INTRO TO DIG SIG PRO
INTRO TO DIG SIG PRO E C E 467
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This 5 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 40 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
ECE39 467667 Introduction to Digital Signal Processing Lecture 6 Chapter 2 z Transforms for DSP Motivation for use Extend the power of frequency domain analysis Recall for LSl system Yle39w Xle39 Hle w 2xnequot n Xhne39j quot Converges for all stablesystems n n may not converge Examgle xn a un Recall from Lecture 3 DTFT Xel ixWe kquotn Xane j l n0 n0 1 Ewe 1m iff 39ae l lt1 0 1 aequot dlt1 To handle case of all gt 1 introduce a quotconvergence factor rquotquot where r gt0 ie consider DTFTxnr leltngtr quotefimquot Zlar e 1 1 ar 1e j Valid if a r je D lt 1 lar 1l lt 1 la lt r This leads to definition of z transform ECE 39i 467 667 Introduction to Digital Signal Processing aixmii zxmirquot no where zre r 2 0 Values of z for which the Z converges are collectively called the region of convergencequotROC of Xz The ROG is a region bounded by circles centered at the origin in the complex zplane N0Ei 39 If the ztransform converges for r1 Xzr1 DTFTxn Xej z 2 el Examgle revisited z transform of xn a un Ekl z lzlgtlal oo on n H Xz Zanz 23 1 39 nO n20 Z 1 I 93 Xz has a quotzeroquot at 20 and a pole at za j l N ROC of Xz ECE 1 467667 Introduction to quotDigital Signal Processing If a lt1 then theiRO39C includes the circle where 2 1 where r1andzequot I 39 gt If a1 lt 1 then X e w 1 jg 21239 ejm a 1ae j e 39 Same as when Xequot Was derived directly L a1 z 1 Example 3mm 3a un Izlgt1 Zplane 7R0C 0 Examgle Leftsided signal goes from n ooto n 2 some finite value xn bquotu n 1 T1 for n 1gt 39Ogt n21gtns 1 391 1 b n 22gt Xz Zbquotz39n Z J 39noo Z ECE T 467 667 Introduction to Digital Signal Processing Letmn N 2 m 0 Z m Xz j j 1 b gob 1 1 Elt1 1 b b lzlltbl Example 2sidecl signal extends from n oo to n oo xn a un bquotu n 1 Combine previous results to get Xz XZ Z Z z a z b ROC Overlap of 21 gt a and 2 lt b z plane ECE T467 667 Introduction to Digital Signal Processing39 Examgle finite length signal xn 6n 1 6n 8n 1 a I 1 T n Z 1xnz n z1z 1 ROC r inltzltltgto 39 gmp wnotherfinitell39ength sign 11 39 xnun unN x f1 quotquot 13924 z Z n0 1 24 z z39m39 z 1 R00 z gt 0 since xn is quotcausalquot and has finite length Note that Xz can be written 1 1 1 X2 1 H z 22 z
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