bsteinbach

ano passadodiscuss

The power vs voltage thing I think comes from the historical connection of dB to sensing waves. Because you can change impedance to match any sensor, but power is conserved, power was just more often more important to discuss than voltage. I still have to look it up every time whether 10x is 10 or 20 dB.

bsteinbach

ano passadodiscuss

In defense of sliding windows of time series, if not clustering, the principal components analysis of the matrix on page 1 is one robust way to extract a linear model for the system producing the time series. It's useful for finding frequencies of closely spaced oscillators in a time series when least squares fitting would be arduous to parameterize and convergence a battle.

bsteinbach

há 3 anosdiscuss

Not only similar, it's identical to Newton Raphson in ideal arithmetic!

The nth order product approximation for the reciprocal of d = 1 - x is y_n = product_{i=1}{n} (1 + x^{2^(i-1)}) = sum_{i=0}^{2^n - 1} x^i = (1 - x^(2 n)) / (1 - x).

The Newton-Raphson iteration for the reciprocal of d is z_{n+1} = z_n * (2 - (1-x) * z_n).

Induction with the base case z_0 = 1 = y_0 shows the sequences are equal by inserting y_n for z_n in the Newton Raphson iteration: z_{n+1} = (1 - x^(2 n)) * (2 - (1 - x^(2 n))) / (1 - x) = ( 1 - x^(4 n) ) / ( 1 - x) = y_{n+1}.

So I guess you could call it an explicit product formula for the Newton Raphson reciprocal.

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