# Explanation of quantum algorithm for numerical gradient estimation

My question concerns this algorithm: Fast Quantum Algorithm for Numerical Gradient Estimation by Stephen P Jordan https://arxiv.org/pdf/quant-ph/0405146

I am unable to understand this step of the algorithm (bottom right corner of page 1, some of the explanation continues onto page 2).

I would expect the phases to be $$e^{i 2\pi f(\delta)}$$ based on what is described. It seems a lot of steps are skipped so I was hoping someone more familiar with the algorithm could reconstruct it.

The use of $$N$$ is to convert between the actual real numbers in the calculation and the fixed point integer encodings of them. I don't understand why $$l$$ multiplies $$\delta$$ , based on what appears before in the paper it seems that $$l/2$$ is small displacement in the input used for the estimation of the gradient so $$f(\delta/N - l\vec{1}/2)$$ makes sense but in the paper $$l$$ is also multiplied to $$\delta$$.

The $$N$$ mutiplied outside the function application also swaps between real number and fixed point encoding, the division by $$l$$ is from the denominator of the gradient estimation (though I am not sure why it is $$l$$ and not $$l/2$$. I am not sure what the $$m$$ does.