Download Fast transforms: algorithms, analyses, applications by Douglas F. Elliott, K. Ramamohan Rao PDF

By Douglas F. Elliott, K. Ramamohan Rao

This publication has grown from notes utilized by the authors to teach quickly rework sessions. One classification used to be subsidized through the educational division of Rockwell overseas, and one other used to be subsidized by means of the dep. of electric Engineering of The college of Texas at Arlington. a number of the fabric was once extensively utilized in a quick path subsidized through the college of Southern California. The authors are indebted to their scholars for motivating the writing of this publication and for feedback to enhance it.

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7 by dots. 12) The first value in S is § of a rotation around the unit circle, the second value is | of a clockwise rotation around the unit circle starting from the positive real axis, etc. The sampled values of exp( —j2nkn/$) are labeled on the unit circle (Fig. 7) as step numbers 0 , 1 , 2 , . . , 7. The sequence S takes a distinct complex value for each step number. For k = 2 the sampled sequence is x x S 2 = 1^-72712(0/8)^ -j2n2(l/8)^ e e ~ j2n2(2/8)^ ^e ~j27t2(7/8)| (3 |3) Step numbers are 0,2, 4, 6, 0,2, 4, 6.

The inverse discrete Fourier transform (IDFT) is also a summation. 14) with only the first TV coefficients. , X(N — 1)} allow us to solve for only TV unknown transform sequence values. 23) is unity for n = 0, 1,2,...

10-2). 17)? 12-2) F o r example, 5 = 9 ( m o d u l o 4). 10-1) hold a n d show that f P =fiP ( m o d u l o 1) where f a n d f are the frequencies of a n y t w o cosine waveforms in t h e s u m m a t i o n a n d P is t h e period of x{t). 12-2) is equivalent t o requiring t h a t \a — b\ = kc, where k is an integer. 14 Show t h a t t h e &th Fourier series complex coefficient can be written X(k) = \X(k)\e j6k Im • — J X(k) ^^ 7 1 X(k) /2TTfT e" j 2 l T f T S 1 \ /2 Fig. 12 by t. T h e phase shift of a complex F o u r i e r series coefficient d u e t o delay of the time function 28 2 FOURIER SERIES AND THE FOURIER TRANSFORM where 6 = tan [ — sign(k)b /a ].

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