By Douglas F. Elliott, K. Ramamohan Rao

This ebook has grown from notes utilized by the authors to coach quickly rework periods. One type used to be backed by way of the educational division of Rockwell overseas, and one other was once subsidized by way of the dep. of electric Engineering of The college of Texas at Arlington. many of the fabric used to be extensively utilized in a brief direction backed by way of 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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4 \ / 1 \ / 4 \ / \ W \\\ W cos(27r9t) \ / \ / \ \/ \Jr 4 \/ \ / 1 1 Waveforms t h a t have the same values when sampled at ^ s intervals. •|x(k)| W/z 25 Fig. 5 Periodic p r o p e r t y of D F T coefficients for x{n) = acos(2nn/8) + b sin(27i«/8) a n d N = 8. The repetition of coefficients at intervals of TV is the periodic property of the D F T . The reason for the periodic property is also apparent if we look at the exponential factor in the D F T definition. We note that e x p [ - / 2 7 r ( & + lN)n/N] since exp( — jlnlri) expressed = e x p ( - j2nkn/N)exip(- jlnlri) = exp( - j2nkn/N) = 1 for integral values of / and n.

25) as P approaches infinity. 28) is the Fourier transform of x(t). The function X(f) can be either real or complex valued and will be called the spectrum of the signal x(t). 27) defines a meaningful function would require a lengthy mathematical digression [T-3]. 28) and seeing if a well-defined answer results for X(f). Transforms required for F F T analysis are derived in the following sections, and the derivations of many other transforms are outlined in the problems. The signal x(i) can be recovered from its spectrum X(f) using the inverse Fourier transform.

Interval + Q/2 to — Q/2 and zero elsewhere. 87) F o r the system of Fig. 6 the input x(t) is the four delta functions with amplitude i a t / = + 4 and + 5 Hz. 87) is illustrated in Fig. 8 for k = 3 (f= 6). This figure shows that the convolution results in the sum of four values, two of which are zero. 28 sinc[(f-6)/2] Fig. 8 Functions for c o m p u t a t i o n of the convolution. 7 23 TABLE OF FOURIER TRANSFORM PROPERTIES Other Fourier series coefficients are computed for k = 0, ± 1, ± 2 , . .