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1

See "ePeriodicity: Mining Event Periodicity from Incomplete Observations" by Zhenhui Li, Jingjing Wang, and Jiawei Han (2013); preprint at https://faculty.ist.psu.edu/jessieli/Publications/tkde14.pdf


2

The trigonometric functions are essentially exponential. Thus, a doubling of the argument corresponds to a squaring of the function (in a sense). In this case, it can be seen by applying the angle addition formula: $$ \begin{aligned} \cos( 2\theta ) &= \cos( \theta + \theta ) \\ &= \cos(\theta)\cos(\theta) - \sin(\theta)\sin(\theta) \\ &= \cos^...


2

If it helps any, generate a unit amplitude sinewave at 1 Hz and its square: Then the sinewave and its square look like this: You can see the DC component: the averaged value of the squared sinewave (averaged over an integer number of periods) is 1/2. And the red sinewave frequency is exactly doubled, so the period is halved. The DC and doubled frequency ...


3

This seems like more of a semantics problem. A signal is periodic with time $T$ if $$x(t+n\cdot T) = x(t), n \in \mathbb{Z}$$ So the signal is periodic in $0.5$ since the for $T = 0.5 \cdot n$ the argument of the cosine is an integer multiple of $2 \pi$. Since it's periodic in $0.5$ it's also periodic in all integer multiples of $0.5$, i.e $1$, $1.5$, $2$...


0

The difference here is that statistical autocorrelation assumes a stationary power signal (so basically an infinite periodic signal) and does a normalisation to $[-1,1]$ and that autocorrelation just takes the finite signal as it is and does the classical convolution of the signal with itself, leading to just one point of perfect overlap and decreasing ...


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