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Unformatted text preview: GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Summer 2008 Problem Set #8 Assigned: Week of 7-Jul-08 Due Date: 14/15-Jul-08 Please check the T-square “Announcements” daily. All official course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. PROBLEM 8.1 *: Try your hand at expressing each of the following in a simpler form: (a) [ δ ( t- . 04) + δ ( t + . 04)]cos(10 πt ) u ( t ) = (b) [cos(30 πt ) u ( t ) + 3 δ ( t- 4) + 2 e- 5 t u ( t )] * δ ( t- 8) = (c) d dt n e- 5 t [ u ( t )- u ( t- 3)] o = (d) t Z-∞ [ δ ( τ- . 04) + δ ( τ + . 04)]cos(10 πτ ) u ( τ ) dτ = Note: use properties of the impulse signal δ ( t ) and the unit-step signal u ( t ) to perform the simpli- fications. For example, recall δ ( t ) = d dt u ( t ) where u ( t ) = Z t-∞ δ ( τ ) dτ = 1 for t ≥ 0 for t < Be careful to distinguish between multiplication and convolution. Convolution is denoted by a “star”, as in x ( t ) * δ ( t- 2) = x ( t- 2) and multiplication is usually indicated as in x ( t ) δ ( t- 2) = x (2) δ ( t- 2). PROBLEM 8.2 *: Do Problem P-9.17 on p. 282 of Signal Processing First , except with the following two modifi- cations: (i) Change the lower limits of the integral from t- 2 to t- 5 and change the upper limit of the integral from t + 2 to t + 5, and (ii) change the input in part (d) from x ( t ) = u ( t + 1) to x ( t ) = u ( t + 4). PROBLEM 8.3 *: Consider the functions x ( t ) = 3 e- 2( t- 3) [ u ( t- 3)- u ( t- 5)] and h ( t ) = 6[ u ( t- 1)- u ( t- 7)] (a) To get your mind thinking in convolutionland, draw three different plots with “ τ ” on the horizontal axis....
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- Spring '08
- Signal Processing