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EURASIP Journal on Applied Signal Processing 2005:1, c 2005 Hindawi Publishing Corporation Subband Array Implementations for Space-ime Adaptive Processing Yimin Zhang Center for Advanced Communications,

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EURASIP Journal on Applied Signal Processing 2005:1, c 2005 Hindawi Publishing Corporation Subband Array Implementations for Space-ime Adaptive Processing Yimin Zhang Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA Kehu Yang Electronic Engineering Research Institute, School of Electronic Engineering, Xidian University, Xi an, Shaanxi , China Moeness G Amin Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA Received 1 January 2004; Revised 30 June 2004 Intersymbol interference (ISI) and cochannel interference (CCI) are two primary sources of signal impairment in mobile communications In order to suppress both ISI and CCI, space-time adaptive processing (SAP) has been shown to be effective in performing spatio-temporal equalization, leading to increased communication capacity as well as improved quality of service he high complexity and slow convergence, however, often impede practical SAP implementations Several subband array structures have been proposed as alternatives to SAP hese structures provide optimal or suboptimal steady-state performance with reduced implementation complexity and improved convergence performance he purpose of this paper is to investigate the steady-state performance of subband arrays with centralized and localized feedback schemes, using different decimation rates Analytical expressions of the minimum mean-square error (MMSE) performance are derived he analysis assumes discrete Fourier transform (DF)-based subband arrays and considers both unconstrained and constrained weight adaptations Keywords and phrases: space-time adaptive processing, subband array, array processing, mobile communications, intersymbol interference, cochannel interference 1 INRODUCION he applications of wireless communications are rapidly expanding from voice transmission to a wide class of multimedia information With such increasing needs, wireless communication systems are developing toward higher-speed digital wireless networks he communication channels are often frequency selective, as a result of long multipath delays relative to the symbol period, causing intersymbol interference (ISI) In many mobile communication systems, the frequency resource is reused, cochannel interference (CCI) represents another source of channel distortion and signal impairment herefore, ISI and CCI are two primary sources that limit the communication capacity and the quality of services in mobile communications While adaptive arrays are effective for spatial processing of CCI suppression; as adaptive equalizers are effective for temporal filtering for ISI reduction, neither of them are effective when both the CCI and ISI are present he use of space-time adaptive processing (SAP) technology is an effective way to perform spatio-temporal equalization that mitigates the above two problems 1, 2] Objectives are to increase the communication capacity and enhance the quality of services A variety of algorithms have been developed for the implementation of the SAP systems, including those based on least-mean square (LMS), recursive least squares (RLS), and sample matrix inversion (SMI) he direct use of SAP system often involves high-dimension space in the joint spatial and temporal domain his, in turn, brings a high complexity and slow convergence rate, rendering the SAP system unattractive his shortcoming has motivated extensive research work for devising alternative implementation 3, 4, 5, 6, 7, 8, 9] Among those methods, subband or frequency-domain arrays offer the amenability of parallel implementation with reduced processing rates in each subband 10, 11] With appropriate power normalization or data self-orthogonalization, subband arrays can achieve improved convergence 12, 13, 14] 100 EURASIP Journal on Applied Signal Processing he subband (including frequency-domain) adaptive arrays can be classified, in terms of their feedback methods, into two classes, namely, centralized feedback and localized feedback In 7], the partial feedback scheme was also introduced as a generalization of the above schemes For the centralized feedback schemes, Compton has shown that the frequency-domain array provides identical steady-state performance of the corresponding SAP system 15] Such equivalence, however, is valid only for the undecimated (window sliding) cases he use of decimation may provide significant system complexity reduction in subband array implementations he analysis of the performance degradation with the use of decimation has been recently considered by ran et al 8] only for ISI without taking the CCI signals into account On the other hand, for the localized and partial feedback schemes, low computations, parallel processing, and faster convergence can be achieved at the cost of suboptimal steady-state performance 3, 7] Although the investigation of localized subband arrays, according to Compton 15], dated back to early 1970s 16], a detailed performance analysis, to our knowledge, was not available until recently 7, 8] In 7], the performance of discrete Fourier transform (DF) filter bank-based subband arrays has been considered for the aforementioned three feedback schemes no decimation is applied In 8], the performance of localized feedback subband array is analyzed for the DF-based subband arrays in the absence of CCI users he results of 8] show that, in a frequency-selective multipath fading environment, the subband array performance depends on the number of subbands, input signal-to-noise ratio (SNR), the source directions-of-arrival (DOAs), and the multipath time delays In addition to the above literature, 3] provides various numerical comparison results between the centralized and localized feedback schemes In this paper, we investigate the performance of DFbased subband arrays with different decimation rates Both unconstrained and constrained subband array structures are considered o consider the minimum mean-square error (MMSE) performance, the reference signal is considered to be available he steady-state performances of subband adaptive arrays with the centralized and localized feedback schemes and different decimation rates are analyzed, and expressions for the MMSE are derived It is shown that decimation compromises the optimum performance for both centralized and localized feedback subband array schemes he convergence performance of different subband array structures is also investigated and compared It is worth noting that there is an extensive literature in frequency-domain equalizations and echo-cancellation methods using single-sensor receivers (see, eg, 17, 18, 19, 20] and references therein) hese methods provide a fundamental development in the theory of subband processing However, important differences exist between singleand multi-sensor systems in both formulations and performances he inclusion of the spatial domain to subband signal processing affects both the processing structure and the performances Single-antenna receivers cannot deal with the cancellation of CCIs In addition, we specifically address the problem of subband arrays with arbitrary decimation rates for both centralized and localized feedback structures he rest of this paper is organized as follows Section 2 introduces the signal model and reviews the analysis of SAP performance Section 3 considers the subband decomposition, and the aliasing issue with the use of decimation Section 4 formulates the subband arrays with both centralized and localized feedback schemes he steady-state performance of different subband array structures is analyzed in Section 5 Section 6 compares the computational complexity between the subband arrays and conventional SAP systems Section 7 considers the convergence performance data self-orthogonalization and the step-size selection are addressed Numerical examples are provided in Section 8 for illustration 2 SPACE-IME ADAPIVE PROCESSING 21 Signal model We consider a base station using an antenna array of N sensors with P users Without loss of generality, the user signal of interest is denoted as s 1 (n) he signals from other users as s p (n), p = 2,, P, form the CCIs to the signal of interest When frequency-selective channels are considered for each user, the received data vector at the array is expressed as x(t) = x 1 (t),, x N (t) ] P = p=1 i= s p (i)h p (t i)+b(t), the superscript denotes matrix or vector transpose, s p (n) andh p (t) are the nth information symbol and the channel response vector (including the pulse shaping) of the pth user, respectively, and b(t) is the additive noise vector he data vector is sampled at t = n + i, is the symbol duration of the signal waveform and is the sampling period he integer ratio of J = / is referred to as the oversampling factor hen, the data vector takes the following discrete-time expression: (1) P x(n + i ) = s p (n d)h p (d + i )+b(n + i ) p=1 d=0 (2) We make the following assumptions (A1)hetimerequiredforthereceivedwaveformassociated with a given transmission path to propagate across the array is much smaller than the inverse of the user signal bandwidth (A2) he user signals s p (n), p = 1, 2,, P, arewidesense stationary (if sampled at the symbol rate, ie, J = 1) or cyclostationary (if sampled at fractionally spaced symbol cycle, ie,j 1) hese signals are independent and identically distributed (i i d) with Es p (n)s p (n)] = 1, E( ) denotes the statistical expectation operator and the superscript denotes the complex conjugate Subband Array Implementations for Space-ime Processing 101 (A3) All channels h p (t), p = 1, 2,, P, are linear timeinvariant, and of a finite duration within 0, (D p +1)], D p are nonnegative integers (A4) he noise vector b(n) is zero-mean and temporally and spatially white with variance σ at each array sensor Under these assumptions, we can stack the J samples within each symbol period resulting in the following NJ 1 vector containing data received at the NJ virtual channels (or extended channels): x(n) = x n] x n ] x n (J 1) ] ] P D p = s p (n d) h p (d)+ b(n), p=1 d=0 (3) h p (n)= h p n] h pn ] h ] ] p n (J 1), b(n) = b n] b n ] b n (J 1) ]] (4) 22 Space-time adaptive processing When a JM-tap FIR filter is used at the output of each array sensor, or equivalently, an M-tap FIR filter is used at each of the NJ virtual channel, we obtain a MNJ 1 vector that contains all the input values at the SAP system at time instant n: x(n) = x (n) x (n 1) x (n M +1) ] (5) Similarly, we define b(n) = (n) b (n 1) b (n M +1) ], s p (n) = sp (n) s p (n 1) ( )] s p n M Dp, H p = ( ) h p (0) hp Dp 0 0 ( ) 0 hp (0) hp Dp 0 0 ( ) 0 0 hp (0) hp Dp hen, we represent all M symbol samples captured at the NJ virtual channels of the SAP as P x(n) = H p s p (n)+b(n) (7) p=1 Denote w as the weight vector corresponding to x(n) hen the output of the SAP becomes (6) y(n) = w H x(n), (8) the superscript H denotes Hermitian (conjugate transpose) operation When a training signal, which is an ideal replica of s 1 (n), is available at the receiver, the optimum weight vector under the MMSE criterion can be provided using the Wiener-Hopf solution: with w opt = R 1 o r o (9) R o = E x(n)x H (n) ], r o = E x(n)s 1 (n v) ], (10) v is a delay 21], which is chosen to minimize the following MMSE: MMSE = E s 1 (n v) w H optx(n) 2 = 1 r H o R 1 o r o (11) Substituting (7) in(10), and using assumption (A2), we have r o = H 1 e v, (12) e v = 0,,0,1,0,,0 ], (13) }{{} v provided 0 v m+d 1 1 hat is, r o is the (v+1)th column of H 1 22] For example, choosing v = 0orv = M + D 1 1 yields only one effective weight for each virtual channel he optimum value of v usually occurs around (M + D 1 )/2 1, but the actual result depends on the channel characteristics ypically, J is chosen as either one or two 23] In addition, it can be shown 21] that, when the channels meet the following conditions: (1) H 1 is full column rank, (2) the columns of H 1 are linearly independent of the columns of H p, p = 2,, P, the selection of M and N satisfying MNJ column rank{h} (14) yields perfect equalization conditions in noise-free scenarios, H = H 1,, H P ] When all H p, p = 1,, P, arefull column rank, the above requirement is equivalent to 1 P M D p, NJ P p=1 NJ P (15) 3 SUBBAND DECOMPOSIION 31 Subband decomposition and subband arrays Subband decomposition and reconstruction of a signal are performed by exploiting a set of analysis and synthesis filters he analysis filters decompose a wideband signal into a set of narrowband subband signal components 24] Highly decorrelated subband signals are often desired in subband decomposition-based equalization problems to ensure faster 102 EURASIP Journal on Applied Signal Processing convergence and reduce the performance loss in localized feedback schemes 7, 21, 25, 26] o achieve effective decorrelation between subband signals, the analysis filters are required to be close to the ideal bandpass filters 5, 26, 27] his necessitates the use of long analysis filters (ie, filters with long taps) and, therefore, is usually not desirable Long analysis filters not only imply a long time delay in the process of subband decomposition and reconstruction of the signals, but also apply a strict condition to the stationarity of the channel More importantly, for nonblind subband array systems, long analysis filters yield ineffective use of the training signals For these reasons, we consider, in this paper, DF-based filter bank, the transform matrix of the analysis filters is square We maintain that long analysis filters remain useful in certain application scenarios such as blind spatio-temporal equalization and echo-cancellation applications, the training signal is not a problem Combining the subband signal processing and array processing results in subband array processing So far, several subband arrays have been proposed for spatio-temporal equalizations 3, 4, 5, 6, 7, 8, 9, 10, 12] For DF-based subband arrays, the performance without decimation has been discussed in 7], as the performance with decimation is analyzed for CCI-free situations in 8] In the latter, only the maximum decimation is considered, that is, the decimation rate is the same as the number of subband bins, resulting in a blockwise subband array scheme In this paper, we deal with more general cases of DFbased subband arrays of arbitrary decimation rates L hat is, for each set of data processed in the subband array processing, L output data of y(n) are used As a result, the processing window slides every L symbols It also implies that the weights are updated every L symbols he decimation rate is chosen between one (ie, no decimation) and the number of subband bins M (ie, maximum decimation), namely, 1 L M 32 Consideration of decimation Oneimportantissuetobeconsideredindecimatedsubband signal processing is the alias problem For simplicity of notation and explanation, we illustrate this problem by using a convolution problem for only one of the array sensors In the time domain, the output at the ith-array sensor amounts to the convolution of a data stream x i (n) and the weight vector w i = w i,1,, w i,q ] o study the effect of decimation, we consider a block of input data expressed by a vector x i (n) = x i (n),, x i (n M+1)],M Q Since the weight vector is updated independently in each block, we adopt the overlap-save method, rather than the overlapadd method 17] 1 Overlap-add method can also be used for frequency-domain processing, but it requires special attention, since it adds up convolution results of different blocks within which the weight vector may assume different values 1 he concepts underlying the overlap-save and overlap-add methods are given in 28, 29] he use of these two methods in subband signal processing is discussed in 17] 0 M 1 0 Q 1 x i (n) 0 Q 1 M 1 M + Q 1 0 Q 1 Q 2 Q 1 M 1 M Q +1 w i x i (n) w i x i (n) w i (Period M) Figure 1: Illustration of alias problem ( denotes the convolution operator) 30] Referring to Figure 1, the convolution of w i and x i (n) yields a new vector of length M + Q 1, of which, only M Q + 1 samples (from the Qth sample to the Mth sample) take full consideration of Q data inputs he rest are incomplete, in the sense that the output samples do not use all Q input data In this case, zero-padded data are used instead When DF-based filter banks are used to construct a subband array, the data vector, along with the weight vector, is transformed into the subband domain After the data vector and the weight vector are multiplied in the transform domain, the result is transformed back to the time domain by using the inverse DF (IDF) For the unconstrained subband array structure, the length of both data and weight vectors is equal to the dimension of DF When we perform the convolution of the M 1 data vector and the M 1 weight vector, the result is a vector of length 2M 1 herefore, when the M-point convolution is obtained from the IDF of the product of the DFs of the data and weight vectors, the first M 1 samples are contaminated by alias, and only the last sample is alias-free As a result, in order to avoid alias, the only choice is L = 1, that is, no decimation is made, which is the case considered in 7] When L 1, alias problem arises and performance degradation occurs For the centralized feedback scheme, the alias is controlled such that the error over the L samples of the output data is minimized Subband Array Implementations for Space-ime Processing 103 It is noticed that the weight vector can be constrained such that, in each virtual channel, only the first Q values of its time-domain equivalence are nonzero, Q M In the constrained subband arrays, the lengths of data and weight vectors as well as the dimension of DF can be different With the use of an M-point DF transform, the convolution of M-tap data and Q-tap weights yields M Q + 1 points of alias-free output samples hat is, the decimation rate can take the value L M Q + 1 without causing an alias problem It is pointed out that the alias-free results are achieved at the cost of reduced number of degrees of freedom from M to Q, which, as we will show later, does not necessarily improve the system performance 4 SUBBAND ARRAYS 41 Formulation of subband array signals In this section, we formulate the expression of a DF-based subband array with M subbands and a decimation factor of L Let the subband decomposition divide the M samples of data sequence at the output of the ith virtual channel, x i (n) = x i (n),, x i (n M +1) ], (16) into M subbands, that is, to form the vector x,i (n) = x (1) i (n),, x (M) i (n) ], (17) i = 1,, NJ, and the superscript (m) denotes the data component at the mth subband x i (n)and x,i (n) are related by the following equation: x,i (n) = x (1),i(n), x,i(n), (2), x (M),i (n) ] = o x i (n), (18) o is the M M DF matrix with its (i, k)th element being o ] i,k = (1/ M)W (i 1)(k 1) M, i, k = 1,, M,and W M = exp( j2π/m) It is noted that o is unitary and symmetric, that is, o H o = H o o = I M and o = o,i M is the M M identity matrix hen, the NJ 1datavectorat the mth subband is obtained as x (m) (n) = x (m),1 (n), x (m),2 (n),, x,nj(n) (m) ] By defining x 1 (n) x 2 (n) = x NJ(n) W 0 M W m 1 M W (m 1)(M 1) M (x (1) x (n) = (n) ) (,, x (M) (n) ) ] (19) (20) as the MNJ 1 signal vector for all the M subbands in the subband array, we can relate x (n) andx(n), defined in (5), by x (n) = x(n), (21) the transform matrix is expressed in the form = o I NJ (22) and denotes the Kronecker product operator It is easy to confirm that is also unitary, that is, H = H = I MNJ 42 Adaptive subband arrays Denote by (w (m) ) the NJ 1 weight vector to the signal vector x (m) (n) at the mth subband, and by w = (w (1) ),, ) ] H the MNJ 1weightvectortotheentiresubband signal vector x (n) he subband output is obtained as the following M 1vector: (w (M) ỹ (n) = ( ) w (1) Hx (1) (n) ( w (M) ) Hx (M) x (1) (n) X (n) = O = X (n)w, (23) (n) x (M) O (n) (24) is an MNJ M matrix he t

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