Technology & Engineering
Fourier Transform Table
The Fourier Transform Table is a reference tool used in signal processing and engineering to convert a function of time or space into a function of frequency. It provides a systematic way to analyze and manipulate complex waveforms by decomposing them into simpler sinusoidal components. The table contains mathematical relationships and formulas for performing Fourier transforms and inverse Fourier transforms.
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9 Key excerpts on "Fourier Transform Table"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Fourier Analysis and Fourier Transform Fourier analysis In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions. Fourier analysis is named after Joseph Fourier, who showed that representing a function by a trigonometric series greatly simplifies the study of heat propagation. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis . In mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transform. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis has a corresponding inverse transform that can be used for synthesis. Applications Fourier analysis has many scientific applications — in physics, partial differential equa-tions, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, optics, diffraction, geometry, and other areas.
eBook - PDFIntroduction to the Senses
From Biology to Computer Science
- Terry R. J. Bossomaier(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
3.5 Fourier Analysis and the Fourier Transform Finally we get to the details of how to analyse a signal into its frequency components, or more generally from one domain to another, reciprocal domain, usually frequency. Fourier Analysis and the Fourier Transform do essentially the same thing. In the first case the domain is finite, akin to the strings on a violin and the frequencies occur in discrete, finite steps. In the latter, the domain is infinite and frequency is a continuous variable. Closely related to the continuous transform is the Discrete Fourier Transform. Most computational work in audio or image processing uses this discrete version. However, the ideas which are important for understanding senses come from the continuous transform and the sampling of continuous signals, so these are the foci of the present chapter. The Fourier Transform (FT) then, is the method of finding the frequency components of a signal. There is a variety of different ways to define it, and we’re going to have to leave the mathematical niceties to the many excellent books specialising in the area, such as Gaskill (1978) and Bracewell (1999). We will use what is sometimes called the real form. It’s actually not the mathematically most elegant, but it has the advantage of relating directly to the equations for a real wave form, for a real function, f (x ) 7 . 7 x will often be either time for an auditory signal, or space for an image; we refer to the domain of f (x ) as the time or space domains and the transform as the frequency domain.
- William Bober, Andrew Stevens(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
239 Chapter 8 Fourier Transforms and Signal Processing 8.1 Introduction In electronic circuits, we are often presented with voltage and currents that repre-sent signals that carry meaningful information. Examples include ◾ audio, including hi-fi and voice ◾ video, including television and surveillance ◾ telemetry from temperature sensors, strain gauges, medical instrumentation, power meters, and the like ◾ radar, loran, and GPS signals for detection and guidance systems on aircraft and ships ◾ modulated signals as AM, FM, VHF, UHF, CDMA (code division multiple access), GSM (Groupe spéciale mobile), and Wi-Fi for broadcast and point-to-point transmission of audio, video, voice, and data There are countless reasons for processing signals. For audio and video, we might want to filter out noise or reduce bandwidth. For telemetry signals, we may want to examine the data from multiple sensors to achieve a goal such as predicting the 240 ◾ Numerical and Analytical Methods with MATLAB weather, reducing an automobile’s fuel consumption, or interpreting sensor data from an electrocardiogram. In the case of communication systems, we can use modulation to multiplex audio or video signals (from radio or TV stations) onto high-frequency carriers, thereby allowing multiple signals to be carried over a common medium (e.g., a coaxial cable or optical fiber) or through the airwaves. In addition, we typically want to account for the link loss and intersymbol interference (ISI) introduced by the channel. The primary mathematical tool used in signal processing is the Fourier trans-form . This transform provides a method for describing a signal in terms of its frequency components (i.e., in the “frequency domain”) instead of in the more familiar and tangible “time domain.” Table 8.1 lists several types of signals and their corresponding frequency ranges (i.e., the range of frequency components that might be present in the waveforms).
- Alasdair McAndrew(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 7 The Fourier Transform 7.1 Introduction The Fourier Transform is of fundamental importance to image processing. It allows us to perform tasks that would be impossible to perform any other way; its efficiency allows us to perform other tasks more quickly. The Fourier Transform provides, among other things, a powerful alternative to linear spatial filtering; it is more efficient to use the Fourier Transform than a spatial filter for a large filter. The Fourier Transform also allows us to isolate and process particular image “frequencies,” and so perform low pass and high pass filtering with a great degree of precision. Before we discuss the Fourier Transform of images, we shall investigate the one-dimensional Fourier Transform, and a few of its properties. 7.2 Background Our starting place is the observation that a periodic function may be written as the sum of sines and cosines of varying amplitudes and frequencies. For example, in Figure 7.1 we plot a function and its decomposition into sine functions. Some functions will require only a finite number of functions in their decomposition; others will require an infinite number. For example, a “square wave,” such as is shown in Figure 7.2, has the decomposition f ( x ) = sin x + 1 3 sin 3 x + 1 5 sin 5 x + 1 7 sin 7 x + 1 9 sin 9 x + · · · (7.1) In Figure 7.2 we take the first five terms only to provide the approximation. The more terms of the series we take, the closer the sum will approach the original function. This can be formalized; if f ( x ) is a function of period 2 T , then we can write f ( x ) = a 0 + ∞ X n =1 a n cos nπx T + b n sin nπx T 149
eBook - PDFSinusoids
Theory and Technological Applications
- Prem K. Kythe(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
3 FourierTransforms Fourier transforms (FTs) constitute the central theme of this book. They are used in every technological application which starts from Chapter 4 onward. The purpose of the Fourier transform is to represent an image in terms of sine and cosine functions (sinusoids). Using these two basic functions, it is easy to perform processing operations, especially frequency domain filtering, which after inversion using inverse Fourier transform (IFT) is converted back to the spatial domain. Using Fourier transform, any digital image can be represented as a weighted sum of sine and cosine functions. These weights are used to reconstruct the image using the inverse Fourier transform. Linear image processing operations can be implemented using convolution in the spatial domain or by filtering in the frequency domain. Some image processing operations perform better if frequency domain solutions are used. 3.1 Definitions Let f ( x ), -∞
eBook - ePubFundamentals of Digital Image Processing
A Practical Approach with Examples in Matlab
- Chris Solomon, Toby Breckon(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
In image processing, we are usually concerned with 2-D spatial distributions (i.e. functions) of intensity or colour which exist in real space – i.e. a 2-D Cartesian space in which the axes define units of length. The Fourier transform operates on such a function to produce an entirely equivalent form which lies in an abstract space called frequency space. Why bother? In the simplest terms, frequency space is useful because it can make the solution of otherwise difficult problems much easier (Figure 5.1). Fourier methods are sufficiently important that we are going to break the pattern and digress (for a time) from image processing to devote some time to understanding some of the key concepts and mathematics of Fourier transforms and frequency space. Once this foundation has been laid, we will move on to see how they can be used to excellent effect both to look at image processing from a new perspective and to carry out a variety of applications. Figure 5.1 Frequency-space methods are used to make otherwise difficult problems easier to solve 5.2 Frequency space: the fundamental idea We will begin our discussion of Fourier methods by summarizing, without any attempt at rigour, some of the key concepts. To stay general, we will talk for the moment of the Fourier analysis of signals rather than images. (1) The harmonic content of signals. The fundamental idea of Fourier analysis is that any signal, be it a function of time, space or any other variables, may be expressed as a weighted linear combination of harmonic (i.e. sine and cosine) functions having different periods or frequencies
- Keith C Brown(Author)
- 2020(Publication Date)
- Royal Society of Chemistry(Publisher)
8 The Fourier TransformThe whole is the sum of the parts. Euclid8.1 Fourier Series†
As NMR/MRI spectroscopists working with modern digital equipment we routinely acquire data in the form of the free induction decay. This is the digitised form of the analogue time domain data passing through the receiver. We wish, of course to look at these data in the frequency domain and also routinely transform the time domain data to frequency domain data for display and further analysis (Figure 8.1 ).Figure 8.1 The Fourier transform of time-domain data to frequency-domain data.This transform of the data is accomplished using the ideas of Jean Baptiste Joseph Fourier (and others). Fourier led a very interesting life to say the least. He was a young man at the time of the French revolution‡ and was very nearly beheaded during the reign of terror led by Robespierre. Eventually, he became acquainted with Napoleon and travelled with him and the French army to Egypt in 1798. Later, Napoleon made him prefect of Grenoble and during this time he met and mentored Jean-Francois Champollion, the person most responsible for the translation of the Rosetta stone,§ which was found in Egypt by Napoleon's army. It was Fourier's studies of heat conduction that brought forward what we now call Fourier analysis and it is this work that he is most associated with.Fourier's basic idea was that the application of infinite sums of sine and cosine functions multiplied by suitable constants can be used to represent any periodic function. To understand what is happening in our spectrometer software to accomplish this remarkable feat we must delve into the mysteries of the Fourier transform.Our analysis of vectors showed us that we could use unit basis vectors multiplied by suitable factors to build up an overall vector that spans the vector space. This is particularly easy to visualise in two- or three-dimensional space. Mathematically, we represent this as:
eBook - PDF- Mark Nixon(Author)
- 2013(Publication Date)
- Newnes(Publisher)
The particular concern is the appropriate sampling frequency of (essentially, the value for N ), or the rate at which pixel values are taken from, a camera’s video signal. 2.3 The Fourier transform The Fourier transform is a way of mapping a signal into its component frequencies. Frequency measures in hertz (Hz) the rate of repetition with time , measured in seconds (s); time is the reciprocal of frequency and vice versa (hertz = 1/second; s = 1/Hz). Consider a music centre: the sound comes from a CD player (or a tape) and is played on the speakers after it has been processed by the amplifier. On the amplifier, you can change the bass or the treble (or the loudness which is a combination of bass and treble). Bass covers the low frequency components and treble covers the high frequency ones. The Fourier transform is a way of mapping the signal from the CD player, which is a signal varying continuously with time, into its frequency components. When we have transformed the signal, we know which frequencies made up the original sound. So why do we do this? We have not changed the signal, only its representation. We can now visualise it in terms of its frequencies, rather than as a voltage which changes with time. But we can now change the frequencies (because we can see them clearly) and this will change the sound. If, say, there is hiss on the original signal then since hiss is a high frequency component, it will show up as a high frequency component in the Fourier transform. So we can see how to remove it by looking at the Fourier transform. If you have ever used a graphic equaliser, then you have done this before. The graphic equaliser is a way of changing a signal by interpreting its frequency domain representation; you can selectively control the frequency content by changing the positions of the controls of the graphic equaliser.
eBook - PDF- John H. Karl(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Application of the Fourier Transform to Digital Signal Processing We have invested in the previous chapter on continuous signal theory because many of the signals that find their way into digital signal processing are thought to arise from some underlying continuous function. In this chapter, we discuss the relationship between these underlying continuous functions and the discrete signals that are produced by their sampling. Fundamental problems will be encountered. A parallel development of the sampling process, in both the time and frequency domains, will clarify these problems and show how to cope with them. However, the deeper insight provided by this development will permit us to evaluate the severity of the limitations encountered in a given circumstance. Finally, after we understand the relationship between the Fourier integral transform and the DFT, new useful ideas will emerge on interpolation, decimation, and modulation. In the preceding chapters, we have pursued a natural course through three kinds of time-frequency transformations. First, evaluation of the frequency response of discrete LSI systems led to the discrete-time/ continuous-frequency description of digital impulse responses and their spectra. Then, by computing this spectra at discrete points in frequency, we were led to the discrete-time/discrete-frequency domain of the DFT. In the last chapter, suitable limits took the DFT over into the Fourier integral transform of continuous time and frequency. To complete our journey into time-frequency transformations, we next address the fourth possibility: continuous time and discrete frequency. 127 7 128 7/ Application of the Fourier Transform Continuous Time, Discrete Frequency: The Fourier Series Discrete spectra are familiar from areas such as spectroscopy, where they are called line spectra.
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