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We generalize the correlation functions of the Clauser-Horne-Shimony-Holt(CHSH) inequality to arbitrarily high-dimensional systems. Based on thisgeneralization, we construct the general CHSH inequality for bipartite quantumsystems of arbitrarily high dimensionality, which takes the same simple form asCHSH inequality for two-dimension. This inequality is optimal in the same senseas the CHSH inequality for two dimensional systems, namely, the maximal amountby which the inequality is violated consists with the maximal resistance tonoise. We also discuss the physical meaning and general definition of thecorrelation functions. Furthermore, by giving another specific set of thecorrelation functions with the same physical meaning, we realize the inequalitypresented in [Phys. Rev. Lett. {\bf 88,}040404 (2002)].Comment: 4 pages, accepted by Phys. Rev. Let

General Correlation Functions of the Clauser-Horne-Shimony-Holt Inequality for Arbitrarily High-Dimensional Systems