Question

Find Z-transform of the following signal $x\left(n\right)=sin(\frac{\pi }{4}n-\frac{\pi }{4}).u(n-1)$.

• A. $X\left(z\right)=\frac{z^{-1}}{\sqrt{2}+2z^{-1}+\sqrt{2}{z}^{-2}}$
• B. $X\left(z\right)=\frac{z^{-2}}{\sqrt{2}-2z^{-1}+\sqrt{2}{z}^{-2}}$
• C. $X\left(z\right)=\frac{z^{-2}}{2\sqrt{2}-4z^{-1}+2\sqrt{2}{z}^{-2}}$
• D. $X\left(z\right)=\frac{z^{-1}}{2\sqrt{2}+4z^{-1}+2\sqrt{2}{z}^{-2}}$

Answer 1

The correct option is B $X\left(z\right)=\frac{z^{-2}}{\sqrt{2}-2z^{-1}+\sqrt{2}{z}^{-2}}$

$sin\left(\frac{\pi }{4}n\right)u\left(n\right)\stackrel{Z.T.}{⟷}\frac{z^{-1}sin\frac{\pi }{4}}{1-z^{-1}\left(2cos\frac{\pi }{4}\right)+z^{-2}}$

$sin\left(\frac{\pi }{4}n\right)u\left(n\right)\stackrel{Z.T.}{⟷}\frac{\frac{z^{-1}}{\sqrt{2}}}{1-z^{-1}(2\times \frac{1}{\sqrt{2}})+z^{-2}}$

$sin\left(\frac{\pi }{4}n\right)u\left(n\right)\stackrel{Z.T.}{⟷}\frac{z^{-1}}{\sqrt{2}-2z^{-1}+\sqrt{2}{z}^{-2}}$

$sin(\frac{\pi }{4}n-\frac{\pi }{4})u(n-1)\stackrel{Z.T.}{⟷}\frac{z^{-2}}{\sqrt{2}-2z^{-1}+\sqrt{2}{z}^{-2}}$