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# Digital Signal Processing ENGR 361

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This 16 page Class Notes was uploaded by Clair Nolan PhD on Friday October 30, 2015. The Class Notes belongs to ENGR 361 at The University of Tennessee - Martin taught by Somsak Sukittanon in Fall. Since its upload, it has received 56 views. For similar materials see /class/232380/engr-361-the-university-of-tennessee-martin in Engineering and Tech at The University of Tennessee - Martin.

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Date Created: 10/30/15

w H 3 K39I t mll l jwtiL 3131mm 1 I iriliil 7 requenuadan hequemymadansa Mex phaseads V dmensaness mesa phasemds Basic Functions Unit sampxc n 5M n UM eiomuh sinom Geometric Sum N2 k aN1 aNz1 E a N2 2 N1 1 a kN1 Examples 205 710 N71 eijwn Types of Discrete time Systems Output at time 11 only depends check if yEnj depends on any Memoryless on input at till 1e y Olt xEn Pj Tac1 21302 ch cl 339 1 1 11 an Sea 111 241 n 342 n aTc1 n bTc2 n T t The shift of input causes the 7 n n 7 n put no and see line lnva rla n shift in output 0 7 y 0 i 7710 ZJ7 L i no 0 1 Output at time 11 only depends y OC 0 n 7 no check if yEnj depends on any auS 1 37 on input at time n and previou no gt O x npcgtsitive number every bounded input reduced I33 I S Ba lt 00 then substitute xEnj Ait39h c0nSta nt Stable 39b unded out ut BIB B d n ln nlty p I I S By lt 00 see inynj is bounded Memoryless Differezllce Difference T ime invariant Causality 0868 Book Example 2 1 a Txn gnxn with gn given b Txn 22n0 xk c Tx n Zi ino xk d Txn xn no 9 Txn exinl f Tx ax b g Txn 39x n h Txn xn 3un 1 e Memoryless Linear Time invariant Causality Stable Convolution Sum yn 93TH hn k oo 0 If xEn and hEn is nite What do we know about yEn 0 What are the starting index and the end index of yEn 0 What is the length of yEn 2 Ways to Compute Convolutional Sum I Pick XI1 or Xn I Pick hn or Xn I Duplicate hn at every non I Bum all values ylnl Properties of Convolution Sum M 1M Mn 00 M w M ynrnh1nh2n kg 34km 7 k M W M m zhmzmwn hm mwwwamhhdmi 0 Cascade TM haw Z hkxnik 76700 Commutative 96 MM MM xfnl W Y Maw 03 Find the Impulse Response yn xn nd M2 1 M1M21kZM xquotk 4 Mn 6n 7 nd quot Absolutely Summable If 2 lhlkllltltgtltgt thenLTIsystemisBIBO k7oo ylnl n M Z xlk k eo yn xMn yn1 xln 1 xn Yquot xln illquot 1 Causal hn 0 1 5 Delta Function 6 n77 nir function zn6n77 mm value Examples xn6n nd5n ndxnl 2 hn 5n1 8n8n 1 3 hn un 5n 8n 1 Lmear Tune 1mm LTD 2M 5M 472776 2M 7246 MW 76 mguenoy Domam Hepxesenmmon of LTI 2M1 quot zu39nmmm my mm M i m Properties of Frequency Response Complex values HM HM 1H1ej HeiwefltHlte quot gt Periodicity 0 Hejw27r hne jw2rn39 MZOO Hejw2quot39 Hej forr an integer e jw2xn e jmne un erjum Examples of Frequency Response Yquot xn quotd1 00 Hew 2 5M nde jwn e jwnd n oo Hef 1 1 M M hm M1M21 15quot 2 4110 m439 0 otherwise 7 1 M2 H 1 quot 7 Iwn e M1M21 2e n 1 sinwM1 M 12EMM1MZ M M1 1 sinwZ 20 Condition for Existence IHequot I Z hke f 5 Z hke i 5 Z hk lt00 k oo k oo k oo Frequency response must exist for stable systems Discrete Time Fourier Transform DTFT xn i Xej DTFT Inverse DTFT Xeiquot i xne7 n oo 1 7 xn Xequotquote quotdw Xej0 X1 7 xO 7 V21 V22 TABLE 21 SYMMETRY PROPERTIES OF THE FOUHIER TRANSFORM Sequence Fourier Tronsfom xn X81 N 1 x n X e39i 2 x39 n x eiw 3 Rexn Xe39 conjugateswimsuit par of Xei 4 ijxn X001 conjugateantisymmetric part of X em 5 xn conjugatesymmetric part XRequot R2Xe quot of xn 6 xn conjugate antisymmetric X1e iJmXe pan of x The following properties apply only when xn is real 7 Any real xn Xe quot Xquote Fourier transform is conjugate symmetric 8 Any real xn XRe XRe J real part is even 9 Any real xn X1021 Xe i imaginary part is odd 10 Any real xn Xei Xe 1 magnitude is even 1 Any real xn ltXequotquot iltXe39 phase is odd 2 xn even part of xn X1421 13 xan odd part of xn lezj TABLE 22 FOURIER TRANSFORM THEOREMS Sequence Fourier Trousform xn X 4 yn Ye 1 axn byn aXe bYei 1 Linearity 2 xn 7 nd nd an integer 2 1quot Xel quot 2 TimeShift 3 ejamnxh Xiwwo 395 Frequencyshift 4 x n 5 nxn 6 xn gtk yn 7 xnyn ATE m 4 Timereversal X2quot if xn real 113131 do X jwYeiw n i Xef Yeiw gtde 2n quot 6 Convolution 397 Window ing Parseval s theorem 8 i lxnz cc 7 2i Xeiquot 12dcu energypreser39ve 7r 4 9 Z xny n Xei Y39ei dw 23 24 Convolunon m Tune Muluphcanon 31 17mg 2M gt39 gt 5M i 2klhlnikl 2 m i wwrrw K2 MM i am y WW Q XENR 3a z 25 Lmss rConatamtrCosL msnt mLfsmms EQ imminxwl M om Ewwww E bnr39uwm m x L Imam E n wz wk u Examples of Diff EQ l 390 yn yn 1xn xn 1 D Heiww w 1 e m 9 yn1 iyln 1 yln a 2 3xn em eum i 1 r yn xln 2xn 11 xn 2 V FIR VS IIR yn j bmxn m 0 FIR Finite duration impulse response the output only depends on input and its delay The length of hn will be nite not in nity Always be stable 0 HR In nite duration impulse response the output depends on its previous delayed output value feedback The length of hn is in nite not always be stable due to feedback 1 film 9 1 e14w yn ym e 1 gym e 2 mu yn xn 2xn 1 xn 2 27 28 1 Hannaquot I huur39lihzse sysia lawman My Linear Phase FIR What do we want from Phase Response Ideally no delay So the phase plot needs to be at a horizontal line at O or pi Why zero phase is not practical Non causal If we need causal we want the delay to be the constant so we can maintain the envelope shape Group delay is constant phase plot is a linear line Example 1 odd length h12321 h 12372 71 Example 2 even length h12 21 h12 72 71 4 Types of Linear Phase FIRS Cenmroi r symmetry mm a M M74 71 Type Symmetry zerosM Length M 2 139 Symmetric Even Odd Camerof r symnulxy T 1 I 1 I I I Z Symmetric Odd Even T 0 AL TM 1 7 3 Amisymmeiric Even Odd is a t 1 53553 4 Antisymmeiric Odd Even T I 342 0 T n 39 4 6 0 m ksymmetrv 1M7 H mm Exam Tasman Hnearphase systems Hype 39I even hM7n mTypeu M edit my MMi n c Type HT M even my rITIM r n 1dTyDE N d M arm hm 4w n Zero Locations in Zplane 0 Again we normale deal with real coef cient 0 Due to the symmetric or anti it produces the conjugate reciprocal similar to Allpass except we get 4 zeros 0 Some types are constrained to have zeros at O or pi or both L3 5quot 0mm Type Constrained Zeros m 2 7T 52 w 3 0 and 7T 3 c A 4 0 a mm 5 quot Yypl al mm m was VT Mneawnase sysmns a Typu x2 Tygz H 4c Type TH a Type N

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