By Duraisamy Sundararajan

This e-book offers entire insurance of sensible Fourier research. It develops the strategies correct from the fundamentals and progressively courses the reader to the complicated issues. It provides the newest and virtually effective DFT algorithms, in addition to the computation of discrete cosine and Walsh-Hadamard transforms. the massive numbers of visible aids reminiscent of figures, stream graphs and move charts makes the mathematical subject effortless to appreciate. furthermore, the various examples and the set of C-language courses (a complement to the ebook) aid drastically in realizing the speculation and algorithms. Discrete Fourier research is roofed first, via the continual case, because the discrete case is less complicated to know and is essential in perform. This ebook can be worthwhile as a textual content for normal or expert classes on Fourier research, and in addition as a supplementary textual content for classes on discrete sign processing, snapshot processing, communications engineering and vibration research.

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**Example text**

1985) Signals, Systems, and Transforms, Prentice-Hall, New Jersey. 1 Determine the amplitude, angular and cyclic frequencies, the period, and phase of the sinusoid. 1. 2 x(n) = - 7 . 5 x(n) = - 1 . 2 Given two adjacent samples and the frequency of a sinusoid, find the polar form of the sinusoid. 1. 3 Express the sinusoid in rectangular form. 5 x(n) = 7 c o s ( f f n - f ) . 4 Express the sinusoid in polar form. 6 i ( n ) = - 3 s i n ( f n ) . 7 x(n) = - 2 c o s ( f n ) . 5 Determine the sinusoid that is the sum of the pair of sinusoids.

T ) ' - ( f - 1 ) ' • • • ' ? 9) The values of these forms of DFT and IDFT results in a better display with the value with index zero in the middle and these forms are also convenient to derive certain derivations. The spectrum of the waveform shown in Fig. 1(d) in the usual format is X(0) = 4, X(l) = V3 - jl, X{2) = 4, X(3) — y/3+jl. The same spectrum in center-zero format is X(—2) — 4. X(-l) = y/3 + jl, X(0) = 4, X{1) = V3 - jl. Getting one format of the spectrum or the signal from the other involves a circular shift by ^ positions (swapping of the positive and negative halves).

7 Find the polar form of three higher frequency sinusoids with the same set of sample values as that of x(n). 8 Find the fundamental cyclic frequency of the sum and the harmonic numbers of the two sinusoids. 9 Find the spectrum of the signal in terms of: (i) amplitude and phase and (ii) real and imaginary parts. 1 Write a program to generate the sample values of a real discrete sinusoid, Acos(jj-n + 6), with period N, phase 6, and amplitude A. 2 Write a program to generate the sample values of a complex discrete sinusoid, Ae^^n+8\ with period N, phase 6, and amplitude A.