Question

The z-transform of cos(\(\frac{π}{3}\) n) u[n] is __________

(a) \(\frac{z}{2} \frac{(2z-1)}{(z^2-z+1)}\), 0<|z|<1

(b) \(\frac{z}{2} \frac{(2z-1)}{(z^2-z+1)}\), |z|>1

(c) \(\frac{z}{2} \frac{(1-2z)}{(z^2-z+1)}\), 0<|z|<1

(d) \(\frac{z}{2} \frac{(1-2z)}{(z^2-z+1)}\), |z|>1

The question was asked in semester exam.

Question is from Properties of Z-Transforms topic in section Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Right option is (b) \(\frac{z}{2} \frac{(2z-1)}{(z^2-z+1)}\), |z|>1

For explanation: Performing z-transform on a^nu[n], we get \(\frac{z}{z-a}\)

∴ x[n] = \(\frac{1}{2} e^{j(\frac{π}{3})n} u[n] + \frac{1}{2} e^{-j(\frac{π}{3})n}u [n]\)

Hence, X (z) =\( 0.5 \left(\frac{1}{1-e^{j(\frac{π}{3})}  z^{-1}} + \frac{1}{1-e^{j(\frac{π}{3})}  z^{-1}}\right)\)

Hence, X (z) = \(\frac{z}{2} \frac{(2z-1)}{(z^2-z+1)}\), |z|>1.