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Let x n ( n = 0, 1, 2, … ) be the samples of an integer-valued input signal.
Let f(x n ) be the samples of a piecewise smooth signal on a uniform grid { x n } n = 0 N − 1 over the interval Ω = [−1,1].
In order to explain the basis of the frequency-estimating procedure, first, let x raw(n) be the samples of a discrete cosine wave in the form {x}_{mathrm{raw}}(n)={A}_0kern0.5em cos left(2pi {f}_0nvarDelta T+{varphi}_0nvarDeltan3em n=0,1,2,dots (1).
Let Y 1, Y 2, …, Y m be the samples of independent identical distributions, and in the condition of normal distribution, Y ¯ = 1 m ∑ i = 1 m Y i, S 2 = 1 m ∑ i = 1 m ( Y i − Y ¯ ) ( Y i − Y ¯ ) ′. are the maximum likelihood estimations of μ, Σ, respectively, if m is big enough and S 2 is positive definite.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com