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 (37) and then form the cosine sum (41) 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 discussed in chapter can be used.