Appendix C

More about Flat Spectrum Chirps

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

DFT(z)=1q1/2k=0q1zkexp(2πjkn/q)

(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:

r(q,p)=1sign[(gcd(q,p)+(q%2)(p%2)1)2]

(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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