10Fall_hw10soln - ECE210 - Fall 2010 - Homework 10 (Prof....

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Unformatted text preview: ECE210 - Fall 2010 - Homework 10 (Prof. Allen) Solutions For all the following problems, it is assumed that f ( t ) ←→ F ( ω ) unless otherwise indicated. 1. Problem 8.1 from text. 2. Problem 8.3 from text. 3. Problem 8.4 from text. 4. Use the "symmetry property" to nd another FT relationship, starting from: f ( t ) = δ ( t- T o ) ←→ e- jωT o = F ( ω ) Solution: We see that F ( t ) = e- jtT o . By the symmetry property, F ( t ) ←→ 2 πf (- ω ) = 2 πδ (- ω- T o ) . By the time scaling property, we obtain the Fourier transform pair of item # 17 from Table 7.2: e jtT o ←→ 2 πδ ( ω- T o ) , 1 where T o can be replaced with ω o . 5. Given a FT pair f ( t ) ←→ F ( ω ) , prove that (a) F * ( t ) ←→ 2 πf * ( ω ) (b) Apply (a) to the FT pair δ ( t- T ) ←→ e- jωT (i.e., from problem 4) Solution: (a) From f ( t ) ←→ F ( ω ) , we know that f ( t ) is the inverse FT of F ( ω ) , so f ( t ) = 1 2 π Z ∞-∞ F ( ω ) e jωt dω f * ( t ) = 1 2 π Z...
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This note was uploaded on 03/05/2011 for the course ECE 210 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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10Fall_hw10soln - ECE210 - Fall 2010 - Homework 10 (Prof....

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