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Input design is an important problem for system identification and has beenwell studied for the classical system identification, i.e., the maximumlikelihood/prediction error method. For the emerging regularized systemidentification, the study on input design has just started, and it is oftenformulated as a non-convex optimization problem that minimizes a scalar measureof the Bayesian mean squared error matrix subject to certain constraints, andthe state-of-art method is the so-called quadratic mapping and inverseembedding (QMIE) method, where a time domain inverse embedding (TDIE) isproposed to find the inverse of the quadratic mapping. In this paper, we reportsome new results on the embeddings/inverse embeddings of the QMIE method.Firstly, we present a general result on the frequency domain inverse embedding(FDIE) that is to find the inverse of the quadratic mapping described by thediscrete-time Fourier transform. Then we show the relation between the TDIE andthe FDIE from a graph signal processing perspective. Finally, motivated by thisperspective, we further propose a graph induced embedding and its inverse,which include the previously introduced embeddings as special cases. Thisdeepens the understanding of input design from a new viewpoint beyond the realdomain and the frequency domain viewpoints.

On Embeddings and Inverse Embeddings of Input Design for Regularized System Identification