Application of High Resolution Direction Finding Algorithms in Mobile Communications Umar Hamid Institute of Communication Technologies, Islamabad, Pakistan
[email protected] Abstract–Mobile communication systems based on 3G and 4G technologies use adaptive antenna arrays to increase their coverage and channel capacity. Adaptive antenna arrays involve direction finding and beamforming algorithms to localize and track both signals i.e. users and interferers. This paper presents simulation and analysis of three high resolution direction finding algorithms namely MUSIC, Root-MUSIC and ESPRIT. These algorithms provide an estimate about the number of incoming signal sources and their angles of arrival on an antenna array. Simulation results have been used to evaluate the performance of these algorithms by varying the antenna array parameters such as number of mobile users, number of antenna elements, time samples acquired and signal-to-noise ratio. Keywords–direction finding, antenna array, MUSIC, RootMUSIC, ESPRIT
I.
INTRODUCTION
Syed Ali Abbas Muhammad Ali Jinnah University, Islamabad, Pakistan
[email protected] MATLAB simulations have been developed for MUSIC (Multiple Signals Classification), Root-MUSIC and ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) algorithms. Results of these simulations have been used for characterizing the linear antenna array with optimal performance.
II. HIGH RESOLUTION DIRECTION FINDING ALGORITHMS A. Data Generation for Antenna Array Several plane wave signals from P sources are incident on a uniform linear array from different angles θi where i = 1, 2 … P. The array consists of Q antenna elements. At any time instant t, the received signal vector Rxd(t) at the antenna array can be written as: Rxd(t) = Σ a(θp) x sp(t)
(1)
Direction finding algorithms are used to identify the angular spread of the sources. They work on the signals received at the output of antenna array and compute the angles of arrival of all the incoming signals. A beamforming processor uses these angles of arrival for calculating the complex weight vectors for beam steering. A beamformer steers the radiation in a particular direction and places the nulls in the interfering directions with the help of known angles of arrival of the sources [1].
Where s(t) is a vector of Px1 incoming sources, and a(θ) is a Qx1 array steering vector for a particular direction θ relative to the array broadside as given below:
Conventional direction finding algorithms are quite vulnerable to the noise of incoming data due to large side lobe levels in the antenna array beam pattern. In addition these algorithms require a large number of antenna elements to resolve the angles of arrival with precise accuracy [2]. This paper focuses on high resolution direction finding algorithms. These algorithms exploit properties of Eigen structures to provide a solution to an underlying estimation problem for a given application. These algorithms apply Eigen value decomposition techniques on the input covariance matrix of received signal to estimate the signal and noise subspaces [3], [4].
Where de is the element spacing and λ is the wavelength of the received signal. The signal vector Rxd(t) can be written as:
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a(θ) = [ 1 e-jα … e-j(N-1)α ]
(2)
Where α is defined as the phase delay across the antenna array and is given below: α = (2π/λ) x de x sin(θ)
Rxd(t) = A x s(t)
(3)
(4)
Where A = [a(θ1) … a(θP)] is a PxQ matrix of steering vectors. The antenna array output Oarray(t) consists of both signal and noise components and can be written as: Oarray(t) = Rxd(t) + w(t)
(5)
Where Rxd(t) and w(t) are assumed to be uncorrelated and w(t) is modeled as white Gaussian process.
B.
D.
Direction Finding with MUSIC
MUSIC is among the most popular algorithms used for estimating the angles of arrival of the incoming signal sources. It was proposed by Schmidt [5] and involves estimating the signal and noise subspaces through Eigen value decomposition of the input covariance matrix as shown in fig. 1 below.
Direction Finding with ESPRIT
The ESPRIT algorithm was proposed by Roy and Kailath [8] and it works by dividing an antenna array of Q elements into two sub-arrays with Q-1 elements. The two sub-arrays are formed with a fixed displacement vector i.e. a fixed distance in a fixed direction. A concept of sub-arrays for ESPRIT algorithm is given in fig. 2 below:
Figure 2 Sub-Arrays Concept in ESPRIT Algorithm
Figure 1 MUSIC Algorithm Flowchart
In this algorithm, the covariance matrix CM consists of P signal Eigen values that form the signal subspace and Q-P noise Eigen values that form the noise subspace. This noise subspace is orthogonal to the steering vectors as shown in fig. 1 above. The MUSIC spectrum shows peaks appearing at the angles of arrival of the signals incident on antenna array. C.
This algorithm exploits the rotational invariance property in the signal subspace by creating two sub-arrays from a single antenna array with a fixed displacement vector as shown in fig. 2 above. The calculations for determining the angles of arrival using ESPRIT algorithm are shown in fig. 3 below:
Direction Finding with Root-MUSIC
MUSIC algorithm involves a comprehensive search through all possible steering vectors to provide an estimate of the incoming angles of arrival. This procedure makes it a computationally intensive algorithm. On the other hand, Root-MUSIC is a model based parameter estimation techniques proposed by Barabell [6]. It is applicable to uniform linear arrays only and involves rearranging the denominator of Pmusic as shown in fig. 1 above, to form a polynomial R(z) on the unit circle. In this way peaks in Pmusic appear as Roots of R(z) either lying on or close to a unit circle [7]. For a true θ, ej(2πdsin(θ)/λ) is a root of R(z). The angles of arrival are calculated by finding the roots, z = z1, z2 … zm, where m = 1, 2 … P, of this polynomial R(z) as follows: θm = sin-1 [ (λ/(2 π de)) arg(zm) ]
(6)
Figure 3 ESPRIT Algorithm Flowchart
The ESPRIT algorithm provides estimation about the angles of arrival in terms of Eigen values as shown in fig. 3 above and involves less computations and storage as compared to MUSIC.
III.
SIMULATION RESULTS AND ANALYSIS
MATLAB simulations have been developed for evaluating the performance of MUSIC, Root-MUSIC and ESPRIT algorithms in terms of providing accurate estimates about the incoming angles of arrival. In these simulations, the user can input parameters such as number of antenna elements (Q), number of mobile users (P), angles of arrival of the mobile users (Theta), number of time samples (Ts) and signal-to-noise ratio (SNR). Scenario 1: This scenario shows the effects of varying the number of antenna elements with fixed number of mobile users, number of time samples and SNR. The values of parameters used are Q = 8 and 16, P = 4, Theta = {-15o, -8o, 0o, 9o}, Ts = 128, SNR = 10 dB.
Table. 1 Analysis of Root-MUSIC and ESPRIT with Q = 8 Theta
Root-MUSIC
ESPRIT
-15o
-14.9764o
-15.0539o
-8o
-7.0388o
-6.2230o
0o
1.9007o
1.3829o
9o
9.3056o
9.9857o
Table. 2 Analysis of Root-MUSIC and ESPRIT with Q = 16 Theta -15
o
Root-MUSIC -14.9634
o
ESPRIT -15.0261o
-8o
-7.9796o
-7.9943o
0o
0.0200o
-0.0027o
9o
9.0131o
8.9823o
Inference on Scenario 1: The spatial resolution of these algorithms improved with increase in number of antenna elements (Q) as shown in fig. 4 and fig. 5 respectively [9], [10] and [11]. Scenario 2: This scenario shows the effects of varying the number of time samples with fixed number of antenna elements, number of mobile users, and SNR. The values of parameters used are Ts = 128 and 256, Q = 8, P = 3, Theta = {-10o, 0o, 12o}, SNR = 10 dB.
Figure 4 Number of antenna elements Q = 8
Figure 6 Number of time samples Ts = 128 Figure 5 Number of antenna elements Q = 16
Tables 1 and 2 below show the comparison of RootMUSIC and ESPRIT algorithms for the simulation scenario mentioned above.
Figure 7 Number of time samples Ts = 256
Figure 8 Number of mobile users P = 5
Tables 3 and 4 below show the comparison of RootMUSIC and ESPRIT algorithms for the simulation scenario mentioned above. Table. 3 Analysis of Root-MUSIC and ESPRIT with Ts = 128 Theta
Root-MUSIC
ESPRIT
o
-9.2721o
0o
0.0469o
0.0244o
12o
11.6038o
11.5949o
-10
o
-9.3195
Table. 4 Analysis of Root-MUSIC and ESPRIT with Ts = 256 Theta -10
o
Root-MUSIC -9.9961
o
ESPRIT -10.0316o
0o
-0.0560o
-0.0395o
12o
11.8737o
12.0146o
Figure 9 Number of mobile users P = 8
Inference on Scenario 3: The MUSIC algorithm is unable to resolve all the signals in the presence of large number of incident signals (P) closer to the number of antenna elements (Q) as shown in fig. 9. Scenario 4: This scenario shows the effects of varying the SNR with fixed number of antenna elements, number of mobile users, and number of time samples. The values of parameters used are SNR = {-10, 10} dB, Q = 8, P = 3, Theta = {10o, 20o, 30o}, Ts = 128.
Inference on Scenario 2: The spatial resolution of these algorithms improved with increase in number of time samples (Ts) as shown in fig. 6 and fig. 7 respectively [9], [10] and [11]. Scenario 3: This scenario shows the effects of varying the number of mobile users with fixed number of antenna elements, number of time samples and SNR. The values of parameters used are Q = 12, P = 5, Theta = {-30o, -22o, -15o, -5o, 3o}, P = 8, Theta = {-40o, -30o, -22o, -15o, -5o, 3o, 11o, 20o}, Ts = 256, SNR = 10 dB.
Figure 10 SNR value -10 dB
Figure 12 Increased number of mobile users in a low SNR situation
Figure 11 SNR value 10 dB
Tables 5 and 6 below show the comparison of RootMUSIC and ESPRIT algorithms for the simulation scenario mentioned above. Table. 5 Analysis of Root-MUSIC and ESPRIT with SNR = -10 dB Theta 10
o
Root-MUSIC 11.3178
o
ESPRIT 10.7428
o
Table 4 below shows the comparison of Root-MUSIC and ESPRIT algorithms for the simulation scenario mentioned above. Table. 7 Analysis of Root-MUSIC and ESPRIT with low SNR -10 dB Theta -48
o
Root-MUSIC o
ESPRIT
-48.3424 *
-48.8873o
20o
21.1347o
20.3730o
-40o
-40.9461o
-40.9438o
30o
30.9762o
29.0924o
-30o
-31.3881o
-31.3054o
-22o
-22.2377o *
-22.4587o
-15o
-14.7468o *
-14.8415o
-5o
-4.1329o
-4.2605o
3o
2.7235o *
2.6663o
11o
11.6717o
11.5122o
20o
19.9272o *
19.7541o
29o
29.1193o *
29.3672o
Table. 6 Analysis of Root-MUSIC and ESPRIT with SNR = 10 dB Theta
Root-MUSIC
ESPRIT
10o
10.2084o
10.3349o
20
o
20.3500
o
20.5909
o
30
o
30.2524
o
30.3895
o
Inference on Scenario 4: The spatial resolution of these algorithms degrades in a low SNR situation as shown in fig. 10 and table 5 respectively. Scenario 5: This scenario shows the effects of increased number of mobile users in a low SNR situation with fixed number of antenna elements, and number of time samples. The values of parameters used are SNR = -10 dB, Q = 14, P = 10, Theta = {-48o, -40o, -30o, -22o, -15o, -5o, 3o, 11o, 20o, 29o}, Ts = 256.
Inference on Scenario 5: In this case Root-MUSIC performs much better than MUSIC and ESPRIT algorithms as shown in table 7. Root-MUSIC is able to resolve most of the incoming signals marked with *. Scenario 6: This scenario shows the effects of increase in number of antenna elements (Q) along with time samples (Ts) on the execution time of these algorithms. The values of parameters used are Q = {4, 8, 16, 32, 64} with Ts = 256 and 512 respectively.
than MUSIC and ESPRIT algorithms as shown in table 7. Fig. 13 and fig. 14 show that increased values of Q and Ts result in increased execution times of all the direction finding algorithms. The execution time of Root-MUSIC increases drastically with large values of Q and Ts.
V.
Figure 13 Execution times (msec) with variable Q for Ts = 256
CONCLUSIONS
This paper presented a detailed analysis of high resolution direction finding algorithms and their application in mobile communication systems using antenna arrays. This paper presented MATLAB simulations of MUSIC, Root-MUSIC and ESPRIT algorithms that allowed the user to change the design parameters of a linear antenna array i.e. Q, Ts, SNR, P along with values of Theta. Simulation results verified that MUSIC algorithm is a preferable choice for direction finding applications having a large antenna array with a large number of time samples. Simulation results also verified that Root-MUSIC algorithm is a preferable choice for direction finding applications detecting a large number of mobile users with narrow angular separation and low SNR environment, keeping the number of antenna elements and time samples fixed. These results can provide help to system engineers involved in design and analysis of linear antenna arrays for mobile communications.
Figure 14 Execution times (msec) with variable Q for Ts = 512
Inference on Scenario 6: The execution time of these algorithms increases with increase in Q and Ts especially when the value of Ts is high as shown in fig. 13 and fig. 14 respectively.
IV.
DISCUSSION ON OBTAINED RESULTS
Fig. 5 and fig. 7 show that increase in number of antenna elements (Q) and time samples (Ts) improve the performance of all the direction finding algorithms in terms of accurately resolving the incoming signals with sharp peaks and minimal errors in detecting the angles of arrival. Fig. 9 shows that increase in number of mobile users (P) with respect to number of antenna elements (Q) results in poor spatial resolution of MUSIC along with significant errors in angle detection of Root-MUISC and ESPRIT algorithms. This issue can be addressed by increasing the values of Q and Ts at the cost of array size and computational requirements. Fig. 10 shows that as the SNR is reduced, the spatial resolution also reduces for almost all the direction finding algorithms. Fig. 12 shows that increased number of mobile users (P) in a low SNR situation degrades the performance of all the direction finding algorithms. In this case increased values of Q and Ts allowed the Root-MUSIC to perform much better
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