By E. Oran Brigham
Here's a new e-book that identifies and translates the fundamental fundamentals of the quick Fourier remodel (FFT). It hyperlinks in a unified presentation the Fourier remodel, discrete Fourier rework, FFT, and primary purposes of the FFT. The FFT is turning into a chief analytical device in such assorted fields as linear platforms, optics, likelihood conception, quantum physics, antennas, and sign research, yet there has consistently been an issue of speaking its basics. hence the purpose of this e-book is to supply a readable and sensible remedy of the FFT and its major functions. In his Preface the writer explains the association of his subject matters, "... each significant inspiration is constructed by means of a three-stage sequential approach. First, the concept that is brought via an intuitive improvement that is often pictorial and nature. moment, a non-sophisticated (but completely sound) mathematical remedy is built to help the intuitive arguments. The 3rd degree contains useful examples designed to check and extend the concept that being mentioned. it really is felt that this three-step process offers which means in addition to mathematical substance to the fundamental homes of the FFT. --- from book's dustjacket
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Extra resources for The Fast Fourier Transform: An Introduction to Its Theory and Application
Example text
New York: McGraw-Hili, 1965. 2. , The Fourier Integral and lIs Applicalions. New York: McGrawHill, 1962. 11-lol>2 T o Show that ! x(~) Prove the following Fourier transform pairs: dh(t) u==v. j27t/H(f) a. ---cJI ~ b. [-j27t1]s(t) 3-7. ' By means of the shifting theorem find the Fourier transform of the following functions: h( ) = A sin [27t/o(1 - ' 0)] a. I 7t(1 - ' 0) b. h(l) = K~(I - ' 0 ) c. h(l) By making a substitution of variable in Eq. (2-28) show that f~~X(I)~(al- In) dl = 3-16. 49 Sec.
1 1 T K HI') HI') HI') HIli T 1 , , -~ 2 61'·'0) '0 '0 ~ 61'·'0) ~ T "' Yo ... I V ·3T ·2T V fI f ·4T r ·5T 000 V ·T A I hit) V A {'I V 4T " 3T V 2T hhl T hU) (\ 5T 000 I I ~ 6(t - nT) h(t) = k6(t) Q) Q) Q) Q + 2A 6(f + fa) {c(/ - ;~) +i{t5(f + fa) H(f) = -i{6(f - fa) H(/) = . ___ ~ 6 f-T H(f) = K Figure 2 ·1I. Fourier transfonn pairs. 1 1 T K HI') HI') HI') HIli T 1 , , -~ 2 61'·'0) '0 '0 ~ 61'·'0) ~ T . -2~)] rio) Ifl::;;f. Ifl>f. + {-[ Q(! + = -tQ(f) Q(f) = sin H(f) =0 . ,- ... -;,;;: ~------~------ 2To Time domain c-~~-cc-c~-;-:::---r=-::"'-'-':;-,---==" Ie -10 A2To H(f) 1 H(I) H(I) H(II 10 Ie 1 1 2To To H(I) Frequency domain J 2\ .
F\ f\ " I" f\, 1 I I = f~~ A cos (27t/ot)e-J2·t. dt = I V V 4- f~~ [e ll•t ,. t'dt v V v Figure 1-9. Fourier transform of A sin (at). = ~ f~~ [e-Il ••(t-t,) + e-Il ••(t·+f)jdt EXAMPLE 2-7 (2-36) Without prooft, the Fourier Iransform of a sequence of equal distant impulse functions is another sequence of equal distant impulses; where arguments identical to those leading to Eq. (2-32) have been employed. 44. = A ~ T{}(f - 10) A + T{}(f + 10) 11 Chap. 2 THE FOURIER TRANSFORM Sec. f) [Eq. % dx (2-41) Since [Eq.