The forward coherent radiation generated by ionic carbon dioxide molecules excited by tunable mid-infrared femtosecond laser pulses was systematically investigated. The coherent radiation corresponding to the ${\tilde{\rm A}}^{2}{\Pi }_{\rm u}$(${\nu }{{'}}=1$)→${\tilde{\rm X}}^{2}{\Pi }_{\rm g}$($ \nu =0 $) transition of the $ {\mathrm{C}\mathrm{O}}_{2}^{+} $ can be effectively created when the mid-infrared laser was in multiple photon resonance with the transition. Furthermore, a pump-probe method was used to investigate the temporal evolution of the radiation process. Based on the suppression of the 337 nm radiation effect induced by probe laser pulses, it was found that the duration of 337 nm coherent radiation was about 0.8 ps, which showed no obvious dependence on the gas pressure. In view of this observation, the nature of this coherent 337 nm radiation was free induction decay radiation induced by multi-photon resonance. This study not only reveals the nature of coherent emission of carbon dioxide, but also points out the universality of free induction decay effect in the resonant interaction of high-field laser with atoms and molecules.