#### Question

(a) A particular discrete-time system has input x[n] and output y[n]. The Fourier transforms of these signals are related by the equation

$Y(e^{jω})=2X(e^{jω})+e^{-jω}X(e^{jω})-dX(e^{jω})/dω.$

(i) Is the system linear? Clearly justify your answer. (ii) Is the system time invariant? Clearly justify your answer. (iii) What is y[n] if x[n]=δ[n]? (b) Consider a discrete-time system for which the transform

$Y(e^{jω})$

of the output is related to the transform of the input through the relation

$Y(e^{jω})=∫_{ω-π/4}^{ω+π/4}X(e^{jω})dω$

. Find an expression for y[n] in terms of x[n].

Verified

#### Step 1

1 of 6

a) i) Consider the signal:

$x[n]=ax_1[n]+bx_2[n]$

where $a$ and $b$ are constants. Then using linearity property of Fourier transform, Fourier transform of $x[n]$ is:

$X(e^{j\omega})=aX_1(e^{j\omega})+b X_2(e^{j\omega})$

Let the response of the system to the signal $x_1[n]$ is $y_1[n]$ and to the signal $x_2[n]$ is $y_2[n]$. Then:

$Y_1(e^{j\omega})=2X_1(e^{j\omega})+e^{-j\omega}X_1(e^{j\omega})-\dfrac{dX_1(e^{j\omega})}{d\omega}$

and

$Y_2(e^{j\omega})=2X_2(e^{j\omega})+e^{-j\omega}X_2(e^{j\omega})-\dfrac{dX_2(e^{j\omega})}{d\omega}$

and for the signal $y[n]$, Fourier transform is:

\begin{align*} Y(e^{j\omega})&=2X(e^{j\omega})+e^{-j\omega}X(e^{j\omega})-\dfrac{dX(e^{j\omega})}{d\omega}\\ &=2\left(aX_1(e^{j\omega})+bX_2(e^{j\omega})\right)+e^{-j\omega}\left(aX_1(e^{j\omega})+bX_2(e^{j\omega})\right)-\dfrac{d}{d\omega}\left( aX_1(e^{j\omega}+bX_2(e^{j\omega}))\right)\\ &=a\left( 2X_1(e^{j\omega})+ e^{-j\omega}X_1(e^{j\omega})- \dfrac{dX_1(e^{j\omega})}{d\omega} \right)+b\left( 2X_2(e^{j\omega})+ e^{-j\omega}X_2(e^{j\omega})- \dfrac{dX_2(e^{j\omega})}{d\omega} \right)\\ &=aY_1(e^{j\omega})+bY_2(e^{j\omega}) \end{align*}

We proved that system is linear!

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