Digital Signal Processing
Digital Signal Processing EL ENG 123
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This 8 page Class Notes was uploaded by Kris Heathcote on Thursday October 22, 2015. The Class Notes belongs to EL ENG 123 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/226764/el-eng-123-university-of-california-berkeley in Electrical Engineering at University of California - Berkeley.
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Date Created: 10/22/15
EE 123 Digital Signal Processing Spring 2007 Lecture 5 7 January 25 evening Lecturer Prof Anant Sahai Scribe SteHen Prince 51 Outline These notes cover the following topics 0 z transform and Region of Convergence o Poles and Zeros o z transform and Region of Convergence Properties 0 lnverting z Transforms Using Partial Piraction Expansion 0 lnverting Irrational z transforms 52 z transform and Region of Convergence 521 z tranforrn de nition Like the DTFT7 the z transform is a tool for representing and analyzing sequences However7 the z transform is a more general representation because it converges for a broader class of sequences It is de ned 00 zmni Z zinw n7oo The mapping between a sequence and its z transform is denoted by This sum is very similar in form to the DTFT In fact7 the z transform is the DTFT of a sequence where the n th entry has been multiplied by the real number and the frequency w is associated with the complex part so that z Z E j When 2 17 the z tranform is equivalent to the DTFT 5 1 m 123 Lecture 5 7 January 25 mm Spring 2mm 522 Convergence of the ethahsfom amean a the ataahsma means that the ahaaaate saaaa as aaseaateay saaaaaaaaaae ex 2 zn s as Jast amp the om aees not wnva39ge eh an seaaaeaaes the ataahsaehh aees not wnva39ge a an seaaaeaees am an wahaes of m Hwevas the Lumkxm cmva eh seqaehaes a maees not A we shan see the canylalt saaaaehce 2 eah he 5a w one wnvagame The set enahaes of m eh whuh the Luansiorm wnva39ges as called the 199011 ofmxwexgence me ha asah LheROCsthen au Zsaaahthat M W aaeaasoahth aw whsaaaaehce the act that any the ahx nte vahle of zaeehhahes toan ah the saaah Men the ROC as plotted ah the complex ayaahe thas aheahs that ymms ah the ROC form mules Demaed aheat the eagh one of the males as the aahat came then the om easts a that seaaaeaee mampae 1 bet as tay w that the ataahsmh eh an whae a gt 1 Th as ah espoaaehtaany Wang seqneme u Gaady the ymhlem wath whveaaehue as due w the sad w haaaatayay the seqaaehae at a z suLh that the ny me wan he yashea aewh ahto wnvagame h we aheese z suLh that the saaah eh n yosxuve as ma w whaexxe we need M gt a Howeteh the negawe em a the Eaan gets muluyhed at pesatave ywas of 1 Now we nokxlga hae cmvexgence a the 16h me ny saae zhtaaatateays we h The mlnum as w aheeee that ehe side of the seqaaehce w be hephesehta at the Luan am Th eaaespehas to haaaatayayahx at ethex hm eh eqea e 1 am yrodvtes the nghvxdai eh aaaasah seaaaehce whde eqea e 1 gm the aeatesaaa eh ahtaaaaasah seqaaehae ee the haghtsam Eaan as aheseh 572 m 123 Lecture 5 January 25 mm Spring 2mm 1m Z QWMZ Delay For meme we heed Z az lt 00 Th 5 a geome aha that only wnva39ges when 1w lt 1 eh m lt a The mm a e geomema genes mm gm 1m Delay 1 1 7 a2quot The ma 5 he ehehe plane Weae the me 121 1h genaal 3 mm sequente Wm have eh we um extend w 00 mampm 2 The euemmh of he ehueeemh pan of he eeqhehee mph 7 11 eeh he mm m he same way he i he e mr Th geomema saAa cmvexga when M lt a 1 a M lt The mm Wench h hesehee he eehehe 1 heeeeeh heheyeh wheexehee h eeeeee he we 5 med by he We of emehe x 1 m 123 Lecture 5 January 25 mm Spring 2mm Gehaaut ah anthems aan mu have ah ROG that mums 0 5m Possibilities for ROC shape 53 Poles and Zeros oh theye we plot he xmwz ed the 17015 x3 aha mm 0 holes as value he the mots of the aehomhem whae X4 0 zem he the m the humamt whae xa o The much of 17015 and law aees not depend eh the we Howevee we may not heme poles 1h emtheht we we uuauy hounded by 17015 574 EE 123 Lecture 5 7 January 25 evening Spring 2007 54 z transform and ECG Properties A few z transform properties will be discussed here 541 Linearity For two sequences and their associated z transforms and ROC7s mm L X1z ROO R1 2M i 992 ROO R2 the linearity property states amln bx n aX1z bX2z ROG contains R11 R12 What happens to the poles when two sequences are added Usually the set of poles is P0lesX1 P0lesX2 Zeros on the other hand cannot be determined without simplifying the z transform An example illustrating this follows Example 3 Let7s nd the z transform of Fiom the rst example and linearity we get 1 1 Xz 1 7 1 7 1 7 i2 1 1 g2 1 Here it obvious that both exponential sequences have a zero at zero However this is not the case of the sum of the sequences 2 7 lz l 22 z 7 7 Xz 1 6 1 1 121 1 52 11 3271 Z 52 g While the poles are the union of the poles of the individual terms the zeros appear at z 0 Because the sequence is causal the ROC extends from the outer pole to in nity 5 5 m 123 Lecture 5 7 January 25 evenzng Spran 2mm Cases with ROC everywhere rt 5 m laung to note the of seqnetma that wnvage or the enme Lylane 1 mee yawenca m m H1 H1 Him 2 G We Irqxqu W 543 Timeshmng property men A may Roe Mew yo temmmmmtmm 00 00 544 Di erentietion property and higherorder poles The dx aenuaxon property Mata 1 dz We nan nee d1 tact w um the Luanst of 4n w uht naquotu7t 5051 on L 2 dz 717W 707W M gt M The repentm m m the dmanmawx appear e double we when plowed 545 Convolution of sequences 1M 22W mummy ROmems Lm Rn EE 123 Lecture 5 7 January 25 evening Spring 2007 55 Inverting ztransforms 551 Long division To obtain the left sided sequence simply divide with the divisor expressed in powers of z The right sided sequence is calculated by dividing by a divisor in powers of 2 1 Example 4 1a2 1 a22 2 liaz lll 1 7 124 a2 124 7 122 2 Zi a 2 The sequence can be read off from the constant coef cients of each term In this example O 1 M1 1 M2 12 Then must be inferred from a partial sequence This method is not preferred because long division must be performed until a pattern is recognized 552 Partial fraction expansion By expanding the polynomial into factors of the denominator we can match the terms to known z transform pairs If the order of the numerator is less than the order of the denominator and the poles are all rst order then this simpli es to a sum of terms with constant numerators w i L Dz k1 17 de l The constant coef cients can be found by multiplying the LHS by the terms denominator and evaluating at z dk Ak 1 dk271X2lzdk 553 Inverting irrational z transforrns Contour integral We can directly evaluate the synthesis equation which unfortunately contains a contour integral 1 7 Xzz 1dz over any R00 2717 5 7 EE 123 Lecture 5 7 January 25 evening Spring 2007 Power Series Expansion If we can express the function as a power series containing 2 then we can read off the sequence directly from the inside of the summation Example 5 Xz lnl 1271 Expanding by power series 1 n1 By grouping the terms this way7 it is easy to see that this power series is exactly the z 1 1 V L transform of a un 7 1 554 Evaluating for one value of 71 If we want to evaluate MO7 and we have 0 for n lt 07 we can assume that all in nities in the ROC act as a single point7 and evaluate Xz as 2 a 00 0 lim Xz If we want to evaluate at mm we can shift in frequency domain by z 1 Subtract 0 2 Shift back by 1 in time multiply by z in frequency 3 Take limit as 2 a 00 This process can be iteratively repeated for n gt 1 5 8
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