Classifications

• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L25/00—Baseband systems
  • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
  • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
  • H04L25/03006—Arrangements for removing intersymbol interference
  • H04L25/03178—Arrangements involving sequence estimation techniques
  • H04L25/03305—Joint sequence estimation and interference removal
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/01—Equalisers
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04B—TRANSMISSION
  • H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
  • H04B1/69—Spread spectrum techniques
  • H04B1/707—Spread spectrum techniques using direct sequence modulation
  • H04B1/7097—Interference-related aspects
  • H04B1/7103—Interference-related aspects the interference being multiple access interference
  • H04B1/7105—Joint detection techniques, e.g. linear detectors
  • H04B1/71055—Joint detection techniques, e.g. linear detectors using minimum mean squared error [MMSE] detector
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L25/00—Baseband systems
  • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
  • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
  • H04L25/03006—Arrangements for removing intersymbol interference
  • H04L25/03178—Arrangements involving sequence estimation techniques
  • H04L25/03331—Arrangements for the joint estimation of multiple sequences
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L25/00—Baseband systems
  • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
  • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
  • H04L25/03993—Noise whitening

Definitions

• the invention generally relates to wireless communication systems, In particular, the invention relates to data detection in such systems.
• ZF zero forcing
• MMSE minimum mean square error
• r is the received vector, comprising samples of the received signal.
• H is the channel response matrix
• d is the data vector to be estimated.
• d may be represented as data symbols or a composite spread data vector.
• the data symbols for each individual code are produced by despreading the estimated data vector d with that code, n is the noise vector.
• (-) H is the complex conjugate transpose (or Hermetian) operation.
• the data vector is estimated, such as per Equation 3.
• d (H H H + ⁇ 2 l) _1 H H r Equation 3
• the present invention has many aspects.
• One aspect of the invention is to perform equalization using a sliding window approach.
• a second aspect reuses information derived for each window for use by a subsequent window.
• a third aspect utilizes a discrete Fourier transform based approach for the equalization.
• a fourth aspect relates to handling oversampling of the received signals and channel responses.
• a fifth aspect relates to handling multiple reception antennas.
• a sixth embodiment relates to handling both oversampling and multiple reception antennas. [0015] BRIEF DESCRIPTION OF THE DRAWING(S)
• Figure 1 is an illustration of a banded channel response matrix.
• Figure 2 is an illustration of a center portion of the banded channel response matrix.
• Figure 3 is an illustration of a data vector window with one possible partitioning.
• Figure 4 is an illustration of a partitioned signal model.
• Figure 5 is a flow diagram of sliding window data detection using a past correction factor.
• Figure 6 is a receiver using sliding window data detection using a past correction factor.
• Figure 7 is a flow diagram of sliding window data detection using a noise auto-correlation correction factor.
• Figure 8 is a receiver using sliding window data detection using a noise auto-correlation correction factor.
• Figure 9 is a graphical representation of the sliding window process.
• Figure 10 is a graphical representation of the sliding window process using a circulant approximation.
• Figure 11 is a circuit for an embodiment for detecting data using discrete Fourier transforms (DFTs).
• DFTs discrete Fourier transforms
• a wireless transmit/receive unit includes but is not limited to a user equipment, mobile station, fixed or mobile subscriber unit, pager, or any other type of device capable of operating in a wireless environment.
• a base station includes but is not limited to a Node-B, site controller, access point or any other type of interfacing device in a wireless environment.
• reduced complexity sliding window equalizer is described in conjunction with a preferred wireless code division multiple access communication system, such as CDMA2000 and universal mobile terrestrial system (UMTS) frequency division duplex (FDD), time division duplex (TDD) modes and time division synchronous CDMA (TD-SCDMA), it can be applied to various communication system and, in particular, various wireless communication systems.
• UMTS universal mobile terrestrial system
• FDD frequency division duplex
• TDD time division duplex
• TD-SCDMA time division synchronous CDMA
• it can be applied to various communication system and, in particular, various wireless communication systems.
• a wireless communication system it can be applied to transmissions received by a WTRU from a base station, received by a base station from one or multiple WTRUs or received by one WTRU from another WTRU, such as in an ad hoc mode of operation.
• n(t) is the sum of additive noise and interference (intra-cell and inter-cell).
• chip rate sampling is used at the receiver, although other sampling rates may be used, such as a multiple of the chip rate.
• Equation 7 results.
• r Hd + n
• r [r(0),-,r(M-l)] ⁇ eC M
• n [n(0),---,n(M-l)] ⁇ eC M h(L- ⁇ ) h(L-2) Kl) A(0) 0 0 h(L-T) h(L-2) ...
• Equation 7 C M represents the space of all complex vectors with dimension M.
• Equation 7 [d(-L + 1), d(-L + 2),...,d(- ⁇ ),d(0), d(T),..., d(N - 1), ⁇ /(N),..., d(N + L- 2) e C N+2L ⁇ 2 - ⁇ N I-l Equation 8 [0037]
• the H matrix in Equation 7 is a banded matrix, which can be represented as the diagram in Figure 1.
• each row in the shaded area represents the vector [h(L -l),h(L -2),...,h(T),h(0)], as shown in Equation 7.
• d is the middle N elements as per Equation 9.
• d [d(0),...,d(N-l)] ⁇ Equation 9
• the first L-l and the last -l elements of r are not equal to the right hand side of the Equation 10.
• the elements at the two ends of vector d will be estimated less accurately than those near the center.
• a sliding window approach is preferably used for estimation of transmitted samples, such as chips.
• a certain number of the received samples are kept in r [k] with dimension N+L-l. They are used to estimate a set of transmitted data d[k] with dimension N using equation 10.
• the window size N and the sliding step size are design parameters, (based on delay spread of the channel (L), the accuracy requirement for the data estimation and the complexity limitation for implementation), the following using the window size of Equation 12 for illustrative purposes.
• N 4N_ x SE Equation 12 SF is the spreading factor. Typical window sizes are 5 to 20 times larger than the channel impulse response, although other sizes may be used.
• the sliding step size based on the window size of Equation 12 is, preferably, 2N_ x SF .
• N s e ⁇ 1,2,... ⁇ is, preferably, left as a design parameter.
• the estimated chips that are sent to the despreader are 2N_ x SF elements in the middle of the estimated d[k] . This procedure is illustrated in Figure 3.
• the system model is approximated by throwing away some terms in the model.
• terms are kept by either using the information estimated in previous sliding step or characterizing the terms as noise in the model.
• the system model is corrected using the kept/characterized terms.
• One algorithm of data detection uses an MMSE algorithm with model error correction uses a sliding window based approach and the system model of Equation 10.
• the estimation of the data such as chips, has error, especially, at the two ends of the data vector in each sliding step (the beginning and end).
• the H matrix in Equation 7 is partitioned into a block row matrix, as per Equation 13, (step 50).
• d f is per Equation 18.
• d f [d(N) d(N + T) ••• d(N + L-2)] T eC H
• g d chip energy per Equation 22.
• E ⁇ d(i)d * (j) ⁇ g d ⁇ ⁇ Equation 22 [0057]
• r is per Equation 23.
• ⁇ r-Jl p d p Equation 23
• d p is part of the estimation of d in the previous sliding window step
• Equation 24 [0059] The reliability of d p depends on the sliding window size (relative to the channel delay span L) and sliding step size.
• FIG. 6 This approach is also described in conjunction with the flow diagram of Figure 5 and preferred receiver components of Figure 6, which can be implemented in a WTRU or base station.
• the circuit of Figure 6 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple IC's, as discrete components or as a combination of IC('s) and discrete components.
• IC integrated circuit
• ASIC application specific integrated circuit
• a channel estimation device 20 processes the received vector r producing the channel estimate matrix portions, H ⁇ H and H y , (step 50).
• a future noise auto-correlation device 24 determines a future noise auto-correlation factor, g d H f JI ⁇ , (step 52).
• a noise auto-correlation device 22 determines a noise auto-correlation factor, E ⁇ a H j, (step 54).
• a summer 26 sums the two factors together to produce ⁇ ! , (step 56).
• a past input correction device 28 takes the past portion of the channel response matrix, H p , and a past determined portion of the data vector, d p , to produce a past correction factor, H ⁇ d ⁇ , (step 58).
• a subtracter 30 subtracts the past correction factor from the received vector producing a modified received vector, r , (step 60).
• An MMSE device 34 uses ⁇ 1 ? H, and r to
• step 62 determines the received data vector center portion d , such as per Equation 21, (step 62).
• the next window is determined in the same manner using a portion of d as d p in the next window determination, (step 64). As illustrated in this
• FIG. 8 This approach is also described in conjunction with the flow diagram of Figure 7 and preferred receiver components of Figure 8, which can be implemented in a WTRU or base station.
• the circuit of Figure 8 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple ICs, as discrete components or as a combination of IC('s) and discrete components.
• IC integrated circuit
• ASIC application specific integrated circuit
• a channel estimation device 36 processes the received vector producing the channel estimate matrix portions, H ⁇ H and H y , (step 70).
• a noise auto-correlation correction device 38 determines a noise auto-correlation correction factor, g d ⁇ L p ⁇ .” + g d ⁇ L f R" , using the future and past portions of the channel response matrix, (step 72).
• a noise auto correlation device 40 determines a noise auto-correlation factor, E ⁇ nn ff ⁇ , (step 74).
• a summer 42 adds the noise auto-correlation correction factor to the noise auto-correlation factor to produce ⁇ 2 , (step 76).
• An MMS ⁇ device 44 uses the center portion or the channel response matrix, H , the received vector, r , and ⁇ 2 to estimate the center portion
• step 78 One advantage to this approach is that a feedback loop using the detected data is not required. As a result, the different slided window version can be determined in parallel and not sequentially.
• the sliding window approach described above requires a matrix inversion, which is a complex process.
• One embodiment for implementing a sliding window utilizes discrete Fourier transforms (DFTs), as follows.
• DFTs discrete Fourier transforms
• ZF zero forcing
• a matrix A c!r e C NxN is a circulant matrix if it has the following form per Equation 28.
• a c;r F ⁇ (A c; -[:,l])F w
• a c ,-[:,1] (a 0 ,a l ,...,a N ) T e C N , i.e. it is the first column of matrix A cir Equation 29
• Columns other than the first column can be used if properly permuted.
• F N is the iV-point DFT matrix which is defined as, for any x e C N , as per Equation 30.
• F ⁇ 1 is the N-point inverse DFT matrix which is defined as, for any x e C N , as per Equation 31. Equation 31
• a ⁇ (•) is a diagonal matrix, which is defined as, for any xe C", as per Equation 32.
• a N (x) diag(F N x)
• Equation 32 The inverse of matrix A cir is expressed, such as per Equation 33.
• a ⁇ F- ⁇ A C Pain[:,1])Ffinity Equation 33
• the following is an application of a DFT based approach to the data estimation process using the sliding window based chip level equalizer.
• the first embodiment uses a single receiving antenna. Subsequent embodiments use multiple receiving antennas.
• Equation 34 h(-) is the impulse response of the channel.
• d(k) is the k th transmitted chip samples that is generated by spreading symbols using a spreading code
• r(-) is the received signal.
• «(-) is the sum of additive noise and interference (intra-cell and inter-cell).
• Equation 36 Based on ML ( M > L ) received signals r(0), • • • , r(M - 1) , Equation 36 results.
• r Hd + n
• r [r(0),- -,r(M -l)f e C M
• n [n(0),- -,n(M -T)] ⁇ e C M h(L -l) h(L -2) •• • h( ⁇ ) h(0) 0 ' 0 h(L -T) h(L -2) ⁇ ⁇ h(T) h(0) 0 H C Mx(M+L- ⁇ )
• the H matrix is Toeplitz.
• the H matrix is block Toeplitz.
• discrete Fourier transform techniques can be applied.
• the Toeplitz/block Toeplitz nature is produced as a result of the convolution with one channel or the convolution of the input signal with a finite number of effective parallel channels.
• the effective parallel channels appear as a result of either oversampling or multiple receive antennas. For one channel, a single row is essentially slide down and to the right producing a Toeplitz matrix.
• Equation (5) The left hand side of equation (5) can be viewed as a "window" of continuous input signal stream.
• the approximated model can be expressed explicitly as per Equation 38.
• r Hd + n h(0) 0 h(T) A(0) • • . h(T) • • . 0
• H h(L -l) j h(0) 0 h(L -T) ' -. Kl) 0 h(L -T)
• Figure 9 is a graphical representation of the sliding window process, as described above.
• Equation 39 neither the matrix R nor the matrix H is circulant to facilitate a DFT implementation.
• Equation 40 In Equation 40, only the first L -1 elements (equations) are approximations of those of Equation 36.
• the matrix H is replaces by a circulant matrix, such as per
• Equation 42 due to the new model, is different than the vector d in Equation 36.
• Equation 42 adds additional distortion to the first L-l element of Equation 39. This distortion makes the two ends of the estimated vector d inaccurate.
• Figure 10 is a graphical representation of the model construction process.
• Equation 45 F- 1 N ⁇ (R crV [:,l])A (H ⁇ [:,l])F r
• Figure 11 is a diagram of a circuit for estimating the data per
• the circuit of Figure 11 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple ICs, as discrete components or as a combination of IC('s) and discrete components.
• IC integrated circuit
• ASIC application specific integrated circuit
• the estimated channel response H is processed by an H determination device 80 to determine the Toeplitz matrix H .
• a circulant approximation device 82 processes H to produce a circulant matrix H e ._ .
• a Hermetian device 84 produces the Hermetian of H rir , H ⁇ _ .
• H Ci> H - and the noise variance ⁇ 2
• R c/ - is determined by a R c/r determining device 86.
• a diagonal matrix is determined by a A M (H" r [:,l]) determining device 88.
• an inverse diagonal matrix is determined by a ⁇ (R C(> [:,1]) determination device 90.
• a discrete Fourier transform device 92 performs a transform on the received vector, r.
• the diagonal, inverse diagonal and Fourier transform result are multiplied together by a multiplier 96.
• An inverse Fourier transform device 94 takes an inverse transform of the result of the multiplication to produce the data vector d .
• the sliding window approach is based on an assumption that the channel is invariant within each sliding window.
• the channel impulse response near the beginning of the sliding window may be used for each sliding step.
• window step size N_. and window size M is per Equation 46, although others may be used.
• ⁇ ymbol e ⁇ 1,2,... ⁇ is the number of symbols and is a design parameter which should be selected, such that M > L . Since M is also the parameter for DFT which may be implemented using FFT algorithm. M may be made large enough such that the radix-2 FFT or a prime factor algorithm (PFA) FFT can be applied.
• PFA prime factor algorithm
• each antenna input, r k is approximated per Equation 47.
• Equations 49 and 50 are estimates of the auto-correlation and cross-correlation properties of the noise terms.
• R ri _ is still a circulant matrix and the estimated data can be determined per Equation 52.
• the noise terms may be correlated in both time and space. As a result, some degradation in the performance may result.
• Multiple chip rate sampling is when the channel is sampled at a sampling rate which is an integer multiple of the chip rate, such as two times, three times, etc. Although the following concentrates on two times per chip sampling, these approaches can be applied to other multiples.
• Equation 53 Equation 53 separates the effective 2-sample-per-chip discrete-time channel into two chip-rate discrete-time channels.
• the matrices He and Ho in Equation 53 are, correspondingly, the even and odd channel response matrices. These matrices are constructed from the even and odd channel response vectors he and ho, which are obtained by sampling the channel response at 2 samples per chip and separating it into the even and odd channel response vectors.
• the channel noise is modeled as white with a variance ⁇ 2 , as per
• Equation 55 results.
• the problem is mathematically similar to the case of the chip-rate equalizer for 2 receive antennas with uncorrelated noise, as previously described.
• the received antenna signals in many implementations are processed by a receive-side root-raised cosine (RRC) filter before being provided to the digital receiver logic for further processing.
• RRC root-raised cosine
• the received noise vector is no longer white, but has a raised-cosine (RC) autocorrelation function.
• RC is the frequency-domain square of a RRC response. Since the RC pulse is a Nyquist pulse, Equation 54 holds, however Equation 55 does not.
• the (i ) t - element of the matrix def 1 till ross — £[ n e n o ] is P er Equation 56. ⁇ Equation 56
• x RC is the unity-symbol-time normalized RC pulse shape.
• ⁇ correlation matrix represent the cross-correlation matrix of the total noise vector and is per Equation 57.
• H ⁇ ; 1 nor H ⁇ H + 1 are Toeplitz and neither can be made Toeplitz through elemental unitary operations (e.g. row/column rearrangements), due to the structure of ⁇ , ( . Accordingly, DFT-based methods based on circulant approximations of Toeplitz matrices cannot be applied here and an exact solution is highly complex.
• Two embodiments for deriving an efficient algorithm for solving this problem are described. The first embodiment uses a simple approximation and the second embodiment uses an almost-exact solution.
• the complexity of the DFT-based approach can be roughly partitioned into 2 components: the processing which has to be performed on every received data set and the processing which is performed when the channel estimate is updated, which is typically done one to two orders of magnitude less frequently then the former operation.
• Equation 61 is the natural order model.
• h e ,i is the i th row of He and h 0 ,i is the i th row of Ho.
• Gi is a 2xiV matrix whose 1 st row is h e ,i and whose 2 nd row is h 0 ,i.
• Gi [x,y] is block-Toeplitz as illustrated in Equation 62.
• Toeplitz matrices are produced. Since the H&r matrix is also banded, the block circulant approximation of H&r is then obtained directly. However, ⁇ b ⁇ is not banded and therefore it is not possible to produce a block-circulant approximation directly from it. Since the elements of A bT tend to 0 as they get farther away from the main diagonal, a banded approximation to ⁇ bT is per Equation 64.
• ⁇ B n and ⁇ . . 0 otherwise Equation 64
• the noise-covariance-bandwidth, Bn is a design parameters that is selected. Due to the decay properties of the RC pulse shape, it is likely to be only several chip. Now ⁇ bT is banded block-Toeplitz and a circulant approximation to it is produced.
• H&r and ⁇ b ⁇ are H&c and ⁇ bC , respectively.
• a block-circulant matrix C is of the form of Equation 65.
• Ci is an NxN matrix and therefore C is an MNxMN matrix
• a MxN (C) is a block diagonal matrix that depends on C and is given by Equation 67.
• a ; .(C) is an NxN matrix.
• ⁇ (kJ) is the M-point DFT of c (A /) and is per Equation 68. Equation 68 [00122] Equations 66-68 specify the block-DFT representation of square block circulant matrices. N 2 DFTs are required to compute ⁇ MxN (C) .
• the MMSE estimator form as per Equation 68 has several advantages. It requires only a single inverse matrix computation and thus in the DFT domain only a single vector division. This provides a potentially significant savings as divisions are highly complex.
• the almost-exact solution has two steps in the most preferred embodiment, although other approaches may be used. Every time a new channel estimate is obtained, the channel filter is updated, (H H ( ⁇ n +FLH H ) '1 is determined). For every data block, this filter is applied to the received data block. This partition is utilized because the channel is updated very infrequently compared to the received data block processing and therefore significant complexity reduction can achieved by separating the overall process into these two steps.
• the DFT of ⁇ n is the DFT of the pulse shaping filter multiplied by the noise variance ⁇ 2 . Since the pulse shaping filter is typically a fixed feature of the system, its DFT can be precomputed and stored in memory and thus only the value ⁇ 2 is updated. Since the pulse-shaping filter is likely to be close to the "ideal" (IIR) pulse shape, the DFT of the ideal pulse shape can be used for ⁇ n , reducing the complexity and is also far away from the carrier.
• IIR ideal
• the "block-DFT" of H needs to be computed. Since the block is of width 2, it requires 2 DFTs. The result is a Nx2 matrix whose rows are the DFTs of he and ho. 2.
• the "block-DFT” of HH H is computed by finding element-by-element autocorrelations and the crosscorrelation ofhe and ho. This required 6N complex multiplies and 2N complex adds: the products of N 2x2 matrices are computed with there own Hermitian transposes. 3.
• the block-DFT of ⁇ n is added, which requires 3N multiplies (scale the stored block-DFT of the RRC filter by ⁇ 2 ) and 3N adds to add the block-DFT of the two matrices. 4.
• An inverse of ⁇ n + HH ff is taken in the block-DFT domain. To do this an inverse of each of the N 2x2 matrices is taken in the block-DFT domain.
• To estimate the total number of operations, consider a Hermitian a b matrix M The inverse of this matrix is given per Equation 70.
• the complexity of processing a data block r of 2N values involves: 2 N-point DFTs; one product of the N-point block-DFTs (filter and data), which required 8N complex multiplies and 4N complex adds; and 1 N-point inverse DFTs.
• L receive antennas 2 channel matrices — one "even” and one "odd” matrix for each antenna result.
• the channel matrices for I th antenna are denoted as H ⁇ , e and H ⁇ , 0 and h ⁇ , e , n and h ⁇ )0) n denote the n th row of such matrix.
• Each channel matrix is Toeplitz and with the appropriate rearrangement of rows the joint channel matrix is a block-Toeplitz matrix, per Equation 71.
• the matrices Gi are the Toeplitz blocks of HbT. Each Gi is a 2 x2V matrix.
• the MMSE estimation formulation is per Equation 73.
• Equation 73 depends on the assumptions that are made for ⁇ n .
• the introduction of the multiple receive antenna introduces an additional spatial dimension.
• ⁇ n ⁇ n,lant ® ⁇ sp Equation 74
• ⁇ n lant is the noise covariance matrix of the noise observed at a single antenna as per Equation 57.
• ⁇ sp is the normalized synchronous spatial covariance matrix, i.e. it is the covariance matrix between the L noise samples observed at the L antennas at the same time normalized to have l's on the main diagonal- ⁇ S> denotes the Kroenecker product.
• ⁇ n is a 2LNx2LN Hermitian positive semi-definite matrix, which is block-Toeplitz with 2 x2 blocks.
• four preferred embodiments are described: an exact solution; a simplification by assuming that the L receive antenna have uncorrelated noise; a simplification by ignoring the temporal correlation of the noise in the odd and even streams from the same antenna; and a simplification by assuming that all 2L chip-rate noise streams are uncorrelated.
• the complexity of DFT-based processing using the circulant approximation may be partitioned into two components: the processing of channel estimation which need not be done for every new data block and the processing of data itself which is performed for every data block.
• the complexity of processing data involves: 2 forward iV-point DFTs; 2LN complex multiplies; and 1 inverse ⁇ T-point DFT.
• the complexity of processing the channel estimate varies for each embodiment.
• the complexity of computing the "MMSE filter" from the channel estimate is as follows: 2L ⁇ T-point DFT's ; N 2Lx2L matrix products + N 2Lx2L matrix additions to compute ⁇ a + FL b ⁇ ⁇ L b ⁇ H ) ; N2Lx2L matrix inverses to compute the inverse of ( ⁇ n + FL bT H bT H ) ; andN2Lx2L matrix products to produce the actual filter.
• a major contributor to the overall complexity of this process is the matrix inverse step in which an inverse of 2 x2 matrices has to be taken. It is precisely this complexity that can be reduced by various assumptions on the uncorrelated nature of the noise, as follows: 1. If it is assumed that the noise is uncorrelated both temporally (odd/even samples) and spatially (across antennas), then ⁇ n reduces to a diagonal matrix and the problem is identical to single-sample-per-chip sampling with 2L antennas with spatially uncorrelated noise. As a result, the operation of matrix inverse simply reduces to a division since all the matrices involved are Toeplitz. 2.
• the matrix inverses involved are those of 2x2 matrices. 3. If it is assumed that a temporal uncorrelation of odd/even streams but a spatial noise correlation is retained, the matrix inverses involved are LxL.

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