Brigham EO. The Fast Fourier Transform. Englewood-Cliffs, NJ: Prentice-Hall, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... iu(k n=2) x g( k n=2) x) x = xe iun x=2 g( k n=2) x) 20 For g k : g( k n=2) x) k = 0; n 1, this is the discrete Fourier transform of the complex sequence (g k ) and can be calculated by the FFT algorithm for u k = 2 k= n x) k = 0; n 1, simultaneously (see e.g. Brigham [10], Chapter 10) This yields an approximation for M W ( V . By the inverse FFT we obtain the density f 1 and from this we can calculate quantiles. By in nite divisibility we calculate M(T) W (T) V (T ) M W ( V and obtain f T for any T 0. In the normal approximation case the ....

Brigham, E.O. (1974) The Fast Fourier Transform. Prentice-Hall. Englewood Clis, N.J..

.... iu(k n=2) x g( k n=2) x) x = xe iun x=2 g( k n=2) x) For g k : g( k n=2) x) k = 0; n 1, the sum is the discrete Fourier transform of the complex numbers g k and can be calculated by the FFT algorithm for u k = 2 k= n x) k = 0; n 1, simultaneously (see e.g. Brigham [20], Chapter 10) This results in an approximation for in (5.43) By the inverse FFT we obtain the density of M (T ) f W (T ) 54 Example 5.24. Exponential Brownian motion with jumps] Here the L evy process is the sum of a Brownian motion with drift ( W (t) t) t 0 , and a compound Poisson ....

Brigham, E.O. (1974) The Fast Fourier Transform. Prentice-Hall. Englewood Clis, N.J.

....methods present rotation invariance characteristic. Table 1 Experiments description The first experiment used FFT frequencies as features. Fourier transform was used due its time shifting property that can be stated as: A shift in time does not alter the magnitude of the Fourier coefficients [13]. Hence, any alignment algorithm needs to be used. Despite Fourier transform aligned the vectors, the classifiers showed be submitted to a more difficult task, mainly because most of the signal energy is presented in low frequencies in the Fourier spectrum. Using multilayer perceptron (MLP) neural ....

Brigham, E. Oran. The Fast Fourier Transform, PrenticeHall, Inc.,Englewood Cliffs, USA, 1974.

....we do have further information. We know that the periodicity function is the inverse Fourier transform of another function which is expected to be close to a sine function, the frequency of which we wish to measure. Therefore, the mathematically justified interpolation uses a sin x=x function [1]. As this sin x=x function seems more complex than a parabola, the equations could be expected to be difficult to solve, if at all tractable. sin x=x f m V sin##Te#f#F 0 ## #Te#f#F 0 # #F 0 ;V# F 0 =#m ##T e ## sin#### where # = v m v m 1 parabolic f m#1 m v m#1 V # b#f # ....

E. Oran Brigham. The Fast Fourier Transform, pages 102--105. Prentice Hall, 1974.

....taking all the time, Duhamel[8] noted that the test in the inner loops of many bit reversals takes a substantial amount of time. He used the close relation between bit reversal and matrix transpose to completely eliminate the test of whether the two elements had been previously exchanged. Brigham[5] took a conceptually simpler approach that, unfortunately, performs muchworse. For each index in the loop, he explicitly computes the bit reversed value in log 2 N steps. He then tests the values to see if the data elements should be exchanged. Not only does this approach use manyinteger ....

.... 78.7 78.7 78.7 78.8 r cooley[31] 30.6 30.7 30.6 30.8 30.9 30.9 30.8 30.9 yong[43] 28.1 28.1 28.1 28.2 28.2 28.2 28.1 28.2 buneman[6] 112.4 112.4 112.4 112.5 112.5 112.3 112.4 112.5 rodrig[28] 87.2 87.1 87.2 87.4 87.1 87.1 87.4 87.5 r rodr[31] 37.4 37.4 37.4 37.5 37.3 37.3 37.4 37.5 brigham[5] 405.2 448.2 491.2 534.2 400.6 440.7 480.7 520.8 duhamel[8] 37.0 37.1 37.1 37.1 37.1 37.1 37.1 37.2 middled[24] 73.7 73.7 73.7 73.7 72.3 72.2 72.2 72.4 cvl143[39] 32.5 32.6 32.4 32.4 8.2 8.3 8.2 8.2 gather 29.3 29.4 29.3 29.3 6.6 6.7 6.7 6.7 scatter[23] 26.3 26.4 26.4 26.3 6.0 6.0 6.0 6.1 ....

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E. O. Brigham, The Fast Fourier Transform,Prentice--Hall, Inc., Englewood Cliffs, N. J., 1974.

....it further. This can be achieved using, for example, Fourier Transformations. Alternatives include Wavelets [Chu92] but are not considered in this thesis. Theory In this subsection only the most important characteristics are summarized. A more detailed description can be found, for example, in [Bri74, PTVF92] Using the linear Fourier transform, a continuous signal can be transformed between its time domain representation, denoted by h(t) and the frequency domain representation H(f ) The respective equations are H(f) Z 1 1 h(t)e 2 ift dt; and h(t) Z 1 1 H(f)e 2 ift df: 3.1) ....

E. O. Brigham. The Fast Fourier Transform. Prentice Hall, Englewood Clis, NJ, 1974.

....model cache or memory contention. If sufficient memory bandwidth is not provided, no processor architecture will provide significant speedup. The benchmarks we used are C versions of the first five Livermore Loops [McMa72] discrete convolution, and the bit reverse access pattern used in the FFT [Brig74]. The benchmarks were compiled on a MIPS R5000 based SGI Indy running IRIX 5.3. We compiled using the cc compiler with the O3 sopt and non shared options. Loop unrolling was performed on all loops. We use, as points for comparison, a single issue uniprocessor with and without prefetching and the ....

E.O. Brigham. The Fast Fourier Transform, Prentice-Hall, Inc., 1974, p. 165.

....added to the AP instruction stream without increasing the execution time, or that the compiler could even determine that prefetches should be added. The right hand side of Figure 5 shows the performance of the bit reverse algorithm, shown in Figure 7,used in the Fast Fourier Transform algorithm [Brig74]. The bit reverse function, which is not shown, is a function that bit reverses a binary number and is executed on the AP and the CMP. The results shown are for a 1024 element array of 32 bit integers. If the array being read in the bit reverse algorithm does not fit in the cache, the miss rate ....

E.O. Brigham. The Fast Fourier Transform, Prentice-Hall, Inc., 1974, p. 165.

....The simulator does not model cache or memory contention and assumes that all instructions hit in an instruction cache. We used C versions of the first five Livermore loops [McMa72] two of our own benchmarks (discrete convolution [Papo80] and the bit reverse access pattern used in the FFT [Brig74]) and Tomcat from SPECfp95. The benchmarks represent a scientific and signal processing workload. A Instruction Mnemonic Description Prefetch PREF Prefetch data into the cache Put Slip Token PUT SLIP Produce a token for the Slip Control Queue Table 2: Prefetch Processor Instruction Extensions ....

E.O. Brigham. The Fast Fourier Transform, Prentice-Hall, Inc., 1974, p. 165.

....is relatively high, but its performance is not very good for high dimensional data. In order to improve the searching performance, techniques that can reduce dimensionality of the features can be used before indexing. For example, we may use KL Transform [17, 24] or Fast Fourier Transform [10] to calculate the most important features out of a high dimensional feature vectors and then produce a low dimensional one for indexing. 5.8 A Relationship Formula Based on all the experimental results, we want to find out the relationship between the efficiency and the tested parameters. In ....

E. O. Brigham. "The Fast Fourier Transform". Prentice Hall, 1974.

....model cache or memory contention. If sufficient memory bandwidth is not provided, no processor architecture will provide significant speedup. The benchmarks we used are C versions of the first five Livermore Loops [McMa72] discrete convolution, and the bit reverse access pattern used in the FFT [Brig74]. The benchmarks were compiled on a MIPS R5000 based SGI Indy running IRIX 5.3. We compiled using the cc compiler with the O3 sopt and non shared options. Loop unrolling was performed on all loops. We use, as points for comparison, a single issue uniprocessor with and without prefetching. The ....

E.O. Brigham. The Fast Fourier Transform, Prentice-Hall, Inc., 1974, p. 165.

.... and the rotation structure of the phase of the owchart of arbitrary FFTs have been recently developed by Demuth [20] Finally, the large number of books and articles that have been appearing are a required reference for analyzing and understanding the evolution of the aforementioned algorithms [11,15,40,59,64 65]. In general, system architecture has been greatly in uenced by the advances in the technological processes of microelectronics, constantly requiring new ideas for the organization of processing [17,76] The most important advantages currently o ered by VLSI (very large scale integration) and WSI ....

E.O. Brigham, \The fast Fourier transform". (Prentice Hall, Englewood Clis, NJ), 1974.

....as follows: F #u; v# # M,1 X x=0 N,1 X y=0 f#x; y#e ,i2## ux M vy N # : 3.9) In practice however, computing this function directly is impractical. The most common method is to use the Fast Fourier Transform (FFT) which drastically reduces the complexity of the DFT calculation (Brigham, 1974). 3.1.7.2 Problems with Fourier Analysis On first inspection, Fourier analysis sounds like a perfect solution to our problem: it takes a 2D source image and returns a frequency domain containing all of the relative spatial frequencies in that image. However, this is not actually what we ....

Brigham, E. O. (1974). The Fast Fourier Transform, Prentice-Hall, inc., Englewood Cliffs, NJ. ISBN 0-13-307496-X.

....Orr Sommerfeld equation eigenfunctions (disturbance velocity profiles) and amplification rates for given base flow velocity profiles, Reynolds number, and frequency, are obtained. 2. 3 Fourier Transform Fourier analysis yields the complex amplitudes of the frequency spectrum of a periodic signal [1]. Since all DNS cases presented in this paper are unsteady, an important attempt to understand and validate results is Fourier Analysis. The following discrete ansatz definesthe Fourier Transform (FT) that is used F(x,y, n LDt ) Dt L Gamma1 X k=0 f (x,y,t 1 kDt)e Gammai 2nk L , 9) ....

E. Oran Brigham, THE FAST FOURIER TRANSFORM, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1 edn., 1974.

....such as the separation of high frequency noise from the curve signal . Most simply, 2 According to the fundamental relationships between Fourier and spatial domain discretization, the FT leads to a periodic repetition of the curve signal in spatial domain. See literature, such as [17] [18] or [19] we can define an upper cut off frequency, jf g j 0:5 from which on all coefficients are set to zero. The resulting frequency window Gamma g (f) is given by Gamma g (f) ae 1 for jf j f g 0 for jf j f g (25) and the denoised control points by D(f) F (f) Psi (f) Gamma ....

O. Brigham. The Fast Fourier Transform. Prentice Hall, 1974.

....of the Sampling Rate Although the method generally does not require expensive filtering in frequency domain, yet we have to specify the correct sampling rate in order to discretize the FTs of our wavelets and scaling functions. The sampling rate is finally derived by Shannons sampling theorem [1]. The following considerations focus on B spline wavelets, but are not restricted to them. Any other type of wavelet, whose support can be computed with the standard second order moment formulae [2] performs as well. In order to derive the theoretically optimal sampling rate as a function of the ....

Brigham, E.O.: "The Fast Fourier Transform". Prentice-Hall Inc., 1974

....performed both numerically and symbolically. Section 2.2 studies vector convolution and shows its equivalence to DFT and also to generalized DFT. Section 2.3 recalls the sine, cosine and some other transforms. Abundant further material and bibliography on transforms and convolution can be found in [Brigham, 1974, Elliott and Rao, 1982, Blahut, 1984, Clausen, 1989, Duhamel and Vetterli, 1990, Press et al. 1992, Van Loan, 1992, Bini and Favati, 1993, Bini and Pan, 1994] 2.1 The Discrete Fourier Transform and Its Inverse The discrete Fourier transform (DFT) of the coeOEcient vector p = p 0 ; p n ....

Brigham, E.O. 1974. The Fast Fourier Transform, Prentice-Hall, Englewood Clioes, New Jersey, 1974.

....Such artifacts are intolerable in realistic still images or animation, so we will endeavor to eradicate them. Aliasing has been studied extensively in the field of digital signal processing. The reader is referred to the elementary [Jackson86] Hamming83] and advanced [Oppenheim Schafer75] Brigham74] Pratt78] Dudgeon Mersereau84] texts for derivations of the theory we summarize in this chapter. Most of the theoretical research on resampling to date has been driven by audio applications, and is consequently limited to 1 D uniform resampling (i.e. affine mappings) Crochiere Rabiner83] ....

....signal varies slowly, but is poor toward the right, where the original signal varies with high frequency. This behavior suggests that reconstruction accuracy is a function of frequency. Aliasing is best understood in terms of a frequency analysis of signals and filters using the Fourier transform [Brigham74] 3.2.1 Frequency Analysis of Aliasing The units of frequency can be either angular frequency in radians per unit distance, denoted , or rotational frequency in cycles per unit distance, denoted f . The two are related by = 2 f . We will use angular frequency . The response of a linear, ....

E. Oran Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.

....aforementioned factors. There exist various methods to process a speech signal. The most commonly applied method is the MFCC analysis [DM80] It is performed on an interval of sampled speech data a so called frame and returns a coefficient vector. The method includes a Fast Fourier Transform [Bri74] the computation of MEL frequency cepstrum coefficients [DM80] and a cepstral analysis [Rab93] The final step of the signal processing stage is often a vector quantization [LBG80] AKCM90] that returns a vector code for every frame. The word modeling stage identifies sequences of vectors or ....

E. O. Brigham. The Fast Fourier Transform. Prentice Hall, 1974.

.... (x)F (y) where F and F Gamma1 denote the discrete Fourier transform and its inverse: F k (x) N Gamma1 X j=0 x j e Gamma2 ijk=N F Gamma1 k (x) 1 N N Gamma1 X j=0 x j e 2 ijk=N Since the input data x and y and the output data z are all purely real, a technique described in [6] is used to reduce both the forward and reverse transforms to complex transforms of one lower order, which dramatically reduces the run time. Multiprecision division and square root extraction are performed using forms of Newton s iteration that require only multiplications, and thus they ....

Brigham, E. O., The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.

....speed of any current supercomputer, this improvement is significant. If the radix 4 algorithm is performed by incrementing k in the inner loop, then all strides are unity as above. It only remains to precalculate the array U of roots of unity in a manner entirely analogous to the radix 2 case: U[1] = 1 U[2] U[3] U[4] U[5] 1; p 2 i p 2; i; Gamma p 2 i p 2 : U[ 4 t Gamma1 Gamma 1) 3 1] U[ 4 t Gamma1 Gamma 1) 3 2] Delta Delta Delta ; U[ 4 t Gamma 1) 3] 1; fi t ; fi 2 t ; Delta Delta Delta ; fi 4 t Gamma1 Gamma1 t : U[ 4 m Gamma1 ....

E. Oran Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, New Jersey, 1974.

.... Gamma j sin( 2 N ) e Gammaj ( 2 N ) and j 2 = Gamma1, requires of the order of N 2 multiplications and N 2 additions to compute it. A Cooley and Tukey FFT consumes significantly fewer computations of the order of N 2 log 2 N multiplications and an equal number of additions [12]. We can express this as c FFT = N 2 log 2 N = N 2 fl; 11) where fl = log 2 N , i.e. 2 fl = N , as it is generally accepted that for digital machine computation N must be highly composite and a power of 2 [3] The basic algorithm for this part of the classification process is then j = 0 ....

E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., USA, 1974.

....significantly lower and more uniform when normalization is used. In Chapter 4, I describe a different type of normalization which results in even lower and more uniform variances. 3.6. 2 Using FFTs to compute convolution The fastest way to compute convolution is via Fast Fourier transforms (FFTs) [Brigham 1974]. The computation involves a transformation, an elementwise multiplication of two vectors, and an inverse transformation. Computing convolution via these three steps takes O(n log n) whereas the obvious implementation of the convolution equation c i = P j a j b i Gammaj takes O(n 2 ) time. ....

Brigham, E. O. 1974. The Fast Fourier Transform. Prentice Hall, Inc., New Jersey.

.... of the first sample from the true zero optical path difference position [19] A second advantage is that it helps average out the effect of any experimental asymmetries of the interferogram due to imperfect flatness or misalignment of components [20] In order to use the FFT algorithm [21] to process the data, the number of data points should be 2 n , with n an integer. For our two sided interferograms, the longest usable scan length is limited to 2 meters (n=32, with the necessary extra data padded with zeros) The highest frequency resolution is therefore [15] Delta = c ....

O. E. Brigham, The fast Fourier transform, Englewood Cliffs, N. J., Prentice-Hall, 1974

....exact reconstruction of F is possible, if we use the sinc function as a reconstruction filter. In this case the rect function will cut off the alias spectra of F . The corresponding Nyquist frequency and Nyquist region are clearly determined by the sampling rate of the discrete Hartley transform [3]. These results will carry over to the three dimensional case where the sinc and rect functions are just the tensor products of their one dimensional versions. Compact Support Reconstruction Filters: The sinc function, however, has infinite support and cannot be used as a reconstruction filter in ....

Brigham, E. Oran. The Fast Fourier Transform. Prentice-Hall Inc. (1974).

.... mechanical, acoustic and optical devices have been invented for performing Fourier transforms [1] including nature s own such as the human ear) Most of them are now of merely historical interest since the arrival of the computer based algorithm known as the fast Fourier transform (FFT) [2,3] which efficiently computes the discrete Fourier transform. The FFT algorithm can also be phrased in terms of quantum dynamics, i.e. in terms of unitary operations performed by a quantum computer on quantum registers. Indeed, all known quantum algorithms employ the quantum version of Fourier ....

E. O. Brigham, The Fast Fourier Transform (Prentice Hall, Englewood Cliffs, N.J., 1974).

....k Gammaj : 5) Q 00 k = N X j=1 D j Q 0 k Gammaj : 6) In the DSB mode, complex in and complex out, coefficients C and D are identical. In the SSB mode we need to apply a Hilbert transform to the signal in the Harris devices along with the low pass filter. The Hilbert Transform is defined (Brigham 1974) as f Hi (x) 1 Z 1 Gamma1 f(x 0 )dx 0 x 0 Gamma x ; 7) 12 which is equivalent to convolving f(x) with ( Gamma x) Gamma1 . The Hilbert Transform of a real function is equivalent to a filter which leaves the Fourier amplitudes unchanged but changes the phases by Sigma =2, ....

Brigham, E. O. 1974, The Fast Fourier Transform , [Prentice Hall : Englewood Cliffs, NJ].

....Fourier transform may of course be economically computed using some variation of the fast Fourier transform (FFT) algorithm. It is most convenient to employ the radix two fast Fourier transform since there is a wealth of literature on how to efficiently implement this algorithm (see [1] [8], and [16] Thus it will be assumed from this point that N = 2 m for some integer m. One useful trick can be employed to further reduce the computational requirement for complex transforms. Note that the input data vectors x and y and the result vector z are purely real. This fact can be ....

....from this point that N = 2 m for some integer m. One useful trick can be employed to further reduce the computational requirement for complex transforms. Note that the input data vectors x and y and the result vector z are purely real. This fact can be exploited by using a simple procedure ([8], p. 169) for performing real to complex and complex to real transforms that obtains the result with only about half the work otherwise required. One important item has been omitted from the above discussion. If the radix 10 7 is used, then the product of two cells will be in the neighborhood of ....

Brigham, E. O., The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.

....observability range space as outlined in the previous section. 3.2.2 Periodic Excitation If we assume the input signal is periodic with period time M , u(t) u(t kM ) t; k = 0; Sigma1; Sigma2; the deterministic part of the output of the system will be stationary. It is well known [1] that the fraction between the M point discrete Fourier transform of the output and the input exactly equals the transfer function G(z) at M equidistant samples around the unit circle. Hence, by use of periodic excitation we can recover the transfer function without leakage effects. The periodic ....

E. O. Brigham. The Fast Fourier Transform. Prentice Hall, New Jersey, 1974.

.... mechanical, acoustic and optical devices have been invented for performing Fourier transforms [1] including nature s own such as the human ear) Most of them are now of merely historical interest since the arrival of the computer based algorithm known as the fast Fourier transform (FFT) [2, 3] which e#ciently computes the discrete Fourier transform. The FFT algorithm can also be phrased in terms of quantum dynamics, i.e. in terms of unitary operations performed by a quantum computer on quantum registers. Indeed, all known quantum algorithms employ the quantum version of Fourier ....

E. O. Brigham, The Fast Fourier Transform (Prentice Hall, Englewood Cli#s, N.J., 1974).

....F (u; v) j M Gamma1 X x=0 N Gamma1 X y=0 f(x; y)e Gammai2 ( ux M vy N ) 3) In practice however, computing this function directly is impractical. The more common method is to use the Fast Fourier Transform (FFT) which drastically reduces the complexity of the DFT calculation (Brigham, 1974). Algorithms to compute the FFT can be found in (Press et al. 1974) 6.2 Problems with Fourier Analysis At first inspection, Fourier analysis sounds like a perfect solution to our problem: it takes a 2D source image and returns a frequency domain containing all of the relative spatial ....

Brigham, E. Oran. 1974. The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall, inc. ISBN 0-13-307496-X.

....associative operator. In these memories the dimensionality of the association space is exponential in the depth of recursion involved. Smolensky suggests placing a hard limit on the depth of recursion in order to keep the size of the association space tractable (e.g. no structure can be more than 4 levels deep) In a later paper Legendre, Miyata, Smolensky [ 16 ] describe a scheme which permits a soft limit on the depth of recursion, though its properties as the limit is approached or exceeded are not clear. In Pollack s [ 27 ] Recursive Auto Associative Memories (RAAMs) items, associations, ....

....This is particularly relevant to the storage capabilities of HRRs because when recursive frames are stored, convolution products e.g. obj cause agt eat , are the storage cues. VIII.B Using FFTs to compute convolution The fastest way to compute convolution is via Fast Fourier transforms (FFT) 4 ] The computation involves a transform, an element wise multiplication of two vectors, and an inverse transform. We can write a b = f 0 ( f( a) fi f( b) where f is a discrete Fourier transform, f 0 is the inverse discrete Fourier transform, and fi is the ....

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E. O. Brigham. The Fast Fourier Transform. Prentice Hall, Inc., New Jersey, 1974.

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Brigham EO. The Fast Fourier Transform. Englewood-Cliffs, NJ: Prentice-Hall, 1974.

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Brigham, E.O. (1974) The Fast Fourier Transform. Prentice-Hall. Englewood Clis, N.J.

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E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, 1974. 136

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E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Inc., Englewood Cliffs, 1974).

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E. O. Brigham, The Fast Fourier Transform. Prentice-Hall, Inc., 1974.

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E. O. Brigham. The Fast Fourier Transform. PrenticeHall, Inc., Englewood Cliffs, New Jersey, USA, 1974.

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E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., 1974.

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E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, 1974).

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Brigham, Oran. The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice Hall, 1974.

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E.O. Brigham. The fast Fourier transform. Prentice-Hall, Englewood Cliffs, N.J., 1974.

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E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., 1974.

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E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., 1974.

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E.O. Brigham. The Fast Fourier Transform. Prentice Hall, Englewood Cliffs, N.J. 1974

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E.O. Brigham. The Fast Fourier Transform. Prentice Hall, Englewood Cliffs, N.J. 1974

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