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The Fourier transforms of input $x[n]$ and output $y[n]$ of a discrete-time system are related by the following equation: $$Y(e^{j\omega}) + (e^{-j\omega})Y(e^{j\omega})= X(e^{j\omega}) - X^{'}(e^{j\omega})$$ Determine if the system is (i) linear (ii) time-invariant.

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  • $\begingroup$ Is $X^\prime\left(e^{j\omega}\right)$ the derivative of $X\left(e^{j\omega}\right)$ with respect to $e^{j\omega}$ or with respect to $\omega$ or the Fourier transform of $x^\prime(t)$, the derivative of $x(t)$ with respect to $t$? $\endgroup$ – Dilip Sarwate Jun 8 at 15:17
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Hint : Use following properties of Discrete-Time Fourier Transform :

  1. If $DTFT\{x[n]\} = X(e^{j\omega})$, then $DTFT\{x[n - n_0]\} = e^{-j\omega n_0}X(e^{j\omega})$

  2. $IDTFT\{j\frac{\partial{}}{\partial\omega}X(e^{j\omega})\} = n x[n]$

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