Inverse of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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 | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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 | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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 | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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 | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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 | Digital Signal Processing - Electronics and Communication Engineering (ECE) at all poles. Mathematically, this may be expressed as

Inverse of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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

Inverse of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

The document Inverse of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Digital Signal Processing.
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FAQs on Inverse of Z-Transform - Digital Signal Processing - Electronics and Communication Engineering (ECE)

1. What is the inverse of the Z-transform in electrical engineering?
Ans. In electrical engineering, the inverse of the Z-transform refers to the process of converting a given Z-transform function back to its original time-domain representation. It allows us to obtain the original discrete-time signal from its corresponding Z-transform function.
2. How is the inverse Z-transform calculated?
Ans. The inverse Z-transform can be calculated using various techniques, such as partial fraction expansion, power series expansion, residue theorem, or using lookup tables. These methods involve algebraic manipulation and complex analysis to determine the original time-domain signal from its Z-transform function.
3. What are the applications of the inverse Z-transform in electrical engineering?
Ans. The inverse Z-transform has several applications in electrical engineering, including digital filter design, system analysis and control, signal processing, and communication systems. It allows engineers to study and analyze discrete-time systems and signals in the time domain.
4. Are there any specific formulas or properties for calculating the inverse Z-transform?
Ans. Yes, there are specific formulas and properties for calculating the inverse Z-transform. Some commonly used formulas include the linearity property, initial value theorem, final value theorem, and the shifting property. These properties provide a convenient way to calculate the inverse Z-transform in different scenarios.
5. Are there any limitations or challenges in calculating the inverse Z-transform?
Ans. Yes, there are certain limitations and challenges in calculating the inverse Z-transform. One common challenge is dealing with complex poles or zeros in the Z-transform function, which requires techniques like partial fraction expansion and residue theorem. Another limitation is that not all Z-transform functions have a closed-form inverse, leading to the need for numerical approximation methods in some cases. Additionally, the accuracy of the inverse Z-transform calculation can be affected by the choice of method and numerical errors.
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