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Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Document Description: Inverse of Z-Transform for Electrical Engineering (EE) 2022 is part of Digital Signal Processing preparation. The notes and questions for Inverse of Z-Transform have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Inverse of Z-Transform covers topics like and Inverse of Z-Transform Example, for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Inverse of Z-Transform.

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If we want to analyze a system, which is already represented in frequency domain, as discrete time signal then we go for Inverse Z-transformation.

Mathematically, it can be represented as;

x(n) = Z−1X(Z)

where x(n) is the signal in time domain and X(Z) is the signal in frequency domain.

If we want to represent the above equation in integral format then we can write it as

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Here, the integral is over a closed path C. This path is within the ROC of the x(z) and it does contain the origin.

Methods to Find Inverse Z-Transform

When the analysis is needed in discrete format, we convert the frequency domain signal back into discrete format through inverse Z-transformation. We follow the following four ways to determine the inverse Z-transformation.

  • Long Division Method
  • Partial Fraction expansion method
  • Residue or Contour integral method

Long Division Method

In this method, the Z-transform of the signal x (z) can be represented as the ratio of polynomial as shown below;

x(z) = N(Z)/D(Z)

Now, if we go on dividing the numerator by denominator, then we will get a series as shown below

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The above sequence represents the series of inverse Z-transform of the given signal (for n≥0) and the above system is causal.

However for n < 0 the series can be written as;

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Partial Fraction Expansion Method

Here also the signal is expressed first in N (z)/D (z) form.

If it is a rational fraction it will be represented as follows;

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The above one is improper when m < n and an ≠ 0

If the ratio is not proper (i.e. Improper), then we have to convert it to the proper form to solve it.

Residue or Contour Integral Method

In this method, we obtain inverse Z-transform x(n) by summing residues of  Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE) at all poles. Mathematically, this may be expressed as

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Here, the residue for any pole of order m at z = β is

Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The document Inverse of Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
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