Prelim 1 Solutions - ECE220 Exam 1 Solutions (CLOSED BOOK)...

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Unformatted text preview: ECE220 Exam 1 Solutions (CLOSED BOOK) Spring 2008 February 21, 2008 PRINT name, netid, lab section: Kevin Problem 1 (10 points total) Using Eulers identity show that 2 cos( ) cos( n + ) = cos( ( n + 1) + ) + cos( ( n 1) + ) (1) ......................................................................... ......................................................................... Recall the inverse Euler formula for cosines, cos( w ) = ( e j w + e- j w ) / 2 Consider the left hand side of the equation (1), LHS = 2( e j w + e- j w ) / 2 (( e j ( w n + ) + e- j ( w n + ) / 2) = 1 2 ( e j ( w n + + ) + e j ( w n + - ) + e- j ( w n + - ) + e- j ( w n + + ) ) = 1 2 ( e j ( w ( n +1)+ ) + e j ( w ( n- 1)+ ) + e- j ( w ( n- 1)+ ) + e- j ( w ( n +1)+ ) ) = cos( ( n + 1) + ) + cos( ( n 1) + ) = RHS 1 PRINT name, netid, lab section: Dr. Hutchins Problem 2 (20 points total) Measurements of a discrete-time signal x [ n ] known to be a sinusoid of the following form x [ n ] = a cos( n + ) are listed in Table 1 below for 0 n 5 n 1 2 3 4 5 x [ n ] 1.8478 0.6141-1.0521-1.9773-1.5099 0.0209 Table 1. Use the identity established in Problem 1 to find values of a , and . ......................................................................... ......................................................................... Rewriting the equation of Problem 1 we see that it is a difference equation of the form: x [ n + 1] = 2 cos( ) x [ n ] x [ n 1] where x [ n ] = cos( n + ). Thus it is capable (given the result of problem 1) of computing the next term of a sinusoidal sequence from the current sample and one past sample. It is an oscillator. Note that the unknown constant in the difference equation is 2 cos( ). Any three consecutive samples of x [ n ] are sufficient to find this constant, and from it, . For example: cos( ) = x [ n + 1] + x [ n 1] 2 = 1 . 0521 + 1 . 8478 2 . 6441 = 0 . 6497 giving = sqrt 3 2 . This frequency defines the essential nature of the system - its poles in fact - its IIR. We now know the frequency at which the system wants to oscillate, and the exact amplitude a and phase are a matter of establishing initial conditions. Since there are two numbers to determine, two values of the sequence are required. It is most convenient to use the smallest values of n : n = 0 and n = 1....
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This note was uploaded on 03/13/2008 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell University (Engineering School).

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Prelim 1 Solutions - ECE220 Exam 1 Solutions (CLOSED BOOK)...

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