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Unformatted text preview: DSP 1 Home Work # 1 Sriram Yarlagadda [email protected] sid # 2196944 PROBLEM1 Given second order differential equation d 2 f ( t ) dt 2 2 o df ( t ) dt o 2 f ( t ) o 2 ( t ) (1) Applying Laplace Transforms on both sides L { d 2 f ( t ) dt 2 } s 2 f ( s ) sf (0) f (0) L { df ( t ) dt 2 } sf ( s ) f (0) L { f ( t )} f ( s ) Since ( t ) represents an impulse function on applying Laplace transform to ( t ) a unit function is obtained L { ( t )} 1 Given initial conditions to be zero Therefore Therefore the above equations transform as follows L { d 2 f ( t ) dt 2 } s 2 f ( s ) L { df ( t ) dt } sf ( s ) L { f ( t )} f ( s ) Substituting these obtained Laplace Transforms in equation 1 i.e. second order differential equation s 2 f ( s ) 2 o sf ( s ) o 2 f ( s ) o 2 1 Taking f(s) common on LHS we have f ( s ){ s 2 2 o s o 2 } o 2 f (0) f (0) f ( s ) o 2 s 2 2 o s o 2(2) finding the roots for denominator equation in denominator is similar to ax 2 bx c roots for ax 2 bx c 0 are b b 2 4 ac 2 a Similarly the roots for equation 2 are 2 1 o o j The equation 2 can be written as follows ) 1 ( ) ( ) ( 2 2 2 2 o o o s s f Applying Inverse Laplace Transforms on both sides } ) 1 ( ) ( { )} ( { 2 2 2 2 1 1 o o o s L s f L(3) L 1 { f ( s )} f ( t ) RHS of equation 3 is of form at a s a L sin } { 2 2 1 Multiplying and Dividing equation 3 by 2 2 ) 1 ( } ) 1 ( ) ( ) 1 ( ) 1 ( 1 { )} ( { 2 2 2 2 2 2 1 1 o o o s L s f L Here s s o and a= ) 1 ( 2 o and From Shifting property L 1 { 1 s a } e at Therefore t e t f o t o o ) 1 sin( 1 ) ( 2 2 Hence the Laplace transform of second order differential equation d 2 f ( t ) dt 2 2 o df ( t ) dt o 2 f ( t ) o 2 ( t ) is t e t f o t o o ) 1 sin( 1 ) ( 2 2 (4) (II) MATLAB is used to plot the f(t) obtained above and the values of and o are chosen such that the system is stable i.e. the poles should be in the left hand side of the S plane MATLAB PROGRAM clc clear all; close all; W=2.4; E=0.4; t=0:0.01:10; a=sqrt(1(E^2)); b=(W/a)*exp(1*W*E.*t).*sin(W*a.*t); plot(t,b) title('Response for Second order differential equation'); xlabel('time period t'); ylabel('amplitudeof response f(t)'); Plot (III) To plot the normalized power spectrum (0 to 60 db scale) and phase spectrum, we compute the Fourier transform of f(t) ....
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 Spring '10
 Dr.RobertoJAcousta
 Digital Signal Processing, Signal Processing, Laplace, order differential equation

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