ECE-34 Fall 0 Review Quesios. A liear ime-ivaria sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Deermie he oupu for he ipu show o he secod row of he diagram. Jusify your aswer. ipu x [] ipu x [] oupu y [] oupu y [] Noe ha x []= x [+]- x [] As a cosequece of he sysem beig LTI, we mus have y []= y [+]- y []. Skech i below. -. Cosider he sigals f() = u()-u(-) ad g() = u(). a) Skech f() ad f(-). f() f(-).5 b) Skech f(-) g() f(-) f(-)g() -
g().. 3. Cosider he sigal g show above. Is i periodic? If so wha is he period? Is i a power or a eergy sigal? If i is a eergy sigal calculae is eergy ad if i is a power sigal calculae is power. The sigal is periodic wih period N=3 sice g(+n)=g() for ay. Sice i is periodic, i mus be a power sigal ad o a eergy sigal (E is ifiie). P=(/3)(++)=3/4. 4. Evaluae he followig quaiies by ispecio: a. cos( s ) ( s 4) ds cos(4) b. ( s )ds u(-) c. k ( k) k 5. Cosider a sysem described by he followig ipu-oupu relaio: y() = u(-)+x(e ). (a) Is he sysem liear? Jusify.
No. If we choose x 0, he y()=u(-) 0. (b) Is i ime ivaria? Jusify. O(x 0 )() = u(-)+x 0 (e ) = u(-)+x(e -0 ) O he oher had, O(x)(- 0 ) = u(-(- 0 ))+x(e - 0 ) u(-)+x(e -0 ). Thus, O is ime varia. (c) Is i memoryless? Jusify. No, because y(0) depeds o x(). (d) Is i causal? Jusify. No, because y(0) depeds o x(). 6. Deermie wheher he followig sysems are sable, you mus show proper jusificaio (i.e., provide a proof i he case of sable ad a couer example oherwise): Take x() = u(), which is bouded (by oe). The, y ( ) u( s) ds u( ). Ad, y() = u(), which is ubouded. Thus, he sysem is o BIBO sable. (b) y ( ) (/ 3){ x( ) x( ) x( )}. Assume x is bouded by M. The, y( ) (/ 3){ M M M } 3M, which is bouded. The sysem is herefore BIBO sable. 7. Suppose ha he sysems i Problem 6(b) is liear ad ime-ivaria. Calculae is impulse respose. Simply replace x by o obai h( ) (/ 3){ ( ) ( ) ( )}. 8. Cosider a sysem described by he followig ipu-oupu relaio: O(x)() = e -x() (a) Is he sysem liear? Jusify. (b) Is i ime ivaria? Jusify. (c) Is i memoryless? Jusify. Yes, O(x)() depeds o he ipu a ime oly. (d) Is i sable? Jusify.
9. Deermie if he followig sigals are periodic or o. Idicae he period if hey are periodic. Jusify your aswer. a) x[] = si(3). b) x[] = si([ /3]). c) x[] = si([ /3]) u[]. d) x() = cos (3) Oce you masered hese quesios, ry o work he ex se of quesio. Le me kow if you have ay quesios (ex page). Review Quesios: Se. A liear ime-ivaria (LTI) sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Plo y (). Jusify your aswer ad clearly mark he axes as eeded. ipu x () ipu x () oupu y () oupu y () Exploi he fac ha x ()= x () + x (+).. a) Is he sysem described i Problem causal? Jusify your aswer. No. Oupu a ime 0 depeds o ipu a imes greaer ha. b) Ca you say if i is sable or o? Jusify your aswer. No. c) Graph x (+), x (-), x (), ad y () (-) \
g() /3 /3 4. a) Graph g(-). b) Is g i periodic? If so wha is he period? Is i a power or a eergy sigal? If i is a eergy sigal calculae is eergy ad if i is a power sigal calculae is power. Periodic wih period 4. 5. Calculae z() = u(+) * u() ad plo z. No covered by his exam. 6. Evaluae he followig quaiies by ispecio: a. s e ( s) ds e b. ( s) ds u( ) k c. ( k)si (k) si ( ) si () 7. Cosider a sysem described by he followig ipu-oupu relaio: y() = x( ). (a) Is he sysem liear? Jusify. Yes. Show ha addiiviy ad homogeeiy hold (b) Is i ime ivaria? Jusify. No. (c) Is i causal? Jusify. No: y(-) depeds o x(4) (d) Is i sable? Jusify. Yes. y will have he boud as ha for x. 8. Fid he derivaive of exp(-a)u(). Usig he produc rule, -a exp(-a)u() + exp(-a) ()
9. Show ha ay sigal ca be decomposed io eve ad odd pars. See ex. 0. Use he homogeeiy propery of a liear sysem O o show ha if x()=0 for all he O(x)() = 0 for all. (Recall ha he oupu sigal O(x)() is wha we also refer o as y().) O(0)() = O(0 x )() = 0 O(x )() = 0, where x is ay ipu. Use his propery o coclude wheher or o he sysem O is liear, where O (x)() = x( s) e ds. 0 No liear sice for example e ds 0.. For a cerai LTI sysem, i is kow ha he ipu u()-u(-4) yields ad oupu of y()= u(). Fid he oupu whe he ipu is u()-u(-8). Hi: Try o wrie u()-u(-8) i erms of a superposiio of u()-u(-4) ad possibly a shifed versios of i.