computing discrete fourier transforms
The Discrete Fourier Transform
Ngày tải lên :
13/09/2012, 09:49
... aperiodic Fourier Series signals that are continious and periodic Discrete Time Fourier Transform signals that are discrete and aperiodic Discrete Fourier Transform signals that are discrete and ... of Fourier Transform is sometimes called the Discrete Fourier Series, but is most often called the Discrete Fourier Transform You might be thinking that the names given to these four types of Fourier ... repeat themselves in a periodic fashion This type of Fourier transform is called the Discrete Time Fourier Transform Periodic -Discrete These are discrete signals that repeat themselves in a periodic...
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Tài liệu Mathematics - Fourier Transforms and Waves pptx
Ngày tải lên :
17/12/2013, 06:15
... Z -TRANSFORMS 1.2 FOURIER SUMS 1.3 FOURIER AND Z-TRANSFORM 1.4 CORRELATION AND SPECTRA 11 Discrete Fourier ... precision, 23 filter, filter nonrealizable, fold, 20 Fourier downward continuation, 33 Fourier integral, 10 Fourier sum, 7, 10, 17 Fourier transform, Fourier transform quefrency, 14 radian, 7, 10 random, ... some two-dimensional Fourier transforms 2.1 FT AS AN INVERTIBLE MATRIX A Fourier sum may be written (2.1) where the complex value is related to the real frequency by This Fourier sum is a way...
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Tài liệu Mathematics - Fourier Transforms and Waves pdf
Ngày tải lên :
22/12/2013, 21:17
... Z -TRANSFORMS 1.2 FOURIER SUMS 1.3 FOURIER AND Z-TRANSFORM 1.4 CORRELATION AND SPECTRA 11 Discrete Fourier ... precision, 23 filter, filter nonrealizable, fold, 20 Fourier downward continuation, 33 Fourier integral, 10 Fourier sum, 7, 10, 17 Fourier transform, Fourier transform quefrency, 14 radian, 7, 10 random, ... some two-dimensional Fourier transforms 2.1 FT AS AN INVERTIBLE MATRIX A Fourier sum may be written (2.1) where the complex value is related to the real frequency by This Fourier sum is a way...
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Tài liệu 01 Fourier Series, Fourier Transforms, and the DFT doc
Ngày tải lên :
27/01/2014, 03:20
... Continuous and Discrete Time Spectra The Discrete Fourier Transform Properties of the Discrete Fourier Series • Fourier Block Processing in Real-Time Filtering Applications • Fast Fourier Transform ... developed set of Fourier methods, including the discrete time Fourier transform (DTFT) and the discrete Fourier transform (DFT), are extensions of basic Fourier concepts that apply to discrete time ... 1 Fourier Series, Fourier Transforms, and the DFT 1.1 1.2 Introduction Fourier Series Representation of Continuous Time Periodic Signals Exponential Fourier Series • The Trigonometric Fourier...
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Fourier transforms in spectroscopy kauppinen j
Ngày tải lên :
17/03/2014, 14:45
... Next, we compute N /2 Fourier transforms of four data using the N Fourier transforms of two data, then the N /4 Fourier transforms of eight data, and so on until we have one Fourier transform of ... number of computed data H is the discrete inverse Fourier transform of h 45 3.3 Discrete spectrum Example 3.3: Show that the discrete inverse Fourier transform of the discrete convolution M−1 hk ... the discrete inverse Fourier transform of the original signal vector h obtained from the transforms of the partial vectors? Solution Let us denote w M = ei 2π/M The discrete inverse Fourier transforms...
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DISCRETE WAVELET TRANSFORMS - A COMPENDIUM OF NEW APPROACHES AND RECENT APPLICATIONS pdf
Ngày tải lên :
23/03/2014, 03:20
... Discrete Wavelet Transform for Image Signals Masahiro Iwahashi and Hitoshi Kiya Chapter A Pyramid-Based Watermarking Technique for Digital Images Copyright Protection Using Discrete Wavelet Transforms ... in [13] The derivation assumes that circular convolution based on fast Fourier transform (FFT) and inverse fast Fourier transforms (IFFT) is used for transforming an image pyramidally This derivation ... Discrete Wavelet Transforms - A Compendium of New Approaches and Recent Applications http://dx.doi.org/10.5772/3424...
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mathematics of the discrete fourier transform
Ngày tải lên :
31/03/2014, 15:21
... from powers of z, and plot examples using Mathematica • The Discrete Fourier Transform (DFT) Derived This chapter derives the Discrete Fourier Transform (DFT) as a projection of a length N signal ... DFT This chapter introduces the Discrete Fourier Transform (DFT) and points out the elements which will be discussed in this reader 1.1 DFT Definition The Discrete Fourier Transform (DFT) of a signal ... representation of continuous-time signal reconstruction from discretetime samples is given DRAFT of “Mathematics of the Discrete Fourier Transform (DFT),” by J.O Smith, CCRMA, Stanford, Winter...
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The discrete fourier transform
Ngày tải lên :
08/06/2014, 20:19
... Now we focus on DT signals for a while The discrete Fourier transform or DFT is the transform that deals with a finite discrete- time signal and a finite or discrete number of frequencies Which frequencies? ... Ex Ey E x Ey Summary We have now covered all the most important transforms: continuous time: Laplace, Fourier, Fourier Series, discrete time: Z, DTFT, DTFS, DFT/FFT The first six are for pencil ... What tool we use to represent periodic discrete- time signals? Since xps [n] is N -periodic, we know from Ch that we can expand it using a discrete- time Fourier series (DTFS): N −1 2π ck e N...
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Fourier Transforms in Radar And Signal Processing_2 pdf
Ngày tải lên :
18/06/2014, 10:05
... relating to Fourier transforms and a set of Fourier transform pairs We then include three illustrations as examples before the main applications in the following chapters 2.2 Notation 2.2.1 Fourier ... (2.13), Y = 1/X , constant Prime indicates differentiation ͬ 22 Fourier Transforms in Radar and Signal Processing Table 2.2 Fourier Transforms Pairs Pair Function Transform Notes 1a (2.6) 1b ␦ (x ... of Fourier analysis are given in Tables 2.1 and 2.2 below The rules are relationships that apply generally to all functions (u and v in Table 2.1) and their transforms (U Table 2.1 Rules for Fourier...
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Fourier Transforms in Radar And Signal Processing_3 pot
Ngày tải lên :
18/06/2014, 10:05
... 32 Fourier Transforms in Radar and Signal Processing This is in the form of a Fourier series, with period 1/X = Y, and the coefficients ... ∞ u ′(x ) = ͵ 2 iyU ( y ) exp (2 ixy ) dy −∞ so u ′(x ) is the inverse Fourier transform of 2 iyU ( y ) 34 Fourier Transforms in Radar and Signal Processing R9b: ∞ U( y) = ͵ u (x ) exp (−2 ... 2 iy exp (− x − 2 ixy ) + 2 iy ͪ |ͬ ∞ iy The Fourier transform of h (x ) is now found to be, using P1b, ͫ 1 ␦( y) + iy ͬ 36 Fourier Transforms in Radar and Signal Processing P2b: From...
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Fourier Transforms in Radar And Signal Processing_4 ppt
Ngày tải lên :
18/06/2014, 10:05
... 54 Fourier Transforms in Radar and Signal Processing Figure 3.14 Rising edge as the difference of two Ramp ... R6a, ͫ ␦( f ) + 2 if ͬ ͭͫ ␦( f ) + 2 if ͬ ͮ [exp (−2 if T1 ) − exp (−2 if T2 )] (3.24) 56 Fourier Transforms in Radar and Signal Processing Using T and ⌬T as given in Figure 3.14, the difference ... ͫR ( f ) e if ⌬T s − R ( f ) e − if ⌬Ts ͬ (2 f )2⌬Ts e −2 if Tr e −2 if Ts (3.30) 58 Fourier Transforms in Radar and Signal Processing As a check, we note that if we used a single rounding...
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Fourier Transforms in Radar And Signal Processing_5 doc
Ngày tải lên :
18/06/2014, 10:05
... form of itself (Figure 5.2) The (inverse) Fourier transform of this (from P3a, R5, R7a, and R8b) is Figure 5.2 Equivalent forms of U ( f ) 92 Fourier Transforms in Radar and Signal Processing ... is the (inverse) Fourier transform of G Expanding the comb function, we have u (t ) = (t ) ⊗ ∑ ␦ (t − rT ′ ) u (rT ′ ) ∑ (t − rT ′ ) u (rT ′ ) = ∑ w r u (rT ′ ) = 96 Fourier Transforms in Radar ... sampling rates are given by r m = 2f u /W (n + 1) and r M = 2f l /W (n − 1), where f u = f + 80 Fourier Transforms in Radar and Signal Processing W /2 and f l = f − W /2; a central rate (very close...
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Fourier Transforms in Radar And Signal Processing_6 ppt
Ngày tải lên :
18/06/2014, 10:05
... 98 Fourier Transforms in Radar and Signal Processing Figure 5.8 FIR interpolation weights with oversampling ... So the magnitudes of the tap weights are | ͩ ͪ| | ͩ ͪ| wr = r− = √2 ͩ ͪ 2 r− 2 (5.18) 100 Fourier Transforms in Radar and Signal Processing Comparing this with (5.5), we see that the weight ... Time Series Figure 5.10 Filter weights with oversampling and trapezoidal spectral gate 101 102 Fourier Transforms in Radar and Signal Processing Figure 5.11 Trapezoidal rounding w r ( ) = [(r...
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Fourier Transforms in Radar And Signal Processing_7 docx
Ngày tải lên :
18/06/2014, 10:05
... case.) We note that (6.5) and (6.6) are in the form of (inverse) Fourier transforms If (t ) and G ( f )* | U ( f ) | are a Fourier pair and so are (t ) and | G ( f ) | | U ( f ) | 2, then ... actually inverse Fourier transforms that are required [see (6.6) and (6.7)] In many cases (using symmetric functions, in particular), there is no distinction between forward and inverse transforms, ... (forward) transforms to i sncr , so, from Rule 4, we have i snc (x ) ⇔ ramp (−y ) = −ramp ( y ) (as ramp is odd)—that is, ramp ( f ) inverse transforms to −i snc (t ) (However, as ramp2 is even, it transforms...
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Fourier Transforms in Radar And Signal Processing_8 pdf
Ngày tải lên :
18/06/2014, 10:05
... equation, the components of which are values of the Fourier transforms of the distorted responses Thus, for the amplitude distortions, we require the transforms of the ramp functions, and these are ... quite so directly achieved This approach still requires Fourier transforms and is presented in Section 7.4 Although we cannot use the Fourier transform of the aperture in the general array case, ... determining the components of the matrix and vector used to obtain the weights The Fourier transform is 161 162 Fourier Transforms in Radar and Signal Processing also useful for general results regarding...
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Fourier Transforms in Radar And Signal Processing_9 doc
Ngày tải lên :
18/06/2014, 10:05
... 164 Fourier Transforms in Radar and Signal Processing a sampled aperture; we can still use the Fourier transform, but the aperture distribution is ... elements: (a) beam patterns; (b) element weights 180 Fourier Transforms in Radar and Signal Processing and the weight function, given by the Fourier transform of (7.28), is a (x ) = (u /U ) comb1/U ... -space) will be the same as the pulse spectra, as the same Fourier relationship holds (Strictly speaking, for the pulse spectra the forward Fourier transform is required, while for the beam patterns...
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Fourier Transforms in Radar And Signal Processing_10 pot
Ngày tải lên :
18/06/2014, 10:05
... summary, 122–23 worst case for, 93 Inverse Fourier transform, 12–13, 33, 135 of aperture distribution, 163 performing, 74 See also Fourier transforms 196 Fourier Transforms in Radar and Signal Processing ... length, 121 tap weights, 121 weights for interpolation, 94 Fourier series, 32 coefficients, finding, concept, representation, 32 Fourier transforms complex, of constant functions, defined, generalized ... Aperture distribution, 162, 169 function, 164–65 inverse Fourier transform, 163 linear array, 182 rect function, 163 193 194 Fourier Transforms in Radar and Signal Processing Autocorrelation...
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Fourier Transforms in Radar And Signal Processing_1 ppt
Ngày tải lên :
18/06/2014, 15:20
... 32 Fourier Transforms in Radar and Signal Processing This is in the form of a Fourier series, with period 1/X = Y, and the coefficients ... ∞ u ′(x ) = ͵ 2 iyU ( y ) exp (2 ixy ) dy −∞ so u ′(x ) is the inverse Fourier transform of 2 iyU ( y ) 34 Fourier Transforms in Radar and Signal Processing R9b: ∞ U( y) = ͵ u (x ) exp (−2 ... 2 iy exp (− x − 2 ixy ) + 2 iy ͪ |ͬ ∞ iy The Fourier transform of h (x ) is now found to be, using P1b, ͫ 1 ␦( y) + iy ͬ 36 Fourier Transforms in Radar and Signal Processing P2b: From...
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Fourier Transforms in Radar And Signal Processing_2 docx
Ngày tải lên :
18/06/2014, 15:20
... 54 Fourier Transforms in Radar and Signal Processing Figure 3.14 Rising edge as the difference of two Ramp ... R6a, ͫ ␦( f ) + 2 if ͬ ͭͫ ␦( f ) + 2 if ͬ ͮ [exp (−2 if T1 ) − exp (−2 if T2 )] (3.24) 56 Fourier Transforms in Radar and Signal Processing Using T and ⌬T as given in Figure 3.14, the difference ... ͫR ( f ) e if ⌬T s − R ( f ) e − if ⌬Ts ͬ (2 f )2⌬Ts e −2 if Tr e −2 if Ts (3.30) 58 Fourier Transforms in Radar and Signal Processing As a check, we note that if we used a single rounding...
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Fourier Transforms in Radar And Signal Processing_3 doc
Ngày tải lên :
18/06/2014, 15:20
... form of itself (Figure 5.2) The (inverse) Fourier transform of this (from P3a, R5, R7a, and R8b) is Figure 5.2 Equivalent forms of U ( f ) 92 Fourier Transforms in Radar and Signal Processing ... is the (inverse) Fourier transform of G Expanding the comb function, we have u (t ) = (t ) ⊗ ∑ ␦ (t − rT ′ ) u (rT ′ ) ∑ (t − rT ′ ) u (rT ′ ) = ∑ w r u (rT ′ ) = 96 Fourier Transforms in Radar ... sampling rates are given by r m = 2f u /W (n + 1) and r M = 2f l /W (n − 1), where f u = f + 80 Fourier Transforms in Radar and Signal Processing W /2 and f l = f − W /2; a central rate (very close...
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