GATE Papers >> EEE >> 2015 >> Question No 45

Question No. 45

Consider a discrete time signal given by

The region of convergence of its Z-transform would be

Answer : (C) the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.

Solution of Question No 45 of GATE 2015 EEE Paper

$\mathrm x\lbrack\mathrm n\rbrack=-(-0.25)^\mathrm n\;\mathrm u(\mathrm n)+(0.5)^\mathrm n\;\mathrm u(-\mathrm n-1)$

Signal $\mathrm x\lbrack\mathrm n\rbrack$ is sum of two signals, one is right sided $\left[\left(-0.25\right)^\mathrm n\;\mathrm u(\mathrm n)\right]$ and other is left sided $\left[\left(0.5\right)^\mathrm n\;\mathrm u\left(-\mathrm n-1\right)\right]$.

The right sided signal will have pole at location with magnitude 0.25. So, ROC is |Z| > 0.25.

The left sided signal will have pole at location with magnitude 0.5 So, ROC is |Z| < 0.5.

So, ROC of X|z| (Z transform of $\mathrm x(\mathrm n)$ will be) 0.25 < |Z| < 0.5.