Discrete and continuous Fourier transforms: analysis, by Eleanor Chu

By Eleanor Chu

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

This available, self-contained publication presents significant interpretations of crucial formulation within the context of purposes, development a great starting place for the appliance of Fourier research within the many diverging and continually evolving parts in electronic sign processing companies. It comprehensively covers the DFT of windowed sequences, a number of discrete convolution algorithms and their purposes in electronic filtering and filters, and plenty of FFT algorithms unified below the frameworks of mixed-radix FFTs and leading issue FFTs. a great number of graphical illustrations and labored examples support clarify the suggestions and relationships from the very starting of the textual content.

Requiring no earlier wisdom of Fourier research or sign processing, this publication provides the root for utilizing FFT algorithms to compute the DFT in quite a few program components.

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18) |Xk |ejφk ej2πfk t . y(t) = k=−n CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS 14 A reminder: The de nitions fk = T1k and ωk = 2πfk may be used to express y(t) in terms of Tk (individual period) or ωk (individual angular frequency) in all forms. Also, when fk = k/T , this fact is commonly recognized wherever fk is used. To convert one form to another, one may use the relationship between the coef cients as summarized below. Relation 1 De n e X0 ≡ 0 when A0 and B0 are missing. For 1 ≤ k ≤ n, Ak ∓ jBk , and f−k = −fk .

6 9 = 3 3 . It can be easily veri ed that y(t + To ) = y(t). , y(t + T ) = y(t). Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk ; of course, one may plot Ak and Bk versus the values of fk if that is desired. 7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function √ y(t) = sin(2πt) + 5 sin(2 3πt) √ is not periodic because f1 = 1 and f2 = 3 are not commensurate.

We do not know how many cycles x ˜(t) has completed over the interval T . Mathematically, the function x ˜(t) interpolating the two samples is no longer unique if the ˜ frequency f is not speci ed. , the frequency we can resolve for x ˜(t) is f˜ = r˜/T = 1/T . When we deal with discrete samples taken from a composite signal, the so-called aliased frequencies are equivalent in the sense that they contribute the same numerical values at the sample points. 5, the signal y(θ) = cos(θ) + 2 cos(3θ) + 3 cos(5θ) cannot be distinguished from x(θ) = 6 cos(θ) based on the two values sampled at θ1 = 0 and θ2 = π, because y(0) = x(0) = 6 and y(π) = x(π) = −6.

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