#### 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

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

That is, the optimal least-squares amplitude estimate may be found by the following steps:

- Multiply the data by to zero the known phase .
- Take the DFT of the samples of , suitably zero padded to approximate the DTFT, and evaluate it at the known frequency .
- Discard any imaginary part since it can only contain noise, by (5.39).
- Divide by
to obtain a properly normalized coefficient of projection
[264] onto the sinusoid
(6.40)

**Next Section:**

Sinusoidal Amplitude and Phase Estimation

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Matlab for Computing Minimum Zero-Padding Factors