Estimation of Sinusoidal Parameters

Sinusoidal parameters are estimated by maximizing the PDF, either (9.13) or (9.12), under the assumption that the AR parameters are known and fixed. To maximize these PDF, we need the derivatives with respect to the paremeters.

Taking the first derivatives of (9.12) w/r to the amplitude parameters $\alpha_1 \cdots \alpha_k$ and $\beta_1 \cdots \beta_k$, we have

$$\left[ \frac{\partial \log p({\bf x})}{\partial \alpha_1}, \frac... ...c{1}{\sigma^2} ({\bf x}-{\bf y})^\prime {\bf A}^\prime {\bf A} {\bf W} {\bf H}.$$

Taking the first derivative of (9.12) w/r to frequency $f_i$, we get

$$\left[ \frac{\partial \log p({\bf x})}{\partial f_1} \ldots \frac... ...c{1}{\sigma^2} ({\bf x}-{\bf y})^\prime {\bf A}^\prime {\bf A} {\bf W} {\bf F},$$ (9.14)

where ${\bf F}$ is the $N\times k$ matrix

$$F_{t,j} = (t-N/2-1) \left\{ \beta_j \cos(2\pi f_i (t-N/2-1)) - \alpha_j \sin(2\pi f_i (t-N/2-1) ) \right\}.$$

We can also obtain the Fisher's Information matrix by taking the negative expected value of the second derivatives. From the above, we can obtain

$${\bf I}_{\bf bb} = \frac{1}{\sigma^2} {\bf H}^\prime {\bf W} {\bf A}^\prime {\bf A}{\bf W} {\bf H},$$

$${\bf I}_{\bf ff} = \frac{1}{\sigma^2} {\bf F}^\prime {\bf W} {\bf A}^\prime {\bf A}{\bf W} {\bf F},$$

$${\bf I}_{\bf f\tilde{b}} = \frac{1}{\sigma^2} {\bf H}^\prime {\bf W} {\bf A}^\prime {\bf A}{\bf W} {\bf F}.$$

All these quantities can be computed in the frequency domain with help of the FFT. Multiplication by matrix ${\bf A}$ is the same as multiplying Fourier coefficient by $A_k$. The ML estimation approach of Section 17.5 can be used. However, some parameters (the amplitudes ${\bf b}=[\alpha_1, \alpha_2, \ldots, \beta_1, \beta_2, \ldots]$) can be determined in one step if the frequencies are assumed:

$$\hat{\bf b} = \left( {\bf H}^\prime {\bf W} {\bf A}^\prime {\bf A... ...f W} {\bf H}\right)^{-1} {\bf H}^\prime {\bf W} {\bf A}^\prime {\bf A} {\bf x}.$$ (9.15)

So, it is only necessary to iterate on the frequencies.