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Fourier Analysis Fourier analysis of spatial and temporal visual stimuli has become common in the last 35 years. For many people interested in vision but not trained in mathematics this causes some confusion. The use of these Fourier methods does not mean that the visual system performs a Fourier analysis. At present it should be understood that this approach is a convenient way to analyze visual stimuli. Jean Baptiste Fourier, a mathematician, showed that any repetitive waveform can be broken down into a series of sine waves at appropriate amplitudes and phases. Fourier analysis may be performed mathematically if the expression f (t) describing the waveform or complex tone.
One of the important ideas for sound reproduction, which arises from Fourier analysis, is that it takes a high quality audio reproduction system to reproduce percussive sounds or sounds with fast transients. The sustained sound of a trombone can be reproduced with a limited range of frequencies because most of the sound energy is in the first few harmonics of the fundamental pitch. A sine wave is a wave of a single frequency. It has a given frequency, amplitude and phase. Fourier's discovery can be illustrated through taking an appropriately chosen set of sine waves and add them together to produce a square wave. Clearly, if it is possible to construct a wave of a particular pattern by adding together appropriately chosen sine waves then the reverse is true as well.
The building of complex waves by combining appropriately chosen sine waves is called Fourier synthesis. The breaking apart of a complex wave into its component sine waves is called Fourier analysis. Fourier analysis studies approximations and decompositions of functions using trigonometric polynomials. Of incalculable value in many applications of analysis, this field has grown to include many specific and powerful results, including convergence criteria, estimates and inequalities, and existence and uniqueness results. Extensions include the theory of singular integrals, Fourier transforms, and the study of the appropriate function spaces.
The Fourier approach to analyzing visual stimuli actually comes under the heading of Linear Systems Analysis. Linear transforms, especially Fourier and Laplace transforms, are widely used in solving problems in science and engineering. The Fourier transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary-value problems (Brigham, 2 - 3) and has been very successfully applied to restoration of astronomical data (Brault and White). The Fourier transform, a pervasive and versatile tool, is used in many fields of science as a mathematical or physical tool to alter a problem into one that can be more easily solved.
Some scientists understand Fourier theory as a physical phenomenon, not simply as a mathematical tool. In some branches of science, the Fourier transform of one function may yield another physical function (Bracewell, 1 - 2). The Fourier transform, in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes (Brigham, 4). The Fourier transform of f (x) is defined as F (s) = f (x) exp (-i 2 xs) dx. Applying the same transform to F (s) gives f (w) = F (s) exp (-i 2 ws) ds.
If f (x) is an even function of x, that is f (x) = f (-x), then f (w) = f (x). If f (x) is an odd function of x, that is f (x) = -f (-x), then f (w) = f (-x). When f (x) is neither even nor odd, it can often be split into even or odd parts. To avoid confusion, it is customary to write the Fourier transform and its inverse so that they exhibit reversibility: F (s) = f (x) exp (-i 2 xs) dx f (x) = F (s) exp (i 2 xs) ds so that f (x) = f (x) exp (-i 2 xs) dx exp (i 2 xs) ds as long as the integral exists and any discontinuities, usually represented by multiple integrals of the form? [f (x+) + f (x-) ], are finite. There are functions for which the Fourier transform does not exist; however, most physical functions have a Fourier transform, especially if the transform represents a physical quantity. Other functions can be treated with Fourier theory as limiting cases.
Many of the common theoretical functions are actually limiting cases in Fourier theory. Usually functions or waveforms can be split into even and odd parts as follows: f (x) = E (x) + O (x) where E (x) = ? [f (x) + f (-x) ] O (x) = ? [f (x) - f (-x) ] and E (x), O (x) are, in general, complex. In this representation, the Fourier transform of f (x) reduces to 2 E (x) cos (2 xs) dx - 2 iO (x) sin (2 xs) dx It follows then that an even function has an even transform and that an odd function has an odd transform. (Bracewell, 14). An important case is that of an Hermitian function, one in which the real part is even and the imaginary part is odd, i. e. , f (x) = f (-x). The Fourier transform of an Hermitian function is even.
In addition, the Fourier transform of the complex conjugate of a function f (x) is F (-s), the reflection of the conjugate of the transform. The cosine transform of a function f (x) is defined as Fc (s) = 2 f (x) cos 2 sx dx. This is equivalent to the Fourier transform if f (x) is an even function. In general, the even part of the Fourier transform of f (x) is the cosine transform of the even part of f (x). The cosine transform has a reverse transform given by f (x) = 2 Fc (s) cos 2 sx ds. Likewise, the sine transform of f (x) is defined by FS (s) = 2 f (x) sin 2 sx dx.
As a result, i times the odd part of the Fourier transform of f (x) is the sine transform of the odd part of f (x). Combining the sine and cosine transforms of the even and odd parts of f (x) leads to the Fourier transform of the whole of f (x): f (x) = CE (x) - i SO (x) where, C, and S stand for -i times the Fourier transform, the cosine transform, and the sine transform respectively, or F (s) = ? FC (s) -? iFS (s) (Bracewell, 17 - 18). Since the Fourier transform F (s) is a frequency domain representation of a function f (x), the s characterizes the frequency of the decomposed co sinusoids and sinusoids and is equal to the number of cycles per unit of x (Bracewell, 18 - 21).
If a function or waveform is not periodic, then the Fourier transform of the function will be a continuous function of frequency (Brigham, 4). It is known that most geophysical signals can be expressed as a decomposition of the signal into sine and cosine functions of different frequencies (also referred to as harmonics). This is called Fourier analysis. This concept is usually exposed in a calculus or physics course where sine and cosine functions expressed as a Fourier series are used to represent a periodic function of time. (In 1822, Joseph Fourier was the first person who attempted to prove the convergence of such a series. ) There are the usual conditions placed on the signal, i.
e. : 1) it cannot be multi valued at any given time, 2) it cannot have an infinite number of discontinuities, or maxima or minima, and 3) it must be bounded within its period. The frequencies of the trigonometric functions are the spectral components of the Fourier series. These frequencies are predetermined by the periodicity, T of the function and are equal to n/T Therefore, the frequency spectrum is composed of discrete line spectra. When a signal is not periodic, the spectrum is not discrete and the Fourier series must be generalized into the Fourier integral or Fourier transform. As long as the integral of the absolute value of the signal converges, the continuous signal s (t) can be expressed as the Fourier integral, where Second equation defines the Fourier transform of s (t); equation 3. 1 is the inverse Fourier transform that recovers s (t) back from S (f). These equations are at the heart of spectral analysis and they are so tightly connected that they are usually called the Fourier transform pair.
It is customary to use a lower case symbol for the time (or space) domain function and an upper case symbol for the corresponding function of frequency. S (f) and s (t) are referred to as the frequency domain and time domain representations of the signal, respectively. In general transform language, the terms in the integrands exclusive of the s (t) and S (f) are called the kernels of the transforms. In the Fourier transform pair the kernels differ only slightly; the sign of the exponent in the Fourier transform is - and in the inverse transform it is +. To save space in writing the Fourier transform pair, the common shorthand expressions S (f) = [s (t) ] and s (t) = - 1 [S (f) ] will be used for the remainder of this web site for the Fourier transform and inverse Fourier transform, respectively, unless the explicit expressions are required. The 2?
appearing in the transform kernels can be included with the frequency f to express the Fourier transform pair in the angular frequency, ? domain (in radians / s ) as and There seemed to have been a gigantic jump from considering the digitizing of s (t) to now expressing it in a Fourier transform pair. Furthermore, it is obvious from the equations that the transform pair are complex functions with the inclusion of i = (- 1) 1 / 2 in the transform kernels. If one simply remembers a couple of things from basic mathematics, the above would make more sense. For one, the transform kernels, e. g. , exp (i 2?
f) are of the general form of Eulers identity, so, From Eulers relationship it can clearly seen that the Fourier transform pair have sine and cosine terms just like a Fourier series does. And, since one knows that integration is the limiting expression of a summation that becomes continuous, it is realized that the Fourier transform is really the expression of a infinite, continuous summation of sine and cosine functions. In fact, the Fourier transform can be expressed using separate sine and cosine transforms. So, Fourier analysis expressed by the Fourier transform is simply the decomposition of a signal into its composite frequency (sine and cosine) components. Rather than the discrete spectral lines (frequencies) appearing in a Fourier series, the Fourier transform has a continuous spectrum to represent a non periodic process. The transform of a signal into its continuous frequency components is familiar to us all in nature when white light passing through a glass prism exposes its color spectrum.
When this happens with rain drops it is called a rainbow. So a rainbow is really natures Fourier transform although one have ever heard anyone call a rainbow a Fourier transform. The Fast Fourier Transform (FFT) is a DFT algorithm developed by Tukey and Cooley in 1965 which reduces the number of computations from something on the order of N 02 to N 0 log N 0. There are basically two types of Tukey-Cooley FFT algorithms in use: decimation-in-time and decimation-in-frequency. The algorithm is simplified if N 0 is chosen to be a power of 2, but it is not a requirement. The Fourier transform, an invaluable tool in science and engineering, has been introduced and defined.
Its symmetry and computational properties have been described and the significance of the time or signal space (or domain) vs. the frequency or spectral domain has been mentioned. In addition, there are important concepts in sampling required for the understanding of the sampling theorem and the problem of aliasing. A popular offspring of Fourier transform (DFT) is the Fast Fourier Transform (FFT) algorithm. The Fourier Transform is a projection of any function onto complex exponential's of the form exp (jan), where w is the frequency. Mathematically, the integral of the product of two functions is an inner product and the complex exponential's are a convenient set of orthogonal basis functions for an arbitrary function space.
That is the Fourier's theorem. Interpreted another way, one can view a sampled signal (i. e. a list of numbers) as a vector of arbitrary dimension.
One can define a vector space containing all such possible sampled signals. Now, what are some reasonable basis vectors which span this space? One convenient basis is the natural basis, which contains the orthogonal unit vector But a projection onto these basis vectors yields little insight. A more useful basis is the normalized exponential's, exp (jan). A projection onto this basis allows us to reconstruct the signal as a sum of exponential's. It is good because complex exponential's are sinusoids.
It is sinusoids - or, at least, one can detect the presence or absence of sinusoids at various frequencies. Thus, such a transform is musically relevant. Most significantly, the operation of filtering is very simple once we are in the frequency domain - in order to change the amount of a certain frequency which is in a signal, we just multiply that coefficient by a constant. The transform described above is very close to the Discrete Fourier Transform. It is not exactly the same, unfortunately, because the DFT is not normalized.
That is, instead of projecting onto exp (jan), one actually projects onto 1 /N exp (jan), where N is the number of samples. The point, however, remains: the DFT is a convenient way of breaking down a signal into its frequency components. Best of all, efficient algorithms (called Fast Fourier Transforms) exist to compute the DFT, which allow us to perform efficient filtering by taking the DFT, multiplying the frequency coefficients, and then reconstructing the signal (i. e. taking the inverse DFT. ) Doing this in the time domain requires linear convolution, which is in general much more time-intensive. Fourier Transform has the following properties: 1.
Scaling Property If {f (x) } = F (s) and a is a real, nonzero constant, then {f (ax) } = f (ax) exp (i 2 sx) dx = |a|- 1 f () exp (i 2 s / a ) d = |a|- 1 F (s / a ). From this, the time scaling property, it is evident that if the width of a function is decreased while its height is kept constant, then its Fourier transform becomes wider and shorter. If its width is increased, its transform becomes narrower and taller. A similar frequency scaling property is given by {|a|- 1 f (x / a ) } = F (as). 2. Shifting Property If {f (x) } = F (s) and x 0 is a real constant, then {f (x - x 0) } = f (x - x 0) exp (i 2 sx) dx = f () exp (i 2 s (+ x 0) ) d = exp (i 2 x 0 s) f () exp (i 2 s) d = F (s) exp (i 2 x 0 s). This time shifting property states that the Fourier transform of a shifted function is just the transform of the unsifted function multiplied by an exponential factor having a linear phase.
Likewise, the frequency shifting property states that if F (s) is shifted by a constant s 0, its inverse transform is multiplied by exp (i 2 xs 0) {f (x) exp (i 2 xs 0) } = F (s-s 0). The Fourier transform, an invaluable tool in science and engineering, has been introduced and defined. Its symmetry and computational properties have been described and the significance of the time or signal space (or domain) vs. the frequency or spectral domain has been mentioned. References: Bracewell, Ron N. , 1965, The Fourier Transform and Its Applications, New York: McGraw-Hill Book Company, 381 pp. Brigham, E.
Oren, 1988, The Fast Fourier Transform and Its Applications, Englewood Cliffs, NJ: Prentice-Hall, Inc. , 448 pp. Cooley, J. W. and Tukey, J.
W. , 1965, An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation, 19, 90, pp. 297 - 301. Gabel, Robert A. and Roberts, Richard A. , 1973, Signals and Linear Systems, New York: John Wiley & Sons, 415 pp. Gaskill, Jack D. , 1978, Linear Systems, Fourier Transforms, and Optics, New York: John Wiley & Sons, 554 pp.
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