Inverse Laplace & Z-Transform - Signals and Systems - Electrical Engineering

Table of Contents
1. Inverse Laplace Transform
2. Relationship between Laplace Transform and Fourier Transform
3. Inverse Z-Transform
4. Relationship between Z-Transform and Discrete-Time Fourier Transform (DTFT)
5. Examples
6. Worked examples of inverse Laplace (simple cases)
7. Conclusion
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Inverse Laplace Transform

Inverse Laplace Transform

There is a one-to-one correspondence between a time-domain signal x(t) and its Laplace transform X(s). Obtaining the signal x(t) when X(s) is known is called the Inverse Laplace Transform (ILT).

Common methods used to obtain x(t) from a known X(s) include the following:

• The complex inversion formula (Bromwich integral).
• Partial fraction expansion for rational functions.
• Residue evaluation of the Bromwich integral.
• Series expansion (Laurent or Taylor) and term-wise inversion.
• Recognition using standard transform tables and properties (time shifting, scaling, differentiation in s, integration in s).

Bromwich (complex inversion) formula

The Bromwich integral expresses the inverse Laplace transform exactly as a complex contour integral. It is the fundamental inversion formula.

In this formula C denotes any vertical line in the complex s-plane that is parallel to the imaginary axis and lies inside the region of convergence (ROC) of X(s). The contour is taken from \sigma - j\infty to \sigma + j\infty where \sigma is chosen so that the line lies within the ROC. The integral is evaluated as a line integral in the complex plane and may be computed by residue summation when appropriate.

Practical methods for inversion

Although the Bromwich integral is the exact expression, the following practical techniques are most often used because they simplify computation for typical engineering transforms.

• Partial fraction method: Express X(s) as a sum of simpler rational terms whose inverses are known from standard tables.
• Residue method: Evaluate the Bromwich integral by closing the contour and summing residues of X(s)e^{st} at appropriate poles (the direction for closing the contour depends on causality and the sign of t).
• Series method: Expand X(s) in a convergent Laurent or Taylor series in an appropriate region (for example in powers of 1/s for large |s|) and invert term-by-term using known pairs.
• Use of transform properties and tables: Match parts of X(s) with known transform pairs and apply linearity, time-shifting, frequency-shifting, and differentiation/integration in s.

Region of Convergence (ROC) and causality

The ROC plays a central role in inversion and in determining the time-domain character of the signal.

• The ROC determines which vertical line may be chosen for the Bromwich contour.
• For right-sided (causal) signals the ROC is to the right of the rightmost pole.
• For left-sided (anti-causal) signals the ROC is to the left of the leftmost pole.
• For two-sided signals the ROC is a vertical strip between poles.
• The imaginary axis (when s = j\omega) must lie inside the ROC if the Fourier transform of x(t) exists.

Relationship between Laplace Transform and Fourier Transform

Laplace transform as a generalisation of the Fourier transform

The complex variable s is written as follows.

When the Laplace transform is evaluated on the imaginary axis and that axis lies inside the ROC, the Laplace transform reduces to the Fourier transform.

Graphically, this corresponds to evaluating X(s) along the vertical line Re(s)=0 (the imaginary axis) provided that line lies within the ROC.

Thus the Fourier transform is a special case of the Laplace transform obtained by setting \sigma = 0 and integrating along the imaginary axis.

Inverse Laplace vs inverse Fourier

If the inversion contour of the Bromwich integral is chosen along the imaginary axis and that axis is inside the ROC, the inverse Laplace transform reduces to the inverse Fourier transform. This is the continuous-time analogue of the Z/DTFT relationship.

The Laplace transform is applicable to a broader class of signals (for example signals that grow or decay exponentially) while the Fourier transform requires convergence on the imaginary axis and typically applies to energy-finite or absolutely summable signals.

Inverse Z-Transform

Inverse Z-Transform

There is a one-to-one correspondence between a discrete-time sequence x[n] and its Z-transform X(z). Obtaining x[n] when X(z) is known is called the Inverse Z-Transform (IZT).

The inverse Z-transform can be written as a contour integral in the complex z-plane.

In this expression C is any closed contour that encircles the origin and lies entirely within the ROC of X(z). In practice the contour is usually taken as a circle |z|=r with radius chosen inside the ROC. Inversion techniques parallel those used for the Laplace transform: partial fractions, residue summation, or series expansion are common choices.

Relationship between Z-Transform and Discrete-Time Fourier Transform (DTFT)

The DTFT is obtained by evaluating the Z-transform on the unit circle, provided the unit circle lies inside the ROC.

If the substitution z = e^{j\omega} is valid (that is, the unit circle |z|=1 lies in the ROC), the inverse Z-transform integral reduces to the inverse DTFT integral and we recover the discrete-time Fourier inversion.

Consequently the Z-transform extends the DTFT by allowing analysis in regions of the complex plane where the DTFT (unit circle) may not converge; this makes the Z-transform a more general tool for discrete-time analysis.

Examples

Illustration that Z-transform may exist while DTFT does not:

The Z-transform of the given sequence is shown below (see image).

However the DTFT of the same sequence does not exist because the infinite summation diverges when evaluated on the unit circle (see image).

This example confirms that the Z-transform can converge for values of |z| other than unity while the DTFT (which requires |z|=1) may fail to exist.

Worked examples of inverse Laplace (simple cases)

Example 1: Invert \\(X(s)=\\dfrac{1}{s+a}\\), where \\(\\Re(a)>0\\).

The corresponding time-domain signal is a standard transform pair.

Therefore the inverse transform is

Here u(t) denotes the unit step function; the ROC \Re(s)>-\Re(a) indicates a right-sided (causal) exponential.

Example 2: Invert a rational transform by partial fractions: \\(X(s)=\\dfrac{s+2}{s^{2}+4s+5}\\).

The denominator can be written as a completed square and the numerator arranged to match standard transform numerators. The algebraic inversion proceeds as follows.

Recognise the standard Laplace pair for a damped cosine:

Comparing terms gives the inverse transform directly.

This result follows because the numerator matches s+a with a=2 and the denominator matches (s+a)^{2}+\omega^{2} with \omega=1.

Conclusion

Key points to remember:

• If the Laplace transform of x(t) is X(s), the inverse Laplace transform is given in principle by the Bromwich integral; in practice inversion is often done by partial fractions, residue summation, series inversion or table recognition.

• If the imaginary axis lies in the ROC of X(s) and the Laplace transform is evaluated along it (s=j\omega), the result is the Fourier transform of x(t).
• The Z-transform and Laplace transform generalise the DTFT and Fourier transform respectively by extending analysis to complex values of the transform variable and allowing convergence for a broader class of signals.
• Choose inversion methods (partial fractions, residues, series expansion, or recognition via tables) based on the algebraic form of X(s) or X(z), the ROC, and on causality properties.

These relationships allow analysis of both continuous-time and discrete-time signals more generally and enable recovery of time-domain signals from transform-domain representations using contour integration or algebraic techniques. For engineering practice, identify the ROC and causality first, then select the algebraic inversion method that fits the form of the transform.