INTRO TO DIG SIG PRO
INTRO TO DIG SIG PRO E C E 467
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This 12 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 467 at Clemson University taught by John Gowdy in Fall. Since its upload, it has received 41 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.
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Date Created: 09/26/15
ECE 467667 Introduction to Digital Signal Processing LECTURE 12 Frequency Response z 39 Consrder Hz ROCzt21gta z a where a lt 1 He Hz 226M elm ewa Geometric interpretation 2 91m 2 plane Kt k Vector representation ECB T467 667 Introduction to Digital Signal Processing 1 K Distance from Z ei to zero at z0 Distance from Z e to pole at za varies with 0 meanKl Leiw lHzx iigi 1 minHz Leia iHZ z 1 1JE OFT For arbitrary m iHzi 1 J12 a2 2acosco Formula for length of 3rd side of triangieii the other two sides and the angle between them is known Phase Response iziej i m39Z 2 arg Hz arg z arg z a de of line comm angle of line connecting z 93 with zero at 20 2 ei with pole at za co in this ease ECE T 467 667 Introduction to Digital Signal Processing M 0 arg za O OO 21 arg Hz m0 E chum 11 Etan 1l 2 a i 2 a 2 arg Hz w 2 33 See below 2 Ajnz quot V W ECE T467 667 Introduction to Digital Signal Processing More General Example Hm kz a 2 bZ b d1 length of 21 a d1 d2 length of 21h d3 length of 21 b n 9 3 81 angle of zra 92 angle of 21h 83 angle of 21b Heim1 lk d1 arg Hejm 9139quot 92 93 arg k W 0 if k is real pos 1 if k is real neg In general lHew1 kl Hidistancefrom zejm to im zero Hidistanoefrom 291quot to jth pole ECE T467667 Introduction to Digital Signal Processing and arg Hequot 2 angle of line connecting ith zero to z e l angle of line connecting jth pole to z ej argk 1 Final note on z transforms ROC form xn tyge 2 gt a rightsided xnO nlt0 rightsided xn 5 O for a lt 21 lt 00 some nlt0 lzl lt a leftsided Xn0 ngt0 leftsided xn 0 for O lt zl lt a some ngtO finite duration xn0 for ngtO and xn i O for at least one n lt O lzlltoo finite duration xnO for nltO and xn 7 O for at least one n gt O zgt0 i i a lt lzl lt twosided ECE T467667 Introduction to Digital Signal Processing 00 finite deviation xn at O 0 lt 2 lt oo for some nlt0 nq for 00 some ngt0 all z xn 0 only for n0 Recall Xz Z xnzquotn n 22x2 zx 1 x03Sl 2 z 4 EDGE T467667 Introduction to Digital Signal Processing Chapter 3 Analoq Filter Desicm Recall ideal Iowpass analog filter H09 l 4 Ideal Low Pass Filterquot l 9 Q C E passbancf stopband Actual realizable Iowpass filter 1 Hon AC Al quotquot quot1quotquot Q Q Q c r lt gt passband stopband transntion band In passband AC g Hj 2 lt1 ln stopband O lt Hj 2l lt Ar Transition band width 9r QC Where QC cutoff frequency Butterworth Filter ECE T 467 667 Introduction to Digital Signal Processing 2r n orderquot of filter no of poles of Hns 2 1 increasing n l l 1 I l I I I l l I Q Q G Let Gn 2 20og101Hnj 2 quotgainquot in db 10Iog1oHn0s2gt12 1 PTquot 1 S20 9 2n 10Io 1 910 20 For 2 ltlt QC GnS2 z 1Olog101 0 39 2n For 2 gtgt 520 GnQ z 1Oog1o 20n Iog C 10Iog10 ECE 391 467667 Introduction to Digital Signal Processing Consider 21 1052c 54521 520n og10 20n db 22 1oog2c Gn 22 a 20n l0g100 40n db 23 1 00096 Gn 23 a 20n lotg1000 60n db gtGn 2 decreases by 20n db per decade for 2 20 Approximate Hj 2 in db 52c 10Sc 1009c Q 2oj 5 4o slope 20 dbldecade 50 slope 40 dbdecade slope 60 dbdecade 2 1 H09 I 1296 39 2 Independent of n In db this is 1010910 301 db NI L ECE T467667 Introduction to Digital Signal Processing Recall for Butterworth filter may 2 1g2 HnumHn39um 2n 1 m 90 I HnUSllH 1049 In general for a stable system H052 Hs S jg and Hs Hj 2 Q 139 Likewise H s H j2 Q s l 9 HnsHn s Hnj 2Hn j 2 I Q 3 1 IHHUQ12 2n S Q 9 1 s J QC 1 2n 1 8 JQC Poles of HnsHns are the 2n roots of 5 Zn 1 1 0 190 Solve for roots 10 ECE T467 667 Introduction to Digital Signal Processing n n 9 2 n even Szn C2 01n1 C2 Czn 90 n odd 1 j S QCZ 2 n even 1 Qc2quot2 n odd In general for a complex value R 2 He 1 621dlt 1 RE Rrie N k01 N1 Therefore for n odd poles of HnsHns are at Sch 2 Qc e k01 2n 1 For n even poles are at s ch ocejk7n k0 2n1 11 ECE 1 467667 Introduction to Digital Signal Processing Examgle n3 splane pole spacing lt IE n 3 starting angle O X J V Poles of stable H3s Poles of H3s Example n4 x x 39 splane pole spacing lt ZE n 4 7 TE 7395 startm an le g 9 2n 8 k J J Y Y Poles of stable H4s Poles of H4s
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