Signal Processing and Linear Systems-B.P.Lathi copy

1 36 1 a n lti discrete time s ystem is specified by a

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Unformatted text preview: "k 1 0.1-4 1 0.2-1 1 0.2-3 i. = 1O/-,s. *' = tan-1(c":W~0.8) - tan-1(c":\l~0.5) 1 2.1-5 ( a) F h = ( b) 2.86 cos (0.5k - 1.6725) T <0: jfl 1 2.2-2 ( e) B W = 0.~87 Hz = 5~0:2 = 10 kHz. ( b) F8 2: 2 h = 100 kHz, 1 2.2-3 H [z] = L 707(1-::-1:lt:2-v'2Z +l) /350 H [ ] = [0.031416Z(Z-0.97474)] 1 2.5-1 ej (0.8"kH.6,,) + e i(1.2"k-L6,,) 1 0.1-3 "Tr/3, No = 6, D r = i [3 + 4 cos(ir) + 2cos(~r)] (a) 1 (b) e - jOko (c) ~ (d) ~ 1 0.2-2 e JU-(l. a (eJ - a) flo 1 ( ) F: 1 2.5-3 + e jO.9d + e jL1"k + e jL5"k] H[z] = :D.. H [e jwT ] = ~ 1 2.5-4 ( a) z2 °i5~~~~+1 ( b) J;8~~:''t-1' K a rbitrary 1 2.6-1 ( a) H[z] = ~~:;:: ( b) IH[e jwT ] = ~Itanwtl ( d) T =25{LS. = 2 7fk 4 sinc ( T) 1 0.3-1 (a) 1 a lOe-jIOn -1 ae , ll 1 0.3-2 () a ( b) e -j 3fl[3+4cosfl+2cos2fl] (b) I -ae- jn c os 0 0 (1 ae , U)2 1 2ae , U cosflo+a2e ,21l l ±ae- jO 1 0.4-1 (a) I[k] = ~sinc2(¥) ( b) F[2k)(fl) = ~ Ln A(fl-;!"n), F[4k)(fl) = 1 /4 10.5-1 [ 2(-O.5)k - ~(-0.8)k + ~(-0.2)k]u[k] 1 0.5-3 1 0.6-1 T he 3 -point O FT is F1 = 8, F2 = d~, F3 = e-j~. F (fl) = 3 e jfl + 3 + 2e- jn (c) T he 8 -point O FT is 8, 6.5355 + jO.707, 3 + j , - 0.5355 + jO.707, - 2, - 0.5355 - jO.707, 3 - j, 6.5355 - jO.707. 1 0.6-3 T he I -point O FT is Fo = 1 a nd F (fl) = 1 (c) T he N o-point O FT is 1, 1, , " ', 1, 1. 1 0.6-7 (a) T he 5-point response is 9, 12, 16, 8 3. F (fl) = (a) T he No-point O FT is No, 0, 0, 0, " ', 0, 0, 1 0.6-8 T he fi-point response is 3 ,8, 14, 9, 5, 3 or T- 7r ,Z z -I' Z2 21 sin 1 .9383%+0.9391 TI Iwl < - 2!. T 1 2.5-5 ( b) H[z] = !:::=::; ( e) H[z] = z~l e:~~I~~~l. 1 2.6-6 0.09057(z+1)2 H[ ] 0.0512(z+1)2 5 H[ ] _ 0 .3203(z-1)' H[ ] - . 2 0.9871z+0.3493 1 2.6-4 z z - z2 L355.+0.6125 1 2.6z - z3 0.9102.2+0.5545. 0.0979 0.3619(z-1)2 [ ]_ 0.0736(z2_1)2 H[z] = z3 0.4169z2+O.4054 1 2.6-7 H z - z4 2.5503z3+2.7876z2 L553z+0.394 1 2.6-8 H[ ] = z 1 2.8-2 (c) F i(fl) = 211(~). [ _(0.5)k + ~(0.8)k]u[k] + ~(2)ku[-(k + 1)] 1 0.6-4 a h[k] = , ,(L5) [sin 2 (k - 5) - sin 1 .6(k - 5)] 1 2.6-3 0.04235(z-1)2 2 .821z3+3.507z2 %4 = ~ L~=-7 t z-k [ ]_ 1 2.6-9 H z - 2 .231z+0.6416 0.4175(z2_L4712z+1)2 %4 1 .857%3+1.538.2:2 0 .9003z+0.3498 0 <0: k <0: lO. ( k o dd), HHam[Z] = ~ L~=-7 t ( 0.54 + 0.46 cos ¥ ) z-k 1 2.8-4 H[z] 1 2.8-5 ( b) T he P eriodic Convolution is 17, 15, 16. h[k] = ~[-0.2885, 0.3140, - 0.3779, 0.5282, - 1.0137, 0, 1.0137, - 0.5282, 0.3779, - 0.3140, 0.2885]. 1 0.6-9 T he o utput y[k] is 1, - 1, 1, 3, - 1, 1, . ... C hapter 1 3 C hapter 1 1 1 1.1-3 (b) -~8[k] + [(2)k - 1(3)k]u[k] 1 3.1-1 X2, X2 = -<PI ( xtl X2 - <P2(X1)X1 + 1 ( nonlinear). (d) [~-7(2)k + ~(3)k]u[k] zz87(;~:tl (e) [ 3(-I)k - 3(2)k + 2k(2)k]u[k] 2, - 1, and 4 1 1.2-2 ( ) -,z(z+1) a ( z-,),) (b) z(z2+ 4z + 1) 1 1.3-3 [~+ ~(2)-k - ~(4)-k]u[k] 1 1.3-8 [ 0.318(e)k - 0.367( y'2)k cos( ¥ k - 0.525)]u[k] 1 1.3-11 (a) 1 2[4- k + 3 (2)-k - 4(3)-k]u[k] (b) 1 2[4-(k-2) + 3 (2)-(k-2) - 4(3)-(k-2)]u[k - 2] (z 1 )4 or z 6+2z5+3z4+z,z3±3z2±2Z±1 (c) - =-[1- (.'!)m] Z-Q. 1...
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