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# INTRO TO DIG SIG PRO E C E 467

Clemson

GPA 3.84

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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 59 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 4 Examgle continued sing N 3m 2 He 1 m sun 2 t Note that sin 03E 2 0 when 9 K7 2 vl 03435 Consider N5 i I 39 gt m Also sin 32 0 when 93 K7E 2 Hg wK2n When dK27t both Sm 325 and sing 9 0 one case of OFK39ZTI For xltlt1 sln x as x or use L39Hopltal39s rule 1 ECE 14676567 Introduction to Digital Signal Processing 5 5 39 sing Plot for N5 1 sm 2 5 1 co 5 5 5 5 Also an He 5 1Oorn2mor r n 4115 3W5 2715quot quot5 39 Td5 27 21r5 3ru5 47c5 n Recall for LSI systems 5 xn 2 919 e He and 39 ECE39 467667 Introduction to Digital Signal Processing xn Acoslmon 1 Hei These are steady state solutions n AHej 1cosmon p arg Hequot lf xn does not extend from n I So ton oo then yn also includes an additional transient response Infigure 113 p48 xn begins at n0 and is 0 for nlt0 Mi 5 4 Figure 113 Steady state and transient rc5ponsc 3 4 2 of a system speci ed by Mn I 1 O8ynl x0010 an input xn cos 0051mun 0 l I 2 3 4 lt Transient gt o Study state Mn Munitude change xn Examgle Design of Ideal lowgass filter Desired Frequency Response HEW 1 l n no O me n 1 1 1 elmquot 139 410m 39 Dc hn 2 J e dm 2 jn mo 9quot 0 1enltncn men n 0 21tjn 7m ECE39 467667 Introduction to Digital Signal Processing 39 39 39 n For n0 hn 1 I1 dm 01 f3 20 C 92 275 me 271 2n n For me E 2 I hn g l quotQ lt non causal n 2 39 lt IIR I f I I I 0 1 2 3 l 6 1 2 sinwcnl C d A h oo onsr er n n n n gt gtsystem is BIBO unstable To obtain a stable approximation truncate hn ie set hn 0 for lngtN where N is finite Frequency response for truncated hn Hej AASAAIAAAAAA yvv VVYVV vvx quotTC TC 1 27 ECE 467667 Introduction to Digital Signal Processing Consider convergence of Fourier Transfrom If lt a then 2hne jmn conver es Hejm n n uniform y to However if 2 hn I gt oo it is still possible for Ethne39l on to 39 n n converge to some Hel although the convergence will not be uniform convergence The previous example ideal low pass filter is such an example Using Fourier Transform to describe and analyze LSl svstems Consider xn hn yn Vln Z xkhn k k new zyltne iwn DTPrim n xkhn ke l MPY by elmquot e l k and interchange order of 2 s 1 ECE fi i467 667 Introduction to Digital Signal Processing yeiw z xke jcokz hn kbjuKnk Y For each k let mnk l hme39j quot Hequotquot gtYei xe39iwHej 39 39 LSI Systern described in frequency domain Parseval39s Relation First define the energy of a discrete time signal E zn xnxn anlxmz If xn is eal then E Z x2n n Now derive Parseval39s relation Write E xnxn lei ei quotdw xquot ndm xer pow am g vJ Z xne j n J X39lequot n xeimfdm 1 Xe 2dm Parseval39s Relation n 6 ECE 27467667 Introduction to Digital Signal Processing Samplino Theorem Let xa t be a bandlimited signal with Xa 0 fer all S2gtQowhere Qois finite Then no information islost in sampling the signal if the sampling rate 1Fsatisfies 1 gt 2f0 where 820 21391f039 units of f0 Hz units of 20 radsec Reconstruction Formula Start with bandlimited 39X t 1 Let xn xat tnT where F gt 42fo Then xat can be reconstructed from its samples using singtukT xaa zxkgt H T k T t kT Also called Interpolation Formula t sing t1wkT At tt1 between samples xat1 Zxk I k 731 ECE 467667 Introduction to Digital Signal Processing Next prove Sampling Theorem and Reconstruction Formula Steps 1 Express xalt lnT in terms of Xaj 2 2 Express xn in terms of Xej using definition of Inverse DTFT 3 Express Xe in terms of X8109 by equating the expressionsof steps 1 and 2 for xn 4 State Sampling Theorem 5 Derive ReconstructionFormula using Results obtained by developing Sampling Theorem Definition of Inverse Fourier Transform of analog signals Definition of DTFT quot

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