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Assignment 4 soultion

Assignment 4 soultion - School of Engineering Science Simon...

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School of Engineering Science Simon Fraser University ENSC-380, Spring 2009 Solution of Assignment 4 Solution of Problem 1. This can be done using two methods: (1) do the convolution first and then take the CTFT of the result, or (2) compute the CTFT of each term seperately to avoid the convolution and then multiply the obtained CTFTs. You could try the first approach as an exercise. Here is the solution obtained using the second method: 5 cos( t ) * δ ( t + 1) ←→ F ( 5 cos( t ) ) F ( δ ( t + 1) ) (1) ←→ F ( 5 cos( t ) ) F ( δ ( t ) ) e j 2 πf time-shifting (2) ←→ 5 2 £ δ ( f - 1 2 π ) + δ ( f + 1 2 π ) / e j 2 πf using FT table (3) (4) Solution of Problem 2. As explained in page 319 of the textbook, if the time-domain signal is real-valued, its CTFT has the property that the behavior for the negative frequencies if the complex conjugate of the behavior for positive frequencies. Now let x ( t ) be a real-valued signal. The magnitude of X ( f ) is | X ( f ) | = X ( f ) X * ( f ) . Then using X ( f ) = X * ( - f ), as explained above, we can show that the magnitude of X ( - f ) is | X ( - f ) | = X ( - f ) X * ( - f ) (5) = X * ( f ) X ( f ) (6) = | X ( f ) | . (7) Solution of Problem 3. (a) The answer can be obtained simply by using the time-shifting property of DTFT: tri[
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