Properties of Laplace & Z-Transform - Signals and Systems - Electrical

Table of Contents
1. Introduction
2. Linearity
3. Differentiation in the time domain
4. Time shift
5. Time scaling and related operations
6. Frequency (s/z) shifting by multiplication in time
7. Convolution and multiplication
8. Differentiation in the transform domain
9. Initial and final value theorems
10. ROC - general remarks and causality
11. Summary of key transform pairs and properties
12. Concluding remarks
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Properties of Laplace and Z-Transform

Introduction

This chapter summarises the standard properties of the bilateral Laplace transform and the bilateral z-transform used in signals and systems. Each property is stated for continuous-time (Laplace) and for discrete-time (z-transform) where a corresponding property exists. Discussion includes the transform formula, region of convergence (ROC) effects, and simple worked illustrations. The aim is to provide a clear, syllabus-aligned reference suitable for undergraduate electrical-engineering students and competitive-exam preparation.

Linearity

Laplace transform

If two signals have Laplace transforms as follows then their linear combination transforms as:

then

The ROC of the linear combination contains at least the intersection of the two ROCs. If the intersection is empty, the linear combination may have no Laplace transform.

Z-transform

For discrete-time sequences the same linearity holds:

then

The ROC of the sum contains at least the intersection of the individual ROCs.

Differentiation in the time domain

Laplace transform (continuous time)

The Laplace transform of a time-derivative follows from integration by parts. Let

Then, applying integration by parts,

and hence

The ROC of \\(sX(s)\\) includes the ROC of \\(X(s)\\) and may extend beyond it depending on convergence at extremes.

Worked example - Laplace of a derivative

Example: Find the Laplace transform of \\(\dot{x}(t)\\) in terms of \\(X(s)\\) and initial conditions.

Sol.

\[ \mathcal{L}\{\dot{x}(t)\} \;=\; \int_{-\infty}^{\infty} \dot{x}(t)\,e^{-st}\,dt \]

\[ = \big[ x(t)e^{-st} \big]_{t=-\infty}^{\infty} + s \int_{-\infty}^{\infty} x(t)e^{-st}\,dt \]

\[ = sX(s) - x(0^-) \]

provided the boundary term vanishes at infinities under the ROC assumptions.

Z-transform (discrete time)

The discrete-time analogue is the relation between sequence multiplication by index and differentiation of the z-transform:

\[ \sum_{n=-\infty}^{\infty} n\,x[n]\,z^{-n} \;=\; -z\,\frac{dX(z)}{dz} \]

This property is useful for computing first moments and for deriving other transform relations. The ROC is the same as that of \\(X(z)\\) except possibly at the origin or infinity depending on the behaviour of the sum.

Time shift

Laplace transform

Time shifting a continuous-time signal to the right or left results in multiplication by an exponential factor in the s-domain. For a shift by \\(t_0\\) we have:

The ROC is shifted accordingly; for bilateral Laplace transforms the ROC is the set of \\(s\\) for which the shifted integral converges.

Z-transform

Shifting a discrete-time sequence by \\(n_0\\) samples multiplies its z-transform by \\(z^{-n_0}\\):

Because of

with ROC equal to the original ROC except for the possible addition or deletion of the origin or infinity.

Multiplication by \\(z^{-n_0}\\) for \\(n_0 > 0\\) introduces poles at \\(z=0\\), which may cancel corresponding zeros of \\(X(z)\\) at the origin. In this case the ROC for the shifted sequence equals the ROC of \\(X(z)\\) but with the origin deleted. Similarly, if \\(n_0 < 0\\)="" the="" roc="" may="" change="" by="" deleting="">

Time scaling and related operations

Laplace transform - time scaling

For continuous time, a scaling of the time axis by a nonzero factor \\(a\\) gives the transform:

then

Where

and

A special case is time reversal when \\(a=-1\\): \\(x(-t)\\) has transform \\((1/|a|)X(s/a)\\) evaluated at \\(a=-1\\) with ROC reflected accordingly.

Z-transform - time expansion, upsampling and downsampling

The continuous-time concept of arbitrary time scaling does not extend directly to discrete-time (integer index) signals. However, two related operations are commonly used:

• Upsampling (insertion of zeroes): For integer \\(k>1\\) define a new sequence by placing \\(k-1\\) zeros between samples of \\(x[n]\\).
• Downsampling (decimation): Keep every \\(k\\)-th sample and discard others (may require anti-aliasing filtering).

One definition of the upsampled sequence is

x^{(k)}[n] = x[n/k] if \\(n\\) is a multiple of \\(k\\)

= 0 if \\(n\\) is not a multiple of \\(k\\)

If

x[n] \u2194 X(z) with ROC = R

then the z-transform of the upsampled sequence relates to \\(X(z)\\) by replacing \\(z\\) with \\(z^k\\) and appropriate scaling of the ROC; formally one obtains copies in the z-domain and ROC changes consistent with those copies.

Frequency (s/z) shifting by multiplication in time

Laplace transform

Multiplying a signal by an exponential scales the s-variable:

\[ e^{\alpha t}\,x(t) \;\longleftrightarrow\; X(s-\alpha) \]

The ROC is shifted: \\( \{ s : s-\alpha \in \mathrm{ROC}(X) \} \\).

Z-transform

Multiplication by an exponential sequence \\(\beta^{n}\\) yields a scaling of the z-variable:

\[ \beta^{n}\,x[n] \;\longleftrightarrow\; X\!\left(\frac{z}{\beta}\right), \qquad \beta
eq 0 \]

The ROC is scaled accordingly: \\( \{ z : z/\beta \in \mathrm{ROC}(X) \} \\).

Convolution and multiplication

Convolution in time corresponds to multiplication in the transform domain, and vice versa:

• Continuous time: If \\(y(t) = x(t) * h(t)\\) then \\(Y(s) = X(s)\,H(s)\\).
• Discrete time: If \\(y[n] = x[n] * h[n]\\) then \\(Y(z) = X(z)\,H(z)\\).

The ROC of the product must include the intersection of the two ROCs of the multiplicands. Poles and zeros of the product and cancellations can modify the ROC; for example, pole-zero cancellation may enlarge the ROC.

Differentiation in the transform domain

Operations that multiply by time in the time domain correspond to differentiation in the transform domain:

• Laplace: \\(t\,x(t) \;\longleftrightarrow\; -\dfrac{dX(s)}{ds}\\).
• Z-transform: \\(n\,x[n] \;\longleftrightarrow\; -z\,\dfrac{dX(z)}{dz}\\).

Repeated multiplication by time/index leads to higher derivatives.

Initial and final value theorems

Laplace transform

Initial value theorem:

\[ x(0^+) \;=\; \lim_{s \to \infty} s\,X(s) \]

Final value theorem:

\[ \lim_{t \to \infty} x(t) \;=\; \lim_{s \to 0} s\,X(s) \]

These hold under appropriate conditions on the poles of \\(sX(s)\\): the final value exists only if all poles of \\(sX(s)\\) (except possibly a simple pole at the origin for special cases) lie in the left half s-plane.

Z-transform

Initial value theorem:

\[ x[0] \;=\; \lim_{z \to \infty} X(z) \]

Final value theorem (for causal sequences):

\[ \lim_{n \to \infty} x[n] \;=\; \lim_{z \to 1} (z-1)\,X(z) \]

Both z-transform theorems require conditions on the pole locations (e.g., for final value, poles inside the unit circle except possibly a simple pole at \\(z=1\\) must be absent) so that the limits exist.

ROC - general remarks and causality

The ROC is a fundamental part of a bilateral transform and determines convergence and the uniqueness of inverse transforms. Important points:

• The ROC of a Laplace transform is a vertical strip, left half-plane, or right half-plane depending on signal type (right-sided, left-sided, two-sided).
• The ROC of a z-transform is an annulus (region between two circles), exterior of a circle, interior of a circle, or entire plane excluding poles.
• A right-sided causal signal has ROC of the form \\(|z|>r\\) for z-transform and \\(\Re(s) > \sigma_0\\) for Laplace; an anti-causal left-sided signal has ROC of the form \\(|z|<>
• The ROC cannot contain poles; it is bounded by poles. Pole-zero cancellations can change but not create poles inside the ROC.

Summary of key transform pairs and properties

If

with ROC = R then the following important properties hold.

1. 
   with ROC = R.
2. 
   with ROC = R.
3. 
4. Multiplication by an exponential: \\(e^{\alpha t}x(t) \leftrightarrow X(s-\alpha)\\) where \\(\Re(s-\alpha) \in \mathrm{ROC}(X)\\).
5. If \\(y(t)=x(t)*h(t)\\) then \\(Y(s)=H(s)X(s)\\) where the ROC of \\(Y(s)\\) contains \\(\mathrm{ROC}(X) \cap \mathrm{ROC}(H)\\) subject to pole/zero cancellations.

If

with ROC = R then

1. 
   with ROC = R except for the possible addition or deletion of infinity from the ROC.
2. The continuous-time concept of time scaling does not directly extend to discrete time; see the upsampling/downsampling discussion above.
3. Other properties of the z-transform (linearity, time shift, modulation, convolution) are analogous to Laplace properties, with ROC adjustments specific to discrete-time indexing.

Concluding remarks

This chapter has restated the principal properties used to manipulate Laplace and z-transforms: linearity, time-differentiation and time-multiplication relations, time shifting, time scaling (continuous time), modulation, convolution and the rules for ROC. Students should practise these properties with concrete signals (exponentials, sinusoids, step and impulse sequences) and verify ROC behaviour to become fluent with transform-domain analysis. Use the initial and final value theorems with care - always check the pole conditions before applying them.