For passive positioning technology, not only the signal transmitting time of the target is not known at all, but also the initial phase of the signal is not known. Therefore, the phase comparison method cannot be used. But if we just use geometry, starting from the basic definition of phase shift detection of distance, it is explained in mathematical form that, because phase shift measurement has the fuzziness of period, the number of wavelengths contained in the observed quantity itself is an unknown variable to be determined, so the positioning equation based on phase shift measurement is unsolvable if we only use existing analytical methods. If the unknown quantity representing the period ambiguity is regarded as an undetermined quantity, the solution of the phase difference localization equation can still be obtained formally. As a mathematical basis, this chapter gives a linear solution method of passive positioning equation based on path difference measurement in two dimensional plane. In the passive positioning problem based on path difference measurement in two-dimensional plane, it is generally necessary to use at least 3 or more measuring stations to collect data to get the path difference between the radiation source and each measuring station. The existing method is to make use of these distance differences to form a set of nonlinear hyperbolic equations about the position of the radiation source, and the coordinate position of the radiation source can be obtained by solving the hyperbolic equations. The author's existing research results show that for the plane multi-station positioning problem, the linear equations can be obtained if the auxiliary equations are constructed by using the existing plane geometric relations on the basis of the path difference measurement.

One-dimensional double-base array is the most basic model of multistation passive positioning system. The study and analysis of this basic model can not only help to improve the design performance of the positioning system, but also help to improve the measurement accuracy of the positioning system. But so far, most of the researches on passive positioning system only arrange the mathematical equations from the perspective of solving the unknown quantity, without in-depth research on the internal correlation between each parameter. In fact, the one-dimensional linear array has many interesting inherent characteristics. Three intrinsic characteristics of one-dimensional double-base arrays are studied and given. Firstly, four methods of proving the midpoint direction finding solution on a single basis are described. Then, the arithmetic characteristics between adjacent paths are given. On this basis, some physical properties are briefly described. And then we can reveal the internal correlation between the difference of the path difference and other motion parameters as well as physical parameters. Finally, the median relationship among the three arrival angles and among the three radial distances is analyzed.

This chapter is the most important one in this monograph. In this chapter, a detection method for phase change rate without phase ambiguity is presented. The results of this study lay a foundation for the realization of unambiguous phase difference localization. First, based on the functional relationship between the distance change rate and the phase shift change rate, the phase difference measurement method of the phase shift change rate is given through the difference approximation of the distance change rate. Then, the expression of phase difference change rate based on the multi-channel phase difference measurement is obtained by using the phase difference measurement of phase shift change rate and differential processing by phase differential rate of change. On this basis, by stripping the time difference term corresponding to the baseline length from the change rate of phase difference, a function representing the different characteristics of the number of wavelengths and the phase difference per unit wavelength length is extracted. The subsequent simulation results show that the variation of the difference function of path difference per unit wavelength length is very regular. The corresponding correction number can be determined directly by distinguishing the range of difference of phase difference, and the range is obtained by the actual measurement. A function expression can be obtained independent of the difference term of the integer of wavelengths as well as equivalent to the difference function of path difference. Finally, the problem of nonfuzzy phase difference measurement for path difference is briefly described. These research results undoubtedly provide a powerful technical support for the practical design of engineering related to phase measurement.

Through a simple angle substitution, we can obtain the corresponding relationship between frequency shift and phase difference. Based on this relationship, the Doppler shift can be obtained by simple phase difference measurement. However, the reason why the function relation between phase shift and frequency shift can produce the expansion efficiency is also due to the relationship, that is obtained by the first order change of distance, between the rate of change of phase shift and the Doppler frequency shift. It's just a simple mathematical generalization of what's already known. Further, based on the two basic modes of angle permutation and firstorder change, various relations between phase shift and frequency shift, as well as their first-order change, can be determined. This mathematical description extends to physical applications, enabling many motion parameters and observations that would otherwise be difficult to detect to be obtained by simple phase shift/difference detection.

In this chapter, the fuzzy-free phase difference direction finding method based on a short baseline array is introduced and three different methods are presented. Firstly, the virtual short baseline direction finding method based on onedimensional double-base asymmetrical array is studied deeply, which is constructed by subtraction of the ratio of different sides between two adjacent baselines. The author's findings show that, although the difference between the lengths of two adjacent baselines is less than half a wavelength, the difference of the integers of wavelengths will not be zero in the direction of partial arrival angles but will jump. In this regard, the correction can be realized by the determination of the sine value of the arrival angle by adopting a method like the fuzzy-free detection analysis of the phase difference rate. The second approach is the orthogonal phase difference direction finding method based on equivalent simulation. It is found in the simulation that the curve shape of the differential function of path difference per unit wavelength obtained after phase jump correction is very similar to that of the cosine function. If the maximum value of the function is used for normalization processing and simple square root processing, then the function obtained is basically equivalent to the cosine function. At this time, it can be proved in principle that the results given are equivalent to the Doppler direction finding technique. Then, using the orthogonal array, the maximum value of the function which cannot be known in the one-dimensional array is eliminated by means of the orthogonal ratio, so the real-time direction finding based on phase difference measurement without phase ambiguity is realized. The third approach is the airborne direction finding method based on Doppler-phase measurement. The study shows that the airborne single-baseline interferometer can achieve high precision direction finding without phase ambiguity after integrating Doppler measurement information. The main method is to directly obtain the wavelength integer solution of the radial distance by comprehensively utilizing the velocity vector equation, Doppler frequency shift and its rate of change. Thus, the integral value of the wavelength contained in the path difference between two adjacent array elements can be given. By means of the phase difference measurement, the value less than the wavelength integer in the path difference can be determined. This chapter also explores the effect of phase difference measurement errors on the difference of wavelength integers. The expression of the wavelength number difference based on the phase difference measurement can also be approximated by the unambiguous phase difference direction finding method based on the virtual short baseline. The root mean square measurement error of the wavelength number difference is derived. Through analysis, it is revealed that the wavelength number difference has little effect on the accuracy of a single baseline phase difference direction finding.

This chapter presents several methods of distance ranging for basic short baseline arrays. These methods are either based on basic array positioning equations or on basic physical definitions, but the final mathematical form of these methods is proved to be the same. From the derivation of the ranging solution, we can see that the one-dimensional double-base array is the basis of the plane positioning analysis. Whether it is based on the Doppler frequency shift signal or the motion parameters, it can be attributed to the positioning solution of the one-dimensional double-base array. 

This chapter first studies the relationship between angle measurement and phase difference measurement. On the one hand, this helps to understand the physical properties, and on the other hand, it helps to simplify the process of error analysis. Then, two kinds of direction-finding methods which can improve the positioning accuracy are presented. One is a double - station cross – directionfinding algorithm involving higher - order geometric parameters. The other is to use the algorithm characteristic of adjacent path difference to virtually expand the baseline length of the detection array, to improve the accuracy of the bi-station direction-finding system. Whereafter, the passive ranging formula for two detection platforms with different motion directions is researched. The analytical formula of target distance based on angle measurement of two carriers is directly derived in different directions and at different speeds in planar polar coordinate system. In the end. Several methods of the azimuth-only estimation of the course of a moving target in a straight line is presented. Its analytical process has nothing to do with the detection of time, and the analytical formula obtained is purely related to the azimuth angle.

On the basis of the existing research, the passive detection method based on Doppler shift is further converted to the passive detection method based on the non-fuzzy phase difference measurement. Based on the phase-frequency conversion, the passive detection methods of two different models, that is a method of detecting a fixed target by a moving station, are presented in this chapter for improving the ranging accuracy. The first one studies the single station passive ranging method which takes the motion trajectory as the baseline of the detection array. The second one studies the ranging method which transforms the single base array into the virtual double base array by the relation between frequency shift and path difference. The research in this chapter provides some reference information on how to improve the performance of single or dual station passive positioning. This chapter proves that the accuracy of the phase difference ranging is better than that of the Doppler frequency difference. The significance of phase-frequency conversion lies in the first use of Doppler positioning method to give the location method which is difficult to obtain by using the phase difference equation, and then through phase-frequency conversion, further obtain better measurement accuracy than Doppler frequency difference measurement. 

In this chapter, two passive detection methods of long baseline fixed bistatic are presented based on the short baseline composite array. The first method is a two-station phase differential direction method with a long baseline. Direction finding and ranging are realized at each station using short baseline phase difference measurement technology. The path difference between two stations is obtained by using the radial distance measured by the two stations, respectively. On this basis, using the midpoint direction finding solution of a single basis, the phase differential direction finding formula of two stations with long baseline can be given directly. The second method is a method that uses a similar recursive method to implement the long baseline multi-station passive detection based on phase difference measurement without ambiguity. The similar recursion method is a method to obtain the long base path difference by using the similarity of triangles according to the path difference of short base line array, and then construct the virtual three-station linear array, from which the location calculation can be carried out directly by using the ranging formula of the double base array. It may be a feasible method to construct a long baseline multi-station phase difference detection system using a short baseline composite array.