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...therefore design , as desired. If the transmitter has knowledge of the channel and noise covariance , it can transmit waveforms having such that (5.9) is maximized using “water filling,” as in, e.g., =-=[44]-=- [then, the maximized (5.9) can be viewed as a measure of capacity of the channel]. Numerical Example 1—Concentrated Likelihood Function: In this numerical example, we study the concentrated likelihoo...

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...delay and Jakes’ Doppler spread sampling in (2.5a) and (2.6). 1) Transmit–Receive Antenna Arrays: The measurement model (2.3) can also be used to describe transmit–receive antenna array systems [29], =-=[30]-=-. Here, the basis functions describe signals sent by the transmitter array. For example, in a slow, flat fading environment (i.e., negligible Doppler effect and 460 IEEE TRANSACTIONS ON SIGNAL PROCESS...

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...del proposed in [27] can also be viewed as a special case of (2.4) with as in (2.5a) (with set to zero), , and . An interesting, physically motivated model for follows by discretizing the Jakes model =-=[28]-=- with the basis-function vector of the form (2.6) where is proportional to the speed of the mobile (and, consequently, to the Doppler spread), and . In general, if a mobile receiver (or mobile transmi...

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... time-delay and Jakes’ Doppler spread sampling in (2.5a) and (2.6). 1) Transmit–Receive Antenna Arrays: The measurement model (2.3) can also be used to describe transmit–receive antenna array systems =-=[29]-=-, [30]. Here, the basis functions describe signals sent by the transmitter array. For example, in a slow, flat fading environment (i.e., negligible Doppler effect and 460 IEEE TRANSACTIONS ON SIGNAL P...

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... addition, to have equal separation for each pair of symbols, we may choose constant for all . Note that the above two conditions are closely related to the recently proposed unitary space–time codes =-=[56]-=- and are tailored for the case where the channel is unknown to the receiver. In the case of a single transmitter and unstructured channel model with negligible delay and Doppler spreads [i.e., when in...

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...of and in (3.1) into the likelihood function (see, e.g., [19, App. A]). This concentrated likelihood function is written in the form of a generalized likelihood ratio (GLR) test statistic (see, e.g., =-=[37]-=- and [38, p. 418] for the definition of GLR) for testing . To find the ML estimates of and for unknown and , substitute and in (3.1a) and (3.1b) by and . The above concentrated likelihood function can...

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...onsider the unstructured channel model with negligible delay and Doppler spreads [i.e., in (2.5)]. Then, the decision statistic becomes closely related to the sample matrix inversion (SMI) beamformer =-=[55]-=-; see [40]. 1) Noncoherent Concentrated-Likelihood Receiver: Consider now a concentrated-likelihood receiver that utilizes only the data containing the currently received symbol (i.e., no training dat...

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..., respectively, whereas and are the corresponding sampling intervals. Since the choices of and , respectively, depend on the bandwidth and duration of the signal , these quantities are known a priori =-=[26]-=-. If we also assume that prior knowledge on the delay and Doppler spreads is available, then we can also determine and (and therefore as well) a priori. In this case, , , and reduce to , , and , respe...

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... spatially correlated noise having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) [2]–[9] or spatially white noise =-=[10]-=-, [11], or both [12]–[16]. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured array responses and model the sign...

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...ls. Unlike (2.5), it is assumed in [21] and [26] that the receiver is perfectly synchronized to the “global” delay and Doppler shift, which are set to zero. The time-varying channel model proposed in =-=[27]-=- can also be viewed as a special case of (2.4) with as in (2.5a) (with set to zero), , and . An interesting, physically motivated model for follows by discretizing the Jakes model [28] with the basis-...

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...el in (2.2) and the noise model from Section II. If the DOA parameters and basis-function parameters are known, the above model is known as the generalized multivariate analysis of variance (GMANOVA) =-=[33]-=-. For known basis functions (i.e., known ), exact and approximate ML methods for DOA estimation are derived in [4]. For spatially and temporally white noise, an ML estimation algorithm for unknown DOA...

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...by a transmitter array with elements. Note also that the model (2.3) allows channel matrix to be of an arbitrary rank, which is of practical importance in transmit–receive antenna array systems [31], =-=[32]-=-. III. MAXIMUM LIKELIHOOD ESTIMATION WITH STRUCTURED ARRAY We present the ML estimation procedure for the structured array response model in (2.2) and the noise model from Section II. If the DOA param...

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...mation , this term reduces to the noncoherent detector in [46]. VII. SYMBOL DETECTION Recently, several authors have proposed incorporating known spatial noise covariance into the receiver design. In =-=[47]-=-–[49], maximum likelihood sequence estimators (MLSE) accounting 466 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 3, MARCH 2002 for the known spatial noise covariance were proposed; lately, the...

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...itted by a transmitter array with elements. Note also that the model (2.3) allows channel matrix to be of an arbitrary rank, which is of practical importance in transmit–receive antenna array systems =-=[31]-=-, [32]. III. MAXIMUM LIKELIHOOD ESTIMATION WITH STRUCTURED ARRAY We present the ML estimation procedure for the structured array response model in (2.2) and the noise model from Section II. If the DOA...

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...e since we can choose a subset of the set of transmit antennas so that the channel is of full rank. The optimal choice of this subset (that maximizes channel capacity) has been considered recently in =-=[54]-=-, and the resulting channel is always of full rank; see [54, Lemma 1]. Define the data matrix containing both the known symbols and currently-received symbol with snapshots. Then, the corresponding ba...

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...se having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) [2]–[9] or spatially white noise [10], [11], or both [12]–=-=[16]-=-. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured array responses and model the signal as a linear combinatio...

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...ul for estimating the locations and radial velocities of moving point targets (for example, in radar) with , , and . Define and . The ML estimates of and for known and are [33, ch. 6.4], [34, ch. 5], =-=[35]-=- (3.1a) (3.1b) where (3.2a) (3.2b) (3.2c) (3.2d) (3.2e) and denotes the identity matrix of size . Note that , , and are functions of only, and is a function of both and . To simplify the notation, we ...

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... model in (2.1) is a special case of (2.2) with , , , diag , , and . The structured array basis-function formulation in (2.2) is very general. It has been used in [19] for EEG/MEG source location, in =-=[4]-=- for DOA estimation (assuming completely known basis functions), and in [20] for radar detection (assuming known DOA’s and basis functions). This model stems from multivariate statistical analysis; se...

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...derived for time-delay estimation in a slow, frequency-selective fading environment. A flat-fading scenario, when the channel matrix degenerates to a vector (i.e., ), is used in [2] and [6] (see also =-=[24]-=- and [25] for related radar work). B. Basis-Function Models Now, we propose the following fairly general basis-function structure to model fast, frequency-selective fading (2.4) where (of size ) and (...

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...d noise having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) [2]–[9] or spatially white noise [10], [11], or both =-=[12]-=-–[16]. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured array responses and model the signal as a linear combi...

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..., we need a suitable parameterization of the array response matrix . For example, we may choose , where , discretize the angular spread of the received signals [21], or use a distributed source model =-=[22]-=-. For communication applications in particular, we are interested in synchronizing the receiver (i.e., estimating the time-delay and Doppler parameters ; see Section II-B) and estimating the channel m...

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...on in Section IV. A. Unstructured Array Model We describe the unstructured array response model in which the whole array response matrix is assumed to be unknown with an arbitrary (known) rank, as in =-=[23]-=-. The unstructured array model is robust compared with an incorrectly-structured array model.1 In addition, it avoids nonlinear DOA parameterization. The unstructured array model can be written as (2....

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...of the angle between the 462 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 3, MARCH 2002 th components of estimated canonical coordinates of the data and basis functions, which is defined (see =-=[36]-=-) as (5.3a) (5.3b) for . Using the Poincaré separation theorem [38, pp. 64–65], maximizing (3.5) with respect to the unstructured array response [of size , and having full rank ] yields (5.4) see also...

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...ructured array basis-function formulation in (2.2) is very general. It has been used in [19] for EEG/MEG source location, in [4] for DOA estimation (assuming completely known basis functions), and in =-=[20]-=- for radar detection (assuming known DOA’s and basis functions). This model stems from multivariate statistical analysis; see the discussion in Section III. In Section II-B, we show how it can be used...

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...the vector signal received by the array at time becomes (2.1) for , where is the vector of DOA parameters (and may contain additional parameters, such as scattering and polarization coefficients; see =-=[17]-=-), and , , and are the complex amplitude, Doppler shift, and time delay for the th path ( ). Further, denotes the (single DOA) array response vector, and is additive noise. Note that in the above mode...

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...ally correlated noise having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) [2]–[9] or spatially white noise [10], =-=[11]-=-, or both [12]–[16]. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured array responses and model the signal as ...

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...known in general). Note that the true model in (2.1) is a special case of (2.2) with , , , diag , , and . The structured array basis-function formulation in (2.2) is very general. It has been used in =-=[19]-=- for EEG/MEG source location, in [4] for DOA estimation (assuming completely known basis functions), and in [20] for radar detection (assuming known DOA’s and basis functions). This model stems from m...

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...oncoherent detection (as expected). The above detector can be viewed as a multivariate extension (accounting for multiple receive antennas and spatially correlated noise) of the multiuser detector in =-=[57]-=-. For one basis function (i.e., ) and one receive antenna (i.e., ), it further reduces to the standard noncoherent detector (see, for example, [58, sec. 5.4]). C. Recursive Implementation We derive a ...

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...tion [i.e., , implying that is a row vector] and one receive antenna (i.e., , implying that is a row vector), and after the monotonic transformation , this term reduces to the noncoherent detector in =-=[46]-=-. VII. SYMBOL DETECTION Recently, several authors have proposed incorporating known spatial noise covariance into the receiver design. In [47]–[49], maximum likelihood sequence estimators (MLSE) accou...

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...ading channels in the presence of spatially correlated noise having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) =-=[2]-=-–[9] or spatially white noise [10], [11], or both [12]–[16]. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured ...

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... For the structured array model in (2.2), we need a suitable parameterization of the array response matrix . For example, we may choose , where , discretize the angular spread of the received signals =-=[21]-=-, or use a distributed source model [22]. For communication applications in particular, we are interested in synchronizing the receiver (i.e., estimating the time-delay and Doppler parameters ; see Se...

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...derived for the real case in [23, (34)]) uses only the right singular vectors of , whereas (5.6c) uses the reduced-rank representation of . Methods for efficiently computing (5.6) when are derived in =-=[42]-=-. When , i.e., the channel matrix is of full rank, its ML estimate is simply the estimated Wiener filter , and the ML estimate of the noise covariance simplifies to [which follow from (3.1) by replaci...

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...ection [39]. When there is only one path, i.e., in (2.1) or in (2.2), (2.4), and (2.5), the concentrated likelihood is obtained by replacing , , and with , , and in (3.3) and (3.5); see also [25] and =-=[40]-=-. If , the concentrated likelihood further reduces to the expression in [24, (16)]. This scenario, which has been analyzed extensively in [24], implies that matched filtering has DOGAND ˇZIC´ AND NEHO...

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...n , this term reduces to the noncoherent detector in [46]. VII. SYMBOL DETECTION Recently, several authors have proposed incorporating known spatial noise covariance into the receiver design. In [47]–=-=[49]-=-, maximum likelihood sequence estimators (MLSE) accounting 466 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 3, MARCH 2002 for the known spatial noise covariance were proposed; lately, they hav...

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...g channels in the presence of spatially correlated noise having unknown covariance. This is unlike most previous work, which typically assumes slow-fading channel (neglecting the Doppler effects) [2]–=-=[9]-=- or spatially white noise [10], [11], or both [12]–[16]. (References to several recent papers dealing with spatially correlated noise will be given later.) We consider structured and unstructured arra...

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..., methods are derived for time-delay estimation in a slow, frequency-selective fading environment. A flat-fading scenario, when the channel matrix degenerates to a vector (i.e., ), is used in [2] and =-=[6]-=- (see also [24] and [25] for related radar work). B. Basis-Function Models Now, we propose the following fairly general basis-function structure to model fast, frequency-selective fading (2.4) where (...

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...or time-delay estimation in a slow, frequency-selective fading environment. A flat-fading scenario, when the channel matrix degenerates to a vector (i.e., ), is used in [2] and [6] (see also [24] and =-=[25]-=- for related radar work). B. Basis-Function Models Now, we propose the following fairly general basis-function structure to model fast, frequency-selective fading (2.4) where (of size ) and (of size )...

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...s then . Several special cases of the above model have been considered in the literature. Full-rank unstructured channel models [i.e., ] are used in [11] and [16] (assuming spatially white noise) and =-=[8]-=- and [9] (assuming spatially correlated 1Structured array model inaccuracies are due to changes in antenna locations, temperature, surrounding environment, gain and phase errors, mutual coupling, quan...

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...L. 50, NO. 3, MARCH 2002 for the known spatial noise covariance were proposed; lately, they have been extended to account for temporally correlated CCI following the vector autoregressive (VAR) model =-=[50]-=-. Here, we derive coherent matched-filter and concentrated-likelihood receivers that utilize estimated channel response and spatial noise covariance matrices (using results of Sections III and V) as w...

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...on used in this paper may also be too simplistic to account for strong CCI. Recently, more realistic RAKE receivers, which account both for spatial and temporal interference, were proposed; see [52], =-=[53]-=-, and references therein. Unlike in this paper, the spatio-temporal noise model in [52] is parametric (thus sensitive to modeling inaccuracy) and complex, requiring exact knowledge of channel paramete...

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...to the space orthogonal to the row space of . This can further be viewed as the overall power arriving from the direction , normalized by the power of the noise only, arriving from the same direction =-=[39]-=-. When there is only one path, i.e., in (2.1) or in (2.2), (2.4), and (2.5), the concentrated likelihood is obtained by replacing , , and with , , and in (3.3) and (3.5); see also [25] and [40]. If , ...

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... time-varying channel estimation. I. INTRODUCTION TO COMBAT fading and suppress interference in wirelesscommunications, both channel and noise properties need to be estimated. In this paper (see also =-=[1]-=-), we propose algorithms and derive performance measures for space–time estimation of fast, frequency-selective fading channels in the presence of spatially correlated noise having unknown covariance....