Question

Determine the z-transform of the signal \[ x(n)=a^n(\sin\omega_0 n)u(n) \]

• A. \[ \frac{az^{-1}\cos\omega_0} {1-2az^{-1}\sin\omega_0-a^2z^{-2}} ,\quad |z|>|a| \]
• B. \[ \frac{az^{-1}\cos\omega_0} {1+2az^{-1}\cos\omega_0-a^2z^{-2}} ,\quad |z|<|a| \]
• C. \[ \frac{az^{-1}\sin\omega_0} {1-2az^{-1}\cos\omega_0+a^2z^{-2}} ,\quad |z|>|a| \]
• D. \[ \frac{az^{-1}\sin\omega_0} {1-2az^{-1}\cos\omega_0-a^2z^{-2}} ,\quad |z|<|a| \]

Hint:

Memorize the standard z-transform pairs of \[ a^n\cos(\omega_0 n)u(n) \] and \[ a^n\sin(\omega_0 n)u(n). \] They appear frequently in competitive examinations.

Answer 1

The Correct Option is C

Concept: The standard z-transform pair is \[ a^n\sin(\omega_0 n)u(n) \] \[ \Longleftrightarrow \frac{az^{-1}\sin\omega_0} {1-2az^{-1}\cos\omega_0+a^2z^{-2}} \] with ROC \[ |z|>|a|. \]

Step 1: Use Euler representation.
\[ \sin(\omega_0 n) = \frac{e^{j\omega_0 n}-e^{-j\omega_0 n}}{2j}. \] Applying the z-transform to each exponential term and simplifying gives \[ X(z) = \frac{az^{-1}\sin\omega_0} {1-2az^{-1}\cos\omega_0+a^2z^{-2}}. \]

Step 2: Determine ROC.
Because the sequence is right-sided, \[ ROC:\quad |z|>|a|. \] \[ \boxed{ X(z) = \frac{az^{-1}\sin\omega_0} {1-2az^{-1}\cos\omega_0+a^2z^{-2}} } \]