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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:

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

x-{\hat x}&\perp& {\hat x}\\
\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}}

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

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

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University