Question

Which of the following justifies the linearity property of the \( z \)-transform? \( x(n) \leftrightarrow X(z) \).

• A. \( x(n) + y(n) \leftrightarrow X(z) Y(z) \)
• B. \( x(n) + y(n) \leftrightarrow X(z) + Y(z) \)
• C. \( x(n) y(n) \leftrightarrow X(z) + Y(z) \)
• D. \( x(n) y(n) \leftrightarrow X(z) Y(z) \)

Hint:

The linearity property of the \( z \)-transform states that \( \mathcal{Z} \{ x(n) + y(n) \} = X(z) + Y(z) \), meaning the transformation preserves addition.

Answer 1

The Correct Option is B

Step 1: The linearity property of the \( z \)-transform states: \[ \mathcal{Z} \{ a x(n) + b y(n) \} = a X(z) + b Y(z) \] where:
- \( x(n) \leftrightarrow X(z) \),
- \( y(n) \leftrightarrow Y(z) \),
- \( a \) and \( b \) are constants.
Step 2: Evaluating the given options:
- (A) Incorrect: The \( z \)-transform of a sum is the sum of individual transforms, not their product.
- (B) Correct: \( x(n) + y(n) \) transforms to \( X(z) + Y(z) \), satisfying linearity.
- (C) Incorrect: The product of signals in time does not result in an addition of their transforms.
- (D) Incorrect: The product of signals in time corresponds to convolution in the \( z \)-domain, not multiplication.