EECS20n, Solution to Midterm 2, 11/17/00

Transcription

1 EECS20n, Soluion o Miderm 2, /7/00. 0 poins Wrie he following in Caresian coordinaes (i.e. in he form x + jy) (a) 2 poins j 3 j 2 + j += j ++j +=2 (b) 2 poins ( j)/( + j) = j (c) 2 poins cos π/4+jsin π/4 =±(cos π/8+jsin π/8) Wrie he following in polar coordinaes (i.e. in he form re jθ ) (a) 2 poins +j = 2e jπ/4 (b) 2 poins ( + j)/( j) = e jπ/2

2 poins Which of he following discree-ime or coninuous-ime signals is periodic. Answer yes or no. If he signal is periodic, give is fundamenal period and sae he unis. Suppose ha for a discree-ime signal, n denoes seconds, and for a coninuous-ime signal, denoes minues. (a) 2 poins n Ins, x(n) =e 2n Periodic NO; (b) 2 poins Reals, x() =e 2 Periodic YES; Period = 2π/ 2 min (c) 2 poins n Ins, x(n) = cos3πn + sin(3πn + π/7) Periodic YES; Period = 4 sec 4 poins Find A, θ, ω in he following expression: A cos(ω + θ) = cos(2π 0, π 4 )+sin(2π 0, π 4 ) So A = 2,ω = 20000,θ =0. = 2cos(2π 0000)

3 3 H (ω) H (ω) π/2 2/2 π/4 ω H2(ω) ω H 2 (ω) 2 2π periodic ω π ω Figure : Plos for Problem poins On Figure plo he ampliude and phase response of he following frequency responses. On your plos carefully mark he values for ω =0and for one oher non-zero value of ω. (a) 4 poins ω Reals, H (ω) =+jω (b) 4 poins ω Reals, H 2 (ω) =+cosω The frequency response is ploed only for ω>0since H(ω) is even and odd. H(ω) is 2 poins Which of H,H 2 can be he frequency response of a discree-ime sysem? (a) H (ω) =[+ω 2 ] /2, H (ω) =an (ω) (b) H 2 (ω) =+cosω, H 2 (ω) =0 H 2 can be he frequency response of a discree ime sysem since i is periodic wih period 2π.

4 4 h n n Figure 2: Impulse and sep response for Problem poins A discree-ime sysem H has impulse response h : Ins Reals given by h(n) = {, n =, 0, 0, oherwise (a) 3 poins Wha is he sep response of H, i.e. he oupu signal when he inpu signal is sep, where sep(n) =,n 0, and sep(n) =0,n < 0? You can give your answer as a plo or as an expression. (b) 3 poins Wha is he frequency response of H? (c) 4 poins Wha is he oupu signal of H for he following inpu signals? i. n, x(n) =cosn ii. n, x(n) =cos(n + π/6) (a) The sep response is n, y(n) = h( )sep(n +)+h(0)sep(n)+h()sep(n ) 0, n 2, n = = 2, n =0 3, n This is also shown on he righ in Figure 2 (b) The frequency response is ω, Ĥ(ω) =h( )e jω + h(0) + h()e jω =+2cosω (c) The response y is i. n, y(n) =[+2cos]cosn ii. n, y(n) =[+2cos]cos(n + π/6)

5 poins (a) 4 poins Find he frequency response for he LTI sysems described by hese differenial equaions (inpu is x, oupu is y) i. ẏ()+0.5y() =x() ii. ÿ()+0.5ẏ()+0.25y() =ẋ()+x() (b) 2 poins Wha is he response of he firs sysem above for he inpu, x() = e j(00+π/4)? (c) 4 poins Find he frequency response for he LTI sysems described by hese difference equaions (inpu is x, oupu is y) i. y(n)+0.5y(n ) = x(n) ii. y(n)+y(n ) y(n 2) = x(n)+x(n ) (a) The frequency response is i. Ĥ(ω) = jω+0.5 ii. jω+ ω jω+0.25 (b) The response is (c) (i) +0.5e jω (ii) +e jω +e jω +0.25e 2jω, y() = Ĥ(00)ej(00+π/4) = 0.5+j00 ej(00+π/4) 00 ej(00 π/4)

6 6 x u n y v n Figure 3: Periodic signals for Problem poins Figure 3 plos wo coninuous-ime periodic signals x and y boh wih period second, and wo discree-ime signals u and v boh wih period 0 samples. The plos are given only for one period. Suppose he exponenial Fouriers Series represenaions of hese signals are given as: Reals, x() = = Reals, y() = = n Ins, u(n) = n Ins, v(n) = X k e jkωx k= k= 9 k=0 9 k=0 Y k e jkωy U k e jkωun V k e jkωvn (a) 2 poins Give he values of ω x =2πrad/sec, ω y =2πrad/sec, ω u = π/5 rad/sample, ω v = π/5 rad/sample. (b) 2 poins Calculae he values of he coefficiens X 0 =0.25, Y 0 =0.25, U 0 =0.4, V 0 =0.4. These are jus he average values ofhe signal over one period. (c) 3 poins Express y as a delayed version of x and v as a delayed version of u., y() =x( 0.5), n, v(n) =x(n 3). (d) 3 poins Express he FS coefficiens {Y k } in erms of {X k } and {V k } in erms of {U k }. k, Y k = X k e jkπ, V k = U k e jk3π/5.

7 7 x - + H(ω) H2(ω) y H3(ω) H(ω) Figure 4: Feedback sysems for Problem poins Figure 4 shows a feedback sysem obained by composing hree LTI sysems. In he figure, H k (ω),k =, 2, 3 is he frequency response of he hree LTI sysems. (a) 5 poins Calculae he frequency response H(ω) of he feedback sysem in erms of he H k. (b) 5 poins Suppose H k (ω) =/( + j2ω) for all k =, 2, 3. Calculae H(0), H() and lim ω H(ω). (a) The frequency response is H(ω) = H (ω)h 2 (ω) +H (ω)h 2 (ω)h 3 (ω) () (b) We have, H k (0) =,H k () = /( + 2j), lim ω H k (ω) =0. Subsiuing ino () gives H(0) = 2,H() = +2jω +(+2jω) 3, lim H(ω) =0 ω

8 8 h s impulse rain y Figure 5: Impulse reponse, sep response, impulserain and response for Problem poins A coninuous-ime LTI sysem has he impulse response Reals, h() = {, < 0.5 0, oherwise (a) 2 poins Skech he impulse response, and mark carefully he relevan poins on your plo. (b) 2 poins Is his sysem causal? Answer yes or no. (c) 2 poins Skech he sep response of his sysem, i.e. he response o he inpu signal sep() =, 0 and =0,<0? (d) 2 poins Consider he inpu signal impulserain, where Reals, impulserain() = k= δ( 2k). Skech impulserain. (e) 2 poins Skech he response of he sysem o impulserain. (a) The impulse response h is shown in Figure 5 (b) The sysem is NOT causal. (c) The sep response s is he inegral of he impulse response as shown. (d) The impulse rain is as shown. (e) Is response is, y() =(h impulserain)() = k= h( 2k)