By N. Benvenuto, et al.,
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Extra resources for Algorithms for Comm. Systs and Their Applns
Since zeros have been added to the regions -T B t \ -T/2 and T/2 B t \ T, the region of integration of Eq. 32) can be reduced to -T/2 B t \ T/2. Um ¼ U m 2T ¼ Z T=2 xðtÞ expðÀj2p ÀT=2 m tÞdt 2T ð2:33Þ Comparing Eqs. 32), the reader will find that they are equal if m = 2 k. Since m and k are both integers, the value Um is equal to Xk when m is even. 42 2 Fourier Series Expansion Fig. 11 Change in the power spectrum due to the change in the period for the computation of Fourier coefficients of waves BT, B2T and B4T.
The coefficient Xk is referred to as the ‘‘amplitude’’ of the complex wave componentexpfÀj2pðk=TÞg, in the same way that Ak and Bk are the amplitudes of cosine and sine waves, respectively. However, since Xk is complex, it is referred to as the complex amplitude’’. The situation may seem to be more complicated but it is not. There is the same number of coefficients Ak’s for positive and negative k’s and the real and imaginary parts are even and odd, respectively. Since expfÀj2pðk=TÞg 40 2 Fourier Series Expansion has a real (even) part and an imaginary (odd) part, the expression of the Fourier series becomes simpler than using sine and cosine functions.
1 showed that there is only one nonzero term, which is the product of cosine functions with the same frequency. The product of cosf2pðk=TÞtg and sinf2pðk=TÞtg is rewritten as ' m 1& k þ m k kÀm cos 2p t cos 2p t ¼ cos 2p t þ cos 2p t : T T 2 T T The integration of this product from t = 0 to T is zero if k 6¼ m. If k = m, the second term of the right-hand side is T/2 since cosf2pð0=TÞtg ¼ 1. Similarly, the product of cosf2pðk=TÞtg and sinf2pðm=TÞtg is rewritten as & ' k m 1 kþm kÀm cos 2p t sin 2p t ¼ sin 2p t À sin 2p t : T T 2 T T The integration of this product from t = 0 to T becomes zero for both cases: k 6¼ m and k = m.
Algorithms for Comm. Systs and Their Applns by N. Benvenuto, et al.,