ps6 - EE 350 PROBLEM SET 6 DUE 29 October 2007 Reading...

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Unformatted text preview: EE 350 PROBLEM SET 6 DUE: 29 October 2007 Reading assignment: Lathi Chapter 3, Sections 3.4, 3.5, 3.7, 3.8, and 3.9 Recitations sections will meet during the week of October 22. Problem 28: (18 points) A signal f ( t ) is said to be periodic if for some positive constant T o f ( t ) = f ( t + T o ) , for all t ≥ . The smallest nonzero value of T o satisfying the last equality is called the fundamental period of f ( t ). If a signal f ( t ) is not periodic, it is called aperiodic . Determine whether or not each of the following continuous-time signals is periodic. If the signal is periodic, determine the fundamental period. 1. (3 points) f ( t ) = cos 2 (10 t + 45 ◦ ) + sin(5 t + 135 ◦ ) 2. (3 points) y ( t ) = f ( t ) ∗ h ( t ), where f ( t ) = e −| t | [ u ( t + 1) − u ( t − 1)] h ( t ) = ∞ X n = −∞ δ ( t − 2 n ) { Hint: Sketch f(t). } 3. (4 points) y ( t ) = f ( t ) ∗ h ( t ), where f ( t ) is periodic with fundamental period T o and h ( t ) is aperiodic. 4. (4 points) f ( t ) = a o + ∞ X n =1 a n cos( n ω o t ) + ∞ X n =1 b n sin( n ω o t ) , where a , a 1 , a 2 , . . . , b 1 , b 2 . . . are real-valued constant coefficients, ω o = 2 π/T o , and T o is some positive real-valued constant. 5. (4 points) f ( t ) = ∞ X n = −∞ D n e j n ω o t , where the D n are complex-valued constant coefficients, ω o = 2 π/T o , and T o is some positive real-valued constant. Problem 29: (30 points) Consider the trigonometric signal set { 1 , cos ω o t, cos 2 ω o t, . . . , cos nω o t, . . . ... ; sin ω o t, cos 2 ω o t, . . . , cos nω o t, . . . ... } , defined over the time interval [ −∞ , ∞ ], where ω o is the fundamental frequency of the set and nω o is called the n th harmonic of the frequency ω o . The constant term 1 is the 0th harmonic of ω o as cos(0 × ω o t ) = 1. Using the results from Problem Set 5, problem 26, we showed in lecture that the trigonometric signal set is mutually orthogonal as < sin( nω o ) t, sin( mω o t ) > = < cos( nω o ) t, cos( mω o t ) > = n 6 = m T o 2 n = m 6 = 0 < sin( nω o ) t, cos( mω o t ) > = f o r a l l n and m . We also stated that this set is complete , in the sense that we can exactly represent any real-valued periodic signal f ( t ) with fundamental period T o as f ( t ) = a o + ∞ X n =1 a n cos( n ω o t ) + ∞ X n =1 b n sin( n ω o t ) , where a , a 1 , a 2 , . . . , b 1 , b 2 . . . are real-valued constant coefficients given by a o = 1 T o Z T o f ( t ) dt a n = 2 T o Z T o f ( t ) cos nω o tdt b n = 2 T o Z T o f ( t ) sin nω o tdt, and ω o = 2 π/T o . In this problem you will see why it is necessary to use an infinite number of harmonics to represent certain periodic signals, and in Problem 30 see why this concept is important in analyzing the transmission of digital signals across a bus....
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This note was uploaded on 07/23/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Penn State.

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ps6 - EE 350 PROBLEM SET 6 DUE 29 October 2007 Reading...

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