Summary of Least Squares Amplitude Estimation

$$\zbox{\hat{A}= \frac{1}{N}\mbox{re}\left\{\hbox{\sc DTFT}_{\omega_0 }\left[e^{-j\phi} x\right]\right\}}$$

The optimal least-squares amplitude estimate may be found by the following steps:

1. Multiply the data $x(n)$ by $e^{-j\phi}$ to zero the known phase $\phi$
2. Take the DTFT of the $N$ samples of $x$ , suitably zero padded, and evaluate it at the known frequency $\omega_0$
3. Discard any imaginary part since it can only contain noise
4. Divide by $N$ to obtain a true coefficient of projection onto the sinusoid $s_{\omega_0 }(n)\isdef e^{j\omega_0 n}$ (Method 3)

Amplitude and Phase Estimation

Multiplying the optimal amplitude estimator by $e^{j\phi}$ suggests the following generalization for including phase:

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

That is, $\hat{{\cal A}}$ is given by the complex coefficient of projection of $x$ onto the complex sinusoid $e^{j\omega_0 n}$ at the known frequency $\omega_0$ .

Proof by the orthogonality principle (Method 3):

The orthogonality principle for linear least squares estimation states that

$$\zbox{\hbox{\emph{the projection error must be orthogonal to the model}.}}$$

That is, if ${\hat x}$ is our optimal signal model (viewed now as an $N$ -vector in $\mathbb{R}^N$ ), then we must have

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

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

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

The optimality of $\hat{{\cal A}}$ in the least squares sense follows from the least-squares optimality of orthogonal projection.

Frequency Estimation

The preceding cases suggest the following sinusoidal frequency estimator:

$$\hat{\omega}_0^\ast = \arg\{\max_{\hat{\omega}_0} \left\vert\hbox{\sc DTFT}_{\hat{\omega}_0}(x)\right\vert\}.$$

That is, the sinusoidal frequency estimate is defined as that frequency which maximizes the DTFT magnitude.

• Given this frequency, the least-squares sinusoidal amplitude and phase estimates are given by $1/N$ times the DTFT evaluated at that frequency

• It can be shown that this is the optimal least-squares estimator for a single sinusoid in white noise

• If the additive white noise is Gaussian, then it is also the maximum likelihood estimator