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I f'"lirf T lethotLtofDiscretesisna[an(islstemAnalvsft McGrawHill,NewYork,
,':j,f i "U[U.frr,u^,l"inf,
* Schater' D s,ta1 srsr tat procestins prenrice Har. En- i II
Steams, S. D. Digital Silnat Analysls. Hay61en, Rochelie park, NJ, ;975. l fr
I enoolrus I :I
I ( ,l) Civcn rhr fJllo\,,ing scquen.e: r(,ri - - 3"u,. tt - tt+ u\n) -.',,uln\. , i\/ fa) Illusrra": rhe sequence I i
I a: ,t, ,lj::;,:IJ:t'"n ,rf co rvcrl.errr c or the z rran.foin, oi Lh: nc!:rriv.: ,r parr. of I i_ L fcr Whrt i. the region of.:onrcrge,rce of rhe z, trarsform.oithe l,osii;ve,rnd zeto ' II 4., a pan or the ie"qucnce?
vr .'l! 'a'rrlrvL.'ru
zLru ( i' Y' {d) WhJt i, rhe region ofconvcr5cnce ot tne ; ltxnsfon,r.jJ thc,:.il(irc seLtuenccl II -> (et Frrd the i transfo[n ol rl). Whar elfect do the lcros oi X(z) hrue on t|ea
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,., ,, : ,,. 6 ua 5rurrrj or l/i./. eynar e ect do the lcros oi x(1) hJve on tl)ei rrgior ol .'onv, rgenc. I tne pole"l I{lj Il }(/,./ js a unir :am;lc r.sponse ot_ a:ystem. is the syslcrn r:a.tsal? Slrbie.. II| 2.2 For lhrpa.i5 bclo\1. rell uhich one har a betterchance of,.onv,-.rging and rellrrlry I(a)Diiciete-timeFouiief.transI.rm.orztransfofin,'..
,11
lz' :qU- ,)
rb; Lrolace lian.forn or (o'ttlnuc,u\ 'ime Fourier t,.ansfomr.[.2,:) fi"O rhe ? rra lsform oi rhc lnllow ng dricrete-lirne signair, remcriberi0g ro in(lude
tne reelon oJ convergcnce lor eacn.ra) t'nt
- ) urnt I 3rl)" rr1,rr
(b) xQr) : 3i.- i),' u(n) - X3y, u(-; , D(c) x(rr) = (-3)',rr(n) + 5(+)-t,a(-/r -
l)(d) r0i) : 3,:-)" u(n) + 2(4),'u(-n
- I) + 5ij(n)
(e) -(n) = 3( ,' u(n) + S(i),'at,n
- t).(f) -(x) - 3(il" ul-n
- I) + 5(1)', rrl)
(g) "r(n) = 3a-os' u(n) + 4(2.)" u( n
- t).
(h) :(n) = ';E(n) + 5(t? -
1) - +6(n - 2)(i) .(r) = 2l;(n - 3) , 26(n + 3) :
2.4 In Scction 2.1 .2, a nurtber of inrpcrlarrt propertir:s oI the Z transform rvcrr pre.. sented $,irhour proof. Prove the foilowiirg: (a) iineariry, (b) lranslatiqn, (c) rnulri-
pliralion by an exponential, (d) multiplicarion by a ramp, (e) convolution (tilrre.
don,at.rr. {.f, con\o,il{ton oon.ait ,. rf. initidl valuc rhcorrm, arrd th) fina,viluc
It:
II
ll
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i'c rhe ? rransfo--n of lFe lolloql tg ccq c (ec'
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(a) sin Guo.'r) ir(r); (b) cos (aonlu(n); tc) n u(n) ' (d) sin (@od + 0) r(n);(e) cos (toon.1- 0) a(r); (f) sinh (coorr) r'r(n); (8) cosh (cdo,)=a(,)Check your iesults withrthe entries from Tabie 2,1 and'ApPndix Ci ' '
)Fina th. z trrnsfonn ol the foLlowjng iiscrtlte.iime signals,.remeinbering 16 lncludethe region of convergence 1or each onc'. Use the clelinrtions oi rhe z transiorm, lhetables, and the properlies.(a) x(n) = u1,
- 21
(b).r(n): n u(n '1)(c) t(ti) = e'r" u(n - 1) .(e) x(n) = (t1 - 1)2 e " u(n -(f)
-r(n) = (4 + 0.5)(i)^ r(,r)G) r(r) : O:25u(- - l) +
1)
tlt" urn, -
nt)\" u\n) L 1jr" llrn I)
gion of convergence a 2,(a) U.-e
'": diris.on metlroC to e\al .ale.irfrt) at,r - 0 and a = - L(b) Check your value at t = 0 by rrsing the initial vaiue proPertyGiven x(z) : z(z + 1)/(j - 3z + 2), find r(n) for the iollowing recorverlence usinp lhe method indic lledi(a) ROC 1-l ) 2 by panial frlction..rbr ROC :, < I by panial frrciion.(c) ROC I < ly' < zay padial fractions(d) Roc lzl >'2 bt-ivision--Ie) ROC -l < I oy divi ionIn ev:.uatirg rhe invrrse Z transfornr b) Ihe complex inversion integtal, aintegral m'rst be compr,ted. Evaluate the rollowjng typical coniour inlegral(onrodr c:.pecrficd:ta) $.d: c l:l -t).r' , i ,(b.r I 'a: r : J.'1 -- 3J:rc/ 9--,4, ,:l!'z
2.16
2.17
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:-273- 61's1 a. catrselsFtei sPexified by itFriyslerl1 funetion ll(z) as f{,iiorvs:
2.30 Given ]l(z): (32-2 + 2z-t + 5)/(6't-2 + 4z-t + I)(a) Find a difference equation realization ifr!(r) is ihe input and /r) is the output'
for a caus;rl sysrem wiLl_ H'zl as its system lunction
(b) Is the systea BIBO stable? Why or vhy not? : '(c) Find the unit sample response i(n).:(d) Find a difierence equation reaiizatidn of fhe system. Let ,!(,r)rbe the inpui i;d
)(n I be thc oulput.{e) li lhe sy\tcrn ILr,ear or nonlinear?tf1 I' the
'ystr:rt' shifl invari rnl? Why (,r why notl"4^/( 2Jzlfh, impulse response of a discrete-lin"e system tbat is linear and shjft'nvoriant is
\:/Eiven ty 1 ..,.,lgiven by r '. ..-l .r1:1 h(ni- ti\" ul'1t + r- ji'k /?)
(a) Is this a causal or noncausal syst"mi why?(a) Is this a causal or noncausal syslem? Why?(b) Is the iyste,m an ionnite impulse resFonse (IIR) or finite impulse response (FIR)
sy: tem?(cl Is lhr sysrem BIBO 5lable? Why or why not?(d) Fild the Fourier traosform ofr(r).(e) Find the Z transform of h(n). Be sure to include region of cotrvergence(f) Give a difference equaliol.rea]ization of the syslem. .- . j ..-.(g) Fird the ilequency response of the systcm and put in nagnjtude and angle
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(h) Find the erergy in the sequence i(ni. IFind tl.e frequ:nc1 responte fo- the syst,j'n functLon Hlr) givcn b)
1-22*322H(4=n,_2,+tFind both the :oagnitude and phase of the f.equency response aod comment onyour n:sults. Show that we also have unity magnitude aI ali frequencies for,ry(z)of the follorvrn;; general form:
H(, = (a i bz r cz?)f(c t bz'f az2)A fille. with this l1(z) is called an all pass Aiter' Cascaded sections of all Pass flllersare also ali pass and arc sometimes used lbr adjustlng the phase response of a givenclisc:ere-lime 6tler.
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Tutorial sheet No.3 DSP 3'd year control cmd system Dept. all BranchesLeclurcr: Dr.Hamid
Q1: Given the sequence x(n): 1-1 ,2 "1,4\, compute and plot its amplitude and
phase spectrum.
Q2: Compute the DFT for the sequence given in the above question.
Q3: Repeat question 2 using decimation in time FFT .
Q4:Compute frequency spectrum X(K) for the sequence x(n):{ Z,1, -1, -2, -l,l "
111
compute FFT method to compute DFT for the sequence {0, l, 2, 1,0}.
compute FFT method to compute DFT for the sequence {-1,1,-2,-l,l}.QE- Drcsien a 3-tal FIR,Low pass filter with a cutoff frequency of 1,600 IIz and a
sampling rate of 8,000 IIz using Fourier Transform method. Determine thetransfer ftlirction and difference equation of the designed FIR system, andcompnte and ploi the magnitude frequency response for C) : 0, n/4 , nl2 ,3ttl4, n radians-
Q5c-Deqjgr a s-tap FIR Low pass filter with a lower cutoff frequency of 1,600 I{2," _:l
an upper cutoff liequency of 1,800 IIz, and a sampling rate of 8,000 Hz using .Determine the transfer function and difference equation of tle desigled FIRsystem, and compute and plot the magnitude frequency response for f: Z50Hz, 500112" l250Hz,lkHz, 1.5kllz .
o5.
Q6: