This case is the same as the LORAF3 subspace tracking, so requires only 2r− 1 unitary Givens plane rotations to reduce the Rx(t − 1) to Rx(t). In order to find the matrix Hw(t) = QHx(t)Vy(t), we previous consider the conjugated transposed form of (4.44) to
get ⎡
Thus, the recursive form of Hw(t) can be deduce as follow
⎡
The final step is equivalent to the last approach above.
The complete implementing procedure is summarized in Table 4.2. Attractively, this technique utilizes only O(Nr) computational complexity each update in time, so it is faster than the last one.
4.3 Fast Adaptive ESPRIT Algorithm with O(Nr) Com-plexity Using OPAST Subspace Tracking
The adaptive ESPRIT algorithms described above can help us to handle many problems about signal parameters estimation exactly, particularly for the spatial problem of estimat-ing the DOA of signals. Since the techniques of adaptive signal processestimat-ing are applied,
Table 4.2: FAST ADAPTIVE ESPRIT ALGORITHMO(Nr) USING LORAF 3
Input from Subspace Tracker: Q(t−1) ; G(t) z⊥(t)
the computational complexity can be low down successfully. It was proved that the per-formances of these two approaches are satisfactory [4]. Especially, the second technique reduce the computational complexity toO(Nr) at each time step, so the cost of compu-tation is saving very much.
In this section, we propose a new adaptive ESPRIT technique utilizing the OPAST subspace tracking. This method is very simple and is developed intuitively without any complex principle. We only consider the description presented in Section 4.1, and repre-sent the processing of our technique in the following text.
The first step is the same as other approaches. When we obtain the signal subspace WO(t) ∈ C2N×r from the OPAST subspace tracking, we divide it into two split submatri-ces Vx(t) ∈ CN×rand Vy(t) ∈ CN×ras
Simultaneously, we separate the vector p(t) ∈ C2N×1 given from (3.24) into two split subvectors as and (4.48) together, we can get two updating recursions as
Vx(t) = Vx(t − 1) + px(t)qH(t) (4.49) Vy(t) = Vy(t − 1) + py(t)qH(t) (4.50) where q(t) ∈ Cr×1 is come from the OPAST algorithm. Obviously, (4.49), and (4.50) can be regarded as the update form of the split signal subspaces respectively effected by the subarrays, ZX and ZY. Review the basic concept described in Section 4.1, it is not difficult to find the effort shown in (4.8). There is an instinctive idea about that we can obtain an adaptive processing form with low complexity by directly updating the matrix products VxHVx and VxHVy . For the purpose of implementing the idea, we define two
matrix products H1(t) ∈ Cr×r and H2(t) ∈ Cr×ras
H1(t) = VxH(t)Vx(t) (4.51) H2(t) = VxH(t)Vy(t) (4.52) Then we substitute the equation (4.49) into (4.51) to obtain the form as
H1(t) = [Vx(t − 1) + px(t)qH(t)]H[Vx(t − 1) + px(t)qH(t)]
= H1(t − 1)
+ VxH(t − 1)px(t)qH(t) + q(t)[px(t)]HVx(t − 1)
+ p1(t)Qr(t) (4.53)
where p1(t) ∈ Cand Qr(r) ∈ Cr×rare defined as
p1(t) = [px(t)]Hpx(t) (4.54)
Qr(t) = q(t)qH(t) (4.55)
It is successful that we find an updating recursion of the matrix product H1(t). We further apply (4.49), (4.50) and (4.52) to do the similar work to obtain the form as
H2(t) = [Vx(t − 1) + px(t)qH(t)]H[Vy(t − 1) + py(t)qH(t)]
= H2(t − 1)
+ VxH(t − 1)py(t)qH(t) + q(t)[px(t)]HVy(t − 1)
+ p2(t)Qr(t) (4.56)
where p2(t) ∈ R has the relation as
p2(t) = [px(t)]Hpy(t) (4.57) Consequently, the purpose for finding the direct updating recursions is achieved. The final step is that utilize (4.53) and (4.56) to find the matrix product Ψh(t) ∈ Cr×ras
Ψh(t) = [H1(t)]−1H2(t) (4.58)
Thus, the desired phase delay or wanted signal parameters can be estimated by imple-menting the eigenvalue decomposition of Ψh(t). It is worth to note that the initial values of H1(0) and H2(0) must be set suitably. Since the initial value of signal subspace matrix WO(0) has been set in the OPAST algorithm, we must compute the matched H1(0) and H2(0) as
However, our technique is very simple and intuitive without introducing any complex concepts of ‘QR-reduction’ and ‘Givens plane rotation’. Although this approach is also requiredO(Nr) computational complexity per time update, the amounts of computations of this technique is practically less than the approaches described in the last section. The effort of computation costs saving is more obvious when the signal source number r is more large. This is why we call it ‘fast adaptive ESPRIT’. Furthermore, our simulations demonstrate the fact that the performance of our approach is identical to the first presented method withO(Nr2) complexity above. In the next chapter, we will show the results of simulations and do a discussion of comparing the amounts of computations and storage sizes with three techniques. We will also exhibit the summary of our adaptive ESPRIT approach in Table 4.3.
Of course, our technique is also suitable to apply the PAST subspace tracking. The amounts of computations are slightly less than our technique using the OPAST subspace tracking since the inherent property in the algorithm for subspace tracking. However, it lacks the advantages of the OPAST subspace tracking. Thus, we will do not consider our technique utilizing the PAST subspace tracking in next chapter.
Table 4.3: FAST ADAPTIVE ESPRIT ALGORITHM O(Nr) USING OPAST
Chapter 5
SIMULATION RESULTS AND COMPARISON
In this chapter, computer simulations for DOA estimation by Matlab program demonstrate the applicability and the performance of the adaptive ESPRIT algorithm utilizing OPAST subspace tracking which we propose in Section 4.3. Simultaneously, we also simulate the two adaptive ESPRIT algorithms using LORAF subspace trackers described in Section 4.2 for comparison. For simplicity, the discussed three adaptive ESPRIT algorithms are called ESPRIT-OPAST, ESPRIT-LORAF2, and ESPRIT-LORAF3 respectively. Simula-tion results show that ESPRIT-OPAST has the tracking performance almost identical to ESPRIT-LORAF2 and ESPRIT-LORAF3. Then we also compare the required computa-tional complexity and memory sizes to realize each adaptive ESPRIT algorithm.
5.1 Simulation Results
Consider the desired parameters ϕk = ω0δ sin(θk)/c for DOA estimation shown in Sec-tion 2.2, the c is the speed of light obviously. If we assume that the frequency f0 = ω0/(2π) is 150M Hz and the displacement δ is equal to 2m, we can obtain a relation ϕk= 2π sin(θk) = 2πνk. Hence, for simplicity in our simulations, our task is to estimate νk = sin(θk) instead of θk. The received data is generated according to the data model described in Section 2.2 ((2.1) to (2.9)), when the number of source r and the SNR are
given. Furthermore, the forgetting factors α and β in all algorithms are set equal to 0.98 in the overall simulations here.
In the first experiment, the signal sources with constant νk are observed. Consider three cases, one signal sources collected by 6 sensor doublets, two signal sources collected by 10 sensor doublets, and four signal sources collected by 50 sensor doublets. Each case is simulated in both conditions, 3dB SNR and 0dB SNR. The simulation results are show in Figure 5.1 to Figure 5.6. We can find that the tracking curves of three algorithms are very close.
In the second experiment, we compare the performance of the algorithms in tracking two signals with crossed phase delay νkby 10 sensor doublets. Figure 5.7 and Figure 5.8 show that the signal sources with the larger slope and the shorter snapshots in two kinds of environments,3dB SNR and−3dB SNR respectively. Then observe the signal sources with the smaller slope and the longer snapshots. All three algorithms are simulated in both 3dB SNR and−3dB SNR, and exhibit their results in Figure 5.9 and Figure 5.14.
The results show that three algorithms have almost the same performance as each other.
Finally, We now consider two cases for that four signal sources impinging on the array composed of 50 sensor pairs. One case is tracking the suddenly varied phase delay νk, and the other one is tracking the smoothly varied one. Both two kinds of noise conditions, 3dB SNR and −3dB SNR, are set in each case for three algorithms. Figure 5.15 to Figure 5.26 display the simulation results. From the figures, we can find that three approaches almost have the same performance since the tracking curves of three algorithms are almost overlapped.
However, ESPRIT-OPAST has the performance nearly identical to ESPRIT-LORAF2 and ESPRIT-LORAF3.