Chapter 6Chapter 6 - J.C. Bose University Of Science And .

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Chapter edu.tw03 573115203-5731152Original PowerPoint slides prepared by S. K. Mitra4-1-1 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform The DTFT provides a frequency-domain representation ofdiscrete-time signals and LTI discrete-time systems Because of the convergence condition, in many cases, theDTFT of a sequence may not exist, thereby making itimpossible to make use of such frequency-domaincharacterization in these cases A generalization of the DTFT defined byleads to the z-transform z-transform may exist for many sequences for which theDTFT does not exist Use of z-transform permits simple algebraic manipulations4-1-2Original PowerPoint slides prepared by S. K. Mitra The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform For a given sequence g[n], its z-transform G(z) is definedas:where z Re(z) j Im(z) is a complex variable If we let z r ejω, then the z-transform reduces to The above can be interpreted as the DTFT of themodified sequence {g[n]r n} For r 1 (i.e., z 1), zz-transformtransform reduces to its DTFT,provided the latter existspof unityy The contour z 1 is a circle in the z-planeradius and is called the unit circle4-1-3Original PowerPoint slides prepared by S. K. Mitra The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform Like the DTFT, there are conditions on the convergenceof the infinite series For a given sequence, the set R of values of z for whichits z-transformz transform converges is called the region ofconvergence (ROC) From our earlier discussion on the uniform convergenceof the DTFT, it follows that the series converges if {g[n]r n} is absolutely summable, i.e., ifOriginal PowerPoint slides prepared by S. K. Mitra4-1-4 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform In general, the ROC R of a z-transform of a sequenceg[n] is an annular region of the z-plane:where Note: The z-transform is a form of a Laurent series andi an analyticisl ti ffunctionti att every pointi t ini ththe ROC Example – Determine the z-Transform X(z) of the causalsequence x[n] αn μ[n] and its ROC Now The above power series converges to ROC is the annular region z α Original PowerPoint slides prepared by S. K. Mitra4-1-5 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform Example – Determine the z-Transform μ(z) of the unitstep function μ[n] can be obtained fromby setting α 1: Note: The unit step function μ[n] is not absolutelysummable, and hence its DTFT does not convergeuniformlyyOriginal PowerPoint slides prepared by S. K. Mitra4-1-6 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform Example – Consider the anti-causal sequencey[n] αnμ[ n 1] Its zz-transformtransform is given by ROC is the annular region z α Original PowerPoint slides prepared by S. K. Mitra4-1-7 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform Note: the z-Transforms of two sequences αnμ[n] and αnμ[ n 1] are identical even though the two parentsequences are different Only way a unique sequence can be associated with a ztransform is by specifying its ROC TheTh DTFT G(eG( jω) off a sequence g[n][ ] converges uniformlyiflif and only if the ROC of the z-transform G(z) of g[n]includes the unit circle The existence of the DTFT does not always imply theexistence of the zz-transformtransformOriginal PowerPoint slides prepared by S. K. Mitra4-1-8 The McGraw-Hill Companies, Inc., 2007

z Transformz-Transform Example – the finite energy sequencehas a DTFT given bywhich converges in the mean-square sense However, hLP[n] does not have a z-transform as it is notabsolutely summable for any value of rOriginal PowerPoint slides prepared by S. K. Mitra4-1-9 The McGraw-Hill Companies, Inc., 2007

Commonly Used z-Transformz Transform PairsOriginal PowerPoint slides prepared by S. K. Mitra4-1-10 The McGraw-Hill Companies, Inc., 2007

Rational z-Transformz Transform In tthee case oof LTI ddiscrete-timesc ete t e systesystemss weeaareeconcerned with in this course, all pertinent z-transformsare rational functions of z 1 That is, they are ratios of two polynomials in z 1 The degree of the numerator polynomial P(z) is M andthe degree of the denominator polynomial D(z) is N An alternate representation of a rational z-transform is asa ratio of two polynomials in z:Original PowerPoint slides prepared by S. K. Mitra4-1-11 The McGraw-Hill Companies, Inc., 2007

Rational z-Transformz Transform A rationalat o a z-transformt a s o cacan be aalternatelyte ate y writtentte infactored form as At a root z ξl of the numerator ppolynomialyG(ξ(ξl), and asa result, these values of z are known as the zeros ofG(z) At a root z λl of the denominator polynomial G(λl) ,and as a result, these values of z are known as thepolesl off G(z)G( )Original PowerPoint slides prepared by S. K. Mitra4-1-12 The McGraw-Hill Companies, Inc., 2007

Rational z-Transformz Transform ConsiderCo s de Note G(z) has M finite zeros and N finite poles If N M there are additional N M zeros at z 0 (theorigin in the z-plane)poles at z 0 ((the If N M there are additional M N porigin in the z-plane)Original PowerPoint slides prepared by S. K. Mitra4-1-13 The McGraw-Hill Companies, Inc., 2007

Rational z-Transformz Transform Examplea p e – tthee z-transformta sohas a zero at z 0 and a pole at z 1Original PowerPoint slides prepared by S. K. Mitra4-1-14 The McGraw-Hill Companies, Inc., 2007

Rational z-Transformz Transform Apphysicalys ca interpretationte p etat o oof tthee coconceptscepts oof popoleses aandd zerose oscan be given by plotting the log-magnitude 20log10 G(z) for The magnitude plotexhibits very large peaksaround the poles of G(z)(z 0.4 j 0.6928) It also exhibits verynarrow and deep wellsaround the location ofthe zeros (z 1.21 2 j 1.2)1 2)Original PowerPoint slides prepared by S. K. Mitra4-1-15 The McGraw-Hill Companies, Inc., 2007

ROC of a Rational z-Transformz Transform ROCOC oof a z-transformt a s o issaan importantpo ta t coconceptcept Without the knowledge of the ROC, there is no uniquep between a sequenceqand its z-transformrelationship The z-transform must always be specified with its ROC Moreover,, if the ROC of a z-transform includes the unitcircle, the DTFT of the sequence is obtained by simplyevaluating the z-transform on the unit circle There is a relationship between the ROC of the ztransform of the impulse response of a causal LTIdiscrete time system and its BIBO stabilitydiscrete-time The ROC of a rational z-transform is bounded by thelocations of its polesOriginal PowerPoint slides prepared by S. K. Mitra4-1-16 The McGraw-Hill Companies, Inc., 2007

ROC of a Rational z-Transformz Transform Example – the z-transform H(z) of the sequence h[n] ( 0.6)nμ[n] is given by z 0.6 Here the ROC is just outside the circle going through thepoint z 0.6 A sequence can be one of the following types: finitelength, right-sided, left-sided and two-sided TheTh ROC ddependsd on ththe ttype off ththe sequence off iinterestttOriginal PowerPoint slides prepared by S. K. Mitra4-1-17 The McGraw-Hill Companies, Inc., 2007

ROC of a Rational z-Transformz Transform Example – Consider a finite-length sequence g[n] definedfor M n N, where M and N are non-negative integersand g[n] Its z-transform is given by Note:o e G(G(z)) hasas M popoleses aat z aandd N popoleses aat z 0 As can be seen from the expression for G(z), the ztransform of a finite-lengthg bounded sequenceqconvergesgeverywhere in the z-plane except pos

z-Transform In general, the ROC R of a z-transform of a sequence g[n] is an annular region of the z-plane: where Note: The z-transform is a form of a Laurent series and i l ti f ti t i t i th ROCis an analytic function at every point in the ROC Example– Determine the z-Transform X(z) of the

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