Question

Example – Right-Sided Exponential Sequence (1) + Consider x[n]=d"u[n]. Because it is nonzero only for n > 0, this is an example of a right-sided sequence. X(z) = Ža"u[n]=" = {(az")"

Answer 1

Right sided exponential sequence can be solve in below ways

right-sided exponential sequence Consider the signal x[n] = a" u[n]. This has the z-transform x() = a*a[n]==" =Z=y". n=- n=0 Convergence requires that È laz=1" <00, n =0 which is only the case if |az-1|< 1, or equivalently | | > |a). In the ROC, the

series converges to X(-) = ](az=1)" = 1-a2-1 a 1x1>\al, n=0 since it is just a geometric series. The Z-transform has a region of convergence for any finite value of a. z-plane Im unit circle “* Re The Fourier transform of x[n] only exists if the ROC includes the unit circle, which requires that ſal < 1. On the other hand, if |a| > 1 then the ROC does not include the unit circle, and the Fourier transform does not exist. This is consistent with the fact that for these values of a the sequence a” u[n] is exponentially growing, and the sum therefore does not converge.

Same Example Alternative Way x[n] =aľu[n] = X(z) = {a’unk" = {(az")" n =-00 1-a {(az?) = (az ?) – (az 1) 121>2 • For the term with infinite exponential to vanish we need |az|<1 = lal < 121 - Determines the ROC (same as the previous approach) • In the ROC the sum converges to x(z) = Ś(az") = 1 - 2 1-az-1 n =0