Algorithms And Data Structures

Discrete and continuous Fourier transforms: analysis, - download pdf or read online

By Eleanor Chu

ISBN-10: 1420063634

ISBN-13: 9781420063639

ISBN-10: 1420063642

ISBN-13: 9781420063646

Lengthy hired in electric engineering, the discrete Fourier remodel (DFT) is now utilized in a number of fields by using electronic pcs and quick Fourier remodel (FFT) algorithms. yet to properly interpret DFT effects, it truly is necessary to comprehend the center and instruments of Fourier research. Discrete and non-stop Fourier Transforms: research, functions and quick Algorithms provides the basics of Fourier research and their deployment in sign processing utilizing DFT and FFT algorithms.

This available, self-contained e-book offers significant interpretations of crucial formulation within the context of functions, construction an excellent beginning for the applying of Fourier research within the many diverging and consistently evolving components in electronic sign processing organisations. It comprehensively covers the DFT of windowed sequences, a number of discrete convolution algorithms and their purposes in electronic filtering and filters, and lots of FFT algorithms unified less than the frameworks of mixed-radix FFTs and top issue FFTs. various graphical illustrations and labored examples support clarify the thoughts and relationships from the very starting of the textual content.

Requiring no earlier wisdom of Fourier research or sign processing, this ebook provides the root for utilizing FFT algorithms to compute the DFT in a number of program parts.

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Extra resources for Discrete and continuous Fourier transforms: analysis, applications and fast algorithms

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5 Frequency−Domain Plot 10 5 3 A1 = 5 0 −5 −10 A = −7 2 0 1 2 3 4 Frequency f = 1, 2, 3 cycles per unit time Now, with the time-domain plot and the frequency-domain plot of x(t) both available, we see that when x(t) is composite, the frequency content of x(t) can no longer be deciphered from the time-domain plot of x(t) versus t one cannot visually decompose the graph of x(t) into its component graphs. The reverse is also true: the time-domain plot shows the behavior of x(t), which cannot be inferred from the frequency plot alone.

3. 3 Analog frequency grids and corresponding digital frequency grids. 26 = N∆t, N = 11) −5/T −4/T −3/T −2/T −1/T 0 1/T 2/T 3/T 4/T 5/T Digital Frequency Grid: ∆F = ∆f∆t = 1/N (N = 2n+1 = 11) 0 −5/N 5/N = 5/11 Note that after t is absorbed into Fk , we can only refer to the th sample x in the discrete-time domain, so the spacing is = 1, and the reciprocity relation F = 1 N is satis ed . To drive home this last point, we only need to evaluate A0 + x(t) = 2 n Ak cos 2πfk t + Bk sin 2πfk t k=1 CHAPTER 2.

2. The sequence {x } is not periodic if we cannot express its digital frequency F as a rational fraction. We demonstrate the different cases by several examples below. 0125 = 1/80, so the given sequence is periodic with period N = 80 (samples). In this case, we have K = 1, so the N samples are equally spaced over a single period of its envelope function. 10. 35 = 7/20, so the given sequence is periodic with period N = 20 (samples). In this case, we have K = 7, so the N equispaced samples span seven periods of its envelope function.

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Discrete and continuous Fourier transforms: analysis, applications and fast algorithms by Eleanor Chu

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