Inverse Z transform

Inverse Z transform

By Poojan Gohel June 02, 2020

To convert a system frequency domain to Discrete time domain we can used Inverse Z transform.

The representation can be shown as

as we know that x(n) is a signal in time domain and X(z) is a signal in frequency domain.

So basically there are many methods for inverse z transform and here are a few...

Direct Division Method
Partial Fraction Expansion

Direct Division method

This method is useful when it is useful to obtain the closed-form expression of the inverse z transform or have the desire to find the t several terms only. Also, It works if we have an equation containing both numerator and denominator.

We simply need to divide the equation with the highest power of z in the equation. After getting an equation which contains inverse z on numerator and denominator both we can just divide the equations and get the time-domain sequence of the signal. See the example below...

In the above example as shown we need to find up to 4 values of k. Hence we divided until we get a term with z^-4. Simple as that.

Partial Fractions

Find the inverse z-transform of

$X(z)=\frac{1+2z^{-1}+z^{-2}}{1-3z^{-1}+2z^{-2}}$

where the ROC is |z|>2. In this case M=N=2, so we have to use long division to get

$X(z)=\frac{1}{2}+\frac{\frac{1}{2}+\frac{7}{2}z^{-1}}{1-3z^{-1}+2z^{-2}}$

Next Factor the denominator,

$X(z)=2+\frac{-1+5z^{-1}}{\left ( 1-2z^{-1} \right )\left (1-z^{-1} \right )}$

Now do partial-fraction expansion,

$X(z)=\frac{1}{2}+\frac{A_{1}}{1-2z^{-1}}+\frac{A_{2}}{1-z^{-1}}=\frac{1}{2}+\frac{\frac{9}{2}}{1-2z^{-1}}+\frac{-4}{1-z^{-1}}$

Now each term can be inverted using the inspection method and the z-transform table. Thus, since the ROC is |z|>2,

$x[n]=\frac{1}{2} \delta [n]+\frac{9}{2}2^{n}u[n]-4u[n]$

Thank you & see you in the next one.

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