Question
Download Solution PDFIf Z(un) = f(z), then Z(a − nun) (where Z denotes z-transform) equals to :
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
The z-transform of a scaled sequence follows the scaling property of z-transforms, which states that if \( Z(u_n) = f(z) \), then \( Z(a^{-n}u_n) = f(az) \). This property is crucial for analyzing discrete-time signals with exponential scaling.
Calculation:
Given:
1. Original z-transform: \(Z(u_n) = f(z) \)
2. Scaled sequence: \(a^{-n}u_n\)
Solution:
1. Apply the scaling property of z-transforms:
\( Z(a^{-n}u_n) = f\left(\frac{z}{a}\right) \)
2. This result comes from the fundamental scaling property where multiplication by \( a^{-n} \) in the time domain corresponds to scaling the z-domain variable by \( \frac{1}{a} \).
Final Answer:
The correct transformed expression is 3) \( f\left(\frac{z}{a}\right) \).
Last updated on Jun 26, 2025
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