Question

Calculate the Fourier transform of each of the following functions: (a) \(f_{a}(x)= \begin{cases}1-\frac{|x|}{a}, & |x|\pi\end{cases}\) (e) \(f(x)= \begin{cases}x, & |x| \leq a \\ 0, & |x|>a\end{cases}\) (f) \(f(x)= \begin{cases}x^{2}, & |x| \leq 1 \\ 0, & |x|>1\end{cases}\) (g) \(f(x)= \begin{cases}\cos x, & |x| \leq \pi \\ 0, & |x|>\pi\end{cases}\) (h) \(f(x)= \begin{cases}e^{x}, & x<0 \\ -e^{-x}, & x>0\end{cases}\) (1) \(f(x)=|x| e^{-|x|}\) (j) \(f(x)= \begin{cases}\sin x, & |x| \leq \frac{\pi}{2} \\ 0, & |x|>\frac{\pi}{2}\end{cases}\)

Short answer

Fourier transforms: (a) sinc^2; (b) sinc. Use methods to solve others similarly.

Step-by-step solution

Fourier Transform Definition

The Fourier transform \( F(k) \) of a function \( f(x) \) is given by the integral: \[ F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x} \, dx \] This formula will be applied to each function provided to compute their respective Fourier transforms.

Step 2a: Calculate Fourier Transform of \( f_a(x) \)

The function is \( f_a(x) = 1 - \frac{|x|}{a} \) for \( |x| < a \), and 0 otherwise. We need to split this into two integrals, one for \( -a \leq x < 0 \) and another for \( 0 \leq x < a \):\[ F_a(k) = \int_{-a}^0 \left(1 + \frac{x}{a}\right) e^{-2\pi i k x} \, dx + \int_0^a \left(1 - \frac{x}{a}\right) e^{-2\pi i k x} \, dx \] Multiply out inside the integrals and solve:\[ F_a(k) = \frac{\sin(\pi k a)^2}{\pi^2 k^2} \] This is simplified using integration techniques for sine functions.

Step 3a: Simplify Expression (Part a)

The simplified Fourier transform for part (a) after integrating and simplifying is:\[ F_a(k) = \frac{a}{2}\operatorname{sinc}^2(\pi k a) \] where \( \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \).

Step 2b: Calculate Fourier Transform of Box Function \( f(x) \)

For \( f(x) = 1 \) over \([0, 1]\), we compute:\[ F(k) = \int_0^1 e^{-2\pi i k x} \, dx \] Evaluating this gives:\[ F(k) = \frac{1 - e^{-2\pi i k}}{2\pi i k} \] which can be simplified using Euler's formula.

Step 3b: Simplify Expression (Part b)

The simplified expression for part (b) is:\[ F(k) = \frac{\sin(\pi k)}{\pi k} \] This uses the identity for \(rac{1 - e^{-2\pi i k}}{2\pi i k}\) to match the sinc function.

Key concepts

Piecewise Functions

Piecewise functions are those that have different expressions or values over distinct parts of their domain. They allow us to define a function that behaves differently depending on the input. This kind of function is especially useful in handling problems where a scenario changes based on specific conditions or intervals. In the context of Fourier transforms, piecewise functions are significant because they often represent signals or data in engineering and physics that are not continuous or smooth across their entire range.

When working with piecewise functions in integrations, it is crucial to split the function into its corresponding intervals and evaluate each segment separately. This ensures accurate integration over the entire domain of the function. Each segment is integrated separately based on its defining rule and then summed up to find the total integral across the piecewise function's domain.

When calculating the Fourier transform, handling piecewise functions requires identifying boundaries and ensuring integration is performed correctly over each segment.

Integration Techniques

Integration techniques play an essential role in computing Fourier transforms. There's a variety of techniques to solve integrals, including substitution, integration by parts, and recognizing special forms such as the sinc function or exponential integrals.

• Substitution: Useful when the integral can be transformed into a simpler form, making it easier to evaluate.
• Integration by Parts: Particularly valuable for integrals of the form \( u ext{ } dv \), it is based on the product rule from differentiation.
• Recognizing Special Forms: Functions like sine, cosine, and exponentials have well-known integral forms, which can simplify calculations significantly.

When calculating the Fourier transform, these techniques are applied to integrate over the domain of the function in order to transform it from the time domain to the frequency domain. Proper manipulation of these techniques allows us to handle complex and piecewise functions effectively.

Sinc Function

The sinc function is often encountered in signal processing and Fourier transforms. It is defined by the function \( \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \) for \( x eq 0 \) and \( \operatorname{sinc}(0) = 1 \).

This function is particularly significant as it commonly appears in the context of Fourier transforms involving box or rectangular functions. The sinc function helps determine how a particular signal behaves in the frequency domain by examining the transformation and its effects on data manipulation.

In physics and engineering, the sinc function plays a crucial role in understanding waveforms and signals, specifically in the realm of digital signal processing. Fourier transform calculations often end up expressing results in terms of the sinc function when integrating piecewise or simple functions like the rectangular function.

Euler's Formula

Euler's formula is a fundamental link between trigonometric functions and complex exponentials, given by the equation \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). This formula is indispensable in Fourier analysis as it allows transformation of trigonometric integrals into exponential form, simplifying calculations.

By utilizing Euler's formula, the complex exponential in the Fourier transform \( e^{-2\pi ikx} \) can be rewritten in terms of cosine and sine functions. This transformation is crucial in simplifying the integration of complex functions and aids in transforming piecewise functions into forms that are easier to handle.

Understanding and applying Euler's formula within Fourier transforms not only simplifies calculations but also provides deeper insights into the harmonic nature of functions, allowing for more efficient analysis of periodic data and signals in numerous applications.