# Appendix C

# More about Flat Spectrum Chirps

Let us define the q-point DFT of a sequence **z** as follows:

$\text{DFT}\left({\displaystyle \overrightarrow{\text{z}}}\right)=\frac{1}{{\text{q}}^{1/2}}{\displaystyle {\displaystyle \sum _{\text{k}=\text{0}}^{\text{q}-1}}{\text{z}}_{\text{k}}\mathrm{exp}\left(-2\text{\pi jkn/q}\right)}$ |
(C.1) |

FSC is a complex sequence of a unit envelope with DFT of a unit envelope.

We found that FSCs have another remarkable property, e.g., applying to an FSC the DFT twice yields the same FSC. The proof is numerical. Specifically, a Matlab script was written that, for given q and p, it computes several functions of p and q. The first function is:

$\text{r}\left(\text{q},\text{p}\right)=1-\text{sign}\left[{\left(\mathrm{gcd}\left(\text{q},\text{p}\right)+\left(\text{q}\%2\right)\cdot \left(\text{p\%2}\right)-1\right)}^{2}\right]$ |
(C.2) |

In Equation (C.2), gcd stands for the greatest common divisor of two natural numbers, and % denotes computing a remainder of division one integer by another. The function r equals 1 for mutually prime p and q ...

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