This paper describes the parameter convergence properties of an adaptive system to identify a single-input single-output plant model. It is demonstrated that, by using power spectrum analysis, the persistency of excitation (PE) condition in order to guarantee the exponential stability of the adaptive control system can be transformed into the positive definite behavior for the auto-correlation function matrix of adaptive signal. The existence of parameter nominal values can be analyzed by this condition and the convergence rates of parameter are determined by examining the auto-correlation function. We may use the sufficient richness (SR) of input spectrum instead of the PE condition to analyze the parameter boundedness. It can be shown that the eigen values of the auto-correlation function are always related with adaptive gain, input amplitude and positions or numbers of input spectra. In each case, the variation of parameter convergence rate can be also verified.