Question

If \(F(2)\) is the Fourier transform of \(f(x)\), then the Fourier transform of \(f(\alpha x)\) is ........

Short answer

The Fourier transform of \( f(\alpha x) \) is \( \frac{1}{|\alpha|} F\left(\frac{2}{\alpha}\right) \).

Step-by-step solution

Understanding the Fourier Transform

The Fourier transform of a function \( f(x) \), denoted as \( F(k) \), is given by the integral \( F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx} dx \). This represents the frequency components of the original function, \( f(x) \).

Introducing the Scaling Property

There is a Fourier transform property regarding scaling. If a function \( f(x) \) is scaled by a factor \( \alpha \), becoming \( f(\alpha x) \), the Fourier transform of this scaled function is adjusted by a factor of \( |1/\alpha| \) and a change in variable.

Applying the Scaling Property

The Fourier transform of \( f(\alpha x) \) is \( \frac{1}{|\alpha|} F\left(\frac{k}{\alpha}\right) \). This means the function in frequency space is compressed or stretched based on \( \alpha \).

Deriving the Result

Since the problem gives \( F(2) \) as the Fourier transform of \( f(x) \), for \( f(\alpha x) \), the transform becomes \( \frac{1}{|\alpha|} F\left(\frac{2}{\alpha}\right) \), where \( 2 \) is substituted for \( k \) in \( F(k) \).

Key concepts

Scaling Property

The scaling property is an important aspect of the Fourier Transform, as it dictates how transformations affect the frequency spectrum of a function. When we apply the Fourier transform to a scaled version of a function, like \( f(\alpha x) \), where \( \alpha \) represents a scaling factor, the resulting transform also scales in frequency space. This is crucial for signal processing and physics.

• For a function \( f(x) \), its Fourier transform is \( F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx} dx \).
• When the function is scaled by \( \alpha \), it becomes \( f(\alpha x) \).
• The Fourier transform of the scaled function is \( \frac{1}{|\alpha|} F\left(\frac{k}{\alpha}\right) \).

This result implies that the function's frequency components are compressed or expanded depending on whether \( \alpha \) is greater or less than 1. If the scaling factor \( \alpha \) is greater than 1, the function is compressed, making the frequency spectrum wider. Conversely, if \( \alpha \) is less than 1, the function stretches out, and its spectrum narrows.

Frequency Components

Frequency components are an essential part of the Fourier Transform, which helps to break down signals into individual sinusoidal components. This decomposition reveals the frequencies that make up the overall signal.

• Every function \( f(x) \) can be thought of as being made up of various sine and cosine functions with different frequencies and amplitudes.
• The Fourier transform calculates these components, transforming \( f(x) \) into \( F(k) \), a function of frequency \( k \).
• These components provide insight into the signal's behavior, such as identifying dominant frequencies and filtering processes.

Understanding frequency components allows engineers and scientists to analyze signals in many fields, from communications to acoustics. Whether used for analyzing audio signals or for the study of periodic phenomena, frequency analysis via Fourier Transforms is essential.

Integral Transform

An integral transform is a mathematical operation that transforms a function into another form, often to simplify certain calculations or to express a function in a different domain. The Fourier Transform is one such integral transform, significant for converting functions from the time domain to the frequency domain.

• The integral representation of the Fourier Transform is \( F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx} dx \).
• This transformation simplifies the analysis of complex systems by working on frequency components rather than time or spatial components.
• Integral transforms like the Fourier Transform are widely applied in engineering, physics, and applied mathematics for solving differential equations and analyzing signals.

The beauty of the Fourier Transform as an integral transform lies in its ability to deconstruct and reconstruct functions, providing detailed insight into the frequency content of signals, which is vital for applications in image processing, signal processing, and more.