Question: Sum of sinusoids time series model. Suppose that \(z_{1}, z_{2}, \ldots\) is a time series. A very common approximation of the time series is as a sum of \(K\) sinusoids

The \(k\) th term in this sum is called a sinusoid signal. The coefficient \(a_{k} \geq 0\) is called the amplitude, \(\omega_{k}>0\) is called the frequency, and \(\phi_{k}\) is called the phase of the \(k\) th sinusoid. (The phase is usually chosen to lie in the range from \(-\pi\) to \(\pi .\) ) In many applications the frequencies are multiples of \(\omega_{1}\), i.e., \(\omega_{k}=k \omega_{1}\) for \(k=2, \ldots, K\), in which case the approximation is called a Fourier approximation, named for the mathematician Jean-Baptiste Joseph Fourier. Suppose you have observed the values \(z_{1}, \ldots, z_{T}\), and wish to choose the sinusoid amplitudes \(a_{1}, \ldots, a_{K}\) and phases \(\phi_{1}, \ldots, \phi_{K}\) so as to minimize the RMS value of the approximation error \(\left(\hat{z}_{1}-z_{1}, \ldots, \hat{z}_{T}-z_{T}\right)\). (We assume that the frequencies are given.) Explain how to solve this using least squares model fitting.

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