Joint Angle and Delay Estimation for 2D Active ... - IEEE Xplore

Globecom 2013 - Signal Processing for Communications Symposium 1

Joint Angle and Delay Estimation for 2D Active Broadband MIMO-OFDM Systems Yi Zhu †, Lingjia Liu †, and Jianzhong (Charlie) Zhang ‡ †Department of Electrical Engineering and Computer Science, University of Kansas, USA. ‡Wireless Communications Laboratory, Samsung Research America - Dallas, USA.

Abstract—Mobile data traffic is expected to have an exponential growth in the future. In order to meet the challenge as well as the form factor limitation at the base station, 2D “Massive MIMO”, combined with OFDM, has been proposed as one of the enabling technologies to significantly increase the spectral efficiency of a broadband wireless system. In 2D broadband MIMO-OFDM systems, a base station will rely on the spatial information extracted from uplink sounding reference signals to perform downlink MIMO beam-forming. Accordingly, multi-dimensional parameter estimation of a ray-based multipath wireless channel becomes crucial for such systems to realize the predicted capacity gains. In this paper, we study joint angle and delay estimation for 2D broadband MIMO-OFDM systems and analytically assess its estimation performance. To be specific, a tensor-based standard ESPRIT algorithm is naturally applied to estimate the corresponding 3D channel parameters. Further, we give out the closedform mean square error expressions for DoA estimation. It is found that the dimensionality of the antenna array at the base station, as well as the implementation of OFDM, play an important role in determining the estimation performance. Simulation is conducted to evaluate the performance of the tensor-based ESPRIT algorithm, and the empirical results match very well with that from the derived analytical expressions. These insights will be useful for designing practical 2D broadband MIMOOFDM systems in future mobile wireless communications.

I. I NTRODUCTION Rarely have technical innovations changed everyday life as rapidly and profoundly as mobile wireless communications. According to the International Telecommunication Union (ITU) [1], the number of mobile wireless subscriptions globally reached 6.8 billions in 2013, almost as many as the world population 7.1 billions. As a result, in February 2013, Cisco Systems predicted a staggering 66% compound annual growth rate (CAGR) for global mobile data traffic from 2012 to 2017 [2]. This is an 13-fold increase in wireless traffic over a five-year period. To meet the increasing traffic demand, other than reallocating radio spectrum to wireless providers, spectral efficiency will need to be improved significantly. Multipleinput-multiple-out (MIMO) technology, together with Orthogonal frequency-division multiplexing (OFDM), offer efficient ways to increase the spectral efficiency of a mobile broadband communication system [3]. The high data rate wireless transmission scheme OFDM converts a frequency-selective MIMO channel into a parallel collection of frequency flat subchannels, which is beneficial for detection and channel estimation, further the capacity enhancement. It is shown in [4]

978-1-4799-1353-4/13/$31.00 ©2013 IEEE

that unprecedented spectral efficiency and promising system throughput can be obtained through the combination of these two powerful techniques. Because of the sensitivity of MIMO algorithms with respect to the channel matrix properties, channel modeling is particularly critical to assess the performance of underlying MIMOOFDM systems. The parametric channel model could be adopted by performing virtual direction-of-arrival (DoA) and direction-of-departure (DoD) estimation of resolvable paths. It provides a simple geometric interpretation of the scattering environment to characterize the two key MIMO channel metrics: ergodic capacity and diversity level [5]. Despite the advantage of reducing the number of estimation parameters, it is shown in [6] that channel estimation based on DoA and DoD provides the best performance in terms of the error bound. Due to the form factor limitation, two-dimensional (2D) MIMO-OFDM systems are introduced regarding elevation and azimuth domain to fit a large number of antenna elements on the base station in reality. For a base station equipped with such planar arrays, it needs to know the corresponding multi-dimensional channel knowledge. There are many existing subspace-based methods such as MUSIC, ESPRIT and matrix pencil to estimate the one-dimensional (1D) DoA under parametric channel models. However, its counterpart in 2D, together with delay estimation is not yet well explored. The TST-MUSIC algorithm proposed in [7] have great performance in estimating the DoAs and delay of a wireless multi-ray channel, but it can only solve the problem in the case of one close parameter and the complexity is still prohibitively high. In [8], the authors just show the M-dimensional estimation of spatial frequencies using tensor modeling without mentioning the individual physical parameters estimation, which are crucial for practical MIMO transceiver design. Hence, in our paper, we define a natural tensor-based system model for the application of joint angle and delay estimation (JADE). Further, closed-form expressions regarding DoA estimation errors are derived based on perturbation theory. By jointly estimating the channel parameters, DoA estimation could give us accurate spatial information about the fourdimensional (4D) underlying physical channel which is crucial for transmit precoding. A detailed performance comparison between empirical tensor-based approach and numerical evaluation will be given under various antenna configurations. It shows us how dimensionality and practical implementation will impact the channel estimation performance of a 2D MIMO-OFDM system.

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II. DATA M ODEL A typical 2D “Massive MIMO” system with an M1 × M2 antenna array at the base station can be shown in Fig. 1 [9]. In

λ denotes the wavelength. αℓ and τℓ are the complex amplitude and time delay of the ℓth path, respectively. Since time delay can be mapped to phase shift in frequency domain through a Fourier transform, the channel transfer function H(f ) with dimension M1 × M2 is obtained

Z

H(f ) =

P ∑

αℓ a(uℓ , f )aT (vℓ , f )e−j2π(f −fc )τℓ ,

ℓ=1

where a(uℓ , f ) = [1, e−juℓ (f ) , . . . , e−j(M1 −1)uℓ (f ) ]T and a(vℓ , f ) = [1, e−jvℓ (f ) , . . . , e−j(M2 −1)vℓ (f ) ]T . The solid angles of path ℓ are denoted by uℓ (f ) = k(f )d cos(θℓ ) and vℓ (f ) = k(f )d sin(θℓ ) cos(ϕℓ ), where k(f ) is the frequency dependent wavenumber, represented as k(f ) = 2πf c . fc is the reference frequency of the frequency band of interest. Accordingly, we can express the channel frequency response at the qth subcarrier as, 1 ≤ q ≤ Nc

d

Ɵ d

X

Y

ɸ

h

hm

Fig. 1.

H(fq ) =

Model of 2D “Massive MIMO” System

P ∑

αℓ a(uℓ , fq )aT (vℓ , fq )e−j2π(q∆f −fc )τℓ .

(1)

ℓ=1

this particular system, a base station is at the height of h while a mobile station is at the height of hm . The antenna array at the base station is a planar array placed in the X-Z domain with M1 antenna elements vertically and M2 antenna elements horizontally. The spacing between adjacent sensors is assumed to be d, without loss of generality, we fix it to the critical half wavelength distance in order to avoid the spatial spectrum alias. For simplicity, throughout the paper, we assume that there is only one transmit antenna at a mobile station. In the 2D “Massive MIMO” system, instead of mechanical down-tilting the antenna array towards the mobile station, the base station could also perform digital beam-forming in both elevation and azimuth domain, because 2D DoA estimation will provide the base station some channel knowledge on the downlink. In reality, the propagation situation in a wireless communication system is rather complicated. The uplink sounding reference signals usually go through scattering, reflection, refraction, and diffraction before they reach the base station. For a multipath scenario, a wireless channel is usually modeled by a finite number of rays, each parameterized by a complex amplitude, angle and time delay. Following our line of work in [10], we extend it to a practical broadband MIMO-OFDM system with Nc subcarriers. Sub-channels are defined as the spatial channels from one transmit antenna to another receive antenna. Each subchannel consists of P independent paths, and each of which has the corresponding elevation DoA θℓ , azimuth DoA ϕℓ and delay τℓ . Here, the channel is assumed to be static within each OFDM block period, but it may vary from one block to another. Thus for each OFDM block, the discrete time channel impulse response is shown to be h(t) =

P ∑

αℓ a(uℓ )aT (vℓ )δ(t − τℓ ).

ℓ=1

Here, a(uℓ ) = [1, e−juℓ , . . . , e−j(M1 −1)uℓ ]T and a(vℓ ) = [1, e−jvℓ , . . . , e−j(M2 −1)vℓ ]T . uℓ , vℓ represent two spatial fre2πd quencies, expressed as uℓ = 2πd λ cos θℓ , vℓ = λ sin θℓ cos ϕℓ ,

Here, ∆f = 1/T0 is the subcarrier separation in the frequency domain, and T0 denotes the OFDM symbol period. It is clearly seen from (1) that, the array steering vector now is frequency-dependent. The main difficulty in extending the signal subspace approach for processing wide-band signals comes from the fact that the signal subspace at one frequency will be different from that at another frequency. In order to apply the low-complexity ESPRIT algorithm, we first decompose the array output vector in the temporal domain into nonoverlapping narrow-band spectral components by using Discrete Fourier Transform (DFT). And then narrow-band signal subspace processing can be performed over individual bins [11]. Hence, if the time duration for DFT is sectioned into M3 subintervals (M3 is the total number of snapshots in the usual terminology of array signal processing), we can transform the broadband system problem to a combination of individual narrow-band processing results. It is at this point to collect the last exponential term in (1) into a M3 × 1 vector containing delay, the third parameter of interest. The resulting received measurement data can be explicitly written as ym1 ,m2 ,m3 ,i =

P ∑

αℓ si e−j(m1 −1)uℓ (fq ) e−j(m2 −1)vℓ (fq )

ℓ=1 −j(m3 −1)ϖℓ (fq )

e

(2)

+ wm1 ,m2 ,m3 ,i ,

where m3 = 1, 2, . . . , M3 . si denotes the transmitted data symbols, i = 1, 2, . . . , K where K is the number of symbols. 2π τℓ is the newly defined “spatial” frequency, ϖℓ (fq ) = M 3 which we got intuition from [12] that the last exponential term carrying delay shares exactly the same form as the other two spatial frequencies. Since our received signal is referenced by 4 indices, the most common way to handle this multi-dimensional measurement data is to stack dimensions into a highly structured matrix. Hence, we stack the three “spatial” dimensions into rows and signal snapshots along columns. Specifically, let xℓ,i = αℓ si , the matrix-based data model is shown to be

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Y = AX + W,

(3)

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where Y, W ∈ C M1 M2 M3 ×K are the received data matrix and stacked noise matrix, respectively. A ∈ C M1 M2 M3 ×P represents the array manifold matrix, containing all the information regarding desired parameters. X is the P × K matrix of attenuated transmitted data. However, this stacked data model cannot exploit the essential structure of the original received signal. Thus, we define a tensor-based system model to collect all the samples into one multi-dimensional array (MDA), the 4D observation data Y can be expressed as: Y = A ×4 X T + W

(4)

Here, Y ∈ C M1 ×M2 ×M3 ×K denotes the measurement samples of received signal, while W ∈ C M1 ×M2 ×M3 ×K collects all the noise samples. A ∈ C M1 ×M2 ×M3 ×P is referred to as the array steering tensor, which can be expressed as A = I4,P ×1 A(1) ×2 A(2) . . .×3 A(3) , I4,P is the rank-4 identity tensor with each mode a P × P identity matrix. The array response matrices in each mode are shown to be (1)

= [a(u1 , fq ), a(u2 , fq ), . . . , a(uP , fq )]

(2)

= [a(v1 , fq ), a(v2 , fq ), . . . , a(vP , fq )]

A A

A(3) = [a(̟1 , fq ), a(̟2 , fq ), . . . , a(̟P , fq )] where a(̟ℓ , fq ) = [1, e−j̟ℓ (fq ) , . . . , e−j(M3 −1)̟ℓ (fq ) ]T . As we have the identity relationship between matrix and tensor, e.g., A = [A]T(R+1) , the tensor-based data model can be equivalently transformed to the matrix-based data model in frequency domain. Most approaches and results obtained from matrix point of view are readily applied for the tensor case. III. A NALYTICAL P ERFORMANCE

OF

JADE E STIMATION

For a wireless communication system, there always exists various perturbations when we are trying to estimate the signal subspace from the received data Y . We have shown that under matrix case, the subspace decomposition of the perturbed data can be performed by a singular value decomposition (SVD) !   ˆs H
Σˆs 0 V . Y = Uˆs Uˆo H 0 Σˆo Vˆo

The columns in Uˆs are the left singular vectors correspond to P largest singular values in Σˆs with dimension P × P , spanning the estimated signal subspace; while the vectors in Uˆo correspond to the other singular values in Σˆo , spanning the estimated noise subspace. Vˆs and Vˆo are the corresponding right singular vectors. Due to the noise presented in the received data, we have ˜s and Uˆo = Uo + U ˜o , Uˆs = Us + U where Us denotes the true signal subspace and U˜s are the perturbations of estimated signal subspace. Uo is the true noise subspace, while U˜o is the corresponding estimation error. In the high signal-to-noise ratio (SNR) regime, we can use a first-order perturbation expansion to get a linear approximation of the perturbed subspaces as follows [13]

Here, Γ and Υ are matrices whose norms are of the order of ||Y ||. The matrix norm here can be any submultiplicative norm, such as the Frobenius norm. Accordingly, explicit expressions for mean square error (MSE) of DoA estimation can be obtained using ESPRIT-type methods, a common model with shift invariance property is given by J1 AΦ = J2 A, where selection matrices Ji , i = 1, 2 stand for the first and last M2 − 1 rows of the M1 M3 blocks of the array steering matrix for maximum overlapping. Φ is a diagonal matrix whose entries are the phase shift of adjacent elements horizontally. Since the array steering matrix A spans the same subspace as Us , there exists an invertible matrix T such that Us = A T . Note that, this holds approximately when the received signal is noise corrupted or with a finite number of measurements. Suppose we have the transition matrix Λ = T −1 ΦT , J1 U s Λ = J2 U s .

(5)

Using appropriate least square type of methods, we will be able to solve equation (5) to obtain the estimated spatial frequency of vℓ , which are the eigenvalues of Λ. Another two spatial frequencies can be estimated in the same way. The main result in [13], which is the closed-form MSE expressions derived from first-order perturbation analysis is written as
H −1 ∆vℓ = Im kℓ (J1 Us )† (J2 λ−1 ℓ − J1 )Uo Uo W Vs Σs jℓ ,

where λℓ = ejvℓ . jℓ and kℓ are the ℓth left and right orthonormal eigenvectors of Λ, e.g., jℓ = T −1 eℓ , eℓ = [0, . . . , 1, . . . , 0]T has a 1 in its ℓth position. † denotes the matrix pseudo-inverse. In general, this method can be applied to multi-dimensional cases [14]. A tensor-based subspace estimate will be achieved by a truncated Higher-order SVD (HOSVD) of Y, writtern as [15] ˆ [s] U ˆ [s] ˆ [s] ˆ −1 U [s] ≈ Sˆ ×1 U 1 ×2 2 ×3 U3 ×4 Σ[s] ,

(6)

ˆr ∈ where Sˆ ∈ C p1 ×p2 ×p3 ×P is the truncated core tensor and U Mr ×pr C for r = 1, 2, 3 are the matrices of dominant r-mode singular vectors. pr denotes the rank of r-mode singular matrix ˆ [s] represents the diagonal matrix containing unfolding. Here, Σ the P dominant singular values of Y on its main diagonal. Since the HOSVD is conducted along SVDs of the separate matrix unfoldings, we may apply the same approach in [13] to find closed-form expressions of perturbation expansion for higher-order subspace estimate. Assume the 3D array steering tensor features shiftinvariance in each of its modes, we can directly apply multidimensional standard ESPRIT algorithm to write the shiftinvariance equations as

˜s = Us + Uo Γ and U ˜o = Uo + Us Υ. U

3302

(1) ×4 Θ (2) A×2 J1 ×4 Φ (3) A×3 J1 ×4 Ψ

A×1 J1

(1)

= A×1 J2

(2)

= A×2 J2 =

(3) A×3 J2 ,

(7)

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where Θ = diag[ϑ1 , . . . , ϑP ], Φ = diag[ϕ1 , . . . , ϕP ] and Ψ = diag[ψ1 , . . . , ψP ] are the corresponding eigen-matrices for three desired parameters, ϑℓ = exp(−j 2πd λ cos θℓ ), ϕℓ = (r) 2π cos φ sin θ ) and ψ = exp(−j ∈ exp(−j 2πd ℓ ℓ ℓ λ M3 τℓ ) . Ji (Mr −1)×Mr R , i = 1, 2 represent the selection matrices for the rth mode under maximum overlapping. Since the array steering tensor approximately spans the same vector space as the estimated signal subspace U [s] , similar to the matrix case, we have A≈U

[s] ¯ ×4 T

for some P × P nonsingular transform matrix T¯. Substitute the above relation back to (7), [s] (1) ×1 J1 ×4 Θ [s] (2) U ×2 J1 ×4 Φ [s] (3) U ×3 J1 ×4 Ψ

U

[s] (1) ×1 J2 [s] (2) U ×2 J2 [s] (3) U ×3 J2 .

≈U ≈ ≈

Proof sketch: The elevation angle θℓ , azimuth angle φℓ and path delay τℓ are related to the three spatial frequencies through
    λuℓ arccos θℓ  2πd    ℓ g(Θℓ ) =  φℓ  =  arccos 2πdλvsin . θℓ M3 ̟ℓ τℓ 2π

T

Consequently, we can see that an improved signal subspace estimate will lead to a better estimation performance. Applying least square type of methods, estimation of parameters of interest are obtained through the standard ESPRIT algorithm with low complexity and high estimation accuracy. Note that, we still need to conduct a joint Schur decomposition or a simultaneous digitalization algorithm to ensure the correct pairing of different paths over modes. The natural extension to R = 3 explicit expressions of MSE for spatial frequencies is given in Lemma 1. Lemma 1. For the case of 2D JADE application based on a uniform planar array of size M1 × M2 , assume the perturbation term is circular symmetric white noise with zero mean and variance σ 2 . Let M = M1 M2 M3 , the MSE of the ℓth spatial frequency in the rth mode, r = 1, 2, 3, is given by E[((∆uℓ )r )2 ] =

perturbation term is circular symmetric white noise with zero mean and variance σ 2 . The MSEs of the corresponding elevation and azimuth angle estimations are 1 var (θℓ ) = (∆uℓ )2 2 2 π sin θℓ 2 2 (∆uℓ ) π cos2 θℓ cos2 φℓ + (∆vℓ )2 var (φℓ ) = . π 2 sin2 θℓ sin2 φℓ

Mr 1 γℓ M (Mr − 1)2

where γℓ = Rˆs (ℓ, ℓ)/σ 2 is the effective received SNR at the base station for the ℓth path, and Rˆs = XX H is the estimated signal covariance matrix. Lemma 1 is proved in [14] as Theorem 8. However, we extend it here to joint angle and delay estimation application, to see the impact of practical antenna design onto estimation performance. Recall that in our model for 2D antenna array, the two spatial frequencies uℓ and vℓ in the steering matrix are related to the elevation and azimuth angles of interest, respectively. Let (∆uℓ )2 and (∆vℓ )2 represent the MSEs of two spatial frequencies for notation simplicity, 1 1 , γℓ M2 M3 (M1 − 1)2 1 1 (∆vℓ )2 = . γℓ M1 M3 (M2 − 1)2

(∆uℓ )2 =

where Θℓ = [θℓ , φℓ , τℓ ] denotes the desired parameter vector. Note that in the parameter range of our interests (e.g., θℓ ∈ [0, π/2] and φℓ ∈ [0, 2π/3]), the above mapping is one-to-one and the antenna spacing between adjacent elements is typically d = λ/2. We can obtain the covariance matrix of the elevation and azimuth angle through a Jacobian operation as   1 − π sin 0 ∂g (Θℓ ) θℓ . = cot θℓ cot φℓ − π sin θ1ℓ sin φℓ ∂Θℓ Therefore, the expressions of MSEs for the elevation and azimuth angle estimation are the (1, 1)th and (2, 2)th element of the covariance matrix respectively.  IV. S IMULATION R ESULTS In this section, we evaluate the tensor-based joint estimation algorithm under various antenna configurations, to see the impact of practical implementation on estimation performance. Assume it is a two paths situation with DoAs [70, 77]◦ for elevation and [45, 60]◦ for azimuth, path fading amplitudes [1, 0.8] and time delay [0.5, 2.1]s. We take a 64 point FFT to demodulate the received data, which is also the the number of subcarriers in the corresponding OFDM system. For a 1D antenna array, it is well known that a larger number of antenna elements will result in a better DoA estimation performance. Accordingly, it seems to be “natural” that a 2×32 antenna array may have better azimuth estimation compared to that of a 4 × 16 antenna array due to the fact that there are more antenna elements in the azimuth domain. We focus on azimuth angle estimation is because, for a fixed number of total antenna elements, the azimuth angle estimation is more crucial for MIMO beam-forming in cellular systems. However, we may see this conclusion actually turns out to be wrong for ESPRIT-type methods in the high SNR regime. It can be simply calculated from Theorem 1 that the corresponding MSEs of azimuth angle estimation for a 4 × 16 antenna array and that for a 2 × 32 structure are (for fixed SNR 10 dB)

Accordingly, we can characterize the MSEs for both elevation and azimuth angle estimation as shown in Theorem 1.

var(φℓ )4×16 = 3.1 × 10−5 , var(φℓ )2×32 = 5.9 × 10−5 .

Theorem 1. For the case of 2D JADE application based on a uniform planar array of size M1 × M2 , assume the

This result suggests that the antenna array with less horizontal elements performs even better, which sounds a little

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Tensor−based Azimuth Angle Estimation

−1

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2 × 32 Azimuth 4 × 16 Azimuth 8 × 8 Azimuth 16 × 4 Azimuth 32 × 2 Azimuth

−2

10

−3

10 MSE (log)

bit counter-intuitive. Hence, we will evaluate the estimation performance thoroughly through computer simulation, under SNR ranging form −6 dB to 24 dB (dynamic range of SNR in a cellular environment). All the results are based on 500 Monte-Carlo runs. The performance of joint DoA estimation given by tensorbased ESPRIT algorithm is evaluated in Fig. 2 and Fig. 3 under various antenna configurations. We can see from Fig. 2 that the elevation angle estimation performance is proportional to the number of antennas vertically, also the MSE decreases as the SNR increases. This matches with our conventional understanding that for 1D antenna array, a larger number of sensors will contribute to a better performance. However, it is interesting to note that the estimation performance of azimuth angle doesn’t scale proportionally to the number of antennas horizontally, as shown in Fig. 3. We observe that the MSE of azimuth estimation of a 2 × 32 array is even larger than that of 4 × 16 and statistically similar to that of 8 × 8. It is already indicated by our simple calculation in the beginning of the section. The reason this phenomenon happens is because azimuth estimation is actually coupled with elevation estimation, more specifically, it depends on the elevation angle estimation performance. In the case of 2 × 32 antenna configuration, the performance of elevation is so poor that it affects the performance of azimuth estimation. Hence, for joint estimation of three parameters, we can still get the fact that the azimuth angle estimation is more vulnerable compared to elevation angle estimation, as observed in [9].

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Fig. 3. Empirical Azimuth Angle Estimation given by tensor-based ESPRIT Algorithm Under Various Antenna Configurations

small perturbation. If SNR tends to infinity, we can show the analytical results agree with those estimation errors through empirical experiment, but have no meaning to practical system design for such a high SNR.

Closed−Form MSE of Elevation Angle Estimation

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2 × 32 Elevation 4 × 16 Elevation 8 × 8 Elevation 16 × 4 Elevation 32 × 2 Elevation

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Tensor−based Elevation Angle Estimation

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2 × 32 Elevation 4 × 16 Elevation 8 × 8 Elevation 16 × 4 Elevation 32 × 2 Elevation

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Fig. 4. Numerical Evaluation of Elevation Angle Estimation Under Various Antenna Configurations

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Fig. 2. Empirical Elevation Angle Estimation given by tensor-based ESPRIT Algorithm Under Various Antenna Configurations

Additionally, Fig. 4 and Fig. 5 evaluate the closed-form expressions of MSE for DoA estimation under various antenna configurations. It can be seen that the trend of empirical estimation errors meet well with the results given by analytical expressions. In order to see their performance matchness, we plot the “best” four curves together in Fig. 6. It is found quite promising to use these closed-form expressions for predicting the angle estimation performance, especially in high SNR scenario. This is because the theoretical analysis based on perturbation expansion is only valid for relatively

In order to show the superiority of tensor-based ESPRIT algorithm, a performance comparison between our proposed method and existing approaches is performed. In Fig. 7, we adopt the matrix-based standard ESPRIT algorithm using shiftinvariance property, by stacking the measurement data into a highly-structural matrix. We still focus on azimuth angle estimation only, and it can be see from Fig. 7 that, the trend of estimation performance is similar to that in Fig. 3. However, the estimate accuracy is improved almost a half magnitude of order. This result matches well with our former analytical derivation, and also coincides with the performance gap observed in Fig. 3 and Fig. 4 in [14].

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Closed−Form MSE of Azimuth Angle Estimation

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2 × 32 Azimuth 4 × 16 Azimuth 8 × 8 Azimuth 16 × 4 Azimuth 32 × 2 Azimuth

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2 × 32 Azimuth 4 × 16 Azimuth 8 × 8 Azimuth 16 × 4 Azimuth 32 × 2 Azimuth

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Fig. 5. Numerical Evaluation of Azimuth Angle Estimation Under Various Antenna Configurations

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R EFERENCES

4 × 16 Empirical Azimuth 4 × 16 Analytical Azimuth 32 × 2 Empirical Elevation 32 × 2 Analytical Elevation

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Empirical and Analytical Performance Comparison

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Fig. 6.

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Fig. 7. Empirical Azimuth Angle Estimation given by matrix-based ESPRIT Algorithm Under Various Antenna Configurations

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Empirical and Numerical Simulation Performance Comparison

V. C ONCLUSION In this paper we analytically characterize the DoA estimation performance based on ESPRIT-type algorithm for 2D MIMO-OFDM systems in high SNR regime. A tensor-based system model concerning joint angle and delay estimation is introduced and the mean square error for both elevation and azimuth angle estimation are identified. For the 3D channel sounding, we can still observe that the azimuth angle estimation actually depends on the elevation angle estimation. It is also found that the performance of DoA estimation depends heavily on the underlying antenna array configuration, as well as the practical implementation of OFDM signalling. This may provide some insight to the system design for future broadband 2D MIMO-OFDM research. VI. ACKNOWLEDGEMENT Y. Zhu and L. Liu’s work is sponsored by Samsung Research America - Dallas (SRA-D) under grant IND0070657. The authors would like to thank Dr. Anding Wang and Dr. Krishna Sayana for useful discussions.

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