z-Transform z-Transform • A generalization of the DTFT defined by • The DTFT provides a frequency-domain ∞ representation of discrete-time signals and X (e jω) = ∑ x[n]e− jωn LTI discrete-time systems n=−∞ • Because of the convergence condition, in leads to the z-transform many cases, the DTFT of a sequence may • z-transform may exist for many sequences not exist for which the DTFT does not exist • As a result, it is not possible to make use of • Moreover, use of z-transform techniques such frequency-domain characterization in permits simple algebraic manipulations these cases 1 2 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra z-Transform z-Transform • Consequently, z-transform has become an important tool in the analysis and design of • If we let z = r e j ω , then the z-transform digital filters reduces to ∞ • For a given sequence g[n], its z-transform jω −n − jωn G(z) is defined as G(r e ) = ∑ g[n]r e n=−∞ ∞ G(z) = g[n]z−n • The above can be interpreted as the DTFT ∑ −n n=−∞ of the modified sequence{g[n]r } where z = Re(z) + jIm(z) is a complex •Forr = 1 (i.e., |z| = 1), z-transform reduces variable to its DTFT, provided the latter exists 3 4 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra z-Transform z-Transform • The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle • From our earlier discussion on the uniform convergence of the DTFT, it follows that the • Like the DTFT, there are conditions on the series convergence of the infinite series ∞ jω −n − jωn ∞ G(r e ) = ∑ g[n]r e ∑ g[n]z−n n=−∞ n=−∞ converges if { g [ n ] r − n } is absolutely • For a given sequence, the set of values of R summable, i.e., if z for which its z-transform converges is ∞ called the region of convergence (ROC) ∑ g[n]r−n < ∞ 5 6 n=−∞ Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 1 z-Transform z-Transform •Example- Determine the z-transform X(z) n • In general, the ROC R of a z-transform of a of the causal sequence x [ n ] = α µ [ n ] and its sequence g[n] is an annular region of the z- ROC ∞ ∞ plane: •Now X (z) = ∑αnµ[n]z−n = ∑αnz−n Rg − < z < Rg + n=−∞ n=0 where 0 ≤ Rg − < Rg + ≤ ∞ • The above power series converges to •Note:The z-transform is a form of a Laurent 1 X (z) = , for α z−1 <1 series and is an analytic function at every 1− α z−1 point in the ROC • ROC is the annular region |z| > |α| 7 8 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra z-Transform z-Transform •Example- The z-transform µ(z) of the unit step sequence µ[n] can be obtained from •Note: The unit step sequence µ[n] is not 1 absolutely summable, and hence its DTFT X (z) = , for α z−1 <1 1− α z−1 does not converge uniformly by setting α = 1: 1 •Example- Consider the anti-causal µ(z) = , for z−1 <1 1− z−1 sequence n • ROC is the annular region 1< z ≤ ∞ y[n] = −α µ[−n −1] 9 10 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra z-Transform z-Transform •Its z-transform is given by −1 ∞ n −n −m m •Note: The z-transforms of the two Y (z) = ∑− α z = − ∑α z sequences α n µ [ n ] and − α n µ [ − n − 1 ] are n=−∞ m=1 identical even though the two parent ∞ −1 −1 −m m α z sequences are different = −α z ∑α z = − −1 m=0 1− α z • Only way a unique sequence can be 1 −1 associated with a z-transform is by = −1 , for α z <1 1− α z specifying its ROC • ROC is the annular region z < α 11 12 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 2 z-Transform z-Transform z-Transform •Example- The finite energy sequence jω • The DTFT G ( e ) of a sequence g[n] sin ωcn converges uniformly if and only if the ROC hLP[n] = πn , − ∞ < n < ∞ of the z-transform G(z) of g[n] includes the has a DTFT given by unit circle jω ⎧1, 0 ≤ ω ≤ ωc • The existence of the DTFT does not always H LP (e ) = ⎨ imply the existence of the z-transform ⎩0, ωc < ω ≤ π which converges in the mean-square sense 13 14 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Table 6.1: Commonly Used z- z-Transform Transform Pairs • However, h L P [ n ] does not have a z-transform as it is not absolutely summable for any value of r • Some commonly used z-transform pairs are listed on the next slide 15 16 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Rational z-Transforms Rational z-Transforms • In the case of LTI discrete-time systems we •The degree of the numerator polynomial are concerned with in this course, all P(z) is M and the degree of the denominator pertinent z-transforms are rational functions polynomial D(z) is N −1 of z • An alternate representation of a rational z- • That is, they are ratios of two polynomials transform is as a ratio of two polynomials in in z − 1 : z: P(z) p + p z−1 + .... + p z−(M −1) + p z−M p zM + p zM −1 + .... + p z + p G(z) = = 0 1 M −1 M G(z) = z(N −M ) 0 1 M −1 M −1 .... −(N −1) −N N N −1 .... D(z) d0 + d1z + + dN −1z + dN z d0z + d1z + + dN −1z + dN 17 18 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 3 Rational z-Transforms Rational z-Transforms • A rational z-transform can be alternately • At a root z = ξ of the numerator polynomial l written in factored form as G(ξ ) = 0, and as a result, these values of z l M −1 are known as the zeros of G(z) p0∏ (1−ξ z ) G(z) = l=1 l • At a root z = λ of the denominator N −1 l d0∏ (1− λ z ) polynomial G ( λ ) → ∞ , and as a result, l=1 l l M these values of z are known as the poles of p0∏ (z −ξ ) = z(N −M ) l=1 l G(z) d N (z − λ ) 0∏l=1 l 19 20 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Rational z-Transforms Rational z-Transforms •Example- The z-transform • Consider M 1 (N −M ) p0∏ =1(z −ξ ) µ (z) = , for z >1 G(z) = z l l −1 N 1− z d0∏ (z − λ ) l=1 l has a zero at z = 0 and a pole at z = 1 •NoteG(z) has M finite zeros and N finite poles •IfN > M there are additional N − M zeros at z = 0 (the origin in the z-plane) •IfN < M there are additional M − N poles at z = 0 21 22 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Rational z-Transforms Rational z-Transforms • A physical interpretation of the concepts of poles and zeros can be given by plotting the log-magnitude 20 log 10 G ( z ) as shown on next slide for 1− 2.4 z−1 + 2.88 z−2 G(z) = 1− 0.8 z−1 + 0.64 z−2 23 24 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 4 ROC of a Rational Rational z-Transforms z-Transform • Observe that the magnitude plot exhibits •ROC of a z-transform is an important very large peaks around the points concept z = 0.4 ± j 0.6928 which are the poles of • Without the knowledge of the ROC, there is G(z) no unique relationship between a sequence • It also exhibits very narrow and deep wells and its z-transform around the location of the zeros at • Hence, the z-transform must always be z =1.2 ± j1.2 specified with its ROC 25 26 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra ROC of a Rational ROC of a Rational z-Transform z-Transform • Moreover, if the ROC of a z-transform • The ROC of a rational z-transform is includes the unit circle, the DTFT of the bounded by the locations of its poles sequence is obtained by simply evaluating • To understand the relationship between the the z-transform on the unit circle poles and the ROC, it is instructive to • There is a relationship between the ROC of examine the pole-zero plot of a z-transform the z-transform of the impulse response of a • Consider again the pole-zero plot of the z- causal LTI discrete-time system and its transform µ(z) BIBO stability 27 28 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra ROC of a Rational ROC of a Rational z-Transform z-Transform •Example- The z-transform H(z) of the sequence h [ n ] = ( − 0 . 6 ) n µ [ n ] is given by 1 H(z) = , 1+ 0.6 z−1 z > 0.6 • In this plot, the ROC, shown as the shaded area, is the region of the z-plane just outside the circle centered at the origin and going • Here the ROC is just outside the circle through the pole at z = 1 29 30 going through the point z = −0.6 Copyright © 2005, S.