軟體無線通訊器多工系統之分析設計(II)

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行政院國家科學委員會專題研究計畫成果報告

軟體無線通訊器多工系統之分析設計(2/2)

計畫類別：個別型計畫

計畫編號： NSC91-2219-E-011-004-

執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日執行單位：國立臺灣科技大學電子工程系

計畫主持人：方文賢共同主持人：陳俊才

報告類型：完整報告

處理方式：本計畫可公開查詢

中華民國 92 年 10 月 30 日

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摘要

在先前的研究中，以小波為基礎我們提出了廣義旁波帶消除器，於本計畫中，我們進一步將小波子頻帶拆解技術應用於時間的拆解，以進一步增進天線陣列無線通信系統中之調適濾波器之收斂速度。此外，為了易於在即時系統上實現，降低調適處理的運算複雜度是非常重要的，尤其是在應用時間的子頻帶拆解時。為了達成此一目的，我們應用選擇性部分區塊更新法來減少調適濾波器的維度。由已完成的模擬結果可驗證我們所提出的方法可以產生與其他方法相近的束波圖，但卻可以大大地降低運算複雜度，並使調適濾波器更快收斂。

同時，在本計畫中，我們更進一步應用小波濾波器群組來設計分碼多工多使用者偵測偵測器。為了有效利用空間處理技術與時間處理技術，我們提出一個低複雜度的空─時處理架構，此一架構由先前所提及之以小波為基礎之束波器與以小波為基礎之多使用者偵測器串接而成。由於串聯結構之運算複雜度較低，我們將採用串聯結構來設計以小波為基礎之線性偵測器，此線性偵測器將由一個束波器和一個多使用者偵測器串接而成。針對要在即時系統實現的考量，在此我們亦將應用選擇性部分區塊更新法在束波器和多使用者偵測器上。此外，由於束波器的運算負載遠大於多使用者偵測器的運算負載，在此我們另考慮將週期性部分更新法應用在束波器上以平衡束波器與多使用者偵測器的運算負載。相關的模擬結果驗證我們所提出方法的可行性。

關鍵字：廣義旁波帶消除器、小波子頻帶拆解、選擇性部分區塊更新法、收斂速度、束波圖、分碼多工線性偵測器、束波器、多使用者偵測、週期性部分更新法。

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ABSTRACT

A recently introduced wavelet-based partially adaptive beamformer with the generalized sidelobe canceller (GSC) as the underlying structure has become a promising choice. In this project, we further employ the wavelet subband decomposition in the temporal domain to further enhance the convergence speed of the adaptive processing. In addition, to facilitate real-time implementations, it is of importance to mitigate the computational load in the weight adaptive process, es- pecially with the introduction of the temporal subband decomposition. To achieve this, we employ the block-selective partial update scheme to reduce the dimension of adaptive weight vectors. As verified by furnished simulation results, the proposed approach can produce comparable beampattern, but with substantially reduced computational overheads and faster convergence speed, compared with previous works.

Furthermore, the technique of wavelet subband filter banks is further employed in this project to design a multiuser detector in code division multiple access (CDMA) systems. To fully exploit the spatial processing and temporal processing, a low complexity spatio-temporal processing scheme is addressed as well, which combines the aforementioned wavelet-based beamformer and multiuser detector in a cascaded structure. To be amenable to real-time implementations, the block- selective partial update scheme is again also utilized in both of the spatial filtering and temporal filtering stages. Furthermore, since the computational burden of the beamformer is substantially higher than that of the succeeding multiuser detecion (MUD), a periodic partial update scheme is used in the beamformer stage to further reduce the computational complexity in order to balance the computational load between these two stages. Relevant simulations are also provided to verify the proposed approach.

Keywords: generalized sidelobe canceller, wavelet subband decomposition, block- selective partial update scheme, convergence speed, beampattern, CDMA linear detector, beamforming, multiuser detection, periodic partial update scheme.

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Chapter 1 INTODUCTION

In the emerging code division multiple access (CDMA) systems, how to efficiently mitigate the co-channel interference (CCI), which is caused by the frequency reuse, is a notable problem. The CCI limits the quality and capacity of the wireless communication system so that if we can alleviate the CCI, system performance can then be improved.

An efficient approach to mitigate the effect of CCI is to introduce an extra spatial dimension, i.e. add in antennas array at the receivers/transmitters. A space-time receiver processes signal samples both in space and time and, as shown in various research reports [1, 2, 3, 4, 5], can indeed improve network capacity, coverage, and quality by reducing CCI while enhancing diversity and array gain.

This is due to the fact that the additional spatial dimension can increase the ability of interference cancellation in a way that is not possible with single antenna receiver. The desired signal and CCI almost always arrive at the antenna array with distinct and often well-separated spatial signatures, and the spatial diversity allow the receiver to employ this difference to reduce CCI. The spatial dimension can also be used to enhance other aspects of receivers and transmitters performance.

For example, in the receiver, the antenna array can used to enhance the array gain and improve signal-to-thermal noise ratio, increase diversity gain, and even suppress intersymbol interference (ISI). In the transmitter, the spatial dimension can enhance array gain, improve transmit diversity, and reduce delay spread at the subscriber end. Thus, the space-time transmitter can use spatial selectivity to transmit signals to desired subscriber and minimize interferences from other users simultaneously.

As such, space-time processing has attracted much attention recently. For example, Bernstein et al [6] considered the applications of the space-time process- ing schemes to increase the capacity of CDMA systems. They proposed the following receiver configurations: (1) jointly space-time optimum combining (2) cascaded space-time maximum ratio combining (MRC), (3) cascaded optimum combining in space and MRC in time, (4) cascaded space-time optimum combining. Also, Chu et al [7] and Wang et al [8] employed space-time processing to design broadband

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beamformers to suppress the interferences in wireless communications. Yener et al [2] proposed a two-dimensional (2-D) jointly spatial and temporal filtering, alone with a power control scheme. The latter, however, needs higher dimensional optimization and thus requires enormous computational overhead. Additionally, it requires a priori information of both the desired user and the interferences, such as received amplitudes, noise level, DOAs, spreading sequences and timing.

Employing the subband decomposition in wireless communication system has also received great attention recently. For example, Chu et al [7] employed the wavelet filters to form the blocking matrix of the generalized sidelobe canceller (GSC). Liu et al [9] employed the generalized DFT (GDFT) in the GSC. Because of the prewhitening effect of the subband decomposition, employing the GDFT decomposition in the GSC can result in a faster convergence speed. Zhang et al [10] considered using the filter banks to mitigate the multipath fading in adaptive array processing. He proposed a subband adaptive array for the mitigation of both ISI and CCI in digital mobile communications and the subband filter banks are used in the front end of the adaptive array receiver, in which the analysis filters enhance the correlation of multipath signals in each subband by decomposing the received signals into a set of subband signals. The adaptive array in [10] is blind in the sense that no a priori knowledge of the temporal characteristics or spatial characteristics of received signals are required.

However, employing the spatial processing will incur a lot of extra computational overheads and increase the complexity of the system, and thus such space-time processing may become unfeasible in practice. Therefore, how to de- crease the computational complexity is a noteworthy issue. Since the computational complexity of adaptive algorithms is proportional to the number of filter coefficients, to reduce the computational complexity, Douglas et al [11, 12] and Do˘gancay et al [13] have addressed the periodic partial update scheme and the block-selective partial update scheme, respectively, in reducing the computational complexity of adaptive filtering. Both schemes update only a portion of the filter coefficients. However, these partial schemes are only employed in temporal adap- tive filtering applications. For this, Goldstein et al [14] proposed to employ the transform matrix to reduce the computational complexity in the spatial adaptive filtering (beamforming).

In this project, we consider the beamformer with GSC as the underlying structure to suppress the interference in wireless communication systems and employ the wavelet-based subband filter banks to further carry out the temporal subband decomposition to increase the convergence speed. Additionally, to meet the requirement of real time processing, partial schemes are employed to reduce the computational complexity. We begin our research by further employing the wavelet subband decomposition in the temporal domain to further enhance the convergence speed of the adaptive processing and employing the block-selective partial update scheme to reduce the computational complexity of the GSC-based beamformer. Then, we further employ the technique of wavelet subband filter banks to design a multiuser detector in CDMA systems. Furthermore, we com- bine the wavelet-based spatial filtering and the wavelet-based temporal filtering to design a low complexity cascaded spatio-temporal linear detector.

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Chapter 2 WIRELESS COMMUNICATION SIGNAL MODELS

In this chapter, we discuss the signal models employed in this project including broadband signal model and the practical DS-CDMA baseband signal model. Ad- ditionally, we will discuss the CDMA signal models under multipath scenarios.

The broadband data model is used in chapter 4, and the CDMA data model and the multupath effect are employed in chapter 5.

2.1 Broadband Signal Model

The equispaced broadband antenna array with N antennas in which every antenna has J taps in the succeeding tap-delay line. The intertap delay spacing is T and the time delays T_i = (i−1)d sin θs

λ , i = 1, · · · , N , are broadband delays used to presteer the antenna array in a specified direction called the look direction. At time instant n, the output signal y(n) can be expressed as

y(n) = w^Hx(n) (2.1)

where

x(n) = ^h x^T(n) x^T(n − 1) · · · x^T(n − J + 1) ⁱ^T (2.2) x(n − i) = ^h x₁(n − i) x₂(n − i) · · · x_N(n − i) ⁱ^T (2.3)

w = ^h w^T₁ w^T₂ · · · w^T_J ⁱ^T (2.4)

w_i = ^h w1i w2i · · · wN i

i_T

(2.5) Suppose that there are I interferences in the wireless communication environment, the received data at n^th snapshot can be represented as

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x(n) = s_s(n)a_s(θ_s) +

XI

i=1

s_i(n)a_I(θ_i) + n (2.6) where s_s(n) denotes the waveform of the desired user.

as(θs) = ^h e^j(1−n⁰^)µ^s e^j(2−n⁰^)µ^s · · · e^{j(N −n}⁰^)µ^s ⁱ^T (2.7) is the steering vector of the desired user with arrival angle θ_s, where n₀ is the reference antenna, µ_s = 2πf d sin θs

λ in which λ is the carrier wavelength, d is the dis- tance between the antennas, and f ∈ [fl, fu] with fl and fu being, respectively, the lowest frequency and highest frequency of the desired user is the relative temporal frequency. s_i(n), i = 1, · · · , I, denote the waveforms of the interferences and are assumed to be complex Gaussian distributed with power σ_Ii², i = 1, · · · , I.

a_I(θ_i) =^h e^j(1−n⁰^)µⁱ e^j(2−n⁰^)µⁱ · · · e^{j(N −n}⁰^)µⁱ ⁱ^T i = 1, 2, · · · , I (2.8) are the corresponding steering vectors of the interferences from θi, where µi =

2πf d sin θi

λ . n is the additive white noise and is complex Gaussian distributed with power σ_n².

2.2 DS-CDMA Signal Model

In the DS-CDMA system, each user is assigned a unique signature sequence. In this project, the Gold Code with processing gain J is employed. Assume that the base station employs an antenna array with N elements, the received signal vector at the output of the antenna array can be expressed as

x(t) =

XK

k=1

√pkbksk(t)e^j2πf^c,k^ta(θk) + n(t) (2.9)

where p_kis the received signal power, b_kis the information bit, s_k(t) is the signature waveform, f_c,k is the carrier frequency, and a(θ_k) is the steering vector of the k^th user. The signature waveforms and the steering vectors are all normalized.

Without loss of generality, the first user is assumed to be the desired user and the others are interferences. The signature waveforms can be expressed as

s_k(t) =

XJ

j=1

s^j_kψ(t − (j − 1)T ), k = 1, 2, · · · , K (2.10)

where ψ(t) is the chip waveform, T is the chip duration and s^j_k = 0 or _ks¹_k_k.

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2.2.1 Baseband Signal Model

The baseband signal model in DS-CDMA system can be obtained by multiplying the complex carriers e^−j2πf^c,k^tin (2.9), and the received signal after modulation at the output of the antennas array can be expressed as

x(t) =

XK

k=1

√p_kb_ks_k(t)a(θ_k) + n(t) (2.11)

The received signals are filtered by the chip matched filters and sampled under the chip rate 1/T . Therefore, the signal received during the l^th chip interval can be expressed as

x_j =

XK

k=1

√p_kb_ks^(j)_k a(θ_k) + n_j j = 1, 2, · · · , J (2.12)

which expresses the signal sampled by the chip matched filters at the l^th chip interval across the N antennas array elements. The signal collected during one bit duration T_b = JT can be expressed as

X =

XK

k=1

√p_kb_ka(θ_k)s^T_k + N (2.13)

where (.)^T denotes the matrix transpose, X = [x1, x2, · · · , xJ]^T, a(θk) = [a⁽¹⁾(θk), a⁽²⁾(θ_k), · · · , a^{(N )}(θ_k)]^T, s_k = [s⁽¹⁾_k , s⁽²⁾_k , · · · , s^(J)_k ]^T denotes the spreading sequence of the k^th user, and N = [n1, n2, · · · , nJ] denotes the additive white Gaussian noise (AWGN) in baseband.

We can rewrite (2.13) in more compact notation as

X = ABDS^T + N (2.14)

where A = [a(θ₁), a(θ₂), · · · , a(θ_k)], B = diag(√ p₁,√

p₂, · · · ,√

p_k), D = diag(b₁, b₂, · · · , b_k), and S = [s₁, s₂, · · · , s_k].

2.2.2 Multipath Scenario

In wireless communication, the signal is usually affected by objects (scatters) in the environment such as buildings, trees, and terrain features, and thus produces the multipath phenomenon. In the multipath phenomenon, the received signals contain both delay spreads and angle spreads. Additionally, when the users are moving, they also contain the Doppler spread. In classical outdoor cellular systems, the delay spread is in the order of 0-10 microseconds, the angle spread ranges from 2 to 60 degrees, and the Doppler spread from 5 to 200 Hz [1].

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The received signal is now given by

x(t) =

XK

k=1 Lk

X

l=1

√pkβkl(t)bk(t − τkl)sk(t − τkl)e^j(2πf^c,k^+φ^kl^)ta(θkl) + n(t) (2.15)

where Lk denotes the number of paths of the k^th user, and τkl is the delay time, φ_kl is the shifted phase, θ_kl is the changed DOA, and β_kl is the fading amplitude of the l^th path of the k^th user.

After sampling, we can rewrite (2.15) as

X = ABGDS^T + N (2.16)

where AN ×L=[a(θ1,1), a(θ1,2), a(θ1,3), · · ·, a(θK,LK)], BL×L=diag(β1,1√

p1, β1,2√ p1, β1,3√

p1, · · ·, βK,LK

√pK), GL×L= diag(e^j(2πf^c,1^+φ^1,1^)nT, e^j(2πf^c,1^+φ^1,2^)nT, e^j(2πf^c,1^+φ^1,3^)nT,

· · ·, e^j(2πf^c,K^+φ^K,LK^)nT), D_L×L = diag(b₁, b₁(τ₁₂), b₁(τ₁₃), · · · , b_K(τ_KL_K)), and S_J×L

= [s₁, s₁(τ₁₂), s₁(τ₁₃), · · ·, s_K(τ_KL_K)], in which b_k(τ_kl) and s_k(τ_kl) denote the ver- sions of the information bit and spreading sequence which are delayed τkl of the l^th path of the k^th user, respectively.

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Chapter 3 REVIEW OF PREVIOUS WORKS

In this chapter, we discuss some previous works which are relevant to our research. First, we introduce the spatial filters (beamformers), and some popular design approaches which can suppress the interferences by incorporating a priori spatial information. Next, we present some well-known MUD techniques, which play the roles of the temporal filtering. Then, we review some combined spatio- temporal filtering schemes, such as the optimum and suboptimum spatio-temporal MMSE detector [2], and the jointly and cascaded space-time optimum diversity combining schemes [6]. Finally, we introduce some partial schemes to alleviate the computational complexity.

3.1 Beamforming

In this section, we discuss the essence of beamforming, which plays the role of spatial filtering. We review some widespread optimum beamformers. Also, we consider real-time implementation of these beamformers which can adjust the weight vectors according to the characteristics of the outside environment.

A widespread approach to the design of optimum beamformers is to mini- mizing the output power Po = E[y(n)y^H(n)] = w^HRxw under a set of judiciously chosen linear constraint. Therfore, the determination of the weight vector can be posed as the following minimization problem

minw w^HR_xw subject to C^Hw = f (3.1) where C is the constraint matrix and f is the corresponding gain vector.

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The above can be solved by the Lagrange multiplier given by

w^HR_xw + λ(C^Hw − f) (3.2)

Minimizing the Lagrangian with respect to w^∗ and solving for Lagrange multiplier, we can obtain

w = R⁻¹_x C(C^HR⁻¹_x C)⁻¹f (3.3) Such a determination of w is referred to as the linear constrained minimum variance beamforming(LCMV)[15]. The constrained beamforming provides a better resolution when compared to the classical beamforming. The technique uses some of the degrees of freedom to form a beam in the desired look direction while simultaneously using the remaining degrees of freedom to form nulls in the direction of interferences. However, the LCMV suffers the following drawbacks: (1) It fails when other signals are correlated with look directional signal; (2) It requires the computation of a matrix inverse which need higher computational overheads.

As an alternative implementation, we can decompose w into two parts: one is the constraint subspace and the other orthogonal to it, i.e.,

w = wq− Bwa (3.4)

where w_q is an N × 1 weight vector given by

w_q= C(C^HC)⁻¹f (3.5)

with B being an N × (N − m) matrix, referred to as the blocking matrix, is orthogonal to C, and w_a is an (N − m) × 1 weight vector. Using this decomposi- tion, the LCMV beamforming can be reformulated as the following unconstrained optimization problem

minwa (w_q− Bw_a)^HR_x(w_q− Bw_a) (3.6)

Such a decomposition scheme given in (3.4) is referred to as the generalized sidelobe canceller (GSC)[16] structure, as shown in Fig.3.1.

The optimum solution of w_a can be readily shown to be

w_a = (B^HR_xB)⁻¹B^HR_xw_q (3.7)

3.2 Multiuser Detection

In contrast to the beamformer, the MUD is an effective approach to perform the temporal filtering to mitigate interferences by using the temporal information in

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Figure 3.1: Block diagram of a generalized sidelobe canceller (GSC).

CDMA communication systems. Assume that the number of users and the length of spreading sequence are K and J, respectively, in a single path binary phase shift keying (BPSK) real channel and the baseband received signal can then be expressed as

x(t) =

XK

k=1

√p_kb_ks_k(t) + n(t), t ∈ [0, T ] (3.8)

where p_kis the received signal power, b_kis the information bit, s_k(t) is the spreading sequence of the k^th user, and n(t) is the AWGN with variance σ².

We present the conventional detector which consists of a bank of single user matched filters, as shown in Fig. 3.2. The output signal of the k^th matched filter can be expressed as

x_k =

Z _T

0 x(t)s_k(t)dt k = 1, · · · , K

= < x(t), sk(t) >

= √

pkbk < sk, sk > +^X

i6=k

√pibi < si, sk > +nk (3.9)

If we define the cross correlation coefficient ρ_ij =< s_i(t), s_j(t) > and assume

||s_k||² = 1, then we can rewrite (3.9) as xk=√

pkbk+^X

i6=k

√pibiρik+ nk (3.10)

where n_k = ρ^R₀^T n(t)s_k(t)dt.

For a compact notation, we rewrite the received signal vector x = [x1, x2, · · · , xK]^T

as x = RBd + n (3.11)

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⋅

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⋅

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⋅

#

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Figure 3.2: The conventional detector with a bank of single user matched filters where R = [ρ_ij] is the normalized cross correlation matrix, B = diag[√

p₁,√

p₂, · · · ,√ p_K] is the amplitude matrix, d = [b₁, b₂, · · · , b_K]^T is the information bit vector, and n = [n₁, n₂, · · · , n_K]^T is the AWGN vector.

There are some problems existing in the conventional MUD. First, when the number of interferences increases, the multiple access interference (MAI) also increases. Second, if the power level of each user in the received signal is different, the weaker users may be overwhelmed by stronger ones. This phenomena, referred to as the near-far effect, is caused by the fact that the transmitters have different geographical locations relative to the receiver or the channel fading.

3.3 Partial Schemes

To meet the requirement of real time implementations, the computational complexity should be limited in an acceptable range. Although the additional spatial dimension can enhance the performance of the communication system, it also substantially increases the overall computational complexity. In this section, we introduce some partial schemes, including the block-selective partial update scheme, and the periodic partial update scheme, all of which adjust only portion of the adaptive weight vector to mitigate the computational complexity.

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3.3.1 Block-Selective Partial Update Scheme

Do˘gancay et al [13] proposed block-selective partial update scheme which reduces the computational complexity of the NLMS algorithm by updating only a subset of the filter coefficients at every iteration. The approach first partitions the input vector x(k) and the coefficient vector w(k) into M blocks of length L = J/M

x(k) =^h x^T₁(k) x^T₂(k) · · · x^T_M(k) ⁱ^T w(k) =^h w^T₁(k) w₂^T(k) · · · w_M^T (k) ⁱ^T

where the coefficient vector blocks w₁(k), w₂(k), · · · , w_M(k) denote candidate sub- sets of w(k) that need to be updated at time instant k. Assuming we only want to update K blocks out of M at every iteration. Let S_K = {s₁, s₂, · · · , s_K} denotes a K-subset with M members of the set {1, 2, · · · , M }, and S be the collection of all K-subsets, i.e., SK ∈ S. The selection of blocks can be carried out by considering the following constrained minimization problem:

SminK∈S min

w_SK(k+1)||w_S_K(k + 1) − w_S_K(k)||²

subject to w^T(k + 1)x(k) = d(k) (3.12) where

w_S_K(k) =^h w^T_s₁(k) w^T_s₂(k) · · · w_s^T_K(k) ⁱ^T

If the blocks which are needed to update at each iteration are given and fixed, the above can be solved by using the Lagrange multiplier given by

J_S_K(k) = 1

2||w_S_K(k + 1) − w_S_K(k)||²+ λ(d(k) − w^T(k + 1)x(k)) (3.13) Minimizing the Lagrangian with respect to w(k+1) and solving for the Lagrangian multiplier, we can obtain

w_S_K(k + 1) = w_S_K(k) + µ

||xSK(k)||²x_S_K(k)e(k) (3.14) where

x_S_K(k) =^h x^T_s₁(k) x^T_s₂(k) · · · x^T_s_K(k) ⁱ^T The number of unique K-subsets of {1, 2, · · · , M } is

Ã M B

!

= M!

B!(M − B)! (3.15)

To choose the K weight blocks which are needed to be updated, we need to find K coefficient blocks with the minimum squared-Euclidean-norm update. The block

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selection problem can be formally written as:

S_K = arg min

SK∈S||w_S_K(k + 1) − w_S_K(k)||²

= arg min

SK∈S||x_S_K(k)e(k)

||x_S_K(k)||²||²

= arg max

SK∈S

X

s∈SK

||x_S_K(k)||² (3.16)

3.3.2 Periodic Partial Update Scheme

Douglas et al [11] employed the periodic partial update scheme [12] in the temporal adaptive filtering to mitigate the computational complexity. This scheme only update the same portion of the adaptive filter coefficients at every time instant.

The coefficient updates are given by

w_i(k + 1) =







wi(k) + µe(l)x(l − i + 1)

, if (k + i)modN = 0 and l = Nbk/Nc w_i(k) , otherwise

(3.17)

where e(k) = d(k)−w^T(k)x(k), w(k) =^h w₁(k) w₂(k) · · · w_J(k) ⁱ^T is the coef- ficient vector of the adaptive filter at time k, x(k) =^h x₁(k) x₂(k) · · · x_J(k) ⁱ^T is the input signal vector, d(k) is the desired response signal, e(k) is the error sig- nal, and b·c denotes the truncation operation. For N = 1, this algorithm reduces to the LMS algorithm.

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Chapter 4 PARTIALLY ADAPTIVE BEAMFOMERS VIA

WAVELET-BASED JOINT SPATIO-TEMPORAL

SUBBAND DECOMPOSITION

In this chapter, we first reviewed a recently introduced wavelet-based partially adaptive beamformer [7] with the GSC [16] as the underlying structure has become a promising choice. The beamformer in [7] employs a set of P -regular, M-band wavelet filters in the design of the blocking matrix involved. This new beamformer is simple, as it does not require computationally demanding eigendecomposition [17] or extra filter banks [18]. In addition, it offers satisfactory performance with fast convergence characteristic compared with previous works. To further speed up the convergence rate of [7] , in this chapter we carry out a temporal subband decomposition of the output of the blocking matrix via the wavelet filters. As such, the input signals are now decomposed in both of the spatial and temporal subbands via the wavelet filters. Furthermore, in light of the fact that the input signals in general occupy only a portion of the spatial or temporal subbands, we also truncate some insignificant subbands to reduce the overall complexity. We attempt to pick the optimal spatio-temporal subband components for adaptive processing under a prescribed criterion. Such a joint processing scheme can be expected to effectively disregard those insignificant spatio-temporal subband components and thus will not substantially degrade the overall performance.

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4.1 GSC-Based Spatio-Temporal Subband De- composition

In [7] , it was suggested to use an N × (^j^{N −M P}_M ^k+ 1)(M − 1) ( bαc denotes the largest integer smaller than or equal to α), blocking matrix B given by

B^T =







H1

H₂ ...

H_{M −1}





 with (4.1)

H_m =







h_m(0) · · · h_m(MP − 1) · · · 0^T_M hm(0) · · · 0^T_M

... . .. ... ...

0^T_M · · · · hm(MP − 1)





 (4.2)

where {hm(0), · · · , hm(MP − 1)}, m = 1, · · · , M , is a set of P -regular M-band wavelet filters [19]. It has also been observed in [7] that such a construction of B satisfies the constraint ¯C^TB = 0, where ¯C = [c₀, · · · , c_d−1] and c_i = [1, 2ⁱ, · · · , Nⁱ]^T if the derivative constraints are employed, as required and that such a wavelet- based beamformer, as shown in Fig. 4.1, calls for less complexity and yields faster convergence speed of adjusting w_a compared with those based on other blocking matrices.

It is noteworthy that such a blocking matrix also in the meanwhile carries out a spatial subband decomposition of the input. In light of this, [8] has considered a partial scheme by disregarding some insignificant ssubband components to reduce the computational overhead.

As discussed in [18, 20] , temporal subband adaptive filters in general provides an improvement in convergence rate over the one without. Therefore, to further speed up the converge behavior of the above adaptive beamformer, we consider to perform a temporal subband decomposition of the output of the blocking matrix, as shown in Fig. 4.2, via a set of wavelet filters. As the above, the temporal decomposition can also be readily implemented via the following J × (^j^{J−M P}_M ^k+ 1)M transformation matrix

D^T =







H₀ H₁ ...

HM −1





 (4.3)

where the lower part of D^T is the same as B^T in (4.2) and H₀ is constituted by the scaling filter {h₀(0), · · · , h₀(MP − 1)} in the same manner as (4.2).

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−

Figure 4.1: The structure of the GSC broadband beamformer, where L = (^j^{N −MP}_M ^k+ 1)(M − 1) for the wavelet-based approach

4.2 Proposed Partial Scheme

The temporal subband decomposition, however, increases the overall computational load. To mitigate this, we can jointly truncate some insignificant spatio- temporal subbands in light of the fact that the interferences in general occupy only a portion of the spatial or temporal subbands. Since only those negligible information is disregarded, the performance does not degrade too much, but the computations will be substantially reduced. To achieve this, we can partition the adaptive filters into several blocks and then select the blocks, which contain crucial information, to be updated.

Suppose that the proposed scheme employs M_s-band wavelet filters for the spatial subband decomposition and M_t-band for temporal subband decomposition, we can in total partition the adaptive filter into M_t(^j^{N −M}_M^s^P^s

s

k+1)(M_s−1) blocks, in which wj,iis of dimension (^j^J−M_M_t^t^P^t^k+1) , 1 ≤ i ≤ (^j^{N −M}_M_s^s^P^s^k+1)(Ms−1), 1 ≤ j ≤ Mt, and denotes the weight vector associated with the output ui(k) after passing the i^th spatial subband and the j^th temporal subband, where u_i(k) denotes the i^th row of B^T[x(k), · · · , x(k − J + 1)]. Also, we intend to update only a prescribed K

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Figure 4.2: (a) The proposed subband GSC structure and (b) the M-band wavelet filterbank structure

out of the available spatial-temporal subbands. Let S denote the collection of all possible subsets of the chosen spatio-temporal subbands, where

S = {S₁, S₂, · · · , S_M_t}

in which S_j dentes the possible subsets of the selected spatio subbands associate with the j^th temporal subband. For example, if sj = {j1, · · · , jaj} ∈ Sj, then only the j₁^th, · · · , j_a^th_j spatial subbands associated with the j^th temporal subband are chosen. Also, ^P^M_j=1^t a_j = K by assumption.

We can extend the technique addressed in Sec. 3.3.1. It follows that, for every possible choice of (s₁, · · · , s_M_t), the updating of the temporal filters can be posed as the following constrained optimization problem:

w_sjmin(k+1)||w_s_j(k + 1) − w_s_j(k)||²

subject to w^H_j (k + 1)z_j(k) = d_j(k), j = 1, 2, · · · , M_t (4.4) where k inside the parenthesis is introduced to emphasized the time step, z_j(k)=

[u₁(k)H^H_j , · · ·, u_L(k)H^H_j ] with L = (^j^{N −M}_M^s^P^s

s

k+ 1)(M_s − 1), d_j(k) is the j^th temporal subband component of w^H_q x(k), w_j(k)=[w_j1(k),· · ·,w_jL(k)] in Fig. 2 (a), and w_s^T_j(k)=[w_j^T₁(k),w_j^T₂(k),· · ·,w^T_j

aj(k)]^T. The above can be solved by using

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the Lagrange multiplier given by

J_s_j(k) = ||w_s_j(k + 1) − w_s_j(k)||²+ λ(d_j(k) − w^H_j (k + 1)z_j(k)) (4.5) Minimizing the Lagrangian with respect to w_s_j(k + 1),

∂J_s_j(k)

∂w_s_j(k + 1) = 0 We can obtain

w_s_j(k + 1) = w_s_j(k) +λ

2z_j(k) (4.6)

Invoking the constraint in (4.4), we can obtain λ

2 = e^∗_j(k)

||z_s_j(k)||² (4.7)

Therefore

wsj(k + 1) = wsj(k) + µ

||z_s_j(k)||²zsj(k)e^∗_j(k), j = 1, · · · , Mt (4.8) Note that (4.8) is equal to performing the normalized LMS algorithms over each temporal subband. To choose the K weight vector, we can again invoke the mini- mal disturbance principle by choosing the k spatio-temporal subbands by

(s₁, · · · , s_M_t) = arg min

sj ∈Sj ,1≤j≤M2P_Mt

j=1sj =K

||w_s_j(k + 1) − w_s_j(k)||²

= arg min

sj ∈Sj ,1≤j≤MtP_Mt

j=1sj =K

||z_s_j(k)e_j(k)

||z_s_j(k)||² ||²

= arg max

j=1sj =K

||z_s_j(k)||²

|e_j(k)| (4.9)

where we have used (4.8).

To save computation, we may drop the denominator |e_j(k)| and only consider (s₁, · · · , s_M_t) = arg max

j=1sj =K

||z_s_j(k)||² (4.10)

which is like taking the power of the subband components and only these principal subband components are taken into account in the updates.

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4.3 Simulations and Discussions

Some simulations are conducted in this section to verify the proposed approach.

A broadband beamformer of equispaced N elements spaced one half wavelength apart with each element having J taps is employed and the GSC with the second order derivative constraints is utilized to determine beamformer weights.

Four approaches are employed for comparison: (i) singular value decomposition (SVD)-based B [17] , (ii) wavelet-based B [7] , (iii) wavelet-based B with a truncation of spatial subbands [8] , and (iv) proposed wavelet-based B along with extra temporal subband decomposition followed by a dynamic spatio-temporal subband selection addressed above. For simplicity, the subband selection scheme is based on (4.10), although both yield similar results. The learning curves based on these four approaches with the normalized LMS algorithm as the adaptive scheme are furnished to assess their convergence characteristics. All of the results are based on 300 independent runs.

As for the computational complexity, our comparison is based on the total number of adaptive coefficients and the number of multiplications per iteration, which dictate the hardware complexity and the overall computational complexity, respectively, of the beamformers. For the wavelet-based approach, assume that P_s regular Ms wavelet filters are employed for spatial decomposition, and Pt regular M_twavelet filters are employed for temporal decomposition. As such, the approach (i) requires (N −d)J adaptive coefficients and N(N −d)+2(N −d)J multiplications, where d is the number of constraints. The approach (ii) requires (^j^{N −M}_M^s^P^s

s

k + 1)(M_s− 1)J adaptive coefficients and N(^j^{N −M}_M_s^s^P^s^k+ 1)(M_s− 1) + 2(^j^{N −M}_M_s^s^P^s^k+ 1)(Ms− 1)J multiplications, whereas (iii) requires QJ adaptive coefficients and N(^j^{N −M}_M_s^s^P^s^k+ 1)(Ms− 1) + 2QJ multiplications, where Q < (^j^{N −M}_M_s^s^P^s^k+ 1)(Ms− 1) denotes the number of selected spatial subbands. The proposed one requires K(^j^J−M_M^t^P^t

t

k+1) adaptive coefficient and N(^j^{N −M}_M^s^P^s

s

k+1)(M_s−1)+2K(^j^J−M_M^t^P^t

t

k+ 1)) multiplications. These expressions are summarized in Tables 4.1 and 4.2.

number of taps needed to be updated

(i) (N − d)J

(ii) (^j^{N −M}_M^s^P^s

s

k+ 1)(M_s− 1)J

(iii) QJ

Proposed K(^j^J−M_M_t^t^P^t^k+ 1)

Table 4.1: Comparisons of the number of taps needed to be updated per iteration based on various approaches.

Experiment

In this experiment, N = 36, J = 16, and the interference environment con-

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total number of multiplications required per iteration

(i) N(N − d) + 2(N − d)J

(ii) N(^j^{N −M}_M^s^P^s

s

k+ 1)(M_s− 1) + 2(^j^{N −M}_M^s^P^s

s

k+ 1)(M_s− 1)J (iii) N(^j^{N −M}_M_s^s^P^s^k+ 1)(M_s− 1) + 2QJ

Proposed N(^j^{N −M}_M^s^P^s

s

k+ 1)(M_s− 1) + 2K(^j^J−M_M^t^P^t

t

k+ 1))

Table 4.2: Comparisons of the total number of multiplications required per iteration based on various approaches.

-80 -60 -40 -20 0 20 40 60 80

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Angle

Gain(dB)

(i) (ii) (iii) proposed method

Figure 4.3: Comparison of the beampatterns

sists of two jammers with arrival angles −56^oand 59^o and with INR’s (interference to noise ratios) 33 dB and 36 dB, respectively, whereas the SNR (signal to noise ratio) is 10 dB. The parameters M_s = 12, M_t = 8, P_s = P_t = 2, and K = 16 are employed. The comparison of the beampatterns and the convergence rates based on these four approaches are as shown in Figs. 4.3 , 4.4 respectively.

From Fig. 4.3, we can observe that the proposed approach yields satisfactory patterns, in which deep nulls are formed in the directions of interferences. The beampatterns based on other approaches also produce similar results. Also, we can note from Fig. 4.4 that the approach (i) possesses the slowest convergence rate, (ii) and (iii) have close convergence characteristics, and the proposed one is the fastest. Based the parameters used and the above expression, the number of adaptive coefficients and the number of multiplications per iteration required are, respectively, 544 and 2312 in (i), 352 and 1496 in (ii), 144 and 1080 in (iii), and 32 and 856 in (iv) in this experiment, which imply that the computational