A Hybrid Adaptive Antenna Array - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 5, MAY 2010

A Hybrid Adaptive Antenna Array Xiaojing Huang, Member, IEEE, Y. Jay Guo, Senior Member, IEEE, and John D. Bunton, Senior Member, IEEE

Abstract—Owing to the excessive demand on signal processing and space constraint, a full digital implementation of a large adaptive antenna array at millimeter wave frequencies is very challenging. Targeted at long range high data rate point-topoint link in the 70/80 GHz bands, a novel hybrid adaptive antenna array which consists of analogue subarrays followed by a digital beamformer is presented in this paper to overcome the digital implementation difficulty. Two subarray configurations, the interleaved subarray and the side-by-side subarray, are proposed, and two Doppler resilient adaptive angle-of-arrival estimation and beamforming algorithms, the differential beam tracking (DBT) and the differential beam search (DBS), are developed. Simulation results on the DBT and DBS performance are provided using a 64 element hybrid planar array of four 4 by 4 element subarrays with the two subarray configurations, respectively. Recursive mean square error (MSE) bounds of the developed algorithms are also analyzed and compared with simulated MSEs. Index Terms—Adaptive antenna array, subarray, beamforming, angle-of-arrival estimation, and mm-wave communications.

I. I NTRODUCTION

A

DAPTIVE antenna arrays have found a wide rage of applications and are becoming the essential parts of the wireless communications systems [1–4]. The use of adaptive antenna arrays for millimeter wave (mm-wave) long range high data rate mobile and ad hoc communications networks is particularly critical due to the limited output power of the monolithic microwave integrated circuits (MMIC) [5,6]. Combining multiple antennas, each of which has its own low noise amplifier (LNA) or power amplifier (PA), to form an antenna array not only increases the communications range but also enables the smart antenna technology to be applied to optimize the system performance. There are several frequency bands available at mm-wave frequencies for high data rate applications, such as the 60 GHz (57-64 GHz), 70 GHz (71-76 GHz) and 80 GHz (81-86 GHz) bands [7–10]. According to the mm-wave propagation characteristics, the 60 GHz band is located at the atmospheric absorption peak, and thus is only suitable for short range transmission [10]. In addition, for most indoor 60 GHz applications, the transmission channels will incur multipath propagation [8]. However, the 70/80 GHz bands (known as the E-bands) are located in an atmospheric window where the attenuation is mild, thus permitting multiple kilometer Manuscript received July 8, 2009; revised November 23, 2009 and March 3, 2010; accepted March 7, 2010. The associate editor coordinating the review of this paper and approving it for publication was C. Yang. The authors are with the Wireless and Networking Technologies Laboratory, CSIRO ICT Centre, Cnr Vimiera and Pembroke Roads, Marsfield, NSW 2122, Australia (e-mail: {Xiaojing.Huang, Jay.Guo, John.Bunton}@csiro.au). The research is partially funded by Boeing Company. Digital Object Identifier 10.1109/TWC.2010.05.091020

communications range [7,10]. With increased transmission power after using antenna array, there is an opportunity to realize even longer range point-to-point line-of-sight (LOS) links which are the targeted applications, such as those providing wireless connectivity between aircrafts and/or between aircrafts and ground vehicles or control stations. Since the antenna elements in an array must be placed closely together to prevent grating lobes, the analogue components, such as the LNA or PA and the down or up converter associated with each antenna element, must be tightly packed behind the antenna element. This space constraint appears to be a major engineering challenge at mm-wave frequencies. For example, at 72 GHz frequency, the required element spacing is only about 2 mm. With the current MMIC technology, the practical implementation of such a digital antenna array remains very difficult [11,12]. Another issue with pure digital beamformers is the excessive demand on real time signal processing for high gain antennas. To achieve an antenna gain of over 30 dB, for instance, one may need more than 1000 antenna elements. This makes most beamforming algorithms impractical for commercial applications. Furthermore, to perform wideband digital beamforming, each signal from/to an antenna element is normally divided into a number of narrowband signals and processed separately, which also adds to the cost of digital signal processing significantly. Therefore, a full digital implementation of large, wideband antenna array at mm-wave frequencies is simply unrealistic [13]. Finally, although multipath is not a major concern for the above mentioned LOS applications, the relative movement between transmitters and receivers will bring other technical challenges such as fast Doppler frequency shift and time-varying angleof-arrival (AoA) of the incident beam. In this paper, a novel hybrid adaptive receive antenna array is proposed to solve the digital implementation complexity problem in large arrays for long range high data rate mmwave communications. In this hybrid antenna array, a large number of antenna elements are grouped into analogue subarrays. Each subarray uses an analogue beamformer to produce beamformed subarray signal, and all subarray signals are combined using a digital beamformer to produce the final beamformed signal [14]. Each element in a subarray has its own radio frequency (RF) chain and employs an analogue phase shifting device at the intermediate frequency (IF) stage. Signals received by all elements in a subarry are combined after analogue phase shifting, and the analogue beamformed signal is down-converted to baseband and then converted into digital domain. In this way, the complexity of the digital beamformer is reduced by a factor equal to the number of elements in a subarray. For example, for a 1024 element hybrid array of 64 subarrays each having 16 elements, only 64 inputs

c 2010 IEEE 1536-1276/10$25.00 ⃝

HUANG et al.: A HYBRID ADAPTIVE ANTENNA ARRAY

to the digital beamformer are necessary, and the complexity is reduced to one sixteenth for algorithms of linear complexity, such as the least mean squire (LMS) algorithm[1,2], as well as digital hardware cost. The digital beamformer estimates the AoA information to control the phases of the phase shifters in the analogue subarrays and also adjusts the digital weights applied to the subarray output signals to form a beam. It should be noted that the subarray technology has been used over the past decades [3, 15–17, 24]. Major ideas include employing a time delay unit to each phased subarray for bandwidth enhancement, and eliminating phase shifters in the subarray for applications requiring only limited-field-of-view. The proposed hybrid antenna array concept differs in that it is a new architecture allowing the analogue subarrays and the low complexity digital beamformer to interact with each other to accommodate the current digital signal processing capability and MMIC technology, thus enabling the implementation of a large adaptive antenna array. The AoA estimation and beamforming algorithms suitable for the proposed hybrid antenna architecture are also significantly different from the conventional ones [18–23]. First, since the inputs to the digital beamformer are the analogue beamformed signals which are obtained based on a previously estimated AoA and an estimated AoA must be fed back to the analogue beamformer, the adaptive AoA estimation in the digital beamformer must be recursive in nature. Due to the architecture difference, most conventional AoA estimation techniques can not be directly applied to the hybrid array. Second, since only the LOS incident beam needs to be considered in our targeted applications, the AoA estimation can be performed more efficiently than many conventional AoA estimation techniques. It is well known that AoA estimation techniques can be classified into two categories, i.e., the spatial spectral based and the parametric methods. The conventional spatial spectral based algorithms, such as the Bartlett beamforming [18] and the various subspace-based methods (e.g., MUSIC [20,21]), need to obtain a spectrum of the AoAs and then find the spectral peaks, whereas the parametric methods, e.g., the ones based on maximum likelihood (ML) principle [22,23], require multi-dimensional search to solve an optimization problem in which the global convergence may not be guaranteed. These techniques are too costly to use in a complexity-reduced digital implementation of a large array. Third, the proposed algorithms make use of the phase difference between adjacent received subarray signals to estimate the AoA information, which not only removes the necessity of a known reference signal or signal synchronization, but also leads to a Doppler resilient solution. The rest of the paper is organized as follows. In Section II, two types of hybrid antenna arrays with interleaved and side-by-side subarray configurations respectively are proposed and the received subarray signal models are given. Section III presents and compares the two corresponding AoA estimation algorithms, referred to as differential beam tracking (DBT) and differential beam search (DBS). Section IV formulates the AoA estimation as a phase estimation problem in the presence of recursive nuisance parameters and derives a recursive mean square error (MSE) bound for the estimation. Section V provides simulation results to demonstrate the performance

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d

d

d

4d 4d

4d

Fig. 1. Hybrid antenna arrays of interleaved linear subarrays (top left) and planar subarrays (top right) versus hybrid antenna arrays of side-by-side linear subarrays (bottom left) and planar subarrays (bottom right). Different subarrays are shown in different fill patterns.

of the proposed algorithms. Finally, conclusions are drawn in Section VI. II. H YBRID A RRAY C ONFIGURATIONS AND S IGNAL M ODELS A. Hybrid Array of Subarrays A subarray is a subset of elements in an antenna array [3]. In the presented research, each element in a subarray is connected to an analogue phase shifter. The received signals from individual elements after phase shifting are combined to produce the output signal of the subarray, which is called analogue beamforming. The proposed hybrid array can be constructed by two or more such subarrays. The subarrays can be interleaved or placed side-by-side with each other. Some examples of the hybrid arrays of subarrays are illustrated in Fig. 1, where the distance between adjacent elements in a subarray is referred to as the subarray element spacing and the distance between corresponding elements in adjacent subarrays is termed the subarray spacing. For the hybrid arrays of interleaved subarrays shown in Fig. 1 the subarray element spacing is 2d and the subarray spacing is d, whereas for the hybrid arrays of side-by-side subarrays shown in Fig. 1 the subarray element spacing is d and the subarray spacing is 4d. The subarray output signals are converted into digital signals at baseband via analogue-to-digital converters (A/D). Then, digital beamforming is performed to control the phase shifters in the subarrays as well as the digital weights associated with respective subarray output signals. The hybrid beamformer structure is illustrated in Fig. 2 using the linear array of two side-by-side subarrays, where the RF and downconversion devices are not shown for simplicity. B. Received Signal Models Denoting the received signal of the mth subarray, 𝑚 = 0, 1, ⋅ ⋅ ⋅ , 𝑀 − 1, where 𝑀 is the total number of subarrays, as 𝑠𝑚 (𝑡), and the received information-bearing signal as 𝑠˜ (𝑡),

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 5, MAY 2010 ~ s (t )e j 2πf Dt

where 𝑃𝑠 (𝜃, 𝜙) =

α0

α1

α2

α3

α0

α1

α2

α3

Phase Shifter

Phase Shifter

Phase Shifter

Phase Shifter

Phase Shifter

Phase Shifter

Phase Shifter

Phase Shifter

Analogue Beamformer

A/D

Adaptive AoA Estimation Weight

Weight

w0∗

w1∗

Digital Beamformer

𝑠 [𝑛] =

s[n]

𝑀−1 ∑ 𝑚=0

Fig. 2. Illustration of hybrid beamformer structure with two side-by-side linear subarrays.

the received subarray signal can be expressed as 𝑠𝑚 (𝑡) =˜ 𝑠 (𝑡) 𝑒𝑗2𝜋𝑓𝐷 𝑡

𝑁 −1 ∑

𝑃𝑖,𝑚 (𝜃, 𝜙)

𝑖=0

⋅ 𝑒𝑗 [ 𝜆𝑐 (𝑋𝑖,𝑚 sin 𝜃 cos 𝜙+𝑌𝑖,𝑚 sin 𝜃 sin 𝜙)+𝛼𝑖,𝑚 ] + 𝑛𝑚 (𝑡) 2𝜋

𝑚 = 0, 1, ⋅ ⋅ ⋅ , 𝑀 − 1

2𝜋

(3) is the overall radiation pattern of a subarray. In general, a two-dimensional subarray can have 𝑁 = 𝑁𝑥 × 𝑁𝑦 elements, where 𝑁𝑥 and 𝑁𝑦 are the numbers of elements placed along x-axis and y-axis respectively. The location (𝑋𝑖 , 𝑌𝑖 ) of the ith element, 𝑖 = 𝑖𝑦 𝑁𝑥 +𝑖𝑥 , is given by 𝑋𝑖 = 𝑋0 + 𝑖𝑥 𝑑𝑒𝑥 , 𝑖𝑥 = 0, 1, ⋅ ⋅ ⋅ , 𝑁𝑥 − 1, and 𝑌𝑖 = 𝑌0 + 𝑖𝑦 𝑑𝑒𝑦 , 𝑖𝑦 = 0, 1,⋅ ⋅ ⋅ , 𝑁𝑦 − 1, where 𝑑𝑒𝑥 and 𝑑𝑒𝑦 are the subarray element spacings along x-axis and y-axis respectively, and (𝑋0 , 𝑌0 ) is the location of the element numbered 𝑖 = 0. Finally, denoting the digital weight applied to the mth ∗ subarray signal as 𝑤𝑚 , the overall digital beamformed output signal at 𝑡 = 𝑛𝑇 , where 𝑇 is the sampling period, is

s1 (t )

A/D

𝑃𝑖 (𝜃, 𝜙) 𝑒𝑗 [ 𝜆𝑐 (𝑋𝑖 sin 𝜃 cos 𝜙+𝑌𝑖 sin 𝜃 sin 𝜙)+𝛼𝑖 ]

𝑖=0

Analogue Beamformer

s 0 (t ) Feedback

𝑁 −1 ∑

(1)

where 𝑓𝐷 is the Doppler frequency shift due to the relative movement between the transmitter and the receiver, 𝑁 is the total number of elements in a subarray, 𝑃𝑖,𝑚 (𝜃, 𝜙) is the radiation pattern of the ith element located at (𝑋𝑖,𝑚 , 𝑌𝑖,𝑚 ) in the mth subarray, 𝜃 and 𝜙 are the zenith and azimuth angles respectively, 𝜆𝑐 is the wavelength of the RF signal, 𝛼𝑖,𝑚 is the phase shifted by the ith phase shifter of the mth subarray, and 𝑛𝑚 (𝑡) is the additive white Gaussian noise presented at the output of the mth subarray. Furthermore, ignoring the mutual coupling between elements and other antenna imperfection, we assume that all subarrays are the same, i.e., 𝑃𝑖,𝑚 (𝜃, 𝜙) = 𝑃𝑖 (𝜃, 𝜙) and 𝛼𝑖,𝑚 = 𝛼𝑖 . Also, we assume that the number of subarrays can be expressed as 𝑀 = 𝑀𝑥 × 𝑀𝑦 , where 𝑀𝑥 and 𝑀𝑦 are the numbers of subarrays placed along x-axis and yaxis respectively, and that the locations of the ith elements in the mth subarray, 𝑚 = 𝑚𝑦 𝑀𝑥 + 𝑚𝑥 , are arranged such that 𝑋𝑖,𝑚 = 𝑋𝑖 + 𝑚𝑥 𝑑𝑠𝑥 , 𝑚𝑥 = 0, 1, ⋅ ⋅ ⋅ , 𝑀𝑥 − 1, and 𝑌𝑖,𝑚 = 𝑌𝑖 + 𝑚𝑦 𝑑𝑠𝑦 , 𝑚𝑦 = 0, 1, ⋅ ⋅ ⋅ , 𝑀𝑦 − 1, where 𝑑𝑠𝑥 and 𝑑𝑠𝑦 are the subarray spacings along x-axis and y-axis respectively, and (𝑋𝑖 , 𝑌𝑖 ) is the location of the ith element of the subarray numbered 𝑚 = 0. Then, (1) can be simplified as 𝑠𝑚 (𝑡) =˜ 𝑠 (𝑡) 𝑒𝑗2𝜋𝑓𝐷 𝑡 𝑃𝑠 (𝜃, 𝜙) 𝑠 𝑠 2𝜋 ⋅ 𝑒𝑗 𝜆𝑐 (𝑚𝑥 𝑑𝑥 sin 𝜃 cos 𝜙+𝑚𝑦 𝑑𝑦 sin 𝜃 sin 𝜙) + 𝑛𝑚 (𝑡) (2)

∗ 𝑤𝑚 𝑠𝑚 (𝑛𝑇 ) =

𝑀−1 ∑

∗ 𝑤𝑚 𝑠𝑚 [𝑛]

(4)

𝑚=0

where 𝑠𝑚 [𝑛] = 𝑠𝑚 (𝑛𝑇 ) is the sampled subarray signal in the digital domain. C. Subarray Radiation Pattern Without loss of generality, we assume that the antenna array consists of isotropic elements with omni-directional radiation patterns, i.e., 𝑃𝑖 (𝜃, 𝜙) = 1. Then, the radiation pattern expressed in (3) is the same as the array factor of the subarray. When the phase shifts of the elements in a subarray are chosen as 𝛼𝑖 = − 2𝜋 𝜆𝑐 (𝑋𝑖 sin 𝜃0 cos 𝜙0 + 𝑌𝑖 sin 𝜃0 sin 𝜙0 ), which allows the main beam of the array to be directed towards the direction represented by the angles (𝜃0 , 𝜙0 ), and (𝑋0 , 𝑌0 ) is chosen so that the subarray is centered about the origin of the x-y coordinates, the normalized radiation pattern of a subarray can be expressed as the well known form 𝑃𝑠 (𝜃, 𝜙) 𝑃¯𝑠 (𝜃, 𝜙) = 𝑁𝑥 𝑁𝑦 ( ) sin 𝑁𝑥 𝜆𝜋𝑐 𝑑𝑒𝑥 (sin 𝜃 cos 𝜙 − sin 𝜃0 cos 𝜙0 ) ( ) = 𝑁𝑥 sin 𝜆𝜋𝑐 𝑑𝑒𝑥 (sin 𝜃 cos 𝜙 − sin 𝜃0 cos 𝜙0 ) ) ( sin 𝑁𝑦 𝜆𝜋𝑐 𝑑𝑒𝑦 (sin 𝜃 sin 𝜙 − sin 𝜃0 sin 𝜙0 ) ( ) . (5) ⋅ 𝑁𝑦 sin 𝜆𝜋𝑐 𝑑𝑒𝑦 (sin 𝜃 sin 𝜙 − sin 𝜃0 sin 𝜙0 ) For a two-dimensional hybrid array of interleaved subarrays, we have 𝑑𝑒𝑥 = 𝑀𝑥 𝑑 and 𝑑𝑒𝑦 = 𝑀𝑦 𝑑, where 𝑑 is the element spacing of the hybrid array, which is also the same as the subarray spacing along x-axis or y-axis, i.e., 𝑑𝑠𝑥 = 𝑑𝑠𝑦 = 𝑑. For a two-dimensional hybrid array of side-by-side subarrays, we have 𝑑𝑒𝑥 = 𝑑𝑒𝑥 = 𝑑, whereas the subarray spacings along x-axis and y-axis are 𝑑𝑠𝑥 = 𝑁𝑥 𝑑 and 𝑑𝑠𝑦 = 𝑁𝑦 𝑑, respectively. III. AOA E STIMATION AND B EAMFORMING A. Differential Beam Tracking (DBT) Let’s first consider the hybrid array of interleaved subarrays. From the subarray output signal model (2), it is observed that the inputs to the digital beamformer are affected by the subarray radiation pattern which is determined by the


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AoA (𝜃, 𝜙) as well as an initial AoA estimate (𝜃0 , 𝜙0 ). Therefore, any further AoA estimation can only be made based on a previously estimated AoA, which means that the AoA estimation for the hybrid array must be recursive in nature. It is also observed that, at a given initial AoA, the AoA information can be estimated from the phase difference between the adjacent subarray output signals provided that 𝑃𝑠 (𝜃, 𝜙) ∕= 0. Thus, by taking the cross-correlations of any two adjacent subarray output signals along x-axis and y-axis, denoted as 𝑅𝑥 and 𝑅𝑦 respectively, and assuming that the noise components are independent, we have { } 𝑅𝑥 =𝐸 𝑠∗𝑚𝑦 𝑀𝑥 +𝑚𝑥 (𝑡) 𝑠𝑚𝑦 𝑀𝑥 +𝑚𝑥 +1 (𝑡) } { 2𝜋 𝑠 2 2 (6) =𝐸 ∣˜ 𝑠 (𝑡)∣ ∣𝑃𝑠 (𝜃, 𝜙)∣ 𝑒𝑗 𝜆𝑐 𝑑𝑥 sin 𝜃 cos 𝜙 and 𝑅𝑦 =𝐸

{ {

𝑠∗𝑚𝑦 𝑀𝑥 +𝑚𝑥 2

=𝐸 ∣˜ 𝑠 (𝑡)∣

}

(𝑡) 𝑠(𝑚𝑦 +1)𝑀𝑥 +𝑚𝑥 (𝑡)

∣𝑃𝑠 (𝜃, 𝜙)∣ 𝑒

3.

}

2𝜋 𝑠 2 𝑗𝜆 𝑑 sin 𝜃 sin 𝜙 𝑐 𝑦

2𝜋 𝑑 sin 𝜃 sin 𝜙 𝜆𝑐

(7)

(9)

contain the AoA information of the incident signal and will take on values in the range [−𝜋, 𝜋). They can be obtained from arg {𝑅𝑥 } and arg {𝑅𝑦 } respectively without any ambiguity, i.e., (10) 𝑢𝑥 = arg {𝑅𝑥 } 𝑢𝑦 = arg {𝑅𝑦 } .

(11)

The obtained 𝑢𝑥 and 𝑢𝑦 can be used to determine the phase shifts in the subarrays. The explicit values of the angles − 𝜋2 ≤ 𝜃 ≤ 𝜋2 and − 𝜋2 ≤ 𝜙 ≤ 𝜋2 are not required though they can be(easily from 𝑢𝑥 and 𝑢𝑦 as ) √ 2determined ( ) 𝑢𝑥 +𝑢2𝑦 𝑢 −1 𝜆𝑐 𝜃 = 𝑠𝑖𝑔𝑛(𝑢𝑥 ) sin and 𝜙 = 𝑡𝑔 −1 𝑢𝑥𝑦 . 2𝜋 𝑑 In the digital domain, the cross-correlations along x-axis and y-axis can be estimated iteratively using the digital subarray output signals sampled at 𝑡 = 𝑛𝑇 as 𝑅𝑥(𝑛) = (1 − 𝜇) 𝑅𝑥(𝑛−1) 𝑀𝑦 −1 𝑀𝑥 −2

+𝜇

∑ ∑

𝑚𝑦 =0 𝑚𝑥 =0

𝑠∗𝑚𝑦 𝑀𝑥 +𝑚𝑥 [𝑛] 𝑠𝑚𝑦 𝑀𝑥 +𝑚𝑥 +1 [𝑛] (12)

𝑅𝑦(𝑛) = (1 − 𝜇) 𝑅𝑦(𝑛−1) +𝜇

𝑀 𝑦 −2 𝑥 −1 𝑀 ∑ ∑ 𝑚𝑥 =0 𝑚𝑦 =0

(𝑛)

Determine the subarray phase shifts 𝛼𝑖

(𝑛) 𝑋𝑖 𝑢(𝑛) 𝑥 +𝑌𝑖 𝑢𝑦 , − 𝑑

where 𝐸 {⋅} denotes ensemble expectation. Note that the Doppler frequency shift 𝑓𝐷 does not affect the cross-correlations 𝑅𝑥 and 𝑅𝑦 since the phase shift introduced by 𝑓𝐷 is the same for all subarray signals. Therefore, the AoA estimation based on 𝑅𝑥 and 𝑅𝑦 will be Doppler resilient. Since 𝑑𝑠𝑥 = 𝑑𝑠𝑦 = 𝑑 and assuming 𝑑 ≤ 𝜆2𝑐 , the following variables 2𝜋 𝑢𝑥 = 𝑑 sin 𝜃 cos 𝜙 (8) 𝜆𝑐 𝑢𝑦 =

where 0 < 𝜇 < 1 is the updating coefficient. All available subarray outputs are used for the cross-correlation estimation in order to improve the signal-to-noise ratio (SNR). As with any adaptive algorithm, both the convergence rate and variance should be considered when selecting 𝜇. In general, a larger 𝜇 will speed up the convergence but cause larger variance, whereas a smaller 𝜇 will slow down the convergence but reduce the variance. ∗ = 𝑒−𝑗(𝑚𝑥 𝑢𝑥 +𝑚𝑦 𝑢𝑦 ) , After applying the digital weights 𝑤𝑚 𝑚 = 𝑚𝑦 𝑀𝑥 + 𝑚𝑥 , to the subarray output signals, the beamformed signal can be obtained by (4) accordingly. The algorithm for analogue phase shifter control is now summarized as follows. (𝑛) (𝑛) 1. Update 𝑅𝑥 and 𝑅𝑦 { using} (12) and (13); { } (𝑛) (𝑛) (𝑛) (𝑛) 2. Calculate 𝑢𝑥 = arg 𝑅𝑥 and 𝑢𝑦 = arg 𝑅𝑦 ;

𝑠∗𝑚𝑦 𝑀𝑥 +𝑚𝑥 [𝑛] 𝑠(𝑚𝑦 +1)𝑀𝑥 +𝑚𝑥 [𝑛] (13)

=

𝑖 = 0, 1, ⋅ ⋅ ⋅ 𝑁 − 1. Since the above algorithm uses the phase difference between adjacent subarray output signals to obtain the AoA information and to track the AoA adaptively, we call it differential beam tracking. It is a blind algorithm since no knowledge about the reference signal 𝑠˜ (𝑡) is required. B. Phase Ambiguity and Beam Scanning For the hybrid antenna array of side-by-side subarrays, the subarray spacings are 𝑑𝑠𝑥 = 𝑁𝑥 𝑑 and 𝑑𝑠𝑦 = 𝑁𝑦 𝑑. Therefore, the phases of 𝑅𝑥 and/or 𝑅𝑦 can be outside the range [−𝜋, 𝜋), and ambiguity will occur when arg {𝑅𝑥 } and arg {𝑅𝑦 } are used to determine the phases of 𝑅𝑥 and 𝑅𝑦 respectively. To remove this ambiguity and thus obtain the correct AoA information, one can find all the possible 𝑢𝑥 and 𝑢𝑦 values from arg {𝑅𝑥 } and arg {𝑅𝑦 } respectively, and try all the possible combinations of 𝑢𝑥 and 𝑢𝑦 , which represent all the possible beams, to see which combination gives the beam with the maximum output power. The 𝑢𝑥 and 𝑢𝑦 values corresponding to the largest beamformed signal power is used to obtain the AoA information. All the possible 𝑢𝑥 and 𝑢𝑦 values can be determined respectively by 2𝜋𝑝 + arg {𝑅𝑥 } , [ 𝑁]𝑥 [ ] [ ] 𝑁𝑥 𝑁𝑥 𝑁𝑥 𝑝=− ,− + 1, ⋅ ⋅ ⋅ , 0, 1, ⋅ ⋅ ⋅ (14) 2 2 2

𝑢𝑥 (𝑝) =

2𝜋𝑞 + arg {𝑅𝑦 } , 𝑁𝑦 [ ] [ ] [ ] 𝑁𝑦 𝑁𝑦 𝑁𝑦 𝑞=− ,− + 1, ⋅ ⋅ ⋅ , 0, 1, ⋅ ⋅ ⋅ (15) 2 2 2

𝑢𝑦 (𝑞) =

where [.] denotes the operation of taking the integer part of a value. Each combination of 𝑢𝑥 (𝑝) and 𝑢𝑦 (𝑞) represents a possible beam. The phase shifts for given 𝑝 and 𝑞 𝛼𝑖 (𝑝, 𝑞) = −

𝑋𝑖 𝑢𝑥 (𝑝) + 𝑌𝑖 𝑢𝑦 (𝑞) 𝑑

(16)

are applied to control the subarrays towards the selected beam, and the beamformed signal at 𝑡 = 𝑛𝑇 can be obtained

1774


u x (2 ) =

π

u x (1) =

(𝑛)

13π 15

7π 15

arg{R x }

π 3

u x (0 ) =

u x (− 1) = −

π 15

π

u x (− 1) = −

3

u x (− 2 ) =−

t

π 3

u x (− 2 )

11π 15

=−

11π 15

Subframe

−π

Fig. 3.

Scanning Frame

Next Scanning Frame

Illustration of scanning frame and subframes for beam search.

accordingly as 𝑀𝑦 −1 𝑀𝑥 −1

𝑠 [𝑛, 𝑝, 𝑞] =

∑ ∑

𝑒−𝑗 (𝑚𝑥 𝑁𝑥 𝑢𝑥

(𝑛)

(𝑝)+𝑚𝑦 𝑁𝑦 𝑢(𝑛) 𝑦 (𝑞))

𝑚𝑦 =0 𝑚𝑥 =0

⋅ 𝑠𝑚𝑦 𝑀𝑥 +𝑚𝑥 [𝑛] (𝑛)

(17)

(𝑛)

where 𝑢𝑥 (𝑝) and 𝑢𝑦 (𝑞) denote 𝑢𝑥 (𝑝) and 𝑢𝑦 (𝑞) obtained at 𝑡 = 𝑛𝑇 , respectively. To compare the signal powers of different beams, all possible beams are scanned within a period of time. We call this period of time the scanning frame. A scanning frame is divided into subframes for different combinations of 𝑝 and 𝑞. Within each subframe, each beamformed signal power is calculated. At the end of a scanning frame, the beam with the largest signal power is decided as the estimated signal beam. The beam scanning frame and subframe are illustrated in Fig. 3 for a linear subarray with 5 elements (𝑁 = 𝑁𝑥 = 5 and 𝑁𝑦 = 1) and arg {𝑅𝑥 } = 𝜋3 . In this example, there are 𝜋 5 possible beams with 𝑢𝑥 (−2) = − 11𝜋 15 , 𝑢𝑥 (−1) = − 3 , 𝜋 7𝜋 13𝜋 𝑢𝑥 (0) = 15 , 𝑢𝑥 (1) = 15 , and 𝑢𝑥 (2) = 15 according to (14). The scanning frame has 5 subframes and the beams are scanned from 𝑢𝑥 (−2) to 𝑢𝑥 (2) in order. C. Differential Beam Search (DBS) As in the DBT algorithm, the cross-correlations 𝑅𝑥 and 𝑅𝑦 can be still iteratively estimated using (12) and (13) in the digital domain, whereas } the beamformed signal power { 2 can be iteratively estimated in 𝜎𝑠2 (𝑝, 𝑞) = 𝐸 ∣𝑠 [𝑛, 𝑝, 𝑞]∣ a subframe by 𝜎𝑠2(𝑛) (𝑝, 𝑞) = (1 − 𝛽) 𝜎𝑠2(𝑛−1) (𝑝, 𝑞) + 𝛽 ∣𝑠 [𝑛, 𝑝, 𝑞]∣

2

(18)

where 0 < 𝛽 < 1 is the updating coefficient. Since the above iteration is to estimate the average power in a subframe, as a rule of thumb, 𝛽 is selected inversely proportional to the length of the subframe. Combining the above iterations (12), (13), (17) and (18) with the beam scanning scheme, one obtains the adaptive algorithm, called the differential beam search, which proceeds in a subframe for a given combination of 𝑝 and 𝑞 as follows at the 𝑛th iteration. (𝑛) (𝑛) 1. Update 𝑅𝑥 and 𝑅𝑦 using (12) and (13); 2𝜋𝑝+arg{𝑅(𝑛) (𝑛) (𝑛) 𝑥 } 2. Calculate 𝑢𝑥 (𝑝) = and 𝑢𝑦 (𝑞) = 𝑁𝑥 (𝑛) 2𝜋𝑞+arg{𝑅𝑦 } ; 𝑁𝑦

𝑋𝑖 𝑢(𝑛) (𝑝)+𝑌𝑖 𝑢(𝑛) (𝑞)

𝑥 𝑦 , 𝑖 = 3. Determine 𝛼𝑖 (𝑝, 𝑞) = − 𝑑 0, 1, ⋅ ⋅ ⋅ 𝑁 − 1; 2(𝑛) 4. Update 𝑠 [𝑛, 𝑝, 𝑞] and 𝜎𝑠 (𝑝, 𝑞) using (17) and (18); 5. Select 𝑝 and 𝑞 for next subframe. The DBS algorithm is also a blind algorithm since no knowledge of the reference signal is assumed. It can be seen as a generalised DBT in which the beam is locked to the only possible one without ambiguity. The scanning frame can be repeated until the peak power of a beam calculated across multiple scanning frames is sufficiently higher than the powers of other beams (see Section V for more details on the peak power selection). Once the correct beam is determined, the DBT algorithm can then be used to track the change of the selected beam. Due to the delay from the time when the phase shifts are loaded into the phase shifters in subarrays to the time when a change of the beamformed signal is observed, a minimum length of the subframe will be required. D. Comparison between DBT and DBS The DBT algorithm is mainly used for the hybrid array of interleaved subarrays, whereas the DBS algorithm is used for the hybrid array of side-by-side subarrays in the acquisition stage. In terms of digital signal processing complexity, the DBS algorithm is more complicated than the DBT algorithm since DBS needs to search all possible beams to remove the phase ambiguity. In addition, the AoA information of the correct beam is updated for each iteration using DBT, whereas it is only updated once during an entire scanning frame using DBS. Hence, the DBT algorithm has faster convergence than the DBS algorithm and will be particularly useful for tracking fast time-varying AoA. However, in terms of analogue hardware implementation complexity, the interleaved subarray is more difficult to build [24] or expand to form larger arrays due to the complicated corporate feed networks and array architectures. The side-byside subarray is much easier to build in practice. By combining a number of side-by-side subarrays as basic assembly modules, a larger array can be easily formed. Therefore, both digital signal processing and hardware implementation complexities should be taken into consideration for the selection of suitable hybrid array architecture and the corresponding digital beamforming algorithm. If the interleaved subarrays are selected, the DBT algorithm can be applied for both beam acquisition and tracking. If the sideby-side subarrays are selected, the DBS can be applied for acquisition first and the DBT can be then used for beam tracking.

IV. P ERFORMANCE E VALUATION A. Formulation of Blind AoA Estimation Problem The performance of the proposed blind AoA estimation algorithms can be measured by the MSE of the estimated 𝑢𝑥 (or 𝑢𝑦 ). To simplify the analysis, we only consider a linear hybrid array of two interleaved subarrays with 𝑁 = 𝑁𝑥 elements. We assume that the incident beam is pointed at 𝜃 = 0 and the AoA remains unchanged during the estimation. Therefore, the normalized subarray radiation pattern is only a function of the estimated 𝑢 = 𝑢𝑥 = 2𝜋 𝜆𝑐 𝑑 sin 𝜃0 and is thus


1775

sin(𝑁 𝑢) denoted as 𝑃¯𝑠 (𝑢) = 𝑁 sin(𝑢) . We also ignore the Doppler frequency shift since the proposed algorithms are Doppler resilient. Under the above assumption, the received subarray signals in the digital domain at time index 𝑛 can be expressed as ( ) (19) 𝑠0 [𝑛] = 𝑠˜ [𝑛] 𝑃¯𝑠 𝑢(𝑛−1) + 𝑧0 [𝑛]

(25)

( ) 𝑠1 [𝑛] = 𝑠˜ [𝑛] 𝑃¯𝑠 𝑢(𝑛−1) 𝑒𝑗𝑢 + 𝑧1 [𝑛]

(20)

where 𝑠˜ [𝑛] = 𝑠˜ (𝑛𝑇 ) is the reference } sampled at 𝑡 = { signal 𝑛𝑇 with average power 𝜎𝑠2˜ = 𝐸 ∣˜ 𝑠 [𝑛]∣2 , 𝑢(𝑛−1) denotes the variable 𝑢 estimated at time index 𝑛 − 1, 𝑧0 [𝑛] and 𝑧1 [𝑛] are independent complex Gaussian noises with zero mean and with the same noise power 𝜎𝑧2 . Note that, compared ( ) (2), the normalized subarray radiation pattern 𝑃¯𝑠 𝑢(𝑛−1) is used in the above signal models since a scaling factor can be absorbed into 𝑠˜ [𝑛] and thus the SNR in a subarray can be expressed as 𝜎2 𝛾 = 𝜎𝑠2˜ after the subarray is correctly beamformed. 𝑧 The differential signal between 𝑠0 [𝑛] and 𝑠1 [𝑛] is 𝑟 (𝑛) =𝑠∗0 [𝑛] 𝑠1 [𝑛] ( )2 ( ) = ∣˜ 𝑠 [𝑛]∣2 𝑃¯𝑠 𝑢(𝑛−1) 𝑒𝑗𝑢 + 𝑠˜ [𝑛] 𝑃¯𝑠 𝑢(𝑛−1) 𝑒𝑗𝑢 𝑧0∗ [𝑛] ( ) + 𝑠˜∗ [𝑛] 𝑃¯𝑠∗ 𝑢(𝑛−1) 𝑧1 [𝑛] + 𝑧0∗ [𝑛] 𝑧1 [𝑛] ( )2 (21) = ∣˜ 𝑠 [𝑛]∣2 𝑃¯𝑠 𝑢(𝑛−1) 𝑒𝑗𝑢 + 𝑧 [𝑛]

where

( ) ( ) 𝑧 [𝑛] =˜ 𝑠 [𝑛] 𝑃¯𝑠 𝑢(𝑛−1) 𝑒𝑗𝑢 𝑧0∗ [𝑛] + 𝑠˜∗ [𝑛] 𝑃¯𝑠∗ 𝑢(𝑛−1) 𝑧1 [𝑛] + 𝑧0∗ [𝑛] 𝑧1 [𝑛]

(22)

can be approximated as a complex Gaussian ( noise ) 2with zero 2 mean and total noise power 2 ∣˜ 𝑠 [𝑛]∣ 𝑃¯𝑠 𝑢(𝑛−1) 𝜎𝑧2 . From (21), the estimation of 𝑢 is formulated as the phase estimation of 𝑟 (𝑛) based on the observed 𝑟 (𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛, in the presence of random nuisance parameters 𝑠˜ (𝑘) and previously estimated 𝑢(𝑘−1) , 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛. Note that the above analysis can be easily extended to planar hybrid array with multiple subarrays, resulting in similar differential signal models to (21) for the estimation of 𝑢𝑥 or 𝑢𝑦 . The differences are that the subarray radiation pattern will be determined by the actual size of a subarray and the SNR 𝛾 will increase with more subarrays being used in the hybrid array. B. Recursive MSE Bound According to the DBT algorithm based on the obtained differential data set 𝑟 (𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛, the non-coherent estimation of 𝑢 is given by } { (23) 𝑢(𝑛) = arg 𝑅(𝑛) where 𝑅(𝑛) =

𝑛 ∑ 𝑘=1

𝑟 (𝑘) =

𝑛 ∑ 𝑘=1

1 2 2𝜋𝜎𝑅 (𝑛) ⎛ ( )2 ( )2 ⎞ (𝑛) (𝑛) − 𝑅𝑒 {𝑚 } + 𝑅 − 𝐼𝑚 {𝑚 } 𝑅 (𝑛) (𝑛) 𝑟 𝑖 𝑅 𝑅 ⎟ ⎜ ⋅ exp ⎝− ⎠ 2 2𝜎𝑅 (𝑛) (𝑛)

𝑓𝑅(𝑛) ∣˜s,u (𝑅𝑟(𝑛) , 𝑅𝑖 ∣˜ s, u) =

𝑛

( ) 2 ∑

2 ∣˜ 𝑠 [𝑘]∣ 𝑃¯𝑠 𝑢(𝑘−1) 𝑒𝑗𝑢 + 𝑧 [𝑘] 𝑘=1

(24) is complex Gaussian distributed with the joint conditional probability density function (pdf) of its real and imaginary parts [25], given 𝑠˜ [𝑘] and 𝑢(𝑘−1) , 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛, denoted as ˜ s and u, i.e.,

( ) 2 ∑𝑛 2 where 𝑚𝑅(𝑛) = 𝑘=1 ∣˜ 𝑠 [𝑘]∣ 𝑃¯𝑠 𝑢(𝑘−1) 𝑒𝑗𝑢 is the con( ) 2 ∑𝑛 2 2 ditional mean, and 𝜎𝑅 𝑠 [𝑘]∣ 𝑃¯𝑠 𝑢(𝑘−1) 𝜎𝑧2 (𝑛) = 𝑘=1 ∣˜ is the conditional variance of its real or imaginary part. ( (𝑛) ) (𝑛) 𝑢 The( conditional pdf of 𝑢 is thus 𝑓 ∣˜ s , u = (𝑛) 𝑢 ∣˜ s,u ) (𝑛)

𝑓0 𝑢(𝑛) , 𝛾˜s,u

[26] where

( √ )] √ 2 1 −𝛾 [ 𝑒 1 + 4𝜋𝛾 cos 𝑥𝑒𝛾 cos 𝑥 𝑄 − 2𝛾 cos 𝑥 , 𝑓0 (𝑥, 𝛾) = 2𝜋 (26) ∑ (𝑛)

𝛾˜s,u =

∣𝑚𝑅(𝑛) ∣ 2𝜎2 (𝑛) 𝑅

2

=

∣𝑃¯𝑠 (𝑢(𝑘−1) )∣ 2𝜎𝑧2

𝑛 𝑠[𝑘]∣2 𝑘=1 ∣˜

∫∞

2

is the conditional

2 − 𝑡2

𝑑𝑡 is the Q-function. SNR of 𝑅(𝑛) , and 𝑄 (𝑥) = √12𝜋 𝑥 𝑒 Finally, the MSE for the estimation of 𝑢 at time index 𝑛 is ∫ 𝜋 ( )2 ( ) 2 𝑢(𝑛) 𝑓𝑢(𝑛) 𝑢(𝑛) 𝑑𝑢(𝑛) 𝜎𝑢(𝑛) = (27) −𝜋

( ) { ( )} s, u is the unwhere 𝑓𝑢(𝑛) 𝑢(𝑛) = 𝐸˜s,u 𝑓𝑢(𝑛) ∣˜s,u 𝑢(𝑛) ∣˜ conditional pdf of 𝑢(𝑛) . The exact evaluation of 𝜎𝑢2 (𝑛) is practically infeasible since the unconditional pdf of 𝑢(𝑛) involves the expectation over the previously estimated 𝑢(𝑘−1) , which in turn requires the knowledge of the unconditional pdf of 𝑢(𝑘−1) , 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛. Even the widely used modified Cramér-Rao bound (MCRB) analysis [27] is still infeasible since the calculation ) ( 𝑀 𝐶𝑅𝐵 𝜎𝑢2 (𝑛) = 1 { { 2 }} (28) (𝑛) (𝑛) ∂ ln 𝑓𝑅(𝑛) ∣˜s,u (𝑅𝑟 ,𝑅𝑖 ∣˜ s,u) 𝐸˜s,u 𝐸𝑅(𝑛) ∣˜s,u − ∂𝑢2 also requires that the unconditional pdf of 𝑢(𝑘−1) , 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛, is known. To avoid the direct evaluation of the MSE but still obtain a meaningful indication of it, a lower MSE bound of 𝜎𝑢2 (𝑛) , referred to as Recursive MSE Bound, is obtained as (see Appendix) √ (√ ) ( 2 ) 𝛾¯ (𝑛) 𝜋 2 + 1 1 −1 𝛾¯ (𝑛) 𝜋 − (𝑛) 𝑀 𝑆𝐸𝐵 𝜎𝑢(𝑛) = ) 32 sinh ( 𝛾 ¯ 𝜋 𝛾¯ (𝑛) (29) where 𝛾¯ (𝑛) is the average SNR of 𝑅(𝑛) , which is recursively determined by 𝛾¯ (𝑛) = ⎧ 1 𝛾, 𝑛=1   2𝑁 ∫ 𝜋 ( (𝑛−1) ) 2 ⎨ 1 (𝑛−1)

¯ 𝛾¯ + 𝛾 𝑃 𝑢 √ 2 −𝜋 𝑠 (30) (𝑛−1) 𝜋 2 +1  𝛾 ¯  ⎩ ⋅ ( (𝑛−1) (𝑛−1) 2 )3/2 𝑑𝑢(𝑛−1) , 𝑛 > 1 2𝜋 𝛾 ¯ (𝑢 ) +1 √ ( ) and sinh−1 (𝑥) = ln 𝑥 + 𝑥2 + 1 denotes the inverse hyperbolic sine function. Eq. (29) also applies to DBS if the beam scanning scheme is incorporated. For a linear hybrid array of


V. S IMULATION R ESULTS The proposed AoA estimation algorithms are simulated using a planar hybrid array of four interleaved subarrays and a planar hybrid array of four side-by-side subarrays respectively, as shown in Fig. 1. Each subarray has 16 elements arranged as a 4 by 4 matrix and there are 2 subarrays placed along x-axis and y-axis respectively, resulting in a large array of 64 elements. The element spacing is half of the wavelength, i.e., 𝑑 = 𝜆2𝑐 . The incident angles of the received signal are set to 40 degree in zenith and 0 degree in azimuth, corresponding to 𝑢𝑥 = 2.0194 and 𝑢𝑦 = 0 respectively. The SNR per antenna element is set to –10 dB. The reference signal is a realization of a complex Gaussian distributed random signal. The simulation results using DBT for the hybrid array of interleaved subarrays are shown in Fig. 4 and Fig. 5. Fig. 4 shows some realizations of the estimated phases of the cross-correlations along x-axis and y-axis versus the number of iterations with updating coefficient 𝜇 = 0.001 and 0.01 respectively. The estimated phase values with 𝜇 = 0.001 after 400 iterations are 1.979 and 0.016 respectively, corresponding to 39.05 degree in zenith and 0.46 degree in azimuth. Fig. 5 shows the normalized array factors for an interleaved subarray and the overall hybrid array respectively with 𝜇 = 0.001 after 400 iterations. We see that the grating lobe in the subarray is suppressed after combining the four subarrays. The final beamwidth of the hybrid array is similar to that of an interleaved subarray as the aperture of the interleaved subarray is similar to that of the hybrid array. The simulation results using DBS with updating coefficients 𝜇 = 0.001, 0.01 and 𝛽 = 0.25 for the hybrid array of sideby-side subarrays are shown in Fig. 6 to Fig. 9. In this case,

4 3 2 arg(Rx) and arg(Ry)

two side-by-side subarrays with 𝑁 elements, we have sin(𝑁 𝑢 ) 𝑃¯𝑠 (𝑢) = 𝑁 sin 2𝑢 and the incremental of 𝛾¯ (𝑛) in (30) will (2) √ ∫ 𝜋

( 𝑢(𝑛−1) )

2 𝛾 ¯ (𝑛−1) 𝜋 2 +1 1 ¯ ( )3/2 𝑑𝑢(𝑛−1) . be 2 𝛾 −𝜋 𝑃𝑠

𝑁 (𝑛−1) 2𝜋 𝛾 ¯ (𝑢(𝑛−1) )2 +1 Since the average SNR is only updated every 𝑁 samples (assuming a one sample subframe), the convergence rate is reduced by a factor of 𝑁 . Regarding the impact of mutual coupling between antenna elements on the AoA estimation performance, we notice that the effect of mutual coupling can be regarded as a distortion of the subarray radiation pattern and a degraded SNR in the differential signal model (21). This will certainly affect the performance which is eventually determined by the SNR. However, as have been reported recently in the literature on the conventional adaptive antenna array in the presence of mutual coupling [28–30], neither the minimum-mean-squireerror (MMSE) nor the LMS algorithms for beamforming are severely affected by the mutual coupling. Qualitatively speaking, a similar effect will be seen for the hybrid adaptive antenna, i.e., slowing down convergence of the algorithm. Since the SNR 𝛾¯ (𝑛) increases as the iteration number increases, an MSE reached by the proposed algorithm in the ideal condition (i.e., without mutual coupling) can be also reached by the algorithm in a mutual coupling environment with more iteration. A quantitative analysis is beyond the scope of this paper.

1 0 −1 −2

arg(Rx) μ = 0.001

−3

arg(Rx) μ = 0.01

arg(Ry) μ = 0.001 arg(R ) μ = 0.01 y

−4

50

100

150 200 250 Number of Iterations

300

350

400

Fig. 4. Estimated phases of cross-correlations versus the number of iterations using DBT for hybrid array of interleaved subarrays. 1 0.9

Subarray Hybrid Array

0.8 Normalized Array Factor

1776

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −100

−50

0 θ (degree)

50

100

Fig. 5. Normalized array factors of an interleaved subarray and hybrid array (𝜙 = 0).

the correct arg {𝑅𝑥 } and arg {𝑅𝑦 } values are 1.7943 and 0 respectively. The possible 𝑢𝑥 values are –2.6930, −1.1222, 0.4486, 2.0194, and 3.5902, and the possible 𝑢𝑦 values are –3.1416, –1.5708, 0, 1.5708, and 3.1416. There are total 25 different beams to scan, so there are 25 subframes in one scanning frame. The number of samples in one subframe is set to 4 in the simulation and thus one scanning frame has 100 signal samples. Fig. 6 depicts the estimated arg {𝑅𝑥 } and arg {𝑅𝑦 } versus the number of iterations for 4 scanning frames with 𝜇 = 0.001 and 0.01 respectively. Compared with Fig. 4, we see that the convergence is slower than that of DBT using interleaved subarrays. Fig. 7 shows the 𝑢𝑥 and 𝑢𝑦 values when the beams are sequentially scanned with 𝜇 = 0.001. Fig. 8 shows the estimated signal powers for all beams during the beam scanning with 𝜇 = 0.001. We see that the peak power in each scanning frame appears when the beam with the forth value of 𝑢𝑥 and the third value of 𝑢𝑦 is scanned, and thus this beam is detected as the correct one. Note that the final decision on the correct beam can be done after the peak power is significantly higher than the other average powers of different beams to increase the probability of correct decision.


1777

4

6000

3

5000

Estimated Power

1 0

x

y

arg(R ) and arg(R )

2

−1 −2

arg(R ) μ = 0.001

−3

arg(Rx) μ = 0.01

4000

3000

2000

x

arg(Ry) μ = 0.001

1000

arg(R ) μ = 0.01 y

−4

50

100


300

350

0

400

Fig. 6. Estimated phases of cross-correlations versus the number of iterations using DBS for hybrid array of side-by-side subarrays.

Fig. 8. beam.

0

3

300

350

400

Subarray Hybrid Array

0.8 Normalized Array Factor

2

y

1 0

x

u and u


Power profile of different beams. Peak power indicates the correct

0.9

−1

Fig. 7.

100

1

4

0.7 0.6 0.5 0.4 0.3

−2

0.2

−3

0.1

−4

50

50

100


300

350

400

Beam search via 𝑢𝑥 (solid line) and 𝑢𝑦 (dashed line) scanning.

The threshold value can be determined according to the size of a subarray. In the case of this 4 by 4 subarray, since the mainlobe of the array factor can provide a gain of more than 10 dB compared with other sidelobes and considering a 3 dB tolerance, the threshold can be selected as 7 dB (i.e., the peak power is required to be 7 dB higher than other average powers). The larger the subarray is, the higher the threshold will be. The estimated 𝑢𝑥 and 𝑢𝑦 values after 4 scanning frames with 𝜇 = 0.001 for the hybrid array of side-by-side subarrays are 1.970 and −0.022 respectively which correspond to 38.84 degree in zenith and −0.64 degree in azimuth. Finally, Fig. 9 shows the normalized array factors for a side-by-side subarray and the overall hybrid array respectively. We see that after combining the subarrays the beamwidth of the hybrid array is reduced significantly. Compared with Fig. 5, we also see that, though the subarray patterns are different for an interleaved and a side-by-side subarrays, the overall hybrid arrays after the AoA estimation are very similar. This means that the subarray configuration does not affect the overall beamforming accuracy though the beamforming algorithms have different

0 −100

−50

0 θ (degree)

50

100

Fig. 9. Normalized array factors of a side-by-side subarray and hybrid array (𝜙 = 0).

complexities and convergence speeds. Finally, the simulated MSEs for the DBT algorithm under different SNRs per subarray are plotted in Fig. 10, which are obtained after averaging over 100 independent simulations. Note that the subarray SNR of 𝛾=5 dB corresponds to the –10 dB element SNR because the subarray has 16 elements and two subarray cross-correlations are combined for the AoA estimation (thus the gain is 10log(16) dB + 3 dB =15 dB). The calculated recursive MSE bounds are also displayed on the same figure. We see that at lower SNR the MSE bound is rather loose. However, as SNR increases, the recursive MSE bound becomes tighter. VI. C ONCLUSION A hybrid adaptive antenna array with two different subarray configurations and associated AoA estimation and beamforming algorithms are proposed to simplify the implementation of large adaptive antenna arrays, especially at mm-wave frequencies. It has been shown that the proposed DBT and DBS algorithms can estimate the AoA information for hybrid arrays of interleaved and side-by-side subarrays respectively without

1778


4 Simulated MSE Recursive MSE Bound

3.5 3 −5 dB

MSE

2.5 2 1.5

0 dB

1 5 dB

0.5 0

50

100

150 200 250 Number of Iteration

300

350

400

( ) ( ) channel [26], i.e, 𝑓𝑢(𝑛) 𝑢(𝑛) = 𝑓1 𝑢(𝑛) , 𝛾¯ (𝑛) where √ 𝛾 cos 𝑥 1 ) 𝑓1 (𝑥, 𝛾) = ( + ( )3/2 2𝜋 1 + 𝛾 sin2 𝑥 2𝜋 1 + 𝛾 sin2 𝑥 ) ( √ 𝛾 cos 𝑥 𝜋 −1 + tan √ . (32) ⋅ 2 1 + 𝛾 sin2 𝑥 With a Gaussian distributed reference signal, if the sub¯ array ( radiation ) pattern is omni-directional, i.e., 𝑃𝑠 (𝑢) = 1, 𝑓1 𝑢(𝑛) , 𝛾¯ (𝑛) will be the true phase distribution in Rayleigh

fading channel. Since 𝑃¯𝑠 (𝑢) ≤ 1 in practice, which will lead to a reduced SNR, the actual MSE ( in (27) will ) be always greater than that calculated using 𝑓1 𝑢(𝑛) , 𝛾¯ (𝑛) , i.e, ∫ 𝜋 ( )2 ( ) 𝜎𝑢2 (𝑛) ≥ 𝑢(𝑛) 𝑓1 𝑢(𝑛) , 𝛾¯ (𝑛) 𝑑𝑢(𝑛) . (33) −𝜋

Fig. 10. Simulated MSEs and recursive MSE bounds under –5, 0, and 5 dB subarray SNRs.

the knowledge of a reference signal or signal synchronization, and they are also Doppler resilient. The DBT algorithm has lower complexity and converges faster than the DBS algorithm, and is suitable for tracking fast time-varying AoA. The DBS algorithm can be used for beam acquisition in a more practical hybrid array of side-by-side subarrays. The proposed AoA estimation can be formulated as a phase estimation problem under recursive nuisance parameters and a recursive MSE bound is derived to give a meaningful indication of the estimation performance. The effects of mutual coupling and other practical impairments in the hybrid array will be studied in our future research. A PPENDIX R ECURSIVE MSE B OUND

( ) The lower MSE bound is obtained by replacing 𝑓𝑢(𝑛) 𝑢(𝑛) with a known phase distribution determined only by the average SNR of 𝑅(𝑛) , defined by { } (𝑛) 𝛾¯ (𝑛) =𝐸˜s,u 𝛾˜s,u { } { ( 𝑛 2 ) 2 } 1∑ ∣˜ 𝑠 [𝑘]∣

¯ (𝑘−1) 𝑃 𝑢 = 𝐸 𝐸 (𝑘−1)

𝑠 𝑢 2 𝜎𝑧2 𝑘=1 { ( 𝑛 ) 2 } 1 ∑

= 𝛾 𝐸𝑢(𝑘−1) 𝑃¯𝑠 𝑢(𝑘−1) , (31) 2 𝑘=1

and the known pdf of 𝑢(𝑛) is determined according to the nature of the reference signal 𝑠˜ [𝑛]. If 𝑠˜ [𝑛] has a constant phase envelope, the known pdf of 𝑢(𝑛) can be chosen as the ( (𝑛) ) distribution under Gaussian channel [26], i.e, 𝑓 𝑢 = (𝑛) 𝑢 ( (𝑛) (𝑛) ) . However, a reference signal will generally 𝑓0 𝑢 , 𝛾¯ have envelope fluctuations, even if a single carrier Quadrature Phase Shift Keying (QPSK) type modulation is used for data communications, because the proposed algorithms are blind and do not require any signal synchronization. For this reason, 𝑠˜ [𝑛] is generally assumed to be Gaussian distributed and hence ∣˜ 𝑠 [𝑛]∣ in (21) is Rayleigh distributed. The known pdf of 𝑢(𝑛) is thus chosen as the phase distribution under Rayleigh fading

Therefore, the right-hand side ( of (33) ) represents a lower MSE bound, denoted as 𝑀 𝑆𝐸𝐵 𝜎𝑢2 (𝑛) . Furthermore, from (31) 𝛾¯ (1) can be determined as { ( ) 2 } 1

𝛾¯ (1) = 𝛾𝐸𝑢(0) 𝑃¯𝑠 𝑢(0) 2 ( (0) )

2 ∫ 𝜋

1 1

sin 𝑁(𝑢 ) = 𝛾 𝛾 (34)

𝑑𝑢(0) =

4𝜋 −𝜋 𝑁 sin 𝑢(0) 2𝑁 where the initial 𝑢(0) is assumed to be uniformly distributed in [−𝜋, 𝜋), and 𝛾¯ (𝑛) for 𝑛 > 1 can be recursively determined as 𝛾¯ (𝑛) =

∫ 𝜋 ( )2 ( ) 1 1 ¯ (1) (1) (1) 𝛾+ 𝛾 𝑑𝑢(1) + ⋅ ⋅ ⋅ 𝑃𝑠 𝑢 𝑓1 𝑢 , 𝛾¯ 2𝑁 2 −𝜋 ∫ 𝜋 ( )2 ( ) 1 ¯ (𝑛−1) (𝑛−1) + 𝛾 , 𝛾¯ (𝑛−1) 𝑑𝑢(𝑛−1) 𝑃𝑠 𝑢 𝑓1 𝑢 2 −𝜋

=¯ 𝛾 (𝑛−1) ∫ 𝜋 ( )2 ( ) 1 ¯ (𝑛−1) (𝑛−1) + 𝛾 , 𝛾¯ (𝑛−1) 𝑑𝑢(𝑛−1) . 𝑃𝑠 𝑢 𝑓1 𝑢 2 −𝜋 (35)

The MSE bound calculated based on the above recursively determined average SNR is thus called Recursive MSE Bound. At high SNR 𝑓0 (𝑥, 𝛾) in (26) can be approximated as the Gaussian distribution √ ( ) 𝛾 exp −𝛾𝑥2 , −𝜋 ≤ 𝑥 < 𝜋, 𝑓0 (𝑥, 𝛾) ≈ (36) 𝜋 and accordingly 𝑓1 (𝑥, 𝛾) can be approximated as √ 𝛾𝜋 2 + 1 , −𝜋 ≤ 𝑥 < 𝜋, (37) 𝑓1 (𝑥, 𝛾) ≈ 2𝜋 (𝛾𝑥2 + 1)3/2 ∫𝜋 after normalizing it to satisfy the condition −𝜋 𝑓1 (𝑥, 𝛾) 𝑑𝑥 = 1. A closed-form equation of the MSE bound is then obtained as ∫

( ) 𝑓1 𝑢(𝑛) , 𝛾¯ (𝑛) 𝑑𝑢(𝑛) −𝜋 √ ∫ 𝜋 ( )2 𝛾¯ (𝑛) 𝜋 2 + 1 (𝑛) 𝑢(𝑛) ≈ ( )3/2 𝑑𝑢 2 −𝜋 2𝜋 𝛾 ¯ (𝑛) (𝑢(𝑛) ) + 1 √ (√ ) 1 𝛾¯ (𝑛) 𝜋 2 + 1 = sinh−1 𝛾¯ (𝑛) 𝜋 − (𝑛) . 3 𝛾¯ 𝜋 (¯ 𝛾 (𝑛) ) 2 (38)

( ) 𝑀 𝑆𝐸𝐵 𝜎𝑢2 (𝑛) =

𝜋

(

𝑢(𝑛)

)2


R EFERENCES [1] Y. Jay Guo, Advances in Mobile Radio Access Networks. Artech House, Inc., 2004. [2] F. B. Gross, Smart Antennas for Wireless Communications. McGrawHill, 2005. [3] Robert J. Mailloux, Phased Array Antenna Handbook. Artech House, Inc., 2005. [4] D. Rogstad, A. Mileant, and T. Pham, Antenna Arraying Techniques in the Deep Space Network. Wiley-IEEE, 2003. [5] H. Meinel, “Commercial applications of millimeterwaves history, present status, and future trends," IEEE Trans. Microwave Theory Techniques, vol. 43, no. 7, pp. 1639-1653, July 1995. [6] J. I. Agbinya, et al., Advances in Broadband Communication and Networks. River Publishers, 2008. [7] V. Dyadyuk, J. Bunton, J. Pathikulangara, R. Kendall, O. Sevimli, L. Stokes, and D. Abbott, “A multi-gigabit mm-wave communication system with improved spectral efficiency," IEEE Trans. Microwave Theory Techniques, vol. 55, no. 12, pp. 2813-2821, Dec. 2007. [8] H. Singh, J. Oh, C. Y. Kweon, X. Qin, H.-R. Shao, and C. Ngo, “A 60 GHz wireless network for enabling uncompressed video communication," IEEE Commun. Mag., pp. 71-78, Dec. 2008. [9] V. Dyadyuk and Y. J. Guo, “Towards multi-gigabit ad-hoc wireless networks in the E-band," Global Symp. Millimeter Waves (GSMM 2009), Sendai, Japan, Apr. 2009. [10] J. Wells, “Faster than fiber: the future of multi-Gb/s wireless," IEEE Microwave Mag., pp. 104-112, May 2009. [11] H. Zirath, T. Masuda, R. Kozhuharov, and M. Ferndahl, “Development of 60-GHz front-end circuits for a high-data-rate communication system," IEEE J. Solid-State Circuits, vol. 39, no. 10, pp. 1640-1649, Oct. 2004. [12] C. H. Doan, S. Emami, D. A. Sobel, A. M. Niknejad, and R. W. Brodersen, “Design considerations for 60 GHz CMOS radios," IEEE Commun. Mag., vol. 42, no. 12, pp. 132-140, Dec. 2004. [13] T. Do-Hong and P. Russer, “Signal processing for wideband smart antenna array applications," IEEE Microwave Mag., pp. 57-67, Mar. 2004. [14] Y. Jay Guo, J. Bunton, V. Dyadyuk, and X. Huang, “Multi-stage hybrid adaptive antennas," Australian provisional patent, AU2009900371, filed on 2 Feb. 2009. [15] A. P. Goffer, M Kam, and P. R. Herczfeld, “Design of phased arrays in terms of random subarrays," IEEE Trans. Antennas Propagation, vol. 42, no. 6, pp. 820-826, June 1994. [16] R. L. Haupt, “Optimized weighting of uniform subarrays of unequal sizes," IEEE Trans. Antennas Propagation, vol. 55, no. 4, pp.1207– 1210, Apr. 2007. [17] R. J. Mailloux, “Subarray technology for large scanning arrays," in Second European Conf. Antennas Propagation (EuCAP2007), Nov. 2007. [18] H. Krim and M. Viberg, “Two decades of array signal processing research," IEEE Signal Process. Mag., pp. 67-94, July 1996. [19] T. Xia, Y. Zheng, Q. Wan, and X. Wang, “Decoupled estimation of 2-D angles of arrival using two parallel uniform linear arrays," IEEE Trans. Antennas Propagation, vol. 55, no. 9, pp. 2627-2632, Sept. 2007. [20] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and CramerRao bound," IEEE Trans. Acoustics, Speech, Signal Process., vol. 37, no. 5, pp. 72-741, May 1989. [21] M. Viberg and B. Ottersten, “Sensor array processing based on subspace fitting," IEEE Trans. Signal Process., vol. 39, no. 5, pp. 1110-1121, May 1991. [22] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, “Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing," Radar Array Processing, S. Haykin, J. Litva, and T. J. Shepherd, editors, pp. 99-151, Springer-Verlag, 1993. [23] P. Stoica and K. C. Sharman, “Maximum likelihood methods for direction-of-arrival estimation," IEEE Trans. Acoustics, Speech, Signal Process., vol. 38, no. 7, pp. 1132-1143, July 1990. [24] A. Abbaspour-Tamijani and K. Sarabandi, “An affordable millimeterwave beam-steerable antenna using interleaved planar subarrays," IEEE Trans. Antennas Propagation, vol. 51, no. 9, pp. 2193-2202, Sept. 2003. [25] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th edition. McGraw-Hill, 2002. [26] R. Reggiannini, “A fundamental lower bound to the performance of phase estimators over Rician-fading channels," IEEE Trans. Commun., vol. 45, no. 7, pp. 775-778, July 1997. [27] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,"

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IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1391-1399, Feb./Mar./Apr. 1994. [28] Q. Yuan, Q. Chen, and K. Sawaya, “Performance of adaptive array antenna with arbitrary geometry in the presence of mutual coupling," IEEE Trans. Antennas Propagation, vol. 54, no. 7, pp. 1991-1996, July 2006. [29] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: a rigorous network theory analysis," IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1317-1325, July 2004. [30] Z. Huang and C. A. Balanis, “The MMSE algorithm and mutual coupling for adaptive arrays," IEEE Trans. Antennas Propagation, vol. 56, no. 5, pp. 1292-1296, May 2008. Xiaojing Huang (M’99) received his Bachelor of Engineering, Master of Engineering, and Ph.D. degrees from Shanghai Jiao Tong University, Shanghai, China, in 1983, 1986, and 1989, respectively, all in electronic engineering. From 1989 to 1994, Dr. Huang worked in the Electronic Engineering Department of Shanghai Jiao Tong University, where he had been a Lecturer since 1989 and an Associate Professor since 1991. Form 1994 to 1997, he was the Chief Engineer with Shanghai Yang Tian Science and Technology Corporation Limited, Shanghai, China. In 1998, he joined the Motorola Australian Research Centre, Sydney, Australia, as a Senior Research Engineer and had been a Principal Research Engineer since 2003. From 2004 to 2009, he was an Associate Professor in the School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, Australia. In March, 2009, he joined the ICT Centre, Commonwealth Scientific and Industrial Research Organization (CSIRO), Sydney, Australia, as a Principal Research Scientist. His research interests are in communications theory, digital signal processing, and wireless communications networks. Y. Jay Guo (SM’96) is the Director of Broadband for Australia Theme in CSIRO ICT Centre, Australia, and the Director of the Australia China Research Centre for Wireless Communications. From 2005 to January 2010, he served as the Director of the Wireless Technologies Laboratory in CSIRO ICT Centre. Prior to joining CSIRO, Jay held various senior positions in the European wireless industry managing strategic planning and the development of advanced technologies for the third generation (3G) mobile communications systems. Jay has played active roles in the organizing committees of a number of international conferences. He served as Chair of the Technical Program Committee (TPC) of 2010 IEEE WCNC and 2007 IEEE ISCIT. He was Executive Chair of Australia China ICT Summit in 2009 and 2010. He was a Guest Editor of the special issue on “Antennas and Propagation Aspects of 6090GHz Wireless Communications" in IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION . Jay is the recipient of Australian Engineering Excellence Award and CSIRO Chairman’s Medal. He has published three technical books, Fresnel Zone Antennas, Advances in Mobile Radio Access Networks, and GroundBased Wireless Positioning, and authored and co-authored 50 journal papers and over 80 refereed international conference papers. He holds fourteen patents in wireless communications and antennas. He is an Adjunct Professor at Macquarie University, Australia, and a Guest Professor at the Chinese Academy of Science (CAS). He is a Fellow of IET. John D. Bunton received the B.Sc., B.E., and Ph.D. degrees from the University of Sydney, SMIEE. He has been with CSIRO, since 1989, where he is currently a Senior Principal Research Engineer in the ICT Centre. From 1983 to 1988 he was at the Fleurs Radio telescope, Kemps Creek, Australia, where he was responsible for all engineering aspects of the telescope as well as having a significant involvement in the astronomy. At CSIRO he worked in many areas including digital audio, sonar, mining communications and safety, mm-wave imaging, and Gigabit wireless. He is currently project engineer for the Australian SKA Pathfinder and is collaborating in the building of correlators for the SKAMP and MWA radio telescopes. Other current research interests are the Square Kilometre Array (SKA), beamforming systems and rural wireless access.