DSP - Z-Transform Solved Examples

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Example 1

Find the response of the system $s(n+2)-3s(n+1)+2s(n) = \delta (n)$, when all the initial conditions are zero.

Solution − Taking Z-transform on both the sides of the above equation, we get

$$S(z)Z^2-3S(z)Z^1+2S(z) = 1$$

$\Rightarrow S(z)\lbrace Z^2-3Z+2\rbrace = 1$

$\Rightarrow S(z) = \frac{1}{\lbrace z^2-3z+2\rbrace}=\frac{1}{(z-2)(z-1)} = \frac{\alpha _1}{z-2}+\frac{\alpha _2}{z-1}$

$\Rightarrow S(z) = \frac{1}{z-2}-\frac{1}{z-1}$

Taking the inverse Z-transform of the above equation, we get

$S(n) = Z^{-1}[\frac{1}{Z-2}]-Z^{-1}[\frac{1}{Z-1}]$

$= 2^{n-1}-1^{n-1} = -1+2^{n-1}$

Example 2

Find the system function H(z) and unit sample response h(n) of the system whose difference equation is described as under

$y(n) = \frac{1}{2}y(n-1)+2x(n)$

where, y(n) and x(n) are the output and input of the system, respectively.

Solution − Taking the Z-transform of the above difference equation, we get

$y(z) = \frac{1}{2}Z^{-1}Y(Z)+2X(z)$

$= Y(Z)[1-\frac{1}{2}Z^{-1}] = 2X(Z)$

$= H(Z) = \frac{Y(Z)}{X(Z)} = \frac{2}{[1-\frac{1}{2}Z^{-1}]}$

This system has a pole at $Z = \frac{1}{2}$ and $Z = 0$ and $H(Z) = \frac{2}{[1-\frac{1}{2}Z^{-1}]}$

Hence, taking the inverse Z-transform of the above, we get

$h(n) = 2(\frac{1}{2})^nU(n)$

Example 3

Determine Y(z),n≥0 in the following case −

$y(n)+\frac{1}{2}y(n-1)-\frac{1}{4}y(n-2) = 0\quad given\quad y(-1) = y(-2) = 1$

Solution − Applying the Z-transform to the above equation, we get

$Y(Z)+\frac{1}{2}[Z^{-1}Y(Z)+Y(-1)]-\frac{1}{4}[Z^{-2}Y(Z)+Z^{-1}Y(-1)+4(-2)] = 0$

$\Rightarrow Y(Z)+\frac{1}{2Z}Y(Z)+\frac{1}{2}-\frac{1}{4Z^2}Y(Z)-\frac{1}{4Z}-\frac{1}{4} = 0$

$\Rightarrow Y(Z)[1+\frac{1}{2Z}-\frac{1}{4Z^2}] =\frac{1}{4Z}-\frac{1}{2}$

$\Rightarrow Y(Z)[\frac{4Z^2+2Z-1}{4Z^2}] = \frac{1-2Z}{4Z}$

$\Rightarrow Y(Z) = \frac{Z(1-2Z)}{4Z^2+2Z-1}$