Question

If \(\boldsymbol{F}_{e}(f(a x)\\}=k F_{e}(s / a)\), then \(k=\)

Short answer

The value of \(k\) is \(\frac{1}{|a|}\).

Step-by-step solution

Understanding the Problem

The problem involves the transformation of a function within an exponential function, relying on Fourier transform properties. We need to identify how the scaling of an input function by a factor affects its Fourier transform.

Recall the Scaling Property

The Fourier transformation has a property known as the scaling property. It states that if you scale the input of a function by a certain factor, then in its Fourier transform, the frequency domain representation is also affected in a specific manner.

Analyze the Scaling Property

For a function \(f(x)\), the Fourier transform \(F_e(f(x))\) when the function is evaluated at \(f(ax)\) is given by the formula: \[F_e(f(ax)) = \frac{1}{|a|} F_e\left(s/a\right)\]. This transformation is due to the scaling property that involves a factor of \(1/|a|\).

Compare with the Given Expression

According to the problem, we have \(F_e(f(ax)) = k F_e(s/a)\). Based on the scaling property expression \(F_e(f(ax)) = \frac{1}{|a|} F_e(s/a)\), compare this with the given format, thus identifying that \(k = \frac{1}{|a|}\).

Finalize the Value of k

The constant \(k\), which scales the Fourier transform, is determined by the absolute value of the reciprocal of the factor \(a\) that was used to scale the function. Therefore, the value of \(k\) is \(\frac{1}{|a|}\).

Key concepts

Scaling Property

The scaling property of the Fourier transform is a fundamental concept that describes how scaling a function affects its frequency domain representation. Simply put, when you multiply the input of a function by a constant, denoted here as \(a\), it alters how we perceive the function in the frequency domain.

The mathematical expression for this property is given by:

• For a function \(f(x)\), the Fourier transform of \(f(ax)\) becomes \(\frac{1}{|a|} F_e\left(s/a\right)\).
• This means the scaling factor appears as \(\frac{1}{|a|}\) in the frequency domain.

The concept illustrates the conservation of "energy" or "area under the curve" even though frequencies might seem to stretch or compress when scaling is applied. By comparing two expressions of Fourier transforms, one can deduce the effect scaling has on the constant factor, which in our example is identified as \(k = \frac{1}{|a|}\).

This scaling changes the distribution of frequencies in the function, making each point a scaled version of the original. Understanding this helps in signal processing and other applications.

Exponential Function Transformation

An exponential function transformation within the context of Fourier transforms involves analyzing how exponential components in functions transform when applied in various domains. When functions involve terms like \(f(ax)\), the exponential transformation property helps in understanding their behavior under the Fourier transform.

Key insights include:

• The transformation considers changes both in time and frequency domains.
• Scaling in the input function \(f(ax)\) directly influences how the exponential properties manifest in the frequency domain.

Understanding exponential function transformations is crucial when working with signals and systems, as they commonly exhibit exponential characteristics across domains.

Clear comprehension of these transformations facilitates how we manipulate and interpret data through Fourier analysis. Particularly in solving problems where scaling occurs in the argument of an exponential function, grasping these transformations allows us to correctly adjust and apply properties like the scaling factor \(k\).

Frequency Domain Representation

Frequency domain representation is a way to express wave-like functions in terms of their constituent frequencies, instead of focusing on time or space as in time-domain functions. This representation offers valuable insights into the function's properties when analyzed under Fourier transforms.

By understanding frequency domain representations, you gain several benefits:

• It simplifies the mathematical treatment of systems by dealing with frequencies rather than time shifts.
• Analyzing functions like \(f(ax)\), you see their relationship in distributing energy across various frequencies.

The frequency domain representation is especially useful because it reveals how certain operations, such as scaling, impact the spectral content of signals.

In the context of the discussed exercise, understanding the transformation of a function in the frequency domain illustrates the significance of the scaling constant \(k\) as it perturbs the spectral amplitude by \(\frac{1}{|a|}\). This clarity aids in solving complex problems efficiently and accurately, ensuring clean interpretation of functions across both domains.