Analysis of Velocimetry Processing Method Based on Doppler Asymmetric Spatial Heterodyne Spectrometer

The Doppler Asymmetric Spatial Heterodyne (DASH) spectrometer is a new type of high-resolution spectral measurement instrument that has received widespread attention in re-cent years. However, in previous ground experiments, laser sources were often used so that without exposing the premise and limitations of its high measurement accuracy. Therefore, this article provides a detailed introduction to the measured physical quantity transfer process, de-rives the constraints of DASH spectral velocimetry by discrete sampling, spectral offset analysis, and phase deviation analysis methods. Furthermore, it clarifies the impact of back-end signal processing methods on velocity measurement resolution and the range of algorithm errors. With a “spring model” of interference fringes is proposed, which not only provides mathematical model support for the superiority of using phase shift measurement methods to invert Doppler velocity, but also explains the improvement reason of the “asymmetric” structure relative to the original symmetric spatial heterodyne spectrometer. Meanwhile, based on the simulation values of the set parameters, the measurement resolution improvement factor of the interferometric phase measurement method relative to the spatial frequency measurement method is quantitatively calculated. In the field of astronomical remote sensing, due to stellar spectra lack of the monochromaticity and ideal spectral shape of laser sources used in ground experiments, it is necessary to consider whether the specific characteristics of the incident spectrum meet the amplification effect and effective range on the measured physical quantity transmission chain. It provides a reference for the practical application of the DASH spectral velocimetry technology.


Analysis of Velocimetry Processing Method
Based on Doppler Asymmetric Spatial Heterodyne Spectrometer Xiang Peng , Enhai Liu , Shulin Tian, and Rujin Zhao Abstract-The Doppler Asymmetric Spatial Heterodyne (DASH) spectrometer is a new type of high-resolution spectral measurement instrument that has received widespread attention in re-cent years.However, in previous ground experiments, laser sources were often used so that without exposing the premise and limitations of its high measurement accuracy.Therefore, this article provides a detailed introduction to the measured physical quantity transfer process, de-rives the constraints of DASH spectral velocimetry by discrete sampling, spectral offset analysis, and phase deviation analysis methods.Furthermore, it clarifies the impact of back-end signal processing methods on velocity measurement resolution and the range of algorithm errors.With a "spring model" of interference fringes is proposed, which not only provides mathematical model support for the superiority of using phase shift measurement methods to invert Doppler velocity, but also explains the improvement reason of the "asymmetric" structure relative to the original symmetric spatial heterodyne spectrometer.Meanwhile, based on the simulation values of the set parameters, the measurement resolution improvement factor of the interferometric phase measurement method relative to the spatial frequency measurement method is quantitatively calculated.In the field of astronomical remote sensing, due to stellar spectra lack of the monochromaticity and ideal spectral shape of laser sources used in ground experiments, it is necessary to consider whether the specific characteristics of the incident spectrum meet the amplification effect and effective range on the measured physical quantity transmission chain.It provides a reference for the practical application of the DASH spectral velocimetry technology.Index Terms-Analysis resolution, doppler asymmetric spatial heterodyne, interferometric phase measurement, spatial frequency measurement.

I. INTRODUCTION
W ITH the development of science and technology, it can main area for technological competition and expansion in the next century.Therefore, the increasingly desire of humans to expand their space activities, making deep space exploration a hot research direction for major countries [1], [2].Now, China has become a pioneer on par with the United States in aerospace field.In 2021, the Tianwen-1 spacecraft arrived on Mars, marking that China has the ability of long-range space exploration in the universe.The subsequent verification work of Jupiter, Saturn, and even further planet also put forward higher requirements for deep space exploration technology.
Doppler velocimetry based on spectral variation is an important means in current astronomical sounding [3].The Doppler Asymmetric Spatial Heterodyne (DASH) spectrometer, first proposed by American scholar J. M. Harlander and C. R. Englert, which has advantages of compact and sturdy structure, high light flux, high spectral resolution, making it very suitable for deep space exploration and astronomical navigation fields [4], [5].In 2007, the team of J. M. Harlander and C. R. Englert completed the validation of DASH spectral velocimetry technology in the laboratory.And in the following years, high-precision measurements of atmospheric wind fields were achieved on ground-based and airborne detectors [6], [7], [8].As a result, DASH spectroscopy technique has begun to receive significant attention and active expansion from researchers in related fields [9], [10], [11], [12], [13], [14], [15], [16], [17].In the initial DASH spectral velocimetry theory, it introduced measuring the phase shift of interference fringes to reverse the Doppler velocity difference, but did not explain the implementation mechanism and constraint conditions of obtaining high measurement accuracy using interferometric phase measurement method.Subsequently, most of the existing research focused on improving and enhancing the resolution of the front-end optical design, with few questioning the supporting conditions and usage scenarios for their high measurement accuracy.
This article mainly analyzed the sources of high-speed measurement accuracy from the perspective of signal analysis and processing, and derived the implementation mechanism of phase deviation analysis method to achieve high-precision astronomical velocity measurement.The relevant theories provided a basis for the advantages of spatial heterodyne spectral velocimetry, and also pointed out improvement directions for the application of this technology in deep space exploration.

A. The Basic Structure of DASH Spectrometer
The structure of the DASH spectrometer is shown in Fig. 1, in which the blazed gratings of the two interference arms are identical and placed obliquely, and the distances between the two blazed gratings and the beam splitter are not equal, which is the so-called "asymmetric" structure [18].
The incident beam is transformed into parallel light through a collimating lens and projected on two diffraction gratings.After diffraction, the light at a certain wave number returns to its original direction, which is called the Littrow wave number of the corresponding grating.The two emergent wavefronts of the light with Littrow wave number σ L diffracted by the grating are perpendicular to the optical axis; and the light with non-Littrow wave number σ returned through grating diffraction, the propagation direction would have a small angle ±γ with the optical axis.Therefore, the generated interference fringes satisfy the grating equation: Where σ is the wave number of the incident light; m is the diffraction order and typically m = 1; 1/d is the reticle density of the grating.Record the width of the interference fringe period as D, the spatial frequency as k x , and substituted the Littrow wave number σ L and non-Littrow wave number σ into (1), which can be obtained as follows [19]: Record the spectral density of incident light as B(σ), the interferogram signal obtained from the detector can be represented as: Where x is the pixel position, φ = 4π(σ − σ L )Δd is the additional phase difference introduced by a spectrometer; δϕ is the phase difference caused by Doppler frequency shift.

B. Interferometric Phase Measurement Method
According to the linear properties of the time-frequency space, for the case of multiple spectral lines incident, the final waveform received by the detector should be a linear superposition of the interference fringes corresponding to each spectral line.Make the signal go through Discrete Fourier Transform (DFT), separate the characteristic spectral lines and then perform Inverse Discrete Fourier Transform (IDFT).The overall phase value can be obtained by calculating the inverse trigonometric function by the ratio of the imaginary and real parts of the eigenvector: Where the first two items on the left side denote the phase with a zero speed, which can be determined by the initial calibration.
The superscript and subscript 0 indicate that the center position of the image is taken as the reference zero point.
Record the average wavelength in the band as λ 0 (the corresponding wave number is σ 0 ), then the optical path difference of the spectrometer is ΔL = 2Δd, and the fringe interference order is m.If the spectrum is shifted, the fringe would move to the adjacent order, and the order difference is Δm, then there are: Due to the Doppler Effect of light, the shift of signal wavelength satisfies: So the relationship between phase change and velocity can be express as follows: The method of measuring velocity based on the phase shift of interference fringes mentioned above is commonly referred to as Interferometric Phase Measurement (IPM) method.

III. MECHANISM ANALYSIS OF VELOCIMETRY PROCESSING METHOD
Define the wave frequency of light as f, the wavelength as λ, light velocity record as c, and the test speed record as v. Based on the principle of Doppler Effect, it can be known that the direct Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.physical quantity that changes due to velocity modulation is the wave frequency of light.There are: In addition, the wavenumber of light satisfies: So the wave frequency of visible light reaching the 10 ^15 Hz level, far beyond the measure capability of current detector.Therefore, interference is usually used to convert the wave frequency of signal into the spatial frequency of fringes for measurement according to (2).However, the original theory by C. R. Englert team only proposed the measurement method of transforming signal spectrum shift into the phase shift, but did not explain the superiority and necessity of this operation.It is necessary to conduct an in-depth analysis of the high-precision measurement mechanism of DASH spectral velocimetry technology.

A. Resolution of Spectrum Analysis
As is known to all, the most commonly used analytical method for extracting features in interference signals is the DFT method.However, due to the characteristics of discrete sampling, the spectral resolution in the frequency domain space is limited.The definition of spectral interval (reciprocal of spectral resolution) is: Where Fs is the data sampling rate, and N is the number of sampling points entering the transformation calculation.Fs is generally determined by the data acquisition terminal, while N determines the size of the spectrum interval, as shown in Fig. 2. Due to the discreteness and randomness of the sampled data, the spatial frequency of the interference fringe waveform cannot be exactly an integer, which means that the frequency domain sampling point position may not align with the theoretical value of signal energy, resulting in signal energy being distributed in the form of Fourier Expansion on various sampling points, it is called spectrum leakage.The size of the spectral interval further determines the distribution of signal energy in the frequency domain space.Therefore, it is natural to cause system errors when calculate signal characteristics based solely on a few frequency domain sampling points with higher energy.

B. Spatial Frequency Measurement Method
Assuming the pixel width of CCD camera is w, the corresponding spatial data sampling rate is F s = 1/w, and the sampling pixels of the camera record as N (along the x-axis direction).Based on the frequency variation relationship caused by the Doppler Effect, the spatial frequency variation of the interference fringes can be obtained as: Where Δfx can be regarded as a "dimension" in spatial frequency calculation; df is the relative difference of normalized spatial frequency digital quantities, which indicates the total number of interference fringe cycles present on the imaging surface.Assuming the initial state of measurement process is stationary (radial velocity is zero), where v is the actual velocity to be detected.Transform (11), then there are: In addition, the phase deviation between individual pixel intervals w caused by df is denoted as Δϕ x , which causes: Substituting ( 10) and ( 13) into ( 12), after transformation, there are: It is called as Spatial Frequency Measurement (SFM) method.
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C. Comparison of the Two Velocimetry Methods
By comparing the two velocimetry ( 7) and ( 14), it can be concluded that: When taking the value of "w = 13 μm, θ = 14.318 • , Δd = 2 mm" in simulation calculation, there are: From ( 16), it can be seen that the IPM method can improve the resolution by about 301.4 times compared to the SFM method.
Considering the inherent physical manifestations of IPM method, it can be imagined to expand the interference fringes into a "spring" in three dimensions (its projection on the X-Y plane is the energy amplitude curve of the interference fringes).The extension or compression of the spring corresponds to a spatial frequency change of the stripe image, as shown in Fig. 3(considered the intensity attenuation).
When a small stretching or compressing action causes the offset Δϕ of the cross-section of the first coil spring, the offset Δθ = M Δϕ will occur in the M-th coil, which is directly related to the change of spatial frequency.Therefore, a more accurate spatial frequency change value can be obtained by measuring the phase offset at the far end of the spring, which is equivalent to amplifying the signal analysis resolution by M times.The specific measured cycle position M needs to be determined by This is the physical significance of the advantage about IPM method over SFM method.It can also explain the reasons for the improvement of the "asymmetric" structure of the DASH spectrometer compared to the symmetric spatial heterodyne IV.DATA SIMULATION TESTING AND Select a Gaussian distribution spectral emission line with a half width of 0.002 nm as the simulated light source signal, the specific simulation parameters are shown in Table I.
After generating the interference fringe signal, the SFM method and the IPM method are respectively used to invert and analyze the interference waveform data.The simulated Doppler velocities are 6 group in total, set step of 1000 m/s with range of 5000 m/s 10000 m/s.Calculate the interference fringe data by (7) and ( 12) respectively, the simulation results obtained shown in Fig. 4 and Table II.
From the simulation results, it can be seen that at the qualitative the Δv1 is much greater than the error Δv2, indicating that the IPM method indeed has higher resolution accuracy compared to directly measuring the Doppler frequency shift of the stripe signal.At the quantitative level, based on the data of Table II, the proportion of error range (324.5/1.53 = 212) cannot be strictly equal to the resolution ratio described in (16).Analyzed the reason is that there is some data truncation error during the calculation process (systematic error), which amplifies the smaller phase measurement error Δv2 to a certain extent.
It is worth to point out that although it has been proven that IPM method has a significant advantage in improving solution accuracy compared to SFM method, it is necessary to understand the source of this advantage and the prerequisite for resolution amplification.For instance, majority of ground experiments related to DASH use laser light sources, leading to many research conclusions being formed under the assumption of light source conditions with good monochromaticity and energy concentration similar to lasers.However, in the practical aerospace environments, lasers are often unable to be used due to excessively far distance.Specifically, for complex incident spectra, such as the visible light band of stars mostly consisting Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.   of absorption line spectra, the amplification effect of phase measurement may not always satisfy linear laws (360°reversal effect), which limits the actual effective range of this method.
From the above analysis, it can be seen that the SFM method can be regarded as the base of the IPM method.Under the premise of satisfying the phase linear amplification effect described by the "spring model", higher measurement accuracy can be obtained by the IPM method, as shown in Fig. 5.However, it should be noted that amplification from SFM to IPM is not a certain.If assumed IPM to be established consensus of DASH spectral velocimetry technology, may result in significant errors when facing some complex coupled spectral input.

V. CONCLUSION
From the content of the above chapters, it can be seen that the essence of Doppler spectral velocimetry technology comes from the changes in signal spectrum caused by relative velocity.The spatial frequency changes and phase shifts of the interference fringe signal are further derived from corresponding physical relationships [20].For the signal DFT/IDFT analysis, it is the mutual transformation from the discrete sampling points of the time/space domain signal curve to the frequency domain.So, for the inversion analysis of signal features, one is to choose physical quantities with shorter derivative paths under the same resolution conditions for measurement; the second is to pay attention to whether the scaling ratio has an impact on measurement resolution for the derived physical quantities; the third issue is to check the discreteness of sampled data with impact on the resolution ability.
In the majority of introduction about the DASH spectral velocimetry technique, the IPM method is usually directly cited without explaining in detail the reasons and advantages of this approach.This article provides a detailed introduction to the physical quantity transfer process from "Doppler velocity → spectral frequency drift → interference fringe change → spatial frequency offset → interferometric phase shift", and provides the resolution amplification factor for interferometric phase measurement relative to spatial frequency measurement.It proposed the reason for choosing to use IPM method to obtain Doppler velocity in the process of velocity navigation based on DASH spectrometer, which is an explanation of the underlying mechanism and a supplement to the DASH spectral velocimetry theory.

Fig. 2 .
Fig. 2. Structure diagram of spectral interval: (a) spatial domain distribution of single frequency sinusoidal signals; (b) frequency domain distribution of single frequency sinusoidal signals.

Fig. 4 .
Fig. 4. Comparison of data simulation results between the two processing methods.

Fig. 5 .
Fig. 5. Connection and difference between the two processing methods.

TABLE I PARAMETERS
OF THE DATA SIMULATION SETTING calculating the appropriate optical path difference by optical imaging theory.

TABLE II SIMULATION
PROCESSING RESULTS FOR DIFFERENT INPUTS