## 1 Introduction  
Asphal mixtures exhibit viscoelastic behaviors over a wide range of time and temperatures. The responses characterizing the time- and frequency- dependent viscoelastic properties of asphalt mixtures include dynamic modulus, creep compliance and relaxation modulus. These parameters can not only be used to describe the linear viscoelastic properties, but also can indicate the nonlinear viscoelastic properties and failure properties of materials. In fact, these three properties can be interconverted between time- and frequency- domains because linear viscoelastic material properties shared an equivalent mathematical theory, including differential and integral equations [1; 2; 3]. Dynamic modulus is mathematically defined as the maximum dynamic stress divided by the peak recoverable axial strain, and is a key input for the prediction of field rut depth or fatigue crack amount in the current AASHTOWare pavement ME design program. Creep compliance describes the phenomenon that strain decreases with time under constant stress. In contrast, relaxation describes the phenomenon that stress decreases with time under constant strain. Relaxation modulus is required to calculate thermal stress within asphalt pavement structure in finite element analysis and in the pavement design procedure [4]. However, the direct test of relaxation modulus is typically time- and cost-consuming, as the constant-strain is difficult to control in experiments, whereas it is relatively more convenient to measure the dynamic modulus and creep compliance in laboratory with high testing accuracy. Thus, relaxation modulus in most cases is obtained from dynamic modulus or creep compliance by conversion methods. The master curve of dynamic modulus or creep compliance needs to be determined before converting to relaxation modulus. Different mathematical functions can be used to obtain master curve, such as pure power law, generalized power law, modified power law, and Prony series [5]. The pure power law, generalized power law, and modified power law have their own benefits, such as being easy to implement, or can be better used to represent the transition zone of a master curve. However, they were not selected since they are not suitable for characterizing short-term or long-term properties [6; 7]. Prony series was selected in this paper since it has been widely proved to best represent the fitting equations of master curve [8]. Prony series is a mathematical method to fit a set of equidistant data based on a series of damped complex exponentials. It is widely used in linear viscoelastic material functions, since it can significantly improve the computational efficiency attributed to its physical basis in the theory of mechanical models dealing with linear springs and dashpots [9]. The interconversion between dynamic modulus and relaxation modulus was firstly proposed by [PERSON] [3]. Later, [PERSON] and [PERSON] [10] proposed the least squares method to directly fit the Prony series to experimental data. However, during these conversions, negative coefficients cannot be avoided, which may generate oscillations in the master curve. To solve the issue, [PERSON] and [PERSON] [2] introduced constraints to ensure that all the Prony series coefficients are positive so that a smooth master curve could be plotted. Before fitting the Prony series, modification technologies such as pre-smoothing raw data was proved to portray a smoother curve over a wide range of time [4; 11]. Recently, researchers have been developing methods to generate relaxation modulus with no need to use the collocation method or the least squares method, such as semi-analytical algorithm and Levenberg-Marquardt method [12; 13; 14]. The relaxation modulus can be obtained from the creep compliance through numerical or analytical approaches, which have been developed by several researchers [15; 16; 17; 18]. In regards to exact analytical interconversion, the integral solution usually can be achieved in an infinite range.