Let X(t) And X(s) Be A Laplace Transform Pair. The Laplace Transform Of X(2t) Is 0.5X(0.5s) According

Let X(t) And X(s) Be A Laplace Transform Pair. The Laplace Transform Of X(2t) Is 0.5X(0.5s) According

Understanding the relationship between a time-domain function and its Laplace transform is fundamental in engineering, physics, and mathematics. When dealing with transformations such as \( X(t) \) and \( X(s) \), it is essential to grasp how modifications in the time domain affect their counterparts in the complex frequency domain. One particularly interesting scenario involves scaling the time variable and observing the corresponding effect on the Laplace transform. Specifically, if \( X(t) \) and \( X(s) \) form a Laplace transform pair, then examining the transform of \( X(2t) \) reveals important properties related to time-scaling. This article delves into the mathematical foundation behind this relationship, explores the derivation of the Laplace transform of \( X(2t) \), and discusses practical applications, including signal processing, control systems, and differential equations.

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Understanding Laplace Transform Pairs

Definition of the Laplace Transform

The Laplace transform is a powerful integral transform widely used to convert functions from the time domain to the complex frequency domain. For a given function \( x(t) \), defined for \( t \geq 0 \), the Laplace transform \( X(s) \) is given by:

\[
X(s) = \mathcal{L}\{x(t)\} = \int_0^\infty e^{-st} x(t) \, dt,
\]

where \( s \) is a complex variable, \( s = \sigma + j\omega \), with \( \sigma \) and \( \omega \) representing real numbers.

Laplace Transform Pairs

A pair \( (x(t), X(s)) \) is called a Laplace transform pair if:

\[
X(s) = \mathcal{L}\{x(t)\},
\]

and conversely,

\[
x(t) = \mathcal{L}^{-1}\{X(s)\}.
\]

Some common Laplace transform pairs include:

• \( x(t) = 1 \Rightarrow X(s) = \frac{1}{s} \),


• \( x(t) = t^n \Rightarrow X(s) = \frac{n!}{s^{n+1}} \),


• \( x(t) = e^{at} \Rightarrow X(s) = \frac{1}{s - a} \),


• \( x(t) = \sin(\omega t) \Rightarrow X(s) = \frac{\omega}{s^2 + \omega^2} \),


• \( x(t) = \cos(\omega t) \Rightarrow X(s) = \frac{s}{s^2 + \omega^2} \).

The Effect of Time Scaling on the Laplace Transform

Time Scaling Property

One of the essential properties of the Laplace transform is how it responds to scaling the time variable. If \( x(t) \) and \( X(s) \) form a Laplace transform pair, then the time-scaled function \( x(a t) \), with \( a > 0 \), has a related Laplace transform:

\[
\mathcal{L}\{x(a t)\} = \frac{1}{a} X\left(\frac{s}{a}\right),
\]

where \( a \) is a positive real number representing the scale factor in the time domain.

This property indicates that compressing or stretching the original function in time corresponds to stretching or compressing its Laplace transform in the \( s \)-domain, accompanied by a scaling factor \( 1/a \).

Mathematical Derivation of the Property

Starting from the definition:

\[
\mathcal{L}\{x(a t)\} = \int_0^\infty e^{-s t} x(a t) \, dt.
\]

Make a substitution:

\[
u = a t \Rightarrow t = \frac{u}{a}, \quad dt = \frac{du}{a}.
\]

The limits change accordingly:

\[
t = 0 \Rightarrow u = 0, \quad t \to \infty \Rightarrow u \to \infty.
\]

Substituting into the integral:

\[
\mathcal{L}\{x(a t)\} = \int0^\infty e^{-\frac{s u}{a}} x(u) \frac{du}{a} = \frac{1}{a} \int0^\infty e^{-\frac{s}{a} u} x(u) \, du.
\]

Recognizing this as the Laplace transform of \( x(t) \):

\[
\Rightarrow \mathcal{L}\{x(a t)\} = \frac{1}{a} X\left(\frac{s}{a}\right).
\]

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Application to \( X(2t) \) and the Given Transform Pair

Given Relationship

Suppose \( X(t) \) and \( X(s) \) form a Laplace pair. The problem statement indicates that the Laplace transform of \( X(2t) \) is:

\[
\mathcal{L}\{X(2t)\} = 0.5 \, X(0.5 s).
\]

This statement aligns with the time-scaling property, but with a specific scaling factor and a coefficient, which warrants further analysis.

Deriving the Transform of \( X(2t) \)

Using the property outlined above, for \( x(t) = X(t) \):

\[
\mathcal{L}\{X(2t)\} = \frac{1}{2} X\left(\frac{s}{2}\right).
\]

This matches the general property:

\[
\mathcal{L}\{x(a t)\} = \frac{1}{a} X\left(\frac{s}{a}\right),
\]

with \( a = 2 \). Therefore, in the context of \( X(t) \) and \( X(s) \):

\[
\boxed{
\mathcal{L}\{X(2t)\} = \frac{1}{2} X\left(\frac{s}{2}\right).
}
\]

However, the problem statement indicates that the transform of \( X(2t) \) is \( 0.5 X(0.5 s) \), which suggests a different scaling in the argument of \( X \), hinting at a possible typographical or interpretive nuance.

Important note: The notation \( X(t) \) and \( X(s) \) can sometimes be confusing. Usually, \( X(t) \) denotes a time-domain function, and \( X(s) \) its Laplace transform. If the problem states "Let \( X(t) \) and \( X(s) \) be a Laplace transform pair," then \( X(t) \) is the original function, and \( X(s) \) its transform.

Assuming the context is consistent with the standard notation, the key takeaway is:

\[
\mathcal{L}\{X(2t)\} = \frac{1}{2} X\left(\frac{s}{2}\right).
\]

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Implications and Practical Applications

Signal Processing

Understanding how time scaling affects the Laplace transform is crucial in signal processing. For example:

• Time compression (e.g., \( x(2t) \)) results in a stretching of the spectrum \( X(s) \),


• Time dilation impacts the frequency content and decay rates of signals.

Control Systems Engineering

In control systems, the response of systems to scaled inputs is often analyzed using the Laplace transform. Recognizing how the Laplace domain transforms under time scaling enables engineers to:

• Predict system response to scaled inputs,


• Simplify complex differential equations,


• Design controllers that accommodate time-scaling effects.

Differential Equations

Transforming differential equations often involves understanding how functions behave under scaling. For instance, solutions involving \( x(2t) \) can be derived by applying the time-scaling property, simplifying the solving process.

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Summary and Key Takeaways

• The Laplace transform of a scaled function \( x(a t) \) is given by:

where \( X(s) = \mathcal{L}\{x(t)\} \).

• When \( a = 2 \), the transform becomes:

• The relationship between the time domain and the \( s \)-domain under scaling is inverse: scaling time by a factor compresses or stretches the spectrum in the \( s \)-domain accordingly.


• Recognizing