#### Sinusoidal Amplitude Estimation

If the sinusoidal frequency and phase happen to be known, we obtain a simple linear least squares problem for the amplitude . That is, the error signal

 (6.36)

becomes linear in the unknown parameter . As a result, the sum of squared errors

 (6.37)

becomes a simple quadratic (parabola) over the real line.6.11 Quadratic forms in any number of dimensions are easy to minimize. For example, the bottom of the bowl'' can be reached in one step of Newton's method. From another point of view, the optimal parameter can be obtained as the coefficient of orthogonal projection of the data onto the space spanned by all values of in the linear model .

Yet a third way to minimize (5.37) is the method taught in elementary calculus: differentiate with respect to , equate it to zero, and solve for . In preparation for this, it is helpful to write (5.37) as

Differentiating with respect to and equating to zero yields

 re (6.38)

Solving this for gives the optimal least-squares amplitude estimate

 rere (6.39)

That is, the optimal least-squares amplitude estimate may be found by the following steps:
1. Multiply the data by to zero the known phase .
2. Take the DFT of the samples of , suitably zero padded to approximate the DTFT, and evaluate it at the known frequency .
3. Discard any imaginary part since it can only contain noise, by (5.39).
4. Divide by to obtain a properly normalized coefficient of projection [264] onto the sinusoid

 (6.40)

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Sinusoidal Amplitude and Phase Estimation
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