Σήματα (lesson 5)

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Laplace

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<ul><li><p>. Laplace </p><p> . </p><p> 2012 </p></li><li><p> &amp; E2 </p><p>E. Laplace </p><p>.1 Fourier </p><p>Laplace </p><p>.2 Laplace </p><p>E.3 Laplace </p><p>E.4 LTI </p><p>. </p><p>E.5 Laplace </p><p>E.6 </p></li><li><p> &amp; E3 </p><p>E. Laplace </p><p>.1 / Fourier </p><p>/ Laplace </p></li><li><p> &amp; E4 </p><p>E.1 / Fourier / Laplace </p><p> Laplace </p><p> Laplace </p><p> H </p><p> Laplace </p><p> Laplace Fourier </p></li><li><p> &amp; E5 </p><p>E.1 / Fourier / Laplace </p><p> Laplace </p><p> Fourier </p><p> () </p><p> ejt. </p><p> / Fourier </p><p> o / Fourier </p><p> eatu(t), a &gt; 0, </p><p> / Fourier, ejt </p><p> . </p><p>( )st j te e </p></li><li><p> &amp; E6 </p><p>E.1 / Fourier / Laplace </p><p> Laplace </p><p> / </p><p>Fourier. </p><p> Fourier. </p><p> () ejt. </p><p> ( </p><p> ) et. </p><p> () </p><p>2( ) ( )tx t e u t</p><p>2( ) ( ) ( ), 2t t tf t x t e e e u t </p><p>( )f t</p><p>( )x t</p><p>( ) , Re( ) 2t jt j t ste e e e s </p></li><li><p> &amp; E7 </p><p>E.1 / Fourier / Laplace </p><p> Laplace 2( ) ( )tx t e u t ( ) ( ) , 2</p><p>tf t x t e </p><p> ejt f(t) etejt </p><p> . </p><p> &gt; 2 </p></li><li><p> &amp; E8 </p><p>E.1 / Fourier / Laplace </p><p> (bilateral) / Laplace </p><p> / Fourier x(t) </p><p> / Fourier </p><p> / Fourier </p><p> et </p><p> s = + j ( </p><p> j j ), 1</p><p>( ) ( )2</p><p>( ) ( )</p><p>c j st</p><p>c j</p><p>st</p><p>x t F s e dsj</p><p>X s x t e dt</p><p>12</p><p>( ) ( ) jt</p><p>x t X j e d</p><p>( ) ( ) tf t x t e</p><p>( )[ ( ) ] ( ) ( ) ( )t t jt j tx t e x t e e dt x t e dt X j </p><p> F</p><p>12</p><p>( ) ( )t jt</p><p>x t e X j e d</p><p>( )12</p><p>( ) ( ) j t</p><p>x t X j e d </p></li><li><p> &amp; E9 </p><p>E.1 / Fourier / Laplace </p><p>H </p><p> / Laplace </p><p>1( ) ,</p><p>2Re{ } 2</p><p>X ss</p><p>s</p><p>2( ) ( )tx t e u t</p><p>2( ) ( )ty t e u t 1</p><p>( ) ,2</p><p>Re{ } 2</p><p>Y ss</p><p>s</p></li><li><p> &amp; E10 </p><p>E.1 / Fourier / Laplace </p><p> (unilateral) / Laplace </p><p> , / Laplace, </p><p>, . </p><p> , </p><p>/ Laplace . </p><p>/ Laplace </p><p>, </p><p> . </p><p> / Laplace </p><p> Re{s} = . , </p><p> &gt; 0. </p><p>0( ) ( ) stX s x t e dt</p><p>0| ( ) |tx t e dt</p></li><li><p> &amp; E11 </p><p>E. Laplace </p><p>.2 </p><p>Laplace </p></li><li><p> &amp; E12 </p><p>E.2 Laplace </p><p> . </p></li><li><p> &amp; E13 </p><p>E.2 Laplace </p><p> Laplace </p><p> / Laplace </p><p>1 1 2 2</p><p>1 1 2 2 1 1 2 2</p><p>( ) ( ) ( ) ( )</p><p>( ) ( ) ( ) ( )</p><p>x t X s x t X s</p><p>k x t k x t k X s k X s</p><p>L L</p><p>L</p><p>1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t</p><p>00 0 0</p><p>( ) ( ) ( )</p><p>( ) ( ) ( ) ,st</p><p>x t u t X s</p><p>x t t u t t X s e t t</p><p>L</p><p>L</p><p>00</p><p>( ) ( )</p><p>( ) ( )s t</p><p>x t X s</p><p>x t e X s s</p><p>L</p><p>L</p></li><li><p> &amp; E14 </p><p>E.2 Laplace </p><p>( )1 2 (1) ( 1)</p><p>( ) ( )</p><p>( )( ) (0 )</p><p>( )( ) (0 ) (0 ) (0 )</p><p>nn n n n</p><p>n</p><p>x t X s</p><p>dx tsX s x</p><p>dt</p><p>d x ts X s s x s x x</p><p>dt</p><p>L</p><p>L</p><p>L</p><p>1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t</p><p>( )</p><p>( ) ( )</p><p>( )( ) ( 1)</p><p>nn n</p><p>n</p><p>x t X s</p><p>d X st x t</p><p>ds</p><p>L</p><p>L</p><p>0</p><p>0</p><p>( ) ( )</p><p>( )( )</p><p>1 1( ) ( ) ( )</p><p>t</p><p>t</p><p>x t X s</p><p>X sx d</p><p>s</p><p>x d X s x ds s</p><p>L</p><p>L</p><p>L</p><p>( ) ( )</p><p>( )( )</p><p>s</p><p>x t X s</p><p>x tX z dz</p><p>t</p><p>L</p><p>L</p></li><li><p> &amp; E15 </p><p>E.2 Laplace </p><p> / Laplace. </p><p>( ) ( )</p><p>1( ) , 0</p><p>x t X s</p><p>sx at X a</p><p>a a</p><p>L</p><p>L</p><p>1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t</p><p>( ) ( ) ( ) ( )</p><p>( ) ( ) ( ) ( )</p><p>x t X s y t Y s</p><p>x t y t X s Y s</p><p>L L</p><p>L</p><p>( ) ( ) ( ) ( )</p><p>1( ) ( ) ( ) ( )</p><p>2</p><p>x t X s y t Y s</p><p>x t y t X s Y sj</p><p>L L</p><p>L</p></li><li><p> &amp; E16 </p><p>E.2 Laplace </p><p> x(t) dx/dt Laplace, </p><p> . </p><p> x(t) dx/dt Laplace, </p><p> sX(s) </p><p>. </p><p>(0 ) lim ( )s</p><p>x sX s</p><p>1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t1 1 2 2( ) ( )k x t k x t</p><p>0lim ( ) lim ( )t s</p><p>x t sX s </p></li><li><p> &amp; E17 </p><p>E. Laplace </p><p>.3 </p><p> Laplace </p></li><li><p> &amp; E18 </p><p>E.3 Laplace </p></li><li><p> &amp; E19 </p><p>E.3 Laplace </p><p> / Laplace </p><p> Laplace </p><p> , . </p><p> / Laplace , </p><p> s </p><p> m n. m n, ( ) </p><p> (m n). </p><p> / </p><p>Laplace . </p><p>11 1 0</p><p>11 1 0</p><p>( )( )</p><p>( )</p><p>m mm mn n</p><p>n</p><p>b s b s b s bP sX s</p><p>Q s s a s a s a</p><p>1( ) ( )</p><p>2</p><p>c j st</p><p>c jx t F s e ds</p><p>j</p></li><li><p> &amp; E20 </p><p>E.3 Laplace </p><p> kj </p><p> s . </p><p>11 1 0</p><p>11 1 0</p><p>( )( ) ,</p><p>( )</p><p>m mm mn n</p><p>n</p><p>b s b s b s bP sX s m n</p><p>Q s s a s a s a</p><p>1 1</p><p>11</p><p>( )( )</p><p>( ) ( ) ( ) ( )</p><p>r jr rr r</p><p>jj</p><p>kk k kP sX s</p><p>s s s s s s s </p><p>1( ) ( ) ( )r</p><p>js s s </p></li><li><p> &amp; E21 </p><p>E.3 Laplace </p><p> Heaviside </p><p> ( </p><p>) </p><p>. </p><p>11 1 0</p><p>11 1 0</p><p>( )( ) ,</p><p>( )</p><p>m mm mn n</p><p>n</p><p>b s b s b s bP sX s m n</p><p>Q s s a s a s a</p><p>1 2</p><p>1 2 1 2</p><p>( )( )</p><p>( )( ) ( )</p><p>n</p><p>n n</p><p>kk kP sX s</p><p>s s s s s s </p><p>( ) ( )i</p><p>i i s k s X s</p></li><li><p> &amp; E22 </p><p>E.3 Laplace </p><p> , </p><p> short-cuts. </p><p>21 0</p><p>( )( )</p><p>( )( )</p><p>p sX s</p><p>q s s a s a</p><p>21 0s a s a </p><p>21 0</p><p>As B</p><p>s a s a</p></li><li><p> &amp; E23 </p><p>E.3 Laplace </p><p> kj Heaviside </p><p> ci </p><p> Laplace </p><p>( 1)</p><p>1</p><p>1[( ) ( )]</p><p>( 1)!</p><p>ir</p><p>i i</p><p>s </p><p>dc s X s</p><p>i ds</p><p>1 21</p><p>1 2</p><p>1 2</p><p>1 2</p><p>( )( )</p><p>( )( ) ( )( ) ( ) ( ) ( )</p><p>rr r r</p><p>j</p><p>j</p><p>j</p><p>c c cP sX s</p><p>s s s s s s s </p><p>kk k</p><p>s s s </p></li><li><p> &amp; E24 </p><p>E. Laplace </p><p>.4 </p><p> LTI </p></li><li><p> &amp; E25 </p><p>E.4 / LTI </p><p> LTI </p></li><li><p> &amp; E26 </p><p>E.4 / LTI </p><p> / </p><p>Laplace </p><p> - </p><p> . </p><p> Laplace. </p><p>, , </p><p> , </p><p> , </p><p> / Laplace </p><p>( ) ( 1)1 2</p><p>1</p><p>( )( ) (0 ) (0 ) (0 )</p><p>n nn n n</p><p>n n</p><p>d x t d ds X s s x s x x</p><p>dtdt dt</p><p>L</p><p>01 1( ) ( ) ( )</p><p>tx d X s x d</p><p>s s</p><p>L</p></li><li><p> &amp; E27 </p><p>E.4 / LTI </p><p> LTI </p><p> LTI </p><p> / Laplace, </p><p> (s) </p><p> s. </p><p>( ) ( ) ( ) ( )</p><p>( ) ( ) ( )</p><p>x t X s y t Y s</p><p>Y s H s X s</p><p>L L</p><p>( )x t ( )y t</p><p>( )X s ( )Y s( )H s</p><p>( )( )</p><p>( )Y s</p><p>H sX s</p><p>( )( ) | ( ) | Re{ ( )} Im{ ( )}, j H sH s H s e H s j H s s j</p></li><li><p> &amp; E28 </p><p>E.4 / LTI </p><p> LTI </p><p> s = 0 + j </p><p> , .. </p><p> . </p><p> . </p><p> / Laplace. </p><p>( )( ) | ( ) | ( ) | j H j</p><p>s jH s H j H j e</p><p>11 1 0</p><p>11 1 0</p><p>1 2</p><p>1 2</p><p>...( )</p><p>...</p><p> ( )( ) ( ),</p><p> ( )( ) ( )</p><p>m mm mn n</p><p>n</p><p>im</p><p>in</p><p>b s b s b s bH s</p><p>s a s a s a</p><p>zs z s z s z</p><p>ps p s p s p</p></li><li><p> &amp; E29 </p><p>E.4 / LTI </p><p> LTI </p><p> (magnitude) |H(s)| </p><p> 2</p><p>2</p><p>2 8( )</p><p>3 2</p><p>sH s</p><p>s s</p><p> 2</p><p>( )9</p><p>sH s</p><p>s</p></li><li><p> &amp; E30 </p><p>Buy SmartDraw!- purchased copies print this </p><p>document without a watermark .</p><p>Visit www.smartdraw.com or call 1-800-768-3729.</p><p>E.4 / LTI </p><p> LTI </p><p>20</p><p>02 20</p><p>( )</p><p>H s</p><p>s s Q</p><p>Buy SmartDraw!- purchased copies print this </p><p>document without a watermark .</p><p>Visit www.smartdraw.com or call 1-800-768-3729.</p><p>Buy SmartDraw!- purchased copies print this </p><p>document without a watermark .</p><p>Visit www.smartdraw.com or call 1-800-768-3729.</p><p>2</p><p>02 20</p><p>( )s</p><p>H s</p><p>s s Q</p><p>Buy SmartDraw!- purchased copies print this </p><p>document without a watermark .</p><p>Visit www.smartdraw.com or call 1-800-768-3729.</p><p>0</p><p>02 20</p><p>( )</p><p>s</p><p>QH s</p><p>s s Q</p><p>2 2</p><p>02 20</p><p>( ) zs </p><p>H s</p><p>s s Q</p><p>02 20</p><p>02 20</p><p>( )</p><p>s s </p><p>QH s</p><p>s s Q</p><p>Buy SmartDraw!- purchased copies print this </p><p>document without a watermark .</p><p>Visit www.smartdraw.com or call 1-800-768-3729.</p></li><li><p> &amp; E31 </p><p>E.4 / LTI </p><p> LTI </p><p> LTI </p><p> (bounded-input, bounded-output BIBO) </p><p> , . </p><p> LTI BIBO , </p><p>. , BIBO </p><p> . </p><p> LTI </p><p>, </p><p> . , </p><p> , BIBO </p><p>. </p></li><li><p> &amp; E32 </p><p>E. Laplace </p><p>.5 / Laplace </p></li><li><p> &amp; E33 </p><p>E.5 / Laplace </p><p> L : </p><p> C : </p><p> R </p><p> Kircchoff </p><p> Kircchoff </p><p>( )( ) ( ) ( )</p><p>di t t L V s LsI s</p><p>dt </p><p>L</p><p>( ) 1( ) ( ) ( ) ( ) ( )</p><p>d ti t C I s CsV s V s I s</p><p>dt Cs </p><p>L</p><p>( ) ( ) ( ) ( ) t Ri t V s RI s L</p><p>1 1</p><p>( ) 0 ( ) 0k k</p><p>j j</p><p>j j</p><p> t V s </p><p> L</p><p>1 1</p><p>( ) 0 ( ) 0k k</p><p>j j</p><p>j j</p><p>i t I s </p><p> L</p></li><li><p> &amp; E34 </p><p>E.5 / Laplace </p><p> L i(0): </p><p>( )( ) ( ) [ ( ) (0)]</p><p>( ) (0)</p><p>(0)[ ( ) ]</p><p>di t t L V s L sI s i</p><p>dtLsI s Li</p><p>iLs I s</p><p>s</p><p>L</p><p>(0)i</p><p>s</p></li><li><p> &amp; E35 </p><p>E.5 / Laplace </p><p> C (0): </p><p> / Laplace </p><p>( ) 1 (0)( ) ( ) [ ( ) (0)] ( ) ( )</p><p>1[ ( ) (0)]</p><p>d t i t C I s C sV s V s I s</p><p>dt Cs s</p><p>I s CCs</p><p>L</p><p>(0)</p><p>s</p><p>1</p><p>Cs 1</p><p>Cs</p></li><li><p> &amp; E36 </p><p>E. Laplace </p><p>.6 </p></li><li><p> &amp; E37 </p><p>E.6 </p><p> , </p><p> . </p><p> . </p><p> , </p><p> . </p><p>( )X s ( )H s ( )Y s</p><p>( )X s 1( )H s 2( )H s ( )Y s 1 2( ) ( )H s H s( )X s ( )Y s= </p></li><li><p> &amp; E38 </p><p>E.6 </p><p>1( )H s</p><p>2( )H s</p><p>( )X s ( )Y s ( )X s 1 2( ) ( )H s H s ( )Y s= </p><p>( )X s</p><p> ( )G s</p><p>( )H s</p><p>( )Y s ( )X s( )</p><p>1 ( ) ( )</p><p>G s</p><p>G s H s( )Y s= </p></li></ul>