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

Question No. 45 EEE | GATE 2015

Consider a discrete time signal given by 

                     xn=-0.25nun+0.5nu-n-1

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. 

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