2" style="background: #eee;" | Euler's Formula
 +
|-
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| align="right" style="padding-right: 1em;" | [[Complex exponential in terms of sinusoidal signals_ECE301Fall2008mboutin]] || {{:Complex exponential in terms of sinusoidal signals_ECE301Fall2008mboutin}}
 +
|-
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| align="right" style="padding-right: 1em;" | [[Cosine function in terms of complex exponential_ECE301Fall2008mboutin]] || {{:Cosine function in terms of complex exponential_ECE301Fall2008mboutin}}
 +
|-
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| align="right" style="padding-right: 1em;" | [[Sine function in terms of complex exponential_ECE301Fall2008mboutin]] || {{:Sine function in terms of complex exponential_ECE301Fall2008mboutin}}
 +
|-
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! colspan="2" style="background: #eee;" | Other
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|-
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| align="right" style="padding-right: 1em;" | [[sinc function_ECE301Fall2008mboutin]] || {{:sinc function_ECE301Fall2008mboutin}}
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|-
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! colspan="2" style="background: #bbb; font-size: 110%;" | Discrete-Time Domain
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|-
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! colspan="2" style="background: #eee;" | Useful Formulas
 +
|-
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| align="right" style="padding-right: 1em;" | [[DT Fourier Transform_ECE301Fall2008mboutin]] || {{:DT Fourier Transform_ECE301Fall2008mboutin}}
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|-
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| align="right" style="padding-right: 1em;" | [[DT Inverse Fourier Transform_ECE301Fall2008mboutin]] || {{:DT Inverse Fourier Transform_ECE301Fall2008mboutin}}
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|-
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! colspan="2" style="background: #eee;" | DT Fourier Transform Pairs
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|-
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| align="right" style="padding-right: 1em;" | [[:DT Fourier Transform Pair_ECE301Fall2008mboutin]] || {{:DT Fourier Transform Pair_ECE301Fall2008mboutin}}
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|-
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|-
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| align="right" style="padding-right: 1em;" | [[:DT Fourier an_ECE301Fall2008mboutin]] || {{:DT Fourier an_ECE301Fall2008mboutin}}
 +
|-
 +
|}
  
===Series===
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*[[DT Fourier Transform Pair 3_ECE301Fall2008mboutin]] {{:DT Fourier Transform Pair 3_ECE301Fall2008mboutin}}
*[[Finite Geometric Series Formula_ECE301Fall2008mboutin]]{{:Finite Geometric Series Formula}}
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*[[Infinite Geometric Series Formula_ECE301Fall2008mboutin]]{{:Infinite Geometric Series Formula}}
+
 
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===Euler's Formula===
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*[[Complex exponential in terms of sinusoidal signals_ECE301Fall2008mboutin]]{{:Complex exponential in terms of sinusoidal signals}}
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*[[Cosine function in terms of complex exponential_ECE301Fall2008mboutin]]{{:Cosine function in terms of complex exponential}}
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*[[Sine function in terms of complex exponential_ECE301Fall2008mboutin]]{{:Sine function in terms of complex exponential}}
+
 
+
== Discrete-time domain ==
+
 
+
=== Useful Formulas ===
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*[[DT Fourier Transform_ECE301Fall2008mboutin]]  {{:DT Fourier Transform}}
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*[[DT Inverse Fourier Transform_ECE301Fall2008mboutin]]  {{:DT Inverse Fourier Transform}}
+
*put a formula here following syntax described at top of page.
+
 
+
=== DT Fourier Transform Pairs ===
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*[[DT Fourier Transform Pair_ECE301Fall2008mboutin]] {{:DT Fourier Transform Pair}}
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*put a pair here following syntax described at top of page.
+
  
 
=== DT Fourier Transform Properties ===
 
=== DT Fourier Transform Properties ===
*[[DT Fourier Transform Multiplication_ECE301Fall2008mboutin]] {{:DT Fourier Transform Multiplication}}
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*[[DT Fourier Transform Multiplication_ECE301Fall2008mboutin]] {{:DT Fourier Transform Multiplication_ECE301Fall2008mboutin}}
*[[DT Fourier Transform Convolution_ECE301Fall2008mboutin]] {{:DT Fourier Transform Convolution}}
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*[[DT Fourier Transform Convolution_ECE301Fall2008mboutin]] {{:DT Fourier Transform Convolution_ECE301Fall2008mboutin}}
*[[DT Fourier Transform Time Reversal_ECE301Fall2008mboutin]] {{:DT Fourier Transform Time Reversal}}
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*[[DT Fourier Transform Time Reversal_ECE301Fall2008mboutin]] {{:DT Fourier Transform Time Reversal_ECE301Fall2008mboutin}}
*[[DT Fourier Transform Duality_ECE301Fall2008mboutin]] {{:DT Fourier Transform Duality}}
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*[[DT Fourier Transform Duality_ECE301Fall2008mboutin]] {{:DT Fourier Transform Duality_ECE301Fall2008mboutin}}
  
 
=== Parsevel Relationship for DT signals ===
 
=== Parsevel Relationship for DT signals ===
*[[Parsevel Relationship for DT signals_ECE301Fall2008mboutin]]{{:Parsevel Relationship for DT signals}}
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*[[Parsevel Relationship for DT signals_ECE301Fall2008mboutin]]{{:Parsevel Relationship for DT signals_ECE301Fall2008mboutin}}
  
 
== Continuous-time domain ==
 
== Continuous-time domain ==
  
 
=== Useful Formulas ===
 
=== Useful Formulas ===
*[[CT_Fourrier_Coefficients_ECE301Fall2008mboutin]] {{:CT_Fourrier_Coeffiecients}}
+
*[[CT Fourier Transform_ECE301Fall2008mboutin]] {{:CT Fourier Transform_ECE301Fall2008mboutin}}
*[[Time_Shifting_ECE301Fall2008mboutin]] {{:Time Shifting Property}}
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*[[CT Inverse Fourier Transform_ECE301Fall2008mboutin]] {{:CT Inverse Fourier Transform_ECE301Fall2008mboutin}}
*put a formula here following syntax described at top of page.
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*[[CT Laplace Transform_ECE301Fall2008mboutin]] {{:CT Laplace Transform_ECE301Fall2008mboutin}}
 
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*[[CT_Fourrier_Coefficients_ECE301Fall2008mboutin]] {{:CT_Fourrier_Coeffiecients_ECE301Fall2008mboutin}}
*[[CT Energy of a Signal_ECE301Fall2008mboutin]]  {{:CT Energy of a Signal}}
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*[[Time_Shifting_ECE301Fall2008mboutin]] {{:Time Shifting Property_ECE301Fall2008mboutin}}
 
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*[[CT Energy of a Signal_ECE301Fall2008mboutin]]  {{:CT Energy of a Signal_ECE301Fall2008mboutin}}
*[[CT Time-averaged Power of a Signal over an infinite interval_ECE301Fall2008mboutin]]  {{:CT Power of a Signal}}
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*[[CT Time-averaged Power of a Signal over an infinite interval_ECE301Fall2008mboutin]]  {{:CT Power of a Signal_ECE301Fall2008mboutin}}
  
 
=== CT Fourier Transform Pairs ===
 
=== CT Fourier Transform Pairs ===
*[[CT_Fourier Pairs 1_ECE301Fall2008mboutin]]{{:CT_Fourier Pairs 1}}
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*[[CT_Fourier Pairs 1_ECE301Fall2008mboutin| (1)]]{{:CT_Fourier Pairs 1_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 2_ECE301Fall2008mboutin]]{{:CT_Fourier Pairs 2}}
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*[[CT_Fourier Pairs 2_ECE301Fall2008mboutin| (2)]]{{:CT_Fourier Pairs 2_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 3_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 3}}
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*[[CT_Fourier Pairs 3_ECE301Fall2008mboutin| (3)]]  {{:CT_Fourier Pairs 3_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 4_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 4}}
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*[[CT_Fourier Pairs 4_ECE301Fall2008mboutin|(4)]]  {{:CT_Fourier Pairs 4_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 5_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 5}}
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*[[CT_Fourier Pairs 5_ECE301Fall2008mboutin|(5)]]  {{:CT_Fourier Pairs 5_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 6_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 6}}
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*[[CT_Fourier Pairs 6_ECE301Fall2008mboutin|(6)]]  {{:CT_Fourier Pairs 6_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 7_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 7}}
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*[[CT_Fourier Pairs 7_ECE301Fall2008mboutin|(7)]]  {{:CT_Fourier Pairs 7_ECE301Fall2008mboutin}}
*[[CT_Fourier Pairs 8_ECE301Fall2008mboutin]]  {{:CT_Fourier Pairs 8}}
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*[[CT_Fourier Pairs 8_ECE301Fall2008mboutin|(8)]]  {{:CT_Fourier Pairs 8_ECE301Fall2008mboutin}}
 +
*[[CT_Fourier Pairs 9_ECE301Fall2008mboutin|(9)]]  {{:CT_Fourier Pairs 9_ECE301Fall2008mboutin}}
 +
*[[CT_Fourier Pairs 10_ECE301Fall2008mboutin|(10)]]  {{:CT_Fourier Pairs 10_ECE301Fall2008mboutin}}
 +
*[[CT_Fourier Pairs 11_ECE301Fall2008mboutin|(11)]]  {{:CT_Fourier Pairs 11_ECE301Fall2008mboutin}}
 +
*[[CT_Fourier Pairs 12_ECE301Fall2008mboutin|(12)]]  {{:CT_Fourier Pairs 12_ECE301Fall2008mboutin}}
  
 
=== CT Fourier Transform Properties ===
 
=== CT Fourier Transform Properties ===
*[[CT_Fourier_Int/Diff_ECE301Fall2008mboutin]]{{:CT_Fourier_Int/Diff}}
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*[[CT_Fourier_Int/Diff_ECE301Fall2008mboutin]]{{:CT_Fourier_Int/Diff_ECE301Fall2008mboutin}}
 +
 
 
*[[CT Time and Frequency Scaling_ECE301Fall2008mboutin]] : <math>x(at) \leftarrow \rightarrow \frac{1}{|a|}X(\frac{j\omega }{a})\,</math>
 
*[[CT Time and Frequency Scaling_ECE301Fall2008mboutin]] : <math>x(at) \leftarrow \rightarrow \frac{1}{|a|}X(\frac{j\omega }{a})\,</math>
 +
 +
*[[CT Differentiation in Frequency_ECE301Fall2008mboutin]]{{:CT Differentiation in Frequency_ECE301Fall2008mboutin}}
 +
 
*[[CT Convolution_ECE301Fall2008mboutin]] : <math> F(x_1(t)*x_2(t)) = X_1(\omega)X_2(\omega) \!</math>
 
*[[CT Convolution_ECE301Fall2008mboutin]] : <math> F(x_1(t)*x_2(t)) = X_1(\omega)X_2(\omega) \!</math>
 +
 
*[[CT Frequency Shifting_ECE301Fall2008mboutin]] : <math> F(e^{jw0t}x(t)) = X(j(w - w0)) \!</math>
 
*[[CT Frequency Shifting_ECE301Fall2008mboutin]] : <math> F(e^{jw0t}x(t)) = X(j(w - w0)) \!</math>
{{:CT Multiplication Property}}
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*[[CT Time Reversal_ECE301Fall2008mboutin]]{{:CT Time Reversal}}
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{{:CT Multiplication Property_ECE301Fall2008mboutin}}
*[[CT Multiplication Property Mimis Version_ECE301Fall2008mboutin]]{{:CT Multiplication Property Mimis Version}}
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*[[CT Duality Property_ECE301Fall2008mboutin]] :<math> F(x(t)) = X(w) = 2\pi x(-w) \!</math> {{:CT Duality Property}}
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*[[CT Time Reversal_ECE301Fall2008mboutin]]{{:CT Time Reversal_ECE301Fall2008mboutin}}
 +
 
 +
*[[CT Multiplication Property Mimis Version_ECE301Fall2008mboutin]]{{:CT Multiplication Property Mimis Version_ECE301Fall2008mboutin}}
 +
 
 +
*[[CT Duality Property_ECE301Fall2008mboutin]] :<math> F(x(t)) = X(w) = 2\pi x(-w) \!</math> {{:CT Duality Property_ECE301Fall2008mboutin}}
 +
 
 +
*[[CT Conjugate Symmetry_ECE301Fall2008mboutin]] {{:CT Conjugate Symmetry_ECE301Fall2008mboutin}}
  
 
=== Parsevel Relationship for CT signals ===
 
=== Parsevel Relationship for CT signals ===
*[[Parsevel Relationship for CT signals_ECE301Fall2008mboutin]]{{:Parsevel Relationship for CT signals}}
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*[[Parsevel Relationship for CT signals_ECE301Fall2008mboutin]]{{:Parsevel Relationship for CT signals_ECE301Fall2008mboutin}}
  
*[[Parseval's Relation for Aperiodic Signals_ECE301Fall2008mboutin]]{{:Parseval's Relation for Aperiodic Signals}}
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*[[Parseval's Relation for Aperiodic Signals_ECE301Fall2008mboutin]]{{:Parseval's Relation for Aperiodic Signals_ECE301Fall2008mboutin}}
 
   
 
   
 
*put a property here following syntax described at top of page.
 
*put a property here following syntax described at top of page.
 +
 +
== Some Laplace Transform Pairs ==
 +
{{:LaplaceTransforms_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 1_ECE301Fall2008mboutin| (1)]]{{:Laplace Pairs 1_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 2_ECE301Fall2008mboutin| (2)]]{{:Laplace Pairs 2_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 3_ECE301Fall2008mboutin| (3)]]{{:Laplace Pairs 3_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 4_ECE301Fall2008mboutin| (4)]]{{:Laplace Pairs 4_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 5_ECE301Fall2008mboutin| (5)]]{{:Laplace Pairs 5_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 6_ECE301Fall2008mboutin| (6)]]{{:Laplace Pairs 6_ECE301Fall2008mboutin}}
 +
*[[Laplace Pairs 7_ECE301Fall2008mboutin| (7)]]{{:Laplace Pairs 7_ECE301Fall2008mboutin}}

Latest revision as of 06:20, 5 February 2009


Please follow the following model to add your formulas:

Create a page with a descriptive name and type your formula on this page using latex. Then write
{{:name of the page with your formula}}

in the place where you want your formula to appear in this table. (Look at the syntax of the geometric series below for an example.) This will allow other people to refer to your formula later on (by refering to the corresponding page) while still being able to view all formulas on this page.

General Purpose Formulas
Series
Finite Geometric Series Formula_ECE301Fall2008mboutin $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $
Infinite Geometric Series Formula_ECE301Fall2008mboutin $ \sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $
Euler's Formula
Complex exponential in terms of sinusoidal signals_ECE301Fall2008mboutin $ e^{jw_0t}=cosw_0t+jsinw_0t $
Cosine function in terms of complex exponential_ECE301Fall2008mboutin $ cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2} $
Sine function in terms of complex exponential_ECE301Fall2008mboutin $ sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j} $
Other
sinc function_ECE301Fall2008mboutin $ sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta} $
Discrete-Time Domain
Useful Formulas
DT Fourier Transform_ECE301Fall2008mboutin $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $
DT Inverse Fourier Transform_ECE301Fall2008mboutin $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $
DT Fourier Transform Pairs
DT Fourier Transform Pair_ECE301Fall2008mboutin $ e^{jw_0n} \longrightarrow 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $
DT Fourier an_ECE301Fall2008mboutin $ a^{n} u[n], |a|<1 \longrightarrow \frac{1}{1-ae^{-j\omega}} \ $

DT Fourier Transform Properties

Parsevel Relationship for DT signals

Continuous-time domain

Useful Formulas

CT Fourier Transform Pairs

  • (1) $ x(t)= \sum^{\infty}_{k=-\infty} a_{k}e^{jkw_{0}t} \longrightarrow {\mathcal X}(\omega)= 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0})\, $
  • (2)$ x(t)=\sum^{\infty}_{n=-\infty} \delta(t-nT) \longrightarrow {\mathcal X}(\omega)= \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})\, $
  • (3) $ x(t)=\cos(\omega_0 t) \longrightarrow {\mathcal X}(\omega)=\pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] $
  • (4) $ x(t)=sin(\omega_0 t) \longrightarrow {\mathcal X}(\omega)=\frac{\pi}{j} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $
  • (5) $ x(t)=\delta (t)\longrightarrow {\mathcal X}(\omega)=1 \! $
  • (6) $ x(t)= u(t)\longrightarrow {\mathcal X}(\omega)= \frac{1}{jw} + \pi \delta (w) \! $
  • (7) $ x(t)=\delta (t-t_0)\longrightarrow {\mathcal X}(\omega)= e^{jwt_0} \! $
  • (8) $ x(t)=e^{-at}u(t),\text{ where }a\text{ is real,}a>0 \longrightarrow {\mathcal X}(\omega)=\frac{1}{a+j\omega} $
  • (9) $ x(t)=e^{j\omega_0 t} \longrightarrow {\mathcal X}(\omega)= 2\pi \delta (\omega - \omega_0) $
  • (10) $ x(t)=te^{-at}u(t), \text{ where }a\text{ is real,} a>0 \longrightarrow {\mathcal X}(\omega)=(\frac{1}{a+j\omega})^2 $
  • (11) $ x(t)=\left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \longrightarrow {\mathcal X}(\omega)=\frac{2 \sin \left( T \omega \right)}{\omega} $
  • (12) $ x(t)=\frac{2 \sin \left( W t \right)}{\pi t } \longrightarrow \mathcal{X}(\omega)=\left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $

CT Fourier Transform Properties

  • CT_Fourier_Int/Diff_ECE301Fall2008mboutin$ \; \; \; (1)\frac{dx(t)}{dt} \rightarrow j\omega \Chi (\omega)\; \; \; \; \; \; (2) \int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{1}{j\omega}\Chi (\omega) + \pi \Chi (0) \delta (\omega) $

$ F(x(t)y(t))=\frac{1}{2\pi}X(j\omega)*Y(j\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\theta)Y(j(\omega-\theta))d\theta $

$ F(x(t)) = X(w) = 2\pi x(-w) \! $

if

$ \ F(x(t)) = X(w) $

then,

$ \ F(x(t)^*) = X^*(-w) $

Parsevel Relationship for CT signals

  • put a property here following syntax described at top of page.

Some Laplace Transform Pairs

Laplace Transform Pairs
Transform Pair Signal Transform ROC
1 $ \,\!\delta(t) $ $ 1 $ $ All\,\, s $
2 $ \,\! u(t) $ $ \frac{1}{s} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
3 $ \,\! -u(-t) $ $ \frac{1}{s} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $
4 $ \frac{t^{n-1}}{(n-1)!}u(t) $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
5 $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $
6 $ \,\!e^{-\alpha t}u(t) $ $ \frac{1}{s+\alpha} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
7 $ \,\! -e^{-\alpha t}u(-t) $ $ \frac{1}{s+\alpha} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $
8 $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ $ \frac{1}{(s+\alpha )^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
9 $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ $ \frac{1}{(s+\alpha )^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $
10 $ \,\!\delta (t - T) $ $ \,\! e^{-sT} $ $ All\,\, s $
11 $ \,\![cos( \omega_0 t)]u(t) $ $ \frac{s}{s^2+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
12 $ \,\![sin( \omega_0 t)]u(t) $ $ \frac{\omega_0}{s^2+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
13 $ \,\![e^{-\alpha t}cos( \omega_0 t)]u(t) $ $ \frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
14 $ \,\![e^{-\alpha t}sin( \omega_0 t)]u(t) $ $ \frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
15 $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ $ \,\!s^{n} $ $ All\,\, s $
16 $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
  • (1)$ \delta(t) \leftrightarrow 1 $, for all s
  • (2)$ \ u(t) \leftrightarrow \frac{1}{s} $, for Re{s} > 0
  • (3)$ \ -u(-t) \leftrightarrow \frac{1}{s} $, for Re{s} < 0
  • (4)$ \frac{t^{n - 1}}{(n - 1)!}u(t) \leftrightarrow \frac{1}{s^{n}} $, for Re{s} > 0
  • (5)$ - \frac{t^{n - 1}}{(n - 1)!}u(-t) \leftrightarrow \frac{1}{s^{n}} $, for Re{s} < 0
  • (6)$ \ e^{\alpha t }u(t) \leftrightarrow \frac{1}{s + \alpha} $, for Re{s} > $ \ - \alpha $
  • (7)$ \ -e^{\alpha t }u(-t) \leftrightarrow \frac{1}{s + \alpha} $, for Re{s} < $ \ - \alpha $

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva