Question

A discrete-time LTI system with the property such that if the input is

$x\left(n\right)={\left(\frac{1}{2}\right)}^{n}u\left(n\right)-\frac{1}{4}{\left(\frac{1}{2}\right)}^{n-1}u(n-1)$

and output $y\left(n\right)={\left(\frac{1}{3}\right)}^{n}u\left(n\right)$
The impulse response of the system is

• A. $2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{4}\right)}^{n}u\left(n\right)$
• B. $-2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{2}\right)}^{n}u\left(n\right)$
• C. $2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{2}\right)}^{n}u\left(n\right)$
• D. $-2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{4}\right)}^{n}u\left(n\right)$

Answer 1

The correct option is D $-2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{4}\right)}^{n}u\left(n\right)$

Taking DTFT of x(n) we get

$X\left(e^{j\omega }\right)=\frac{1}{1-\frac{1}{2}{e}^{-j\omega }}-\frac{1}{4}\frac{1}{1-\frac{1}{2}{e}^{-j\omega }}{e}^{-j\omega }=\frac{1-\frac{1}{4}{e}^{-j\omega }}{1-\frac{1}{2}{e}^{-j\omega }}$

$Y\left(e^{j\omega }\right)=\frac{1}{1-\frac{1}{3}{e}^{-j\omega }}$

$\frac{Y\left(e^{j\omega }\right)}{X\left(e^{j\omega }\right)}=H\left(e^{j\omega }\right)=\frac{1-\frac{1}{2}{e}^{-j\omega }}{(1-\frac{1}{3}{e}^{-j\omega })(1-\frac{1}{4}{e}^{-j\omega })}$

$H\left(e^{j\omega }\right)=\frac{A}{1-\frac{1}{3}{e}^{-j\omega }}+\frac{B}{1-\frac{1}{4}{e}^{-j\omega }}$

$A=(1-\frac{1}{3}{e}^{-j\omega })H\left(e^{j\omega }\right){|}_{e^{-j\omega }=3}=-2$

$B=(1-\frac{1}{4}{e}^{-j\omega })H\left(e^{j\omega }\right){|}_{e^{-j\omega }=4}=3$

$H\left(e^{j\omega }\right)=\frac{-2}{1-\frac{1}{3}{e}^{-j\omega }}+\frac{3}{1-\frac{1}{4}{e}^{-j\omega }}$

$h\left(n\right)=-2{\left(\frac{1}{3}\right)}^{n}u\left(n\right)+3{\left(\frac{1}{4}\right)}^{n}u\left(n\right)$