By Hamish D. Meikle
Employing the inherent helix within the Fourier remodel expression, this booklet illustrates either Fourier transforms and their houses within the around. the writer attracts on trouble-free complicated algebra to govern the transforms, proposing the guidelines in this type of means as to prevent pages of advanced arithmetic. equally, abbreviations usually are not used all through and the language is stored intentionally transparent in order that the result's a textual content that's available to a wider readership. The remedy is prolonged with using sampled information to finite and discrete transforms, the quick Fourier rework, or FFT, being a different case of a discrete remodel. the appliance of Fourier transforms in statistics is illustrated for the 1st time utilizing the examples operational examine and later radar detection. additionally, an entire bankruptcy on tapering or weighting capabilities is extra for reference. the complete is rounded off by way of a word list and examples of diagrams in 3 dimensions made attainable through state-of-the-art arithmetic courses.
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Extra info for A New Twist to Fourier Transforms
6 The Fourier transform is the form used by Woodward [1, p. 27] and is one of three conventions namely, I f t exp Àj 2p f t dt F f Transform ÀI (1) I f t F f exp j 2p f t df Inverse transform ÀI where f(t) is a time waveform with t in seconds; F(f )pis the spectrum with the variable f Hz; j is À1. A New Twist to Fourier Transforms. Hamish D. Meikle Copyright 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40441-4 24 3 Fourier Transforms The exponential function is recognisable as a helical function from Chapter 2 and the train of ideas leading from the Fourier series to the Fourier transform is described in the next section and the properties of Fourier transforms in the sections that follow.
1 Fourier Series If f(t) is a repeating time function where the magnitude is integrable over its period. Then the Fourier series is given by I f t a0 am cos 2pmt bm sin 2pmt P P 2 m1 (2) where a0 is the steady component of the Fourier series coefficients; am are the cosine coefficients; bm are the sine coefficients; m is an integer number of cycles in P seconds. 9), then I a0 1 ak exp jh exp Àjh 1 bk exp jh À exp Àjh j 2 2 f t k1 I Á a0 1 À ak À jbk exp jh ak jbk exp Àjh 2 2 (10) k1 I kÀI 1 a À jsign kb exp jh jkj 2 k where the sign function is sign k 1; when k !
10]. 1). 15) [2, p. 10]. Multiply both sides of Equation (15) by P to give P=2 f t exp Àj2pkt=P dt PX k (16) ÀP=2 Note that the frequency of the sinusoids with argument 2pkt=P is k/P. As P becomes arbitrarily large, the spacing between the frequencies k/P and (k + 1)/P becomes arbitrarily small, and the frequency becomes a continuous variable. 4 Inverse Transform The signal f(t) may be recovered from its spectrum X(f) using the inverse Fourier transform. The numerator and denominator of the Fourier series with complex coefficients in Equation (13) are multiplied by P to give I f t (20) P X k exp j2pkt=P =P kÀI As P tends to infinity, let the difference of frequencies k/P and (k+1)/P be defined as df, that is df limitP3I k1 k 1 À limitP3I P P P (21) The summation in Equation (20) becomes an integration as the spectral line separation df becomes arbitrarily small.
A New Twist to Fourier Transforms by Hamish D. Meikle