Question

A discrete time system impulse response $h\left(n\right)={\left(\frac{1}{5}\right)}^{n}u\left(n\right).$ If system difference equation is $h\left(n\right)-C.h(n-1)=\delta \left(n\right),$ then the value of C is _______.

• A. 0.2

Answer 1

The correct option is A 0.2

${\left(\frac{1}{5}\right)}^n.u\left(n\right)-C{\left(\frac{1}{5}\right)}^{n-1}u(n-1)=\delta \left(n\right)$

${\left(\frac{1}{5}\right)}^n\left[u\right(n)-{\left(\frac{1}{5}\right)}^{-1}Cu(n-1\left)\right]=\delta \left(n\right)$

$\left[u\right(n)-{\left(\frac{1}{5}\right)}^{-1}Cu(n-1\left)\right]=5^n\delta \left(n\right)$

$\left[u\right(n)-{\left(\frac{1}{5}\right)}^{-1}Cu(n-1\left)\right]=\delta \left(n\right)$

$u\left(n\right)-\delta \left(n\right)={\left(\frac{1}{5}\right)}^{-1}Cu(n-1)$
$u(n-1)={\left(\frac{1}{5}\right)}^{-1}Cu(n-1)$
$1={\left(\frac{1}{5}\right)}^{-1}.C$
$C=\frac{1}{5}=0.2$

Alternate Solution:

$H\left(z\right)=\frac{1}{1-\frac{1}{5}{z}^{-1}}$ .....(i)

$H\left(z\right)-C{z}^{-1}H\left(z\right)=1$

$H\left(z\right)=\frac{1}{1-C{z}^{-1}}$ .....(ii)

Comparing equation (i) and (ii),

$C=\frac{1}{5}=0.2$