Aim at complex function Fourier transform, the complex index expanded form have been discussed in this paper. The paper indicates that the periodical complex function can be used to describe the motion trace of a point in complex plane. Then the engineering application of complex signal frequency spectrum analysis has been discussed in this paper. This technique has been applied to the fields of mechanical engineering and some innovative achievements have been obtained, such as the establishment of new rotation accuracy theory, the suggestion of spectrum analysis for mechanism synthesis and the invention of polar locus generator. This paper also points out some misunderstandings that due to oversight of complex signal in the academia of signal analysis.