Research on new spectral reconstruction solutions for Fourier-transform spectrometer

The spectral reconstruction of Fourier transform spectrometer can be simply achieved by using a Fourier transform or a Fourier cosine transform. However, the traditional Fourier transform solution is carried out in the complex-number field and the result is also a complex-number sequence, which will introduce an extra-phase to the spectrum and lead to the inaccuracy of reconstructed spectral intensity. On the other hand, although researchers use a Fourier cosine transform to avoid the extra-phase problem effectively, this solution has a boundary condition problem which cannot be avoided and may also lead to the inaccuracy of the reconstructed spectral intensity. To solve the problem, an improved Hilbert transform reconstruction solution (IHTRS) and a Fourier conjugated correction reconstruction solution (FCCRS) are developed by analyzing traditional reconstruction solutions. The main thought of IHTRS is using a complex-number sequence to represent the real-number signal, doing the transform in the complex-number field, and extracting the real-number spectrum from the transform result in the end. The main thought of FCCRS is constraining the transform process in the real-number field, using the conjugated property of the Fourier transform, creating the conjugated symmetrical form of the original signal first and acquiring the conjugated symmetrical form of the real spectrum, and extracting the real spectrum from it in the end. The results of the two solutions are compared. By carrying out both the simulation and the experiment using a helium lamp, it can be concluded that the FCCRS is 3 times faster than IHTRS, while the reconstructed spectral intensity accuracy of IHTRS is 29% higher than FCCRS. Both of the two solutions can avoid either the extra-phase problem caused by a discrete Fourier transform (DFT) solution or the boundary condition caused by a discrete cosine transform (DCT) solution effectively and improve the reconstructed spectral intensity accuracy.