## Why I get negative values on using periodogram after Fourier analysis?

sorry for bothering you with this question today. I am trying to analyze wave data that was produced using a wave tank. The period used for the waves is 1.7s, the waves encountered an obstacle at some point and then reflections are expected.

I plotted one of the sensors that are a wave gauge system. And I obtained the next plot:

The x is the time in seconds and the y the wave height in mm. After this I use the code:

ListLinePlot[Abs[Fourier[datcy]], PlotRange -> All] 

And then I get the Fourier transform and I also plot just the data:

I know the plot is reflected and that is why I have a double one, but what I don’t get in wolfram or maybe on signal analysis is how I get a power spectrum that is negative after this:

Periodogram[datcy[[All, 2]], Frame -> True,  

GridLinesStyle -> Directive[Red, Dashed], PlotRange -> All]

It’s on DB?, if that is true then technically there are no negative values and I should interpret as a -150 DB?. I am a bit lost here. Also, have a bit of doubt about why the amplitude of the Fourier plot doest not relate to the amplitude of my wave?.

## Doubts in some inequalities used in asymptotic analysis

I was going through this recent paper, and I had a couple of doubts in the final analysis of complexity of the scheme (you don’t need to go through the whole paper). I included three questions here, as I thought they seem to simple questions.

1. In Pg 17 (last four lines, see after equation 7), there is this inequality that is used (here, $$k(n) = \sqrt{\frac{n\delta(n)}{\log n}}$$ and $$\delta(n) = o(n / \log n)$$):

$$\frac{\binom{n}{a}}{\binom{^n/_k}{^a/_k}^k} \leq (^n/_k)^k$$

Can I know a proof for it?

1. Similarly, in the beginning of Pg 18, how is this possible? (the above inequality is used to get here though, $$(n / k)^{k / 2}$$ thing, and don’t worry about the meaning of $$\mathtt{ss}$$)

$$\mathtt{ss} \leq \sqrt{\binom{n}{\frac{n}{2}-\delta(n)}}\cdot (n / k)^{k / 2} \cdot \binom{k}{2\delta(n)} \cdot 2^{(2\delta(n)+1)n / k} \leq 2^{n / 2 + o(n)}$$

1. Also, this one might be a bit trivial, in Pg 17, one inequality above equation 7, the $$O(n)$$ term is dropped, isn’t that relevant?

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I’ve read a few articles that describe the heuristic detection used by AVs as being either “weight-based” or “rule-based”. The weight-based aspect seems to make sense, but I don’t understand what “rule-based” detection is or how it works.