EECE 340 Homework, Spring 2007

EECE 340-Signals and Systems Homework #7 Problem # 1 Consider the following discrete-time sequence: h(n) = Ar" cos(na@pT +0) u(n), where 0<r<J, a) Find H (z) and specify its region of convergence b) Let a = - and plot the pole-zero diagram for H (z). Problem # 2 Show the following: a) The Z-transform of x'(n)is X"(z’) b) The Z-transform of x(-n)is X(¥4) c) The Z-transform of a"x (n)is X(%) dX(z) d) The Z-transform of nx(n)is -z e) The initial value x(0) of a causal sequence x(n) is given by lim X(z) f) The initial value x(0) for a sequence x(n) such that x(n)=0, n>0 is given by lim X(z). z-0 In part (c), show how the poles and zeros of X (z) are moved. Page 1 of 1 FEEE R40 @) Hl2) = ft sm ewark 7 Sibutlin ~ 1/4. AS al gored, LOWY] 57 nme od r x A} Je JOT ph aye ves, N ia a a D> O2ee! pe” oS) H =o 4=0 One 2 gclen CST 4) = A tot nile Cur = 8) =A) PALE (Tan B71) 2 (ot -2F en lof) oe Mz... eee f n 2 (oT a) | oa rp BOT NB pp eI) /2/)>2z Tul Y aw Pe l2) 2- plane Like fe 2. a) tea rine” A=—o XE) = > rer)” A=— VO? a) Feel Ja Soa A= —te =x — po = Zl # “Cnt DAL) = 2 2b) 2 = S eee" = —go 4=—e ~ 4. S 4(B) = S20 (2S areme” H =~ bo A= —po =Z [ ale cr) | . Sy YOKE pag a pote mgelal 2 2 yy, X (%) A+ a A oy pert LP mae vu Jene! p/2|/=/a/k , LE _ faye [@/ and titated Le , Del 2/ a, An, <4 . im th (ele 2 plane, Ca ttheak > 9) £ LAMY = > viyx” ty a z —- ne ca ee = > LO) (-n ES) = ~ 1) nue” 2 _ vy th bY rr =~ 2 EE px), 4 = - 0 acy y Xl8) = DP xp A) x — 0 ae tim YOR) =P vty) A. 2” 2 —b 00 4=-b0 Sisw BA Lue eed O, Nr 0 / BA sa , “4 Fo