Sinusoidal Amplitude and Phase Estimation

The form of the optimal estimator (5.39) immediately suggests the following generalization for the case of unknown amplitude and phase:

$$\hat{{\cal A}}= {\hat A}e^{j\hat{\phi}} = \frac{1}{N}\sum_{n=0}^{N-1} x(n) e^{-j\omega_0 n} = \frac{1}{N}\hbox{\sc DTFT}_{\omega_0 }(x) \protect$$ (6.41)

That is, $\hat{{\cal A}}$ is given by the complex coefficient of projection [264] of $x$ onto the complex sinusoid $e^{j\omega_0 n}$ at the known frequency $\omega_0$ . This can be shown by generalizing the previous derivation, but here we will derive it using the more enlightened orthogonality principle [114].

The orthogonality principle for linear least squares estimation states that the projection error must be orthogonal to the model. That is, if ${\hat x}$ is our optimal signal model (viewed now as an $N$ -vector in $\mathbb{R}^N$ ), then we must have [264]

$$\begin{eqnarray*} x-{\hat x}&\perp& {\hat x}\\ \Rightarrow\quad \left<x-{\hat x},{\hat x}\right> &=& 0\\ \Rightarrow\quad \left<x,{\hat x}\right> &=& \left<{\hat x},{\hat x}\right>\\ \Rightarrow\quad \sum_{n=0}^{N-1}\overline{x(n)} {\hat A}e^{j(\omega_0 n+\hat{\phi})}&=& N {\hat A}^2 \\ \Rightarrow\quad \sum_{n=0}^{N-1}x(n) {\hat A}e^{-j(\omega_0 n+\hat{\phi})}&=& N {\hat A}^2 \\ \Rightarrow\quad \hbox{\sc DTFT}_{\omega_0 }(x)&=& N \frac{{\hat A}^2}{{\hat A}e^{-j\hat{\phi}}} = N {\hat A}e^{j\hat{\phi}} \end{eqnarray*}$$

Thus, the complex coefficient of projection of $x$ onto $e^{j{\hat \omega}n}$ is given by

$$\hat{{\cal A}}= {\hat A}e^{j\hat{\phi}} = \frac{1}{N} \hbox{\sc DTFT}_{\omega_0 }(x).$$ (6.42)

The optimality of $\hat{{\cal A}}$ in the least squares sense follows from the least-squares optimality of orthogonal projection [114,121,252]. From a geometrical point of view, referring to Fig.5.16, we say that the minimum distance from a vector $x\in\mathbb{R}^N$ to some lower-dimensional subspace $\mathbb{R}^M$ , where $M<N$ (here $M=1$ for one complex sinusoid) may be found by ``dropping a perpendicular'' from $x$ to the subspace. The point ${\hat x}$ at the foot of the perpendicular is the point within the subspace closest to $x$ in Euclidean distance.

Figure: Geometric interpretation of the orthogonal projection of a vector $x$ in 3D space onto a 2D plane ($N=3$ , $M=2$ ). The orthogonal projection ${\hat x}$ minimizes the Euclidean distance $\left \Vert\,x-{\hat x}\,\right \Vert$ .