soln1 - EECS 451 SOLUTIONS TO PROBLEM SET #1 1. Write the...

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EECS 451 SOLUTIONS TO PROBLEM SET #1 1. Write the signal as the sum of 3 scaled and delayed pulses: V 2 τ 2 sin( ωτ/ 2) ωτ/ 2 - e 3 τ 4 V 2 2 τ 4 sin( ωτ/ 4) ωτ/ 4 - e - 3 τ 4 V 2 2 τ 4 sin( ωτ/ 4) ωτ/ 4 = V τ sin( ωτ/ 2) ωτ/ 2 - V τ/ 2 cos(3 ωτ/ 4) sin( ωτ/ 4) ωτ/ 4 . NOTE: F{ real and even } =real and even function checks. 2a. Periodic with period T = 2 π 5 . Note continuous time! 2b. Not periodic since f o = 5 2 π isn’t a rational number. 2c. Not periodic since f o = 1 2 π 6 isn’t a rational number. 2d. Not periodic since f o = 1 2 π 8 isn’t a rational number. Better: 2 cos( n 8 ) cos( 8 ) = cos( n ( π +1) 8 ) + cos( n ( π - 1) 8 ). Neither sinusoid is periodic. 2e. Periodic: f 1 = π/ 2 2 π = 1 4 ; f 2 = π/ 8 2 π = 1 16 ; f 3 = π/ 4 2 π = 1 8 T = LCM [4 , 16 , 8] = 16. 3. f k = 2 πk/N 2 π = k N = k/GCD [ k,N ] N/GCD [ k,N ] N p = N/GCD [ k,N ].
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soln1 - EECS 451 SOLUTIONS TO PROBLEM SET #1 1. Write the...

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