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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 leastsquares amplitude estimate
That is, the optimal leastsquares 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) 
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