Dig Sig Proc&Analys
Dig Sig Proc&Analys EECS 451
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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