INTRO TO DIG SIG PRO
INTRO TO DIG SIG PRO E C E 467
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This 8 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 37 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 21 More on Linear Phase and Definition of Four FIR Filter Types Hequot Hequot lequot u lt Example of linear phase filter since arg Hequotquot Tau lf IDTFT Hej I hn then IDTFrHer hn a iwn Also if input xn 9 Output yn Helme lamejmn Heim eimm lHequot Jxn or scaled timedelayed version of xn where time delay or is independent of frequency 0 Now define Generalized Linear Phase A system has generalized linear phase if its frequency response can be expressed as He quot Ae quotequotquotquot 3 Where Ae is real but may be negative arg He o co B plus contribution of 7 for frequencies where Aequot lt O Mi c rz he 592th reainm 06 gt5 5 ffk a O oschxo as arm burn 4r 3 r 16th is ran in fog Ribs 26 n K hFEJ m4f i VIN qu faxn TATIva VOMIM 72 9x Theoquot VG Mlm z Z 5 wife C93 05 0 HIP ah Txc v PNJ Nut r E wkmor ha P 3ng u AEN 9 49 n6 D Jxrhy 061 R r r94 3 lt lt I IvuwBVW ifh u v I Donn Vb Now consider our quotTypesquot of FIR filters which have generalized linear phasequot Tyge FIR filter hn satisfies symmetry c ondition M is even hn has an old no of terms I Cmurol I symmurv i i i i i 0 L1 M394 n 2 4 39 39kI 0 39U 39 A 039 239 9 Q l n w 39 r w J W quot 2 1 k a J 1 a J a J S 39w2 5 2 0 2 SVllt 3 W 500 339 quotz w 375 Ae zquot E 245039 2 125 i 1 l 0 0 n 39i 31 2quot E 2 Radian frequency a la 4 2 a 2 4 l l l 0 n n 2w 5 2 Radian frequency w b Tyge H FIR filter Magnitude Phase hn satisfies symmetry condition o M is odd hn has an even no of terms Center 0 kg syn imury Radian frequency w b 393 53 H2Jw v a Jwb iJw3 Lad e 4 W I i r v r 6 4 92372 lt szI grow9 39539 2 ya gm 3 SM v 80 45 30T 15 39 5 2 5 Radian frequency w 7 I a 4 2 0 2i 4 JA 1 J 0 1 u g1 21 2 Tyge Ill FIR filter hn satisfies symmetry condition o M is even Cmmol Ivmmnrv I 39 Mi2 24 r Magnitude 08 39 7r 31r 211 0 vol L Rndian frequency w a Phasc i 4 I o colu gt NI N Radian Requech w b Tyge IV FIR filter hn satisfies symmetry condition M is odd Conm q I D E i I 6 E 08 0 J I A 0 1 x 21 21 2 2 Radian frequency w a 10 15 7 a 0 M I 15 L 3 O J I l 0 1 Ir 21 21 2 2 Radian 1requehcyu b o For FIR filter types llV there are inherent constraints on where the zeros of Hz 9 occur For Types I and ll 2 pairs of complex conjugates 4 zeros with r at 1 with the following reciprocal relations 1 212121 211 1 pair of complex conjugates 2 zeros with r1 ejquote je o real pair with r 1 having a reciprocal relationship I I 2121 1 z1orz 1orboth For M odd Type II but not Type l Hz must have zero at Z 1 1 7 gt Type ll is bad for high pass filters Fig 538 p 266 For Types l IV Locations where zeros can occur are the same as those where zeros can occur for Types I and H Locations where zeros must occur are as follows Tyge Ill 2 1 and z 1 bad for low pass and highpass Type IV 2 1 bad for Iowpass