THz-TDS Signal Analysis and Substance Identification

The terahertz time-domain spectroscopy (THz-TDS) imaging system can obtain high-dimensional signals of the substance fingerprint information. It is necessary to process properly to use some signal processing techniques especially for high dimensional signals. As a mathematical description language, geometric algebra (GA) provides not only a powerful algebraic framework for the multi-dimensional vector analysis and computing, but also a unified measurement and geometrical description for different geometric models. On the basis of the GA theories, a new signal analysis method of the THz-TDS is presented. Based on the characteristics of THz-TDS signals, signals are mapped into vectors in the high-dimensional real vector space. The vectors are represented with hyper-complex numbers. We can analyze the vectors using theories of GA. Based on the physical mechanism of the THz-TDS signal analysis, geometric distribution properties and algebraic relationships of THz-TDS signals are deduced. It is demonstrated that every complex refractive index of the sample relates to a unique 2-blade B2, all vectors corresponding to the samples of the same substance are collinear and belong to the intrinsic 2-blade of the substance. In projective interpretation, the 2-blade B2 represents a fixed line and all vectors related to the same substance are along that line. Accordingly, a novel substance identification method based on the relative THz-TDS is presented. In the method, two THz-TDS signals through the samples of the same substance but different thickness are measured. The intrinsic 2-blade B2 of the substance is then determined by the outer product of these two corresponding vectors. Using the conformal split by the fixed bivector B2, each vector corresponding to THz-TDS signals in the vector space Vn can be linearly splitted into vectors in vector spaces V2 and Vn-2. Since that 2-dimensional subspace V2 is the support of B2, the subspace is also a label for substances. So substances of samples can be identified on the magnitudes of projection vectors in that subspace. This method can contribute to the accurate classification and identification, and facilitate the feature extraction. Finally, experiments are presented and show that the substance identification method is feasible and effective.