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# Dig Sig Proc&Analys EECS 451

UM

GPA 3.8

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This 13 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 451 at University of Michigan taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/231537/eecs-451-university-of-michigan in Engineering Computer Science at University of Michigan.

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Date Created: 10/29/15

Recitation 3 EECS 451 Winter 2009 Jan 21 2009 OUTLINE 0 Review of important concepts Lecture 3 4 0 Practise problems Concepts Discrete Time Systems 1 Classi cation of DT systems For Tzn7 0 static memoryless depends only on the present input 0 causal depends only on the present and past inputs 0 timeinvariant Tzn 7 yn 7 k7 where k is an integer 0 linear Ta1m1 ang a1Tx1 a2Tm2n o stable BIBO every bounded input provides a bounded output 2 Linear Time Invariant LTl System 0 completely characterized by the impulse response hn T6n 0 output given by simple convolution operation7 227 znikhk 3 Properties of convolution o Commutative hn hn 71101 71201 01 71101 71201 gtlth1ltngt h2ltngtgt zltngt h1n zltngt W o Associative o Distributive Mn 4 Classi cation of LTl systems by the impulse response 0 Causal The given LTl system is causal if and only if hn 0 Vn lt O 0 Stable BIBO The given LTl system is stable if and only if 227 lt oo 5 Two classes of LTl systems characterized by the impulse response 7 0 Finite Impulse Response FIR system number of non zero hn s are nite 0 In nite Impulse Response HR system number of non zero hn s are in nite 6 The system de ned by linear constant coef cient difference equation 7 221 akym 7 10 Elie 5mm 7 kt o is LTl and causal 0 can be implemented by direct form I or direct form II more ef cient o is recursive and the impulse response is HR if N 2 1 o is non recursive and the impulse response is FIR if N O hn E 1 bM Concepts 2sided Ztransforrn 1 De nition of z transform o For a given DT signal Mn Xz 227 znz where z is complex valued o If mnr is absolutely summable then Xz has a nite value where r 2 ROC Region of Convergence o The set of values of z for which the sequence mnz is absolutely summable ie z E C 227 lt 00 where C is the set of complex numbers 0 Simply put ROC indicates the region of z where Xz is nite o By de nition ROC cannot contain any poles 3 The shape of ROCs o The ROC of an anti causal signal is of the form lt lal o The ROC of a causal signal is of the form gt lal o The ROC ofa two sided signal is of the form lal lt lt o The ROC of a nite length signal is the entire 27space except for z 0 andor 2 00 Problems 1 Determine which of the following systems is static linear time invariant causal stable a Mn 90201 1 b w 8 28 c 227 Mn 7 kpk where pn 710 71Q 1 10 2 Compute the output of the following LTl systems a 90 1 1 1 1 hn 123 0 Mn 171727hn Mn 3 Compute the z transform and the associated ROC s of the following signals Recitation 9 EECS 451 Winter 2009 March 18th 2009 OUTLINE a Review of Discrete Fourier Transform 0 Practice problems Concepts Discrete Fourier Transform 1 For a time limited signal of length N the N point DFT is de ned by N l Xk Z nk jquot n 0 where k r 0 N 1 and the inverse DFT formula is given by 1 N l 2 23n N Z Xkeik 190 wheren0 N1 a Note that the indices range from 0 to N 1 for L n and Xk b The DFT is ef ciently implemented by the Fast Fourier transform FFT c The DFT matrix W whose size is N X N is an orthogonal transformation 2 Relationship to other transformations a If rpm is the N periodic superposition of a time limited signal of length N then the DTFS of rpn is obtained by N 1 Z Pne jna n20 ZIH Ck thus leading to Xk ch b The DFT of wn has the following relationship with X w DTFT and X z z transformation XltkgtXltwgtIw2k XkXzl zeJT where is a timelimited N point signal 3 Circular Symmetries N point circular even is defined by N n 1n for 1 g n g N 1 b Npoint circular odd is de ned by 33N n an for 1 S n S N 1 c If is real we have Xk X k mod N circular Hermitian symmetry 4 Properties of the DFT Suppose N T a Linearity a1x1n a2x2n N7 D ET a1X1k a2X2k b Circular time shift 1 mod N N T e kNXUC c Time reversal n mod N N T X k mod N d Circular frequency shift anej2 k N N325quot Xk k0 mod N e Complex conjugate N T X k mod N and r n mod N N531 XW f Convolution 551n 32n N T X1kX2k g Multiplication m1nc2n NdD fT X1k X2k where denotes Npoint circular convolution h Parseval s theorem 2201 lan2 N T EL Xk2 Problems 1 Compute the DF Ts for the following signals a1na39nfor0 n N 1 b 3371 61 90quot c Mn cosn 2 Given the eight point DFT of the sequence 111 1 0 0 0 0 compute the DFT of the sequences 1391n 10000 111 b 1201 Q0111100 3 Let 33pm be a periodic sequence with fundamental period N Consider the following DFTs N DFT 1300 X10 3N DFT IBM lt X309 Find the relationship between X1k and X3k 4 Let cn be an N point sequence with an Npoint DFT Xk N even Let 1ncn OgngN l y 0 elsewhere and Show that Yk X2k for 0 g k g 2 1 5 Let 2 1 0 2 be a DT signal inputted to a DT lter with frequency response Hw a Compute the output of the lter if hn 1 O 0 1 b What is the output of the lter if you use a 4point DFT and IDFT c Are the answers in the above two parts the same If not how would you use DFTIDFT to get yn PROBLEMS Nquot quot 312 X01 1006th ro Nquot 21Tkh a ej N quotn70 N39l 3 100 7 01 r i Ohm 392 la I aeJ hT ji kcm 2 e N ZDFT gal10420 439 Ng J39Q39Wnk aw 33944 gt102 c N N ho NF aQTHA h lZ TLV he J 339 NeJ N X01 6 quot ice X02 NM k ko 39 I 6 j Z 1N z Z 7M 1 5 1 i O 0 0 0 8 if if M XUL be 3 71 p 539 2 3 1 al 0 i Q 3 139 5 8 3 f 39L1r HHWMHMHWLL 8quot3 G S 11393v1 D l L 5 q S 7 391 i q 39 14 m4 2 1in 392 1L 339 71n 1 Wwd 8 255 395 JL J 3 Xh 1 HLL 2 J a Mia X39Lk C 391Pn5gt it n m v l K 1 a j 4 mm 611 a N beT 3N 3 quot g 23 w X3009 1300 e 3 n20 r N 7139ej 7731 2PM 32L 3M 3 quot 2311 n f C N fr alphaa 3N 710 nrN n2N AMMMN Lam m ZN Nquot 39 erUAsW NA M V 39 4239 3 m g 7V e N quotf g 3971PmM QJ39T lt 0 1 O 0 I 439 J g nww Q ame e N Matt Nso te 1Pn lo Pwocuc will Pw gol N 2 7 mwv 7 PVV Mm 1 er m awn a 2M 39 X301 39 a T Mtg aj 3 Xz 3 Er X z 2r x A a 4 M39 c 3 1 J jw a Mru3 L vx M W N39 Pow LCm NDF7 x k 7dr f 7L 5 4 N w L O M Nquot f m quot39 rm E Wm M m 710 Di 21 Diquot 1 L 1 W p 2TT 2 h 39 MMeJ N gt j N 1130 zz TH 71NL NA 7 i 211 211 le z aU39 39 J 39 39m39NL Z WM 6 N Jr 2 MM 6 N 4 10 I 39 rmNL QJ N N4 Nquot a 2T 2kr M 2172UM 1m 6 J N 910106 J N n20 ManiL Mquot J 2112k n g 7n e N x 2h n o Flame L XML 05125131 L 39 WV 3 9a a 3 h USES 33 n i YE V M LL IE1 o Q m o vw N n w v x M o m u PTA m P um 6 x w 6 ea 2 E ESE 93 r3v x i3 t 32 K i 13 W C 06 R3ltcxg9ltlt E S Recitation 4 EECS 451 Winter 2009 Jan 28 2009 OUTLINE 0 Review of important concepts Lecture 5 6 0 Practise problems Concepts 2sided Ztransforms 1 De nition of z transform o For a given DT signal Mn Xz 227 zn2 7 where z is complex valued o If mnr is absolutely summable7 then Xz has a nite value where r 2 ROC Region of Convergence o The set of values of z for which the sequence mnz is absolutely summable7 ie z E C 227 lt oo7 where C is the set of complex numbers 0 Simply put7 ROC indicates the region of z where Xz is nite o By de nition7 ROC cannot contain any poles 3 The shape of ROCs o The ROC of an anti causal signal is of the form lt lal o The ROC of a causal signal is of the form gt lal o The ROC of a two sided signal is of the form lal lt lt o The ROC of a nite length signal is the entire zispace except for z 0 andor 2 oo 4 Useful z transformation pairs 0 If aw then Xz 2 ROC gt lal 27117 0 If ianu7n 71 then Xz 7 ROC lt lal 5 Properties of 2 transform We have g Xz and ROCX r2 lt lt r1 0 Linearity a1z1n a2z2n g 1le 12X227 R00 2 ROCX1 ROCX2 0 Time shifting mn 7 k g 2 kX27 ROC ROCX except 2 0 or 2 oo o Scaling in the 2 domain a zn g Xf127 ROC laer lt lt lam 0 Time reversal Min g X2 17 ROC i lt lt o Differentiation in the 2 domain g 72 ROC ROCX o Convolution z1n z2n g X12X227 R00 2 ROCX1 ROCX2 0 Correlation z1n 22771 g X12X22 17 R00 2 ROCXIM ROCX2271 6 Useful theorems on 2 transform 0 Initial value theorem lf is causal7 then z0 limZHOOX2 o Multiplication of two sequences and Parseval s relation 7 LTl systems and z transforms o The 2 transform of the impulse response hn is called the system function o DT LTl systems described by LCCDE have a rational 2 transform7 ie 3 o If a signal is outputted by the system when the input signal is 2717 then their 2 transforms are related as Y2 o CausalityStability of the system can be determined from the ROC of Concepts Inverse Ztransform Partial Fraction Expansions 1 BltZgtandMltN 141 o Distinct simple roots XS X Z 0 Multiple simple roots Z if 1137 X2 A12B1 A22Bz o Distinct complex roots 2 22a2 22l72 0 Multiple complex roots X21 14212er31 125322 XV 7 31 2 2 7142 andMEN 0 Divide B2 by A2 to express it as X541 622 11 where the degree of R2 is less than the degree of A2 0 Now apply the appropriate partial fraction expansion to 12 Recitation 2 EECS 451 Winter 2009 Jan 14 2009 OUTLINE 0 Review of Sampling 0 Practise problems Concepts 1 Sampling 0 Sampling Theorem 0 Ideal Sampling and reconstruction 0 Practical issues 0 Interpretation of aliasing in both frequency and time domain Problems 1 A signal Mt cos2007rt is sampled at a rate of ws and stored ls it possible to reconstruct the original signal from the stored samples using an ideal low pass lter when a 1 5 85Hz b f9 800Hz 2 The following analog sinusoidal signal is sampled 400 times a second and each sample is quantized into 256 different voltage levels mat 2 cos5007rt 3cos3007rt the quantization effect

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