Glossary

5.3 Fourier transforms of basic functions and distributions

3 min read•august 7, 2024

Fourier transforms of basic functions and distributions are essential building blocks in harmonic analysis. They help us understand how simple signals behave in the , providing insights into more complex transformations.

These transforms, like the Gaussian and sinc functions, show up everywhere in and physics. Knowing them well is key to tackling tougher problems and grasping the bigger picture of Fourier analysis on the real line.

Basic Functions

Gaussian Function and Exponential Function

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  • defined as f(x)=e−πx2f(x) = e^{-\pi x^2}
    • Also known as the normal distribution or bell curve
    • Fourier transform of the Gaussian function is another Gaussian function
    • Plays a crucial role in probability theory and statistics (central limit theorem)
  • defined as f(x)=eaxf(x) = e^{ax}, where aa is a constant
    • Fourier transform of the exponential function is a shifted delta function
    • Exponential function is its own derivative and integral (up to a constant factor)
    • Used to model exponential growth or decay processes (radioactive decay, population growth)

Rectangle Function and Sinc Function

  • defined as f(x)={1,∣x∣<1212,∣x∣=120,∣x∣>12f(x) = \begin{cases} 1, & |x| < \frac{1}{2} \\ \frac{1}{2}, & |x| = \frac{1}{2} \\ 0, & |x| > \frac{1}{2} \end{cases}
    • Also known as the unit pulse or gate function
    • Fourier transform of the rectangle function is the
    • Used in signal processing to represent a time-limited signal (pulse)
  • Sinc function defined as sinc(x)=sin⁡(πx)πx\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}
    • Normalized form of the sinc function with sinc(0)=1\text{sinc}(0) = 1
    • Fourier transform of the sinc function is the rectangle function
    • Appears in the context of and Fourier transforms (ideal low-pass filter)

Distributions

Dirac Delta Function

  • , denoted as δ(x)\delta(x), is a generalized function or distribution
    • Defined by its properties: ∫−∞∞δ(x)dx=1\int_{-\infty}^{\infty} \delta(x) dx = 1 and δ(x)=0\delta(x) = 0 for x≠0x \neq 0
    • Can be thought of as an infinitely tall, infinitely narrow spike with unit area
    • Fourier transform of the delta function is a constant function (unity)
    • Used to represent an idealized point source or impulse in physics and engineering (point mass, point charge)

Heaviside Step Function and Tempered Distributions

  • , denoted as H(x)H(x), is a discontinuous function defined as H(x)={0,x<01,x≥0H(x) = \begin{cases} 0, & x < 0 \\ 1, & x \geq 0 \end{cases}
    • Also known as the unit step function
    • Fourier transform of the Heaviside step function is the sum of a delta function and a principal value term
    • Used to represent a sudden change or transition in a system (on/off switch, threshold)
  • are a class of distributions that allow for the Fourier transform to be well-defined
    • Includes functions that grow at most polynomially at infinity
    • Examples of tempered distributions: Gaussian function, sinc function, delta function, Heaviside step function
    • Fourier transform of a tempered distribution is another tempered distribution
    • Provides a rigorous framework for studying Fourier transforms in a generalized sense

Key Terms to Review (22)

Continuous fourier transform: The continuous Fourier transform is a mathematical operation that transforms a function defined in the time domain into a function in the frequency domain. This transformation helps to analyze the frequency components of signals and is widely used in various fields, including engineering and physics. It provides a powerful tool for understanding how functions can be represented as sums of sinusoids, revealing their harmonic structure.
Convolution: Convolution is a mathematical operation that combines two functions to produce a third function, expressing how the shape of one function is modified by the other. This operation is crucial in various fields such as signal processing, where it helps to filter signals, and in harmonic analysis, where it connects to Fourier transforms and distributions.
Dirac Delta Function: The Dirac delta function is a mathematical construct that represents a distribution rather than a conventional function, characterized by being zero everywhere except at a single point, where it is infinitely high such that its integral over the entire real line equals one. It acts as an idealized point mass or charge and is crucial in various areas of analysis, particularly in convolution, Fourier transforms, and distribution theory.
Discrete Fourier Transform: The Discrete Fourier Transform (DFT) is a mathematical technique used to convert a sequence of equally spaced samples of a function (often a signal) into a representation in the frequency domain. This transformation is crucial for analyzing the frequency components of discrete signals and plays a key role in applications such as signal processing, filtering, and data compression.
Exponential Function: An exponential function is a mathematical function of the form $f(x) = a imes b^{x}$, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. This type of function is significant because it describes processes of growth and decay, particularly in relation to Fourier transforms, where it serves as a fundamental building block in representing signals and analyzing frequency components.
Fourier series: A Fourier series is a way to represent a periodic function as a sum of simple sine and cosine functions. This powerful mathematical tool allows us to decompose complex periodic signals into their constituent frequencies, providing insights into their behavior and enabling various applications across fields like engineering, physics, and signal processing.
Fourier Transform Pairs: Fourier transform pairs refer to a set of functions where one function is transformed into another through the Fourier transform process. This concept is central to understanding how different functions can be represented in the frequency domain, providing insights into their behavior in both time and frequency representations. The relationship between these pairs is crucial for operations such as differentiation and integration, as well as for transforming basic functions and distributions into their frequency components.
Frequency Domain: The frequency domain is a representation of a signal or function in terms of its frequency components, rather than its time-based characteristics. It allows for the analysis and manipulation of signals by breaking them down into their constituent frequencies, providing insights that are not easily visible in the time domain. This concept is fundamental in various applications such as signal processing, filtering, and harmonic analysis.
Gaussian Function: The Gaussian function is a specific type of exponential function defined by the formula $$g(x) = A e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$, where A is the height, \mu is the mean, and \sigma is the standard deviation. This function is vital in various mathematical and engineering contexts, especially in analyzing signal processing and probability distributions due to its unique bell-shaped curve.
Heaviside Step Function: The Heaviside step function is a discontinuous function defined as zero for negative inputs and one for non-negative inputs. This function is often used in mathematical analysis to represent signals that switch on at a certain point, making it important in the study of Fourier transforms, distributions, and signal processing. Its unique properties allow it to serve as a building block for more complex functions and distributions, linking it to the concepts of basic functions and tempered distributions.
Image analysis: Image analysis is the process of extracting meaningful information from images using various techniques, including mathematical transformations like Fourier transforms. This process plays a crucial role in understanding the structure and patterns within images, making it essential in fields such as computer vision, medical imaging, and remote sensing.
Inverse Fourier Transform: The inverse Fourier transform is a mathematical operation that transforms a frequency-domain representation of a function back into its original time-domain form. This process is crucial for understanding how functions can be reconstructed from their frequency components, allowing insights into both periodic and non-periodic signals.
Linearity: Linearity refers to a property of mathematical operations where the output is directly proportional to the input. In the context of Fourier transforms, linearity ensures that the transformation of a sum of functions is equal to the sum of their individual transformations, preserving the structure of the original functions. This principle is crucial for analyzing signals and functions in harmonic analysis, leading to effective decomposition and reconstruction of data.
Parseval's Theorem: Parseval's Theorem states that the total energy of a signal can be expressed equally in both time and frequency domains, essentially stating that the sum of the squares of a function is equal to the sum of the squares of its Fourier coefficients. This fundamental principle connects various aspects of harmonic analysis, demonstrating how time-domain representations relate to frequency-domain representations, which is crucial for understanding the behavior of signals.
Plancherel's theorem: Plancherel's theorem is a fundamental result in harmonic analysis that establishes the equivalence of the Fourier transform and the L2 norm of a function. This theorem asserts that the Fourier transform preserves the inner product structure of functions in the L2 space, meaning that the energy of a signal is conserved under transformation. It connects deeply with various properties of Fourier transforms, making it crucial for understanding inversion formulas and Parseval's identity.
Rectangle function: The rectangle function, also known as the rectangular or box function, is a piecewise function that takes the value of one within a specified interval and zero elsewhere. This function is crucial in the study of Fourier transforms as it serves as a basic example of how functions can be represented in the frequency domain and demonstrates properties like convolution and modulation.
Sampling Theorem: The Sampling Theorem states that a continuous signal can be completely represented by its samples and fully reconstructed from those samples if it is sampled at a rate greater than twice its highest frequency. This theorem is fundamental in signal processing as it lays the groundwork for understanding how signals can be digitized, analyzed, and manipulated while preserving their essential characteristics.
Schwartz Space: Schwartz space is a collection of smooth functions that decrease rapidly at infinity, along with all their derivatives. This space is essential in harmonic analysis and serves as a foundation for understanding Fourier transforms and tempered distributions, which are central to the study of functional analysis and distribution theory.
Signal Processing: Signal processing refers to the analysis, interpretation, and manipulation of signals to extract useful information or enhance certain features. It plays a crucial role in various applications, such as communications, audio processing, image enhancement, and data compression, by leveraging mathematical techniques to represent and transform signals effectively.
Sinc function: The sinc function, defined as $$\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$$ for $$x \neq 0$$ and $$\text{sinc}(0) = 1$$, is a mathematical function that arises frequently in signal processing and Fourier analysis. It is essential in understanding how differentiation and integration behave in the Fourier domain, as well as in the context of transforming basic functions and distributions. The sinc function serves as the Fourier transform of the rectangular pulse, illustrating the relationship between time-domain signals and their frequency-domain representations.
Tempered distributions: Tempered distributions are a class of generalized functions that extend the notion of ordinary functions and can be differentiated and manipulated in ways similar to classical functions. They are particularly useful in Fourier analysis because they can be transformed using the Fourier transform while still retaining well-defined properties, allowing for the analysis of more complex mathematical objects. This makes them essential in various applications, including solving differential equations and studying signal processing.
Time-shifting theorem: The time-shifting theorem states that if a function is shifted in time, the Fourier transform of that function is multiplied by an exponential factor. Specifically, if you have a function $$f(t)$$ and its Fourier transform $$F( u)$$, shifting the function by $$t_0$$ results in $$f(t - t_0)$$ having a Fourier transform of $$F( u)e^{-i2\pi u t_0}$$. This concept is crucial for understanding how modifications in the time domain affect frequency components.
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