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by: Eloy Ferry


Eloy Ferry
GPA 3.84

John Gowdy

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John Gowdy
Class Notes
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This 11 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 42 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 T467667 Introduction to Digital Signal Processing 39 LECTUR 17 Although the derivations for the bilinear transformation were for a first order system holds for the general case of an N horder system filter Two essential conditions for mapping analog filters to digital filters Condition 1 Poles in the lefthand splane should map to inside the unit circle in the zplaneso that stable causal analog filters map to stable causal digital filters 4 z 1 Check condition 1 for the mapping 5 2 31 sgz lgt Iz1z 1 Tz1 2 I 1 S 2 2 1S39l 39 9 Now substitute 8 o j into A r Af x T T 2 39T 2 1 ojQ Eh 91 2 2 2 2 22v and 121 2 2 1 Cj 2 1IG 2 39 t 2 i 2 B If olt0 AltBgtz2lt1gt izlt1 W4 LH SW51quote inside unit circle 2 Condition 1 satisfied ECE 391 467667 Introduction to Digital Signal Processing Condition 2 jQ axis in the s plane should map to the unit circle in zplane These are the places where the frequency response is defined 2 1 19 If oO lzizz E lZ 12 z1 VJ 2 I gt Condition 2 satisfied in was in 8plane 1 unit circle in zplane 2 Details of the mappinq from jQ axis of s piane to unit circle of zplane QT 1 quot239quot 1m 2 When 5 JQ e I QT Z 1 1 2 has form 26 refeeize QT gt on 26 when 9 tan391 tan391 r So a 2tan n u Examgle for T1 ECE 391 467 667 Introduction to Digital Signal Processing 8711 9 1c 9851 l i 51JZt 1 1 1 r 5Y39T1 0 5b 90 39quot Q gt Analog Filter Ha 9 1 50 90 100 Q Digital Filter Hejco 3 9 N 1 321 E E1 72 D Q HUD 90 1 99100 37E 2 981001 57 5 95100 7571 10 90100 871t 50 50100 977 90 10100 9851 6 100 0 9871 1000 0 99871t 10000 0 99997t EOE T467667 Introduction to 39ligita1 Signal Processing Note For an analog filter with a quotflatquot response in its passband the only effect of the above warping would be to shift the cutoff frequency 1 Haoo Heim I 39 g 10 877 739 21 one period For realizable filter the primary effect of the warping is still on the value of the quotcritical frequencies Hagn Hem kl kii gt 39 k2 l l I 1 1 1 751 7 21C 0 871 D gt Effects of warping on the actual curve of He are not important as long as performance meets a set of designspecs in the desired passband and stopband Eg we don t care exactly whatthe quotripplequot looks like as long as it is restricted in magnitude Steps for designing a digital filter using the bilinear transformation to map an analog filter Given a set of quotcritical digital frequenciesquot m1p2con at which gain conditions k1k2kN are specified 1 Prewarp the mi to get a new set of critical analog frequencies Qizgtan Pi l12N T 2 EOE 391 467667 Introduction to Digital Signal Processing This is the inverse mapping for the bilinear transformation as shown below 3 co 2tan 2 T 2 D 1 tan 2 2 D QT tan 2 2 g tan 9 9 T 2 Design an analog filter Hs which satisfies gain condition ki at 09i for i12N Map the analog filter Hs to a digital filter Hz Hs More on bilinear transformation Example Design a low pass digital filter With the following properties No ripple allowed euse Butterworth filter D1 2 57C 032 751 flail 301 db 15 db ECE 391 46766397 Introduction to Digital Signal Processing 1 Prewarp critical frequencies Q1Etan912 tan257t 2 T 2 Tv v T 2 a 2 48282 9 tan tan375n M 2 T 2 T Design or find in table Hs Since k1 301db QC 21 for Butterworth filter Solve for n 39 k 10 715 1 1 log Jim I 10 n W10 i19412i2 In general when k 301 db first solve for n then solve for QC using J Ti 12 E 10 1 1 10 0 1 From Table 31a the normalized Butterworth filter for n2 is 1 H l s norm 2 J ST Use LP gtLP mapping to get Hs EOE 1 467667 I39ntmdiiction to Digital Signal Processing HSHnorms seipiz c 4 2 1 4 INKS 2 2 22 V sT J I1 Ts 2 Ts4 4 2 4 3 HZHS 21 24 2 SET 12 T2 31quotquot2 1 2f2 T E1724 4 T1z4 T1z4 41z391 41 2 4J 1 2quot X1 2 41 24f 12z 1 2 2 1 22 1 z 2 xI 1 23122 1 z 2 1 22 1 22 2 Hltz 34142135 2 5857865 Impulse Invariance Desiqn Method for Diqital Filters 1 Start with Hs having desired characteristics 2 Find ht 1Hs 3 Let hn httnt sampled version 4 Obtain Hz 3hn ECE 1 467667 introduction to Digital Sigma Processing Examgle N 1 1 Hs Z A causal stablepoles in LH s plane 4 0 N Hz Ake aknT n n0 k1 N m Z Ak E Z 1eakT k1 nO N 1 lz b qkT lt1 all k HZ39 Ak k1 1quot Z 9 k 12 gt e o kT all k Poles at z e kT k 12N Check mm S39p39ane po39es at 3 01k 6k M k 212N Z39Plane poles at z e kT eokTeijT T 2 equotk If 0k ltOthen lzllt1 gtCondition 1 satisfied V J V LH Splane insude unit circle EOE 391 467 667 Int139od391iction to Digital Signal Processing Check Condition 2 Form Sampling Theorem if hn htLtnT then Hej 22 1 2 H jQj 2 7Ek whereQ 13 T k T T Hzonumtcurcienzpiane Hson jn axisin splan 2 Condition 2 satisfied ix 0 r l I T I I I l I i 39u i l i i 27 C 0 7 2n if QmaxT lt m then Hej for lt 7 ECE 391 467667 Introduction to Digitai Signal Processing QmaxI gt 11 Otherwise Hejquotquot is an aliased versio of H jQ Hej r k m a 21 3975 0 7C 275 Qmax T Details of the mapping 27 Q1 3 Q 3 33 z plane splane 1 T 1O EOE 391 467 667 Introduction to Digital Signal Processmg 9 39 Each section of j 39axis of length Q maps onetime aroundunit circle in zplane 11


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