Question

Problem 5.3 (20 Points) A discrete-time, linear time-invariant system H is formed by ar- ranging three individual LTI systems as shown below. LTI LII System 1 System 2 n] > >yn] ATI System 3 Figure 2: The cascaded LTI system H. The frequency response of the individual system H, is as follows: H2 : H el) = -1 + 2e- ja The impulse response of the other individual systems are as follows: Huhn = 0[n] - [n - 1] + [n - 2] - 8n – 3] H3 : h3[n] = 0[n]+68n - 2] + 28[n - 4 (a) Find Hej), the overall frequency response. (b) Sketch the magnitude and phase responses of the overall system H by hand for -23 W S 27. What are the fundamental periods of the magnitude and phase responses?

Answer 1

Answ Gi ven that Ramgng thee ndividual LTI sygtemy ay hown. below a) omiven Cay Caded lT I Sykm H2 H Ha H z)3 H(ew) (HIeSu H2(eu)H3(eu) Given that, H2 (e Su)- 1-es h n2 S(n)S m-1)m-2)-23 (m-3) Apply DT FT heM

24 h3(m) 2()t2s (n-3)-28 (m-4) Apply DT FT thn H3 (eiw)-2t 2e 2e 4w hesu) 2+eju 2e Tar2030 a (e) ies e_ 2 ¢3i@ 2 ¢° 2 e чiw -e42e 2 e (eju) -e е leu) ee 35w we Con eipend it is dalous -lcos w isinu)-0s3w isinzu)

Hleu) - (cos w+Cos 30)+ 5(simwtsingu) [ } )ماذم( | COsw Cos3u) (sinut Sin 3u Simwt Sin3u) ton Closw+Cos 30) H(e*san)l2 |n(en)1-. - 2 [; 0 = )س )دح 31/2 1H (e5u)120 aut w 2 (esw)-o T/2 eiao at w 0-H(esu)2 The usponee 0t magnitude is cy follows

MYMY 2 4 .4-14 -TT IT/2 - 2T31)2 37/2 2T T2 Ti Phase JesPonge is aytollows b Substtuting w Valus ay Same a magnitud ses Pon se we get he es Ponse o Phoe ay follows 2TT 2 TT
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