Inverse Laplace & Z-Transform of Rational Functions - Signals and Systems

Table of Contents
1. Basic idea and standard approach
2. Examples
3. Interpretation of poles, causality and stability
4. Inverse Z-transform of rational functions
5. Worked partial-fraction inversion (detailed steps)
6. Conclusions and key points
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Inverse Laplace transform :: Rational functions

Consider an arbitrary rational polynomial in the Laplace transform variable. Such rational functions arise commonly as system functions or transforms of time-domain signals. The inverse transform is obtained by expressing the rational function as a sum of simpler partial fractions whose inverse Laplace transforms are known.

Basic idea and standard approach

A rational Laplace transform is a ratio of two polynomials in \\(s\\). The poles of the function are the roots of the denominator polynomial and the zeros are the roots of the numerator polynomial. The inverse Laplace transform depends on the pole locations and the region of convergence (ROC). For causal signals or causal LTI systems the ROC is to the right of the right-most pole. For anti-causal signals the ROC is to the left of the left-most pole. For two-sided signals the ROC lies between poles.

To invert a rational Laplace transform one normally does the following:

• Factor the denominator to find poles and factor the numerator to find zeros (if required).
• Perform partial fraction expansion so that each term corresponds to a standard inverse Laplace pair.
• Choose the inverse corresponding to the appropriate ROC (causal or anti-causal or two-sided).
• Write the time-domain signal or impulse response using standard transform pairs (exponentials, sinusoids, polynomials times exponentials, etc.).

Examples

Example 1

Let us consider a specific function in \\(s\\):

Example 2 - LTI system function and ROCs

Consider an LTI system with system function given by a rational function \\(H(s)\\):

If the ROC has not been specified, multiple different ROCs are possible and each ROC corresponds to a different time-domain impulse response. For illustration consider a function with poles at \\(s=-1\\) and \\(s=2\\) and a zero at \\(s=1\\). The possible ROCs and the corresponding impulse responses may be sketched as follows.

Interpretation of poles, causality and stability

Properties of a class of LTI systems can be explained in terms of the locations of the poles in the complex \\(s\\)-plane.

For a causal LTI system with a rational system function \\(H(s)\\) the ROC is to the right of the right-most pole. Consequently, for the system to be BIBO stable, the ROC must include the imaginary axis. That implies the right-most pole of \\(H(s)\\) must lie to the left of the imaginary axis.

In compact form you may write the ROC condition for a causal stable system as

\\[ \text{ROC: } \Re(s) > \Re(p_{\max}) \\]

Inverse Z-transform of rational functions

The inverse Z-transform for a rational function of \\(z\\) follows the same principles as the Laplace case. Factor the denominator to identify poles, perform partial fraction expansion (in powers of \\(z\\) or \\(1/z\\) as appropriate), and then use standard z-transform pairs to obtain the time sequence \\(x[n]\\).

Example 1 (Z-transform)

Consider the z-transform

Worked example - partial fractions in \\(1/z\\)

Consider the z-transform

There are two poles at \\(z=1/4\\) and \\(z=1/3\\). We perform the partial fraction expansion expressed as polynomials in \\(1/z\\) to obtain terms that match standard causal or anti-causal z-pairs. The expansion takes the form:

From the expansion we obtain two partial terms; one with z-transform \\(\\dfrac{1}{1-(1/4)z^{-1}}\\) and the other with z-transform \\(\\dfrac{2}{1-(1/3)z^{-1}}\\). These correspond to geometric sequences in time.

As the ROC is not specified, different ROCs give different inverses. We do not discuss causality and stability here since the given transform may not be a system function. One possible inverse sequence is worked out below; the other possible inverses (corresponding to different ROCs) are left as exercises.

Identification by inspection

Some transforms can be identified immediately from known pairs. For example,

Worked partial-fraction inversion (detailed steps)

The following shows the standard stepwise method to invert a simple rational z-transform by partial fractions. Each line below is a logical step in the algebraic inversion.

Express the given rational function as a sum of simple fractions in powers of \\(z^{-1}\\).

Write the form \\(X(z)=\\dfrac{A}{1-a_1 z^{-1}}+\\dfrac{B}{1-a_2 z^{-1}}+\\cdots\\).

Equate numerators after bringing to common denominator to solve for constants \\(A,B,\\dots\\).

Solve the resulting linear equations for the coefficients \\(A,B,\\dots\\).

Recognise each term \\(\\dfrac{A}{1-a z^{-1}}\\) as the z-transform of the causal sequence \\(A\\,a^{n}u[n]\\).

Assemble the time-domain sequence by summing the inverse pairs for the chosen ROC.

Conclusions and key points

• If the system is causal then the ROC extends from the right-most pole to infinity.
• A system is stable if the ROC includes the imaginary axis and therefore the right most pole of \\(H(s)\\) must be to the left of the imaginary axis.
• A causal system with a rational function \\(H(s)\\) is stable if and only if all poles of \\(H(s)\\) lie in the left half of the \\(s\\)-plane. For z-domain system functions, stability requires that the ROC include the unit radius circle.