Solutions_hw1

Solutions_hw1 - Problem 1 1.1 Partial fraction expansion of...

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Problem 1 1.1 Partial fraction expansion of 200 ( s 2 +25)( s +8) Multyplying the equation with ( s + 8) yields 200 s 2 + 25 = ( s + 8) ± K s - 5 j + K * s + 5 j ² + a a = 200 s 2 + 25 - ( s + 8) ± K s - 5 j + K * s + 5 j ² . Setting s = - 8, you get a = 200 64 + 25 = 200 89 . Analogously, multiply the equation with ( s - 5 j ), remember that s 2 + 25 = ( s - 5 j )( s + 5 j ) and obtain 200 ( s + 5 j )( s + 8) = ( s - 5 j ) ± a s + 8 + K * s + 5 j ² + K Setting s = 5 j , you get K = 20 89 ( - 5 - 8 j ) . Along the same lines you obtain K * = 20 89 ( - 5 + 8 j ) . You can check your results even if you do not own the Symbolic Toolbox: s = tf(’s’); K = -20/89*(5+8*j); Ks = conj(K); a = 200/89; G = K/(s-5*j) + Ks/(s+5*j) + a/(s+8) % Ignore the warnings % Transfer function: % 200 % ------------------------ % s^3 + 8 s^2 + 25 s + 200 % This transfer function is identical to the one we expanded iff the poles, % zeros and dcgain are the same. pole(G) dcgain(G) 1.2 Convolution of sin(5 t ) with e - 8 t We are going to use the notation from the book, so 1 ( t ) denotes the unit step, and F ( s ) = L { f ( t ) } denotes the one-sided Laplace Transform of f ( t ). Thus we also need to assume that both signals are one-sided, i.e. zero for 1
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negative times. We know that L ± 1 ( t ) sin(5 t ) * 1 ( t ) e - 8 t ² = L { 1 ( t ) sin(5 t ) } L ± 1 ( t ) e - 8 t ² = 5 s 2 + 25 · 1 s + 8 = 1 40 200 ( s 2 + 25)( s + 8) (using 1.1) = 1 40 ³ K s - 5 j + K * s + 5 j + a s + 8 ´ , and thus 1 ( t ) sin(5 t ) * 1 ( t ) e - 8 t = 1 40 L - 1 µ K s - 5 j + K * s + 5 j + a s + 8 = 1 40 ( Ke - 5 jt + K * e 5 jt + ae - 8 t ) = 1 40 ³ Re( K ) ( e - 5 jt + e 5 jt ) - Im( K ) j ( e - 5 jt - e 5 jt ) + ae - 8 t ´ = 1 40 ( 2 Re( K ) cos(5 t ) + 2 Im( K ) sin(5 t ) + ae - 8 t ) . Reinserting the values for K , K * and a that we computed in 1.1 yields the result: 1 ( t ) sin(5 t ) * 1 ( t ) e - 8 t = 1 89 ( - 5 cos(5 t ) + 8 sin(5 t ) + 5 e - 8 t ) .
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This note was uploaded on 08/03/2010 for the course ECE PROF. VOLK taught by Professor Volkanrodoplu during the Spring '10 term at UCSB.

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Solutions_hw1 - Problem 1 1.1 Partial fraction expansion of...

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