## 5 Conclusions  
The concept of ERA/DC is extended to nonstationary ambient response in this study. If the ambient excitation on the system can be appropriately expressed as the nonstationary process in the form of a product model with a slowly-varying envelope function and stationary white noise, the system response can be expressed as a nonstationary process coincident with the envelope function of external force. The data correlation between nonstationary excitation and corresponding response can be theoretically shown approximate to stationary data correlation. Afterward, the appropriate extended channel technology is used to build the generalized Hankel matrix containing adequate dynamic information for mode estimation, to extend the applicability of ERA/DC to nonstationary output-only structural modal estimation. The findings of this paper can be concluded as follows.  
(1) This paper deduces that the correlation matrix of excitation and response signals can be made into the form identical with stationary white noise under nonstationary one in the form of a product model. With this characteristic, the ambient response can be directly used for ERA/DC by using channel-expansion technique. It is not necessary to transform ambient response into free-response through additional data processing before modal identification, preventing unnecessary distortion in vibration data to lead errors in parametric estimation of structural system.  
(2) The ambient excitation considered in this paper can be treated as a nonstationary process, which is the chosen nonstationary white noise in the form of a product model of a simple mathematical form and convenient for processing. According to numerical simulation, the structural mode can be identified effectively in most cases. Also, if the  
Figure 19: Mode shapes of the free beam (the solid line is EMA identification result, the dotted line is ERA/DC identification result).  
ambient excitation meets the characteristic of periodic vibration, the method proposed in this paper can be used for identifying modal parameter.  
(3) The ERA/DC is performed directly using ambient response data only for modal parameter identification through the method proposed in this paper, the natural frequency identification result is good. In terms of the identified mode shape, as the response data remains complete, the MAC value approaches 1 meaning the agreement with the actual mode shape is good. In terms of damping ratio, the error changes significantly, adequate response points and time-delayed sampling is required to augment system matrix order and to increase the damping ratio identification accuracy.  
(4) As the nonstationary response only is directly used for modal estiamtion, the identification result may contain structural modes, excitation modes, and fictitious modes. The fictitious mode is eliminated by using the concept of stabilization diagram, and the agreement between the identified mode shapes and actual structural mode shapes is checked by using modal assurance criterion.  
[PERSON] conceived of the proposed idea, designed the experiments and verified the data along with [PERSON]. The draft preparation is originally written by the first and corresponding author [PERSON]. The article is mainly revised by [PERSON], reviewed and edited in part by [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research was supported in part by Ministry of Science and Technology of Taiwan under the Grant MOST 109-2221-E-020-001.  
Not applicable.  
The authors would like to thank anonymous reviewers for their valuable comments and suggestions in revising the paper.  
The authors declare no conflict of interest.