If Z(un) = f(z), then Z(a − nun) (where Z denotes z-transform) equals to :

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NIELIT Scientific Assistant ECE 5 Dec 2021 Official Paper
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  1. \(f\left(\frac{z}{2 a}\right)\)
  2. f (z)
  3. \(f\left(\frac{z}{ a}\right)\)
  4. f (az) 

Answer (Detailed Solution Below)

Option 3 : \(f\left(\frac{z}{ a}\right)\)
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Detailed Solution

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Concept:

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) \( f\left(\frac{z}{a}\right) \).

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