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ete_221_spring2010_finals

# ete_221_spring2010_finals - rect t τ using the Fourier...

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North South University ETE-221 Sec. 1+2 Spring 2010, Final Exam Marks-40 Time- 90min 15th April,2010 Answer all of the following. 1. (a) Define a causal LTI system. (b)Find the convolution of the functions x 1 ( t ) = u ( t - a ) - u ( t + a ) and x 2 ( t ) = Δ( t 2 a ) graphically. You need to explain your choice for the different sections of the graphs in your answer. (c) Find the zero-input response of the system: ( D 4 + 1) y ( t ) = D 3 x ( t ) for the initial conditions x (0) = D 2 x (0) = 0 and Dx (0) = 1. What is the zero state response function for this system? Discuss the asymptotic stability of this system. (4+4+8)=16 2. (a) Prove the duality property of Fourier transformations. (b) Find the fourier transformation of
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Unformatted text preview: rect ( t τ ) using the Fourier transform of u ( t ) and time-shifting property. (c) Using the time-diﬀerentiation property ﬁnd the Fourier transform of Δ( t 2 τ ). 7+8+9=24 Some Properties of Fourier Transformation F ( k 1 f 1 ( t ) + k 2 f 2 ( t )) = k 1 F 1 ( ω ) + k 2 F 2 ( ω ) , F ( F ( t )) = 2 πf (-ω ) , F [ df ( t ) dt ] = jωF ( ω ) F ( f ( t-t )) = F ( ω ) e-jωt , F ( e jta f ( t )) = F ( ω-a ) F ( f 1 ( t ) * f 2 ( t )) = F 1 ( ω ) F 2 ( ω ) , Some Very Basic Fourier Transformation F ( e-at u ( t )) = 1 a + jω , F ( δ ( t )) = 1 F ( u ( t )) = πδ ( ω ) + 1 jω...
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