Serial : 4LS1_A_EC_Signal & Systems_230918

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1 Serial : LS_A_EC_Signal & Syem_8 CLASS TEST (GATE) Delhi oida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar Kolkaa Pana Web: info@madeeay.in Ph: -56 CLASS TEST 8- ELECTROICS EGIEERIG Subjec : Signal & Syem Dae of e : //8 Anwer Key. (c) 7. (c). (d). (a) 5. (b). (d) 8. (b). (c). (c) 6. (d). (a). (d) 5. (c). (b) 7. (c). (b). (c) 6. (d). (a) 8. (c) 5. (b). (a) 7. (a). (a). (d) 6. (a). (b) 8. (b). (c). (b)

2 6 Elecronic Engineering Deailed Explanaion. (c) n z z a u( n) ; z a z a > n z z ( ) un ( ) ; z > z +. (d) f( + ) + 6 L F( ) L ( + ). (a) L[x()] X() e e ( ) ( + ) + e e The fir inegral converge if Re() <, and econd converge if Re{} > herefore ROC < Re() < X() ( ) ( ) ( + ) ( ) + e e e e + ( ) ( ) ( ) ( ) + ;ROC; < Re( ) < ( ) ( + ). (b) 5. (b) E x X(e jω ) x () /6 co 6 J /6 /6 /6 /6 /6 in J J J 6 jω jω jω jω ( e e ) e e ) j + j j + co J X(e jω ) e jω e jω + e jω e jω δ(n) δ(n ) e jω δ(n + ) e jω δ(n ) e jω δ(n + ) e jω o, x(n) δ(n ) δ(n + ) + δ(n ) δ(n + ) x(n) {,,,,}

3 CT-8 EC Signal & Syem 7 6. (a) Given inegral δ ( + ) e ince he hif which i ou of he limi (, ) Hence 7. (c) δ ( + ) e X(z).5 z ; Z >.5 z.5 So, x(n).5 (.5) n u(n) a n, x() (.5) (.5) (b) X(f ) X() f X() f X() f X () f X () f X () f f 5 5 f f max ( ) ample/ec f 5 5 f. (d). Le inpu x [n] produce an oupu y [n] and inpu x [n] produce an oupu y [n]. Then weighed um of oupu i { x [ n]} { x [ n]} py [n] + qy [n] pa + qa The oupu due o weighed um of inpu i { px [ n] + qx [ n]} y [n] a y [n] py [n] + qy [n] So, he yem i non-linear.. The oupu due o delayed inpu i y[n, k] he delayed oupu i a { x[ nk]} y[n k] { x[ nk]} a y[n, k] y[n k] So, he yem i ime invarian.. The oupu depend only on preen inpu. So, he yem i caual.. (c) From he differeniaion propery of Fourier ranform dx() FT ω j X( jω) d x() FT ω Xj ( ω)

4 8 Elecronic Engineering. (a) By uing he ime hifing propery of Fourier ranform 8 ( z ) ( + z ) d x( ) ω FT ω j e X j ω ( ) A B C + + ( z ) ( z ) ( + z ) 8 A( z )( + z ) + B( + z ) + C( z ) By olving above equaion, we ge A B C ( z ) ( + z ) ( z ) ( z ) ( + z ) [() n + 6(n + ) () n + + ( ) n ]u[n]. (b) In ime domain now, If Y(e jω ) jω j( ω/) Xe ( ) Xe ( ) Y(e jω ) x [n] x [n] X(e jω ) n n ( ) Then j( /) Xe ω y[n] n j n/ n e n j n/ ne. (d). x() for < hu he ignal i caual x() i real hu X(jω) i an even ymmeric ignal FT. now Re { Xj ( ω) } even { x( ) } e even {x()} x() + x( ) e. (c) x() e { righ ided x( ) } x() e u() x() co

5 CT-8 EC Signal & Syem e j( / ) j( / ) + e j j j j e e + e e According o he given condiion he oupu i j j j j e e + e e j( / ) j( / ) e + e y y() co co co 5. (c) For yem - : If x() + or, hen y(). Thu, y() x () i alo non-inverible. For yem - : dx() If x() conan, hen y(). Thu, y() i a non inverible yem 6. (d) f() (econd) F() f () area under he curve f() The area under curve ( ) + ( ) + ( ) + ( ) + (/ ) (a) Le x() e a Then fourier ranform X(ω) a a +ω For a, x() e.5 and X(ω) + ω +ω from dualiy propery

6 Elecronic Engineering If, x() X( ω) FT FT X () x( ω) + FT e FT e + FT e + + ( ) FT e.5 ω.5 ω.5 ω.5 ω by comparing wih e p.5 ω hen p. 8. (b) Given finie lengh equence i x[n] {,,,,,,,,5,} x[n] from pareval relaion we have x [ n] n Xk ( ) k k k n Xk ( ) e Xk ( ) x[ n] From he given equence value of i. Xk ( ) k n x[ n] j kn Xk [ ] { } n 7. (a) The energy of he ignal E E x[ n] n 5 8 ( n ) + ( n) + ( n) n n n 6

7 CT-8 EC Signal & Syem E. (c) Given ha he real par of H(e jω ) i H R (e jω ) + αcoω e + e +α. (b) 5 8 n + + n n + n n n n 6 E ( + + 6) + ( ) + ( ) E 76 J jω jω α H R (e jω jω α jω ) + e + e The even par of h[n], which i he invere DTFT of H R (e jω ), i Given h[n] i a caual equence α α h e [n] δ [ n] + δ [ n + ] + δ[ n ] α hence h e [n] δ [ n] + δ[ n ] The equence h[n] he [ n]; n > he []; n n ; n < h[n] δ[n] + αδ[n ] given ha h[n] a n δ[] + αδ[ ] + α α Given and f() g() By uing he ime caling propery LT. + g () G ( ) + LT. x( a) X a a LT. g( ) G F() G

8 Elecronic Engineering A + L F () L F() L L L L 6 +. (a) The IDFT of X(k) i defined a, x[n] k jnk Xk ( ) e ; n, x[n] jnk Xk ( ) e ; n x[] Xk ( ) [ j j] k j + + x[] Xke ( ) [ ( jj ) ( j)( j) ] k k x[] ( ) j k Xke [ ( j)( ) ( j)( ) ] k k j x[] Xke ( ) [ ( j)( j) ( j)( j) ] k 7 x[n],,,. (a) z in c δ( k) k in c δ( k) k k inc k

9 CT-8 EC Signal & Syem k k in in + k k k k an () +. (c) x() n jnω n C e Where ω i he fundamenal frequency. Given x() + co + in The ime period of he ignal co i T T ec The ime period of he ignal in i T T ec T ec T / T T The fundamenal period of he ignal x() i T T T ec Fundamenal frequency ω rad/ ec T ow, x() + co + in j j j j e + e e e + + j j j j + e + e + e e j j j + e + e + e e j j j() j() j() j() j( ω ) j( ω) j( ω) j( ω) C + Ce + Ce + C e + C e value of C, C C C / 5. (b) Given DFT, X(k) {,,, } By he definiion of DFT, X(k) x[ ne ] n jkn

10 Elecronic Engineering jkn The invere DFT, x[n] Xke ( ) ; n,,,..., k Clearly from he queion,. x[n] x[n] a n i x[] k k x[].5 Xke ( ) jnk 6. (d) Given ignal x() can be wrien a + ; x() ; ; oherwie j k Xke ( ) [ () + ( ) + () + ( ) ] energy E ( + ) + ( ) J ( + + ) + (+ ) J energy E J 6 J 5. J 7. (c) From he differeniaion propery of Fourier ranform dx() FT ω j X( jω) d x() FT ω Xj ( ω) By uing he ime hifing propery of Fourier ranform 8. (c) x[] n X() z d x( ) ω FT ω j e X j ω ( ) n aun ( ) az x() e a u() x(nt ) e ant u(nt) at n x[n] ( e ) u( n) n at ( e ) u( n) at e z

11 CT-8 EC Signal & Syem 5. (d) H(z) n hn ( ) z n h(n) for n < ; a he yem i caual o, H () hn ( ) hn ( ) n n Yz () 8 H (z) Xz () z 5z H () 5 8 hn ( ) H() n. (b) Given y[n].5x[n] +.5x[n ] impule repone h[n].5δ[n] +.5δ[n ] h[n], H(e jω ) j + e ω n x[n] co ω j H(ω ) j He + e j He j H(ω ) coω n H( ω) co( ωn+ H( ω)) LTI n n co n co n co n in