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Two ways to compute a spectrum

There are two computationally distinct methods by which we can compute a spectrum: (1) compute all the $s_k$ coefficients from (20) and then form the cosine sum (24) for each $\omega$; and alternately, (2) evaluate $B(Z)$ for some value of Z on the unit circle, and multiply the resulting number by its complex conjugate. Repeat for many values of $Z$ on the unit circle. When there are more than about twenty lags, method (2) is cheaper, because the fast Fourier transform (coming up soon) can be used.