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

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

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

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

Solution

Verified

Step 1

1 of 6

a) i) Consider the signal:

x[n]=ax1[n]+bx2[n]x[n]=ax_1[n]+bx_2[n]

where aa and bb are constants. Then using linearity property of Fourier transform, Fourier transform of x[n]x[n] is:

X(ejω)=aX1(ejω)+bX2(ejω)X(e^{j\omega})=aX_1(e^{j\omega})+b X_2(e^{j\omega})

Let the response of the system to the signal x1[n]x_1[n] is y1[n]y_1[n] and to the signal x2[n]x_2[n] is y2[n]y_2[n]. Then:

Y1(ejω)=2X1(ejω)+ejωX1(ejω)dX1(ejω)dωY_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

Y2(ejω)=2X2(ejω)+ejωX2(ejω)dX2(ejω)dωY_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]y[n], Fourier transform is:

Y(ejω)=2X(ejω)+ejωX(ejω)dX(ejω)dω=2(aX1(ejω)+bX2(ejω))+ejω(aX1(ejω)+bX2(ejω))ddω(aX1(ejω+bX2(ejω)))=a(2X1(ejω)+ejωX1(ejω)dX1(ejω)dω)+b(2X2(ejω)+ejωX2(ejω)dX2(ejω)dω)=aY1(ejω)+bY2(ejω)\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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