Chromatic dispersion compensation is an essential feature of high speed dense wavelength-division multiplexing (DWDM) systems. We propose a dispersion compensator structure whose characteristics meet the optical DWDM system requirements. The proposed structure is based on Fibonacci quasi-periodic multilayer structures composed of layers with large index differences. Studying the dispersive properties of Fibonacci structures with generation numbers $j=3$ and 4, and calculating group delay (GD) and group velocity dispersion (GVD) of their reflection bands, we have demonstrated that to have a smooth GD and almost a constant GVD in each band of a DWDM system, one needs not only to suitably chirp the structure refractive index profile, but also must properly apodize it. We also demonstrate the possibility of achieving high slope GDs and large GVDs by means of high order Fibonacci structures with thicker layers. Finally, by varying the layer dimensions and refractive indices as well as Fibonacci’s order, one can design DWDM dispersion compensators suitable for distances as long as $220\mathrm{km}$ .