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| 序 出版说明 preface. introduction groundwork the fourier transform and fourier's integral theorem conditions for the existence of fourier transforms transforms in the limit oddness and evenness significance of oddness and evenness complex conjugates cosine and sine transforms interpretation of the formulas 3 convolution examples of convolution serial products inversion of serial multiplication / the serial product in matrix notation / sequences as vectors convolution by computer the autocorrelation function and pentagram notation the triple correlation .the cross correlation the energy spectrum 4 notation for some useful functions rectangle function of unit height and base, ii(x) triangle function of unit height and area, a(x) various exponentials and gaussian and rayleigh curves heaviside's unit step function, h(x) the sign function, sgn x the filtering or interpolating function, sinc x pictorial representation summary of special symbols 5 the impulse symbol the sifting property the sampling or replicating symbol iii(x) the even and odd impulse pairs n(x) and ii(x) derivatives of the impulse symbol null functions some functions in two or more dimensions the concept of generalized function particularly well-behaved functions /regular sequences /generalized functions / algebra of generalized functions / differentiation of ordinary functions the basic theorems a few transforms for illustration similarity theorem addition theorem shift theorem modulation theorem convolution theorem rayleigh's theorem power theorem autocorrelation theorem derivative theorem derivative of a convolution integral the transform of a generalized function proofs of theorems similarity and shift theorems / derivative theorem / power theorem summary of theorems 7 obtaining transforms integration in closed form numerical fourier transformation the slow fourier transform program generation of transforms by theorems application of the derivative theorem to segmented functions measurement of spectra radiofrequency spectral analysis / optical fourier transform spectroscopy the two domains definite integral the first moment centroid moment of inertia (second moment) moments mean-square abscissa radius of gyration variance smoothness and compactness smoothness under convolution asymptotic behavior equivalent width autocorrelation width mean square widths sampling and replication commute some inequalities upper limits to ordinate and slope / schwarz's inequality the uncertainty relation proof of uncertainty relation / example of uncertainty relation the finite difference running means central limit theorem summary of correspondences in the two domains 9 waveforms, spectra, filters, and linearity electrical waveforms and spectra filters generality of linear filter theory digital filtering interpretation of theorems similarity theorem / addition theorem / shift theorem / modulation theorem / converse of modulation theorem linearity and time invariance periodicity 10 sampling and series sampling theorem interpolation rectangular filtering in frequency domain smoothing by running means undersampling ordinate and slope sampling interlaced sampling sampling in the presence of noise fourier series gibbs phenomenon / finite fourier transforms / fourier coefficients impulse trains that are periodic the shah symbol is its own fourier transform 11 the discrete fourier transform and the fft the discrete transform formula cyclic convolution examples of discrete fourier transforms reciprocal property oddness and evenness examples with special symmetry complex conjugates reversal property addition theorem shift theorem convolution theorem product theorem cross-correlation autocorrelation.. sum of sequence first value generalized parseval-rayleigh theorem packing theorem similarity theorem examples using matlab the fast fourier transform practical considerations is the discrete fourier transform correct? applications of the fft timing diagrams when n is not a power of 2 two-dimensional data power spectra 12 the discrete hartley transform a strictly reciprocal real transform notation and example the discrete hartley transform examples of dht discussion a convolution of algorithm in one and two dimensions two dimensions the cas-cas transform theorems the discrete sine and cosine transforms boundary value problems / data compression application computing getting a feel for numerical transforms the complex hartley transform physical aspect of the hartley transformation the fast hartley transform the fast algorithm running tune timing via the stripe diagram matrix formulation convolution permutation a fast hartley subroutine 13 relatives of the fourier transform the two-dimensional fourier transform two-dimensional convolution the hankel transform fourier kernels the three-dimensional fourier transform the hankel transform in n dimensions the mellin transform the z transform the abel transform the radon transform and tomography the abel-fourier-hankel ring of transforms / projection-slice theorem / reconstruction by modified back projection the hilbert transform the analytic signal /instantaneous frequency and envelope /causality computing the hilbert transform the fractional fourier transform shift theorem / derivative theorems / fractional convolution theorem / examples of transforms 14 the laplace transform convergence of the laplace integral theorems for the laplace transform transient-response problems laplace transform pairs natural behavior impulse response and transfer function initial-value problems setting out initial-value problems switching problems 15 antennas and optics one-dimensional apertures analogy with waveforms and spectra beam width and aperture width beam swinging arrays of arrays interferometers spectral sensitivity function modulation transfer function physical aspects of the angular spectrum two-dimensional theory optical diffraction fresnel diffraction other applications of fourier analysis 16 applications in statistics distribution of a sum consequences of the convolution relation the characteristic function the truncated exponential distribution the poisson distribution 17 random waveforms and noise discrete representation by random digits filtering a random input: effect on amplitude distribution digression on independence / the convolution relation effect on autocorrelation effect on spectrum spectrum of random input / the output spectrum some noise records envelope of bandpass noise detection of a noise waveform measurement of noise power 18 heat conduction and diffusion one-dimensional diffusion gaussian diffusion from a point diffusion of a spatial sinusoid sinusoidal time variation 19 dynamic power spectra the concept of dynamic spectrum the dynamic spectrograph computing the dynamic power spectrum frequency division / time division / presentation equivalence theorem envelope and phase using log f instead off the wavelet transform adaptive cell placement elementary chirp signals (chirplets) the wigner distribution 20 tables of sinc x, sinc2 x, and exp (-πx2) 21 solutions to selected problems chapter 2 groundwork chapter 3 convolution chapter 4 notation for some useful functions chapter 5 the impulse symbol chapter 6 the basic theorems chapter 7 obtaining transforms chapter 8 the two domains chapter 9 waveforms, spectra, filters, and linearity chapter 10 sampling and series chapter 11 the discrete fourier transform and the fft chapter 12 the hartley transform chapter 13 relatives of the fourier transform chapter 14 the laplace transform chapter 15 antennas and optics chapter 16 applications in statistics chapter 17 random waveforms and noise chapter 18 heat conduction and diffusion chapter 19 dynamic spectra and wavelets 22 pictorial dictionary of fourier transforms hartley transforms of some functions without symmetry 23 the life of joseph fourier index 教师反馈表... |
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