# Determining if the system is linear and time invariant

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.

• 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 $t$? – Dilip Sarwate Jun 8 '20 at 15:17

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