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A dimension separationbased twodimensional correlation interferometer algorithm
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 40 (2013)
Abstract
A twodimensional correlation interferometer algorithm based on dimension separation is proposed for wideband direction finding. The original twodimensional angle searching is divided into 2 onedimensional searching processes in the proposed algorithm. Therefore, the computational complexity is reduced. Meanwhile the introduced interpolation process ensures the direction finding precision. Simulation results demonstrate that compared with conventional correlation interferometer algorithm, the proposed one can offer higher direction finding speed.
1. Introduction
The information contained in the received signal of an array is commonly used to determine the incoming direction of an incident wave[1]. Such a configuration is usually called as the direction finding system. In existing direction finding systems, interferometer has the advantages of high direction finding precision, simple algorithm, and high speed, therefore, it is widely applied in military and civil fields[2, 3].
Phase interferometer calculates the incoming direction with measured space phase differences between receiving elements[4]. The correlation interferometer[5, 6] is the most popular one among phase interferometers. It can reduce the effect from mutual coupling and system error through comparison between measured phase difference vector and phase difference vectors in sample database[7, 8]. For twodimensional direction finding[9], the sample database consists of phase difference vectors. Each of them corresponds to a different azimuth and elevation angle pair. The number of phase difference vectors is the size of sample database, which determines the computational complexity of direction finding algorithm[10]. In order to obtain a preferable direction finding result, the size of sample database should be large enough. Therefore, the involved computational complexity grows. To the best of the authors’ knowledge, the correlation interferometer algorithm based on space angle[11] is the only algorithm that dedicates to reduce the computational complexity of conventional algorithm. Although the computational complexity is reduced through introducing of two space angles, the sideeffect brought to direction finding precision is not investigated.
In modern direction finding environment, there are generally more than one incoming signals. In this case, wideband direction finding system based on multichannel structure is often adopted[12], where correlation interferometer algorithm is used to estimate the incoming direction of signal in each channel[13]. It can be seen that as the channel number grows, the involved computational complexity will increase. The realtime measuring of direction can hardly be guaranteed. To solve the problem mentioned above, a twodimensional correlation interferometer algorithm based on new idea of dimension separation is proposed. Two different similarity functions are chosen to search for azimuth angle and elevation angle, respectively, in this algorithm. Therefore, the original twodimensional searching is divided into 2 onedimensional searching processes. Simulation results demonstrate the effectiveness of the proposed algorithm.
2. Problem formulation
For wideband direction finding system, the frequency range of incoming signal becomes much wider. In order to capture all possible signals, the direction finding system based on multichannel structure is always adopted. Figure 1 shows the system structure. In Figure 1, the received signal is digitalized with an A/D convertor, and then down converted with a DDC. The output of DDC is channelized into N channels. Because the incoming signal may appear in each channel, direction finding should be performed on each channel. The direction finding complexity will grow as the channel number N increases. Therefore, in order to guarantee the realtime of direction finding, it is advisable to decrease the complexity of correlation interferometer. Especially, in the twodimensional direction finding case, where both azimuth and elevation angles should be estimated.
In twodimensional direction finding, circular array is a commonly used array manifold. Consider an Melement uniformly spaced circular array (UCA) as shown in Figure 2, where R is the radius. The incident wave arrives with direction of (ϕ, θ), where ϕ is the azimuth angle and θ is the elevation angle. Consider the origin as the reference point, the received signal of element m can be expressed as
where A is the amplitude, ω_{0} = 2πf_{0} and f_{0} is the frequency of incoming signal, the corresponding wavelength is λ = c/f_{0}. c represents the light speed. φ_{ m } is the phase of element m relative to the reference point which can be expressed as
According to Figure 1, after frequency mixing in RF receiver, A/D sampling and digital downconversion, the output of DDC can be expressed as
where T_{s} is the sampling interval after DDC, ω_{1} is a angle frequency that determined jointly by local oscillators in RF receiver and DDC. Assume Δ = (ω_{0} − ω_{1})T_{s} = 2πμ/D, then:
where D is the decimator factor. When it is input into the multichannel structure, it is filtered with a channelized filter bank. The output of channel k is
where h(n) is the original lowpass filter of channelized filter bank with order P, ω_{ k } = 2πk/D. According to (4), we have
Once y_{ mk }(i) and y_{ mk }(i) are obtained, we can extract the phase difference between elements m and n(≠m) according to the following operation
where H(ω) is the frequency response of original lowpass filter h(n). The phase difference is
It can be seen in (8) that the direction of incident wave resides in the phase differences between elements. Ideally, the incoming direction of incident signal can be determined by any phase difference pair through finding the solution to two equations similar to (8). However, because of the received noise, mutual coupling or system error, different direction finding results will be obtained when using different phase difference pairs. The faced problem thus is how to determine the unique direction of incident wave according to several measured phase differences precisely.
3. Conventional twodimensional correlation interferometer
The phase differences between different elements can form a phase difference vector. Correlation interferometer determines the incoming direction of signal through the comparison of measured phase difference vector and the vectors in sample database. Assume the interested azimuth range is [ϕ_{min}, ϕ_{max}] and the interested elevation range is [θ_{min}, θ_{max}]. Divide the azimuth range with Δϕ and the elevation range with Δθ, which results in P azimuth angle values and Q elevation angle values. P × Q elevation and azimuth pair can be formed. The phase difference vectors corresponding to each angle pair formulate the sample database. Obviously, the size of sample database is P × Q.
Denote the measured phase difference vector as
and the i th phase difference vector in database as Φ(i) ∈ R^{N×1}, where N is the element number in phase difference vector. The correlation process aims to find the phase vector that most similar with the measured vector in the sample database. The angle pair of the most similar vector is considered as the estimated incoming direction. The process can be expressed as
where f(·) is a function that used to measure the similarity between measured phase difference vector and the one in database, which is called as similarity function. Solving (9) is exactly a twodimensional searching process. It can be seen that in order to get the azimuth and elevation angles, a similarity function that is sensitive both to azimuth and elevation angles should be chosen. A cosine similarity function is proposed in[14]. It has an advantage of solving phase ambiguity. According to the cosine similarity function, the similarity between measured vector and sample i in the database is calculated as
where
and Φ_{ n }(i) are the n th element in
and Φ(i). The estimated direction corresponds to the sample vector that maximizes (10).
4. A dimension separationbased twodimensional correlation interferometer
Consider two incident signals with incident angles (ϕ, θ_{1}) and (ϕ, θ_{2}). They share the same azimuth angle but have different elevation angles. If correlation coefficient f_{2}(·) is adopted as the similarity function between measured phase difference vector and sample vector, then
where
represents the phase difference vector corresponds to direction of (ϕ, θ_{ i }), and (·)^{T} means the transposition operation. According to (8), it can be concluded that
Substitute (13) into (12), we have
Consider the common elevation range of (0°, 90°], we have
and
Above equation means the correlation coefficients between sample vector and measured phased difference vectors with the same azimuth are same. Accordingly, it can be concluded that the correlation coefficients between the measured phase difference vector and the vector samples with the same azimuth angle are same. Figure 3 gives an example, where the incoming direction is (105°, 40°). The measured phase difference vector is correlated with each vector sample. It can be seen from Figure 3 that the value of correlation coefficient is unchanged with elevation. Observing the sectional plan of Figure 3, the maximal value of correlation coefficient appears at the azimuth of 105° as shown in Figure 4. Because the sectional plan at different elevation remains the same as Figure 4, the azimuth can be estimated without any information about the elevation. This phenomenon initiates the correlation interferometer based on dimension separation.
4.1 Correlation interferometer based on dimension separation
Since correlation coefficient is only relevant to azimuth angle, the conventional twodimensional searching process can be separated. In the database, the phase difference vectors corresponding to a same elevation angle are chosen, measured phase difference vector is correlated to these sample vectors. The azimuth angle corresponding to the sample vector that maximizes the correlation coefficient can be considered as the estimated azimuth angle. Once the estimated azimuth angle is obtained, the phase difference vectors corresponding to the estimated azimuth angle are chosen. The measured phase difference vector is compared with these sample vectors where the similarity function chosen in this step should be sensitive to the elevation angle. f_{1}(·) in (10) is chosen here. The elevation angle corresponding to the sample vector that has the largest similarity is the estimated elevation angle. Obviously, conventional twodimensional searching process is divided into 2 onedimensional searching processes in this way.
In order to increase the direction finding precision, a twodimensional interpolation[15] is introduced after 2 onedimensional searching processes. Remember P and Q are the number of discrete azimuth and elevation defined in Section Conventional twodimensional correlation interferometer. Assume the index of estimated azimuth angle is p among the sequence {1, 2, …, P}, and the index of estimated elevation angle is q among the sequence {1, 2, …, Q}. In the sample database, the angle pairs that contiguous to the estimated angle pair are the ones with index of (p − 1, q + 1), (p, q + 1), (p + 1, q + 1), (p − 1, q), (p + 1, q), (p − 1, q − 1), (p, q − 1), and (p + 1, q − 1). Combining with the estimated angle pair, a small sample database with nine sample vectors can be formulated as shown in Figure 5, where the similarities between measured phase difference vector and the vectors in small sample database are denoted as s_{ i }, i = 0, 1, …, 8. They are calculated with f_{1}(·). Switch (p, q) to the origin, the index coordinates turn to be the ones in Figure 6. Perform the fitting of quadric surface y = a_{0} + a_{1}x + a_{2}y + a_{3}xy + a_{4}x^{2} + a_{5}y^{2} with these nine points, the coefficients are calculated as follows
The coordinates of surface peak are
Therefore, the azimuth and elevation angles can finally be estimated as
where
and
are the estimated azimuth and elevation angles, respectively, after 2 onedimensional searching processes.
Based on above illustration, the twodimensional correlation interferometer algorithm based on dimension separation can be obtained. The steps are shown as below.
Step 1: Choose an arbitrary fixed elevation angle, and select all of the phase difference vectors corresponding to this elevation angle but different azimuth angles in sample database.
Step 2: Compute the correlation coefficient between measured phase difference vector and all chosen phase difference vectors in Step 1.
Step 3: Choose the azimuth angle corresponding to the phase difference vector which has the maximum correlation coefficient as estimated azimuth angle and denote it as
.
Step 4: Choose all phase difference vectors which share the same azimuth angle that estimated in Step 3.
Step 5: Compute the similarity between the measured phase difference vector and all chosen phase difference vectors in Step 4 according to (10).
Step 6: Choose the elevation angle corresponding to the phase difference vector which has the maximum similarity value as estimated elevation angle and denote it as
.
Step 7: Process the twodimensional interpolation and obtain the final estimated azimuth and elevation angles according to (16)–(18).
Figure 7 shows the flow chart of above algorithm. The azimuth angle is estimated in Steps 1 to 3, where an arbitrary elevation angle is selected because the correlation coefficient is independent of it as shown in Figure 3. The following steps aim to estimate the elevation, where the cosine similarity function is used for its sensitivity to the elevation angle. It is worth mentioning that if Δϕ and Δθ from the database are small enough, running from Steps 1 to 6 is enough to yield a satisfactory result. Therefore, the interpolation process can be avoided. If original sample database contains P × Q sample vectors, there are P × Q times operations that are used to compute the similarities in conventional correlation interferometer algorithm. In the correlation interferometer algorithm based on dimension separation, the involved similarity operation reduces to P + Q times. The direction finding efficiency thus increases.
4.2 Condition for correlation interferometer algorithm based on dimension separation
According to the steps of correlation interferometer algorithm based on dimension separation, the key of it is the correlation coefficient which is independent on elevation angle. Therefore, the sample vectors corresponding to any elevation angle can be used for searching to give the elementary estimated azimuth angle. However, this searching process is only valid in the case of no phase ambiguity, which is also the condition for the proposed algorithm. The surface of correlation coefficient in Figure 3 will change if there is phase ambiguity. Figure 8 shows an example of correlation coefficient in the existence of phase ambiguity. The dimension separation cannot be carried out for the surface which is no longer smooth.
The condition for correlation interferometer based on dimension separation will be analyzed in the following. It is actually the condition for no phase ambiguity. The model in Section Problem Formulation ignores the effect of noise. Considering the additive noise in practical situation, the measured phase difference can be written as
where δ_{m,n} can be modeled as a Gaussian random variable with zero mean and variance of σ_{m,n}^{2}. Assume θ ∈ [θ_{ L }, θ_{ H }] where sin θ > 0, with n > m we can define a positive variable
. The condition for no ambiguity of
for any azimuth angle with probability of 1 − α can be expressed as
where
Substitute (21) into (20), we finally have
where
. For the high signaltonoise ratio (SNR) case, σ_{m,n}^{2} ≪ π^{2}, (22) can be approximated as
Substitute the form of u_{m,n} into (23), we finally have
Therefore, in order to use the correlation interferometer algorithm based on dimension separation, the radius to wavelength ratio must satisfy (24). For example, when α = 0.01, M = 9, R = 0.5, θ_{ L } = 0°, θ_{ H } = 90°, the highest frequency that ensures no ambiguity between contiguous element is depicted in Figure 9 according to (24). Meanwhile, the simulated result averaged among 5,000 Monte Carlo tests is also given, where the investigated phase difference is between elements 1 and 2. It can be seen that the theoretical and simulated curves match well. According to Figure 9, when SNR is 10 dB, dimension separation is valid for signals with frequency lower than 405 MHz. Formula (24) offers the guidance of using correlation interferometer algorithm based on dimension separation in practice.
5. Simulations
In this section, the dimension separationbased correlation interferometer algorithm is compared with conventional one in terms of direction finding precision and efficiency. In order to compare them in the same condition, (10) is used as the similarity function when twodimensional searching or a searching that sensitive to elevation is involved in both algorithms. It is worth mentioning that the change of this function will not influence the comparison results. There are totally three kinds of algorithms to be compared hereinafter. They are conventional correlation interferometer algorithm, dimension separationbased correlation interferometer algorithm without interpolation and dimension separationbased correlation interferometer algorithm with interpolation. For simplicity of description, they are named as Algorithm 1 (conventional correlation interferometer algorithm), Algorithm 2 (dimension separationbased correlation interferometer algorithm without interpolation), and Algorithm 3 (dimension separationbased correlation interferometer algorithm with interpolation).
5.1 Comparison of direction finding precision
Consider a 7element UCA with a radius of R = 1 m. Incidental wave arrives at the array with azimuth angle of 103° and elevation angle of 42°, and the carrier frequency is 70 MHz. Assume the interested azimuth and elevation ranges are [0°, 359°] and [0°, 90°], respectively, and the sample database is obtained with Δϕ = Δθ = 1°. The following results are averaged over 2,000 Monte Carlo runs. Performance of algorithms is evaluated with the root mean square error (RMSE). The average time is enough to represent the statistic characteristics of direction finding result. As Δϕ and Δθ are small, interpolation is not needed. The direction finding precisions of Algorithms 1 and 2 are compared. As mentioned before, the similarity function of (10) is used in Algorithm 1. In Algorithm 2, the correlation coefficient similarity function (11) is used first to realize dimension separation and the similarity function of (10) is used afterwards. The RMSEs of azimuth angle estimate and elevation angle estimate are shown in Figures 10 and11. It can be seen that the precision of Algorithm 2 is consistent with that of Algorithm 1. Although the complexity of Algorithm 2 is reduced compared with Algorithm 1, they offer similar direction finding precision.
In order to investigate the effect of interpolation, assume the interested azimuth and elevation ranges remain unchanged and the sample database is obtained with Δϕ = Δθ = 5°. Algorithms 2 and 3 are compared. The RMSEs of azimuth angle estimate and elevation angle estimate are shown in Figures 12 and13. The RMSEs of azimuth and elevation estimates decrease as SNR increases. As the range of azimuth and elevation is dispersed with 5° and the real incoming angle is (103°, 42°), the RMSEs converge to 2° in Algorithm 1. However, with interpolation the RMSEs of azimuth and elevation estimates remain decrease until SNR reaches 45 dB and converge to a much smaller degree. It can be seen that with interpolation the proposed algorithm can offer higher direction finding precision. The main reason is that Δϕ and Δθ actually determines the minimum direction finding error for a given incoming direction. The interpolation process can break through the minimum error. Therefore, if Δϕ and Δθ are small, the corresponding direction finding precision is satisfactory. Algorithm 2 can be adopted. Otherwise, Algorithm 3 is a better choice.
5.2 Comparison of direction finding efficiency
In order to investigate the speed up of dimension separation, the direction finding efficiency of Algorithm 1 and the one based on dimension separation are compared. In the CPU platform of Celeron G530 with a main frequency of 2.40 GHz, we choose C language as a programming language to investigate the average time over 10,000 runs.
Table 1 compares the direction finding efficiency when the interested azimuth and elevation ranges are [0°, 359°] and [0°, 90°], respectively, and the sample database is obtained with Δϕ = Δθ = 1°. It can be seen that Algorithm 2 needs less direction finding time and thus offers higher direction finding efficiency. The reason is that although in Algorithm 2 both similarity functions of correlation coefficient and cosine are adopted each of them is used for one dimensional searching. However, in Algorithm 1 the similarity function of cosine is used for a twodimensional searching.
Table 2 compares the direction finding efficiency when the interested azimuth and elevation ranges are [0°, 359°] and [0°, 90°] respectively, and the sample database is obtained with Δϕ = Δθ = 5°. As Δϕ = Δθ = 5°, Algorithm 3 is used and the direction finding efficiency of it is compared with that of Algorithm 1. Similarly with the results in Table 1, Algorithm 3 consumes less direction finding time. It is worth mentioning that the efficiencies of Algorithms 2 and 3 are similar because the consumed time of interpolation is much less than that of searching process. Comparing the speed up ratio with that in Table 1, it can be seen that with the increasing of dispersion the speed up ratio will decrease accordingly. The reason is that the size of sample database will decrease when the dispersion increases.
Table 3 compares the direction finding efficiency when the interested azimuth and elevation ranges are [0°, 150°] and [0°, 70°], respectively, and the sample database is obtained with Δϕ = Δθ = 1°. Comparing the speed up ratio with the ones in Tables 1 and2, it can be concluded that the larger the sample database, the more obvious the speed up is.
6. Conclusions
In order to increase the wideband direction finding efficiency, a twodimensional correlation interferometer algorithm based on dimension separation is proposed. Through using correlation coefficient as a similarity function first and another similarity function that is sensitive to elevation angle later, the twodimensional searching is separated into 2 onedimensional searching processes. Therefore, the computational complexity is decreased. In order to guarantee the direction finding precision, a twodimensional interpolation is introduced. Furthermore, the condition for the proposed algorithm is analyzed. Simulation results verify the effectiveness of proposed algorithm and show that the efficiency of the proposed algorithm will increase as the size of sample database grows.
Abbreviations
 RMSE:

Root mean square error
 SNR:

Signaltonoise ratio
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Acknowledgment
This study was supported by the National Natural Science Foundation of China (Grant no. 61101171).
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Keywords
 Correlation interferometer
 Dimension separation
 Correlation coefficient
 Similarity function