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GATE 2024 EE – Question 39

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

If the Z-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the Z-transform of the signal $y[n]=x[2n]$ is

  1. $Y(z)=X(z^2)$
  2. $Y(z)=\dfrac12\left[X\left(z^{-1/2}\right)+X\left(-z^{-1/2}\right)\right]$
  3. $Y(z)=\dfrac12\left[X\left(z^{1/2}\right)+X\left(-z^{1/2}\right)\right]$
  4. $Y(z)=\dfrac12\left[X(z^2)+X(-z^2)\right]$

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Correct answer: (C) $Y(z)=\dfrac12\left[X\left(z^{1/2}\right)+X\left(-z^{1/2}\right)\right]$

Explanation

Downsampling by 2 keeps the even-indexed samples. Define $w[n]=\frac12\left(x[n]+(-1)^nx[n]\right)$ (zero at odd $n$); its transform is $\frac12[X(z)+X(-z)]$. Then $y[n]=w[2n]$ compresses the index, replacing $z$ by $z^{1/2}$: $Y(z)=\dfrac12\left[X(z^{1/2})+X(-z^{1/2})\right]$.