Bluestein's algorithm to evaluate the DFT from $f_o$ to $f_o + k\Delta_F$ Briefly, the convolution between $x(nT) e^{-j2\pi f_o nT} e^{-j \pi \Delta_F Tn^2}$ and $c(nT) = e^{j \pi \Delta_F T n^2}$ multiplied $e^{j \pi \Delta_F T k^2}$ allows me to find the DFT $X(f_k = f_o + k\Delta_F)$.
Let $N_x$ be the number of samples of $x(nT)$, I found that $c(nT)$ must be at least of length $2N_x$ but I don't know why. Can someone tell me how do I choose the number of samples of $c(nT)$?
The convolution in Bluestein's algorithm is a linear convolution, not circular. Since the approach is to use standard FFT's for the convolution operation, the underlying waveforms being convolved must be zero padded out to $M \ge 2N-1$ to compute the linear convolution accurately.