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ee20-hw04_solutions-f10

# ee20-hw04_solutions-f10 - EECS 20 Fall 2010 Homework 4...

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Unformatted text preview: EECS 20 Fall 2010 Homework 4 solutions HW4.1 The forward gain of the inner loop is F ( ! ), and its loop gain is A ( ! ) B ( ! ) F ( ! ). The closed loop frequency response of the inner loop is therefore Forward Gain 1 Loop Gain = F ( ! ) 1 A ( ! ) B ( ! ) F ( ! ) : Call this G ( ! ). The forward gain of the outer loop is G ( ! ) and its loop gain is P ( ! ) G ( ! ). This H ( ! ) = G ( ! ) 1 + P ( ! ) G ( ! ) = F ( ! ) 1 A ( ! ) B ( ! ) F ( ! ) 1 + P ( ! ) F ( ! ) 1 A ( ! ) B ( ! ) F ( ! ) = F ( ! ) 1 + ( P ( ! ) A ( ! ) B ( ! )) F ( ! ) : 1 HW4.2 (a) F ( ! ) = X n 2 Z f ( n ) e in! = X n ;n odd n e in! = e i! 1 X l =0 ( 2 e i 2 ! ) l = e i! 1 2 e i 2 ! : (b) G ( ! ) is recognized as the frequency response of the unit delay. Its impulse response is therefore g ( n ) = ( n 1) : This can be veri ed by checking that e i! = X n 2 Z ( n 1) e in! : (c) The forward gain is F ( ! ) = e i! 1 2 e i 2 ! : 2 (d) The loop gain is kG ( ! ) F ( ! ) = k e i 2 ! 1 2 e i 2 ! : (e) The closed loop frequency response is H ( ! ) = Forward Gain 1 Loop Gain = F ( ! ) 1 + kG ( ! ) F ( ! ) = e i! 1 2 e i 2 ! 1 + k e i 2 ! 1 2 e i 2 ! = e i! 1 ( 2 k ) e i 2 ! : (f) From the formula for H ( ! ), we see that h ( n ) = unless n 0 and n is odd if n = 1 ( 2 k ) if n = 3 ( 2 k ) 2 if n = 5 etc. : Thus X n 2 Z j h ( n ) j = j j 1 X l =0 j 2 k j l : This means that X n 2 Z j h ( n ) j < 1 () j 2 k j < 1 : Thus the closed loop system is BIBO stable if and only if 2 k < 1 and 2 k > 1 ; i.e., if and only if 1 < k < + 1 : 3 HW4.3 (a) < x; x > = n X i =1 x ( i ) x ( i ) £ = n X i =1 j x ( i ) j 2 : This is a sum of nonnegative real numbers, hence it is real and nonnegative. Further, < x; x > equals zero if and only if each j x ( i ) j 2 equals zero, i.e. if and only if each x ( i ) equals zero, i.e. if and only if x is the zero vector. (b) < y; x > = n X i =1 y ( i ) x ( i ) £ = n X i =1 ( x ( i ) y ( i )) £ = ( n X i =1 x ( i ) y ( i )) £ = ( < x; y > ) £ : (c) < x 1 + x 2 ; y > = n X i =1 ( x 1 ( i ) + x 2 ( i )) y ( i ) £ = n X i =1 x 1 ( i ) y ( i ) £ + n X i =1 x 2 ( i ) y ( i ) £ = < x 1 ; y > + < x...
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