Sheet Music

Blind periodically time-varying MMOE channel shortening for OFDM systems

Description
Blind periodically time-varying MMOE channel shortening for OFDM systems
Categories
Published
of 4
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  BLIND PERIODICALLY TIME-VARYINGMMOECHANNEL SHORTENINGFOROFDMSYSTEMS  D. Darsena Parthenope University, Napoli, Italy darsena@uniparthenope.it G. Gelli, L. Paura, F. Verde University Federico II, Napoli, Italy [gelli,paura,f.verde]@unina.it ABSTRACT In this paper, the problem of synthesizing a blind channel shorteningalgorithm for orthogonal frequency-division multiplexing (OFDM)systems is addressed. In particular, a commonly adopted assump-tion in the channel shortening framework is discussed, showing that,when it is violated, the data statistics usually needed for the syn-thesis of the shortening algorithms turn out to be periodically time-varying (PTV) rather than time-invariant. Elaborating on this point,and considering in particular the recently proposed minimum-mean-output-energy (MMOE) blind channel shortening algorithm, it isshown how its synthesis must be modi Þ ed in order to account forthe PTV nature of the data statistics. Numerical results assessing theperformance of the proposed blind PTV-MMOE channel shorteningalgorithm are reported.  Index Terms —  Blind channel shortening, orthogonal frequen-cy-division multiplexing (OFDM), periodically time-varying (PTV) Þ ltering. 1. INTRODUCTION Orthogonal frequency-division multiplexing (OFDM) is a conve-nient and  ß exible choice to achieve high data-rate transmission overdispersive channels, without the need to resort to complicated equal-ization strategies. OFDM systems counteract channel dispersion byinserting a cyclic pre Þ x (CP) at the beginning of each OFDM block.Under the assumption that the CP length  L cp  obeys  L cp  ≥  L h ,where  L h  is the maximum channel order, performing CP removal atthe receiver assures perfect interblock (IBI) suppression. In highly-dispersive channels, however, ful Þ lling condition  L cp  ≥  L h  mightbe impractical. A viable alternative is to preprocess the receivedsignal, before CP removal, by means of a time equalizer (TEQ)aimed at  shortening  the channel up to a length  L eff   ≤ L cp .Several channel shortening algorithms have been proposed in theliterature, both  non-blind   ones (like e.g. the pioneering contribution[1]), which assume knowledge or estimation of the channel impulseresponse (CIR) to be shortened, as well as  blind   ones [2, 3, 4, 5, 6].Channel shortening algorithms are generally implemented as  Þ nite-impulse response (FIR)  Þ lters of order  L e , whose weights are cal-culated by solving quadratic optimization problems, which involveonly second order statistics (SOS) of the received data. In particular,the commonly considered assumption  L g   L e  +  L h  < M   assuresthat the relevant SOS are time-invariant, which entails some simpli- Þ cation in the synthesis and analysis of the shortening algorithms.However, the latter assumption is likely to be violated in systemswith high channel dispersion (large  L h ) and/or small-to-moderatenumber  M   of subcarriers.In this paper, we consider in particular the blind channel short-ening algorithm [6], which is based on the minimum-mean-output-energy (MMOE) criterion. We show that, when  L g  ≥  M  , the SOSneeded for its synthesis exhibit a periodically time-varying (PTV)behaviour, which enforces a PTV structure on the resulting channelshortening  Þ lter. The obtained PTV-MMOE channel shortening al-gorithm can be conveniently implemented by resorting to the Fourierseries representation [7]. Notations:  The Þ elds of complex and real numbers are denoted with C  and  R , respectively; matrices [vectors] are denoted with upper[lower] case boldface letters (e.g., A or a ); the  Þ eld of   m × n  com-plex [real] matrices is denoted as  C m × n [ R m × n ], with  C m [ R m ]used as a shorthand for C m × 1 [ R m × 1 ]; { A } i 1 i 2  indicates the  ( i 1  +1 ,i 2 +1) th element of matrix A ∈ C m × n , with 0 ≤ i 1  ≤ m − 1 and 0  ≤  i 2  ≤  n − 1 ; the superscripts ∗ ,  T  ,  H  , and − 1  denote the con- jugate, the transpose, the Hermitian (conjugate transpose), and theinverse of a matrix, respectively;  0 m  ∈  R m ,  O m × n  ∈  R m × n , and I m  ∈ R m × m denote the null vector, the null matrix, and the identitymatrix, respectively; rep P  { f  [ n ] } =  + ∞ i = −∞ f  [ n − iP  ]  denotes thereplication of period  P   of   f  [ n ] ;  R N  [ n ]  is the rectangular window of length  N  , i.e.,  R N  [ n ] = 1  for  0 ≤  n ≤ N   − 1  and zero otherwise;E [ · ]  and  <  ·  >  denote ensemble and time averaging, respectively; Þ nally,  j   √ − 1  denotes the imaginary unit. 2. OFDMSYSTEMMODELANDSOSANALYSIS Following mostly the framework and notation of [6], we consider anOFDM system with  M   subcarriers, CP length  L cp , and symbol pe-riod  T  . The data symbols  s [ n ] , modeled as an iid zero-mean circularrandom sequence with variance  σ 2 s , are subject to (s.t.) conventionalOFDM precoding, encompassing inverse discrete Fourier transform(IDFT) and CP insertion, obtaining thus the transmitted OFDM se-quence: u [ n ] = M  − 1  m =0+ ∞  i = −∞ s [ iM   +  m ] b m [ n − iP  ] where the  precoding  Þ lters  [8] are given by b m [ n ]   1 √  M   e j  2 πM  mn ,  0 ≤ m ≤ M   − 1 , 0 ≤ n ≤ P   − 1;0  otherwise.(1)The sequence  u [ n ]  undergoes digital-to-analog conversion withsam-pling period  T  c    T/P  , where  P     M   +  L cp , followed by up-conversion for transmission over the physical channel, modeled as alinear time-invariant  Þ lter.At the receiver, after downconversion, and assuming perfect tim-ing and frequency synchronization, the signal is sampled with period 3576978-1-4577-0539-7/11/$26.00 ©2011 IEEE ICASSP 2011  T  c /N  , where  N >  1  is an oversampling factor 1 and the resulting  N  samples are collected in r [ n ] ∈ C N  , which can be expressed as r [ n ] = L h  ℓ =0 h [ ℓ ] u [ n − ℓ ] + w [ n ]  (2)where h [ n ] ∈ C N  contains samples of the baseband equivalent CIR,with  L h  denoting the channel order, and  w [ n ]  ∈  C N  is the noisevector, modeled as a zero-mean circular complex Gaussian randomvector, with autocorrelation matrix  R ww    E [ w [ n ] w H  [ n ]] = σ 2 w I N  , independent from the data symbols.In the sequel, we assume that L cp  < L h  (insuf  Þ cient CP length);in this case, the effect of channel dispersion cannot be completelyeliminated by CP removal. To overcome this drawback, a TEQ canbe employed to shorten the CIR up to a certain length  L eff   ≤  L cp ,thus allowing the use of CP removal to eliminate IBI. The input-output relationship of a FIR TEQ can be expressed as y [ n ] =  f  H  [ n ] z [ n ]  (3)where  f  [ n ]  ∈  C N   ( L e +1) represents the (possibly time-varying)weight vector, with  L e  denoting the equalizer order, and  z [ n ]   [ r T  [ n ] , r T  [ n − 1] ,..., r T  [ n − L e ]] T  ∈ C N  ( L e +1) collects the dataat the TEQ input. It results that z [ n ] =  H  u [ n ] + v [ n ]  (4)where H   h [0]  h [1]  ...  h [ L h ]  0 N   ...  0 N  0 N   h [0]  h (1)  ...  h [ L h ] ...  0 N  ...... ... ... ... ...  0 N  0 N   ... ...  h [0]  h [1]  ...  h [ L h ]  (5)is the  N  ( L e  + 1) × ( L g  + 1)  block Toeplitz channel matrix, with L g   L e + L h ,  u [ n ]  [ u [ n ] ,u [ n − 1] ,...,u [ n − L g ]] T  ∈ C L g +1 ,and v [ n ]  [ w T  [ n ] , w T  [ n − 1] ,..., w T  [ n − L e ]] T  ∈ C N  ( L e +1) .In the following section, we will choose  f  [ n ]  according to theMMOE criterion [6], which involves only SOS of the received data.To this respect, observe that, as widely recognized in the litera-ture (see, e.g., [9, 10]), the multirate  Þ lterbank structure (1) of theOFDM transmitter induces  cyclostationarity  [7] at the transmittedsequence  u [ n ] , and possibly at the TEQ input  z [ n ] . In order tounderstand if   z [ n ]  is actually  wide-sense cyclostationary  (WSCS),we calculate the autocorrelation matrix 2 R zz [ n ]   E [ z [ n ] z H  [ n ]] ∈ C N  ( L e +1) × N  ( L e +1) . Relying on (4), one has: R zz [ n ] =  HR  u  u [ n ] H H  +  σ 2 w I N  ( L e +1)  (6)with  R  u  u [ n ]    E [  u [ n ]  u H  [ n ]]  ∈  C ( L g +1) × ( L g +1) . Observe that R zz [ n ]  is periodic if and only if   R  u  u [ n ]  is periodic, with the sameperiod. Hence, the problem boils down to studying the periodicityof  R  u  u [ n ] , whose entries are given by { R  u  u [ n ] } i 1 i 2  =  R u [ n − i 1 ,i 2 − i 1 ] ,  0 ≤ i 1 ,i 2  ≤ L g  (7) 1 Oversampling [6] is necessary for performing blind channel shortening,and can be substituted by the use of multiple antennas at the receiver. 2 The conjugate correlation matrix  R zz ∗    E [ z [ n ] z T  [ n ]]  is zero due tothe assumption of circularity of the data symbols. where  R u [ n,m ]    E [ u [ n ] u ∗ [ n − m ]]  denotes the autocorrelationof   u [ n ] . By accounting for (1), it can be shown that R u [ n,m ] =  σ 2 s  rep P   M  − 1  k =0 b k [ n ] b ∗ k [ n − m ]  =  σ 2 s , m  = 0; σ 2 s  rep P   { R P  [ n ]R P  [ n − m ] } , m  = ± M  ;0 ,  otherwise . (8)It turns out that  R u [ n,m ] , as a function of   n , is periodic (of period P  ) only for  m  =  ± M   (provided that  P > M  , that is,  L cp  >  0 ).Hence, taking into account that | m | = | i 2 − i 1 |≤ L g  in (7), it resultsthat, when  L g  ≥ M  ,  R  u  u [ n ]  is periodic with period  P  , hence  u [ n ] is WSCS; on the contrary, when  L g  < M  ,  R  u  u [ n ] =  σ 2 s  I L g +1 does not depend on  n , hence  u [ n ]  is wide-sense stationary. The hy-pothesis  L g  < M   was considered in [6] and is commonly adopted inthe channel shortening literature(see, for example, [4]): the resultingTEQs turn out to be time-invariant. However, in highly dispersivechannels (characterized by large values of   L h ) and/or for OFDMsystems with small-to-moderate number  M   of subcarriers, this in-equality is likely to be violated; in this case, the resulting TEQs turnout to be periodically time-varying (PTV). 3. THEPTV-MMOETEQ To shorten the channel up to a certain length  L eff   ≤  L cp , theMMOE TEQ minimizes, with respect to  f  [ n ] , the mean-outputenergy (MOE) at its output:MOE { f  [ n ] }  E [ | y [ n ] | 2 ] =  f  H  [ n ] R zz [ n ] f  [ n ]  .  (9)In order to avoid the trivial solution f  [ n ] =  0 N  ( L e +1) , after express-ing the TEQ input (4) as z [ n ] =  h δ u [ n − δ  ] + L g  d =0 ,d  = δ h d u [ n − d ] + v [ n ]  (10)where  h d  ∈  C N  ( L e +1) is the  ( d  + 1) th column of   H , and 0  ≤  δ   ≤  L g  is a design delay, one can impose the constraint f  H  [ n ] h δ  =  γ  δ [ n ]  ∈  C , which, however, would require knowledgeof   h δ  and, hence, of the CIR to be shortened. In order to imposea fully  blind   constraint, instead, we rely on parameterization  P  1 of   H  (see [6]), which limits  0  ≤  δ   ≤  min[ L e ,L h ] , and set theconstraint  Θ T δ  f  [ n ] =  γ  δ [ n ] , where  Θ δ  ∈  R N  ( L e +1) × N  ( δ +1) is a known  full-column rank matrix [6] satisfying  Θ T δ  Θ δ  =  I N  ( δ +1) ,and  γ  δ [ n ]  ∈  C N  ( δ +1) is a constraint vector, whose choice willbe discussed later. Therefore, the blind PTV-MMOE optimizationproblem is given by min f  [ n ] ∈ C N  ( Le +1)  f  H  [ n ] R zz [ n ] f  [ n ]   s.t.  Θ T δ  f  [ n ] =  γ  δ [ n ]  .  (11)By resorting to the method of Lagrange multipliers, optimization(11) reduces to solving the following couple of equations: R zz [ n ] f  [ n ] =  Θ δ  λ [ n ]  ,  Θ T δ  f  [ n ] =  γ  δ [ n ]  (12)where  λ [ n ]  ∈ C N  ( δ +1) collects the Lagrange multipliers. Since, asdiscussed in the previous section, for  L g  ≥ M  , R zz [ n ]  is a periodicfunction of period  P  , it can be inferred that also  f  [ n ] ,  λ [ n ] , and γ  δ [ n ]  must all be periodic functions in  n  of the same period  P  . By 3577  expanding all the functions in terms of their discrete Fourier series(DFS), and substituting in (12), by straightforward calculations andexploiting the linear independence of complex exponentials, system(12) can be restated as 1 P  P  − 1  ℓ =0  R zz [ m − ℓ ] P   f  [ ℓ ] =  Θ δ  λ [ m ]  (13) Θ T δ   f  [ m ] =   γ  δ [ m ]  (14)where   R zz [ m ] ,   f  [ m ] ,   λ [ m ] , and   γ  δ [ m ] , for  0  ≤  m  ≤  P   −  1 ,represent the  m th DFS coef  Þ cient of  R zz [ n ] , f  [ n ] ,  λ [ n ] , and  γ  δ [ n ] ,respectively. By vertically stacking all the DFS coef  Þ cients  f  [ m ] ,  λ [ m ] , and   γ  δ [ m ] , for  0  ≤  m  ≤  P   − 1 , in  f   ∈  C NP  ( L e +1) ,   λ  ∈ C NP  ( δ +1) , and   γ  δ  ∈  C NP  ( δ +1) , respectively, (13) and (14) can becompactly expressed as Φ zz  f   = ( I P   ⊗ Θ δ )  λ  (15) ( I P   ⊗ Θ T δ  )  f   =  γ  δ  (16)where Φ zz   [ Ψ T  zz [0] , Ψ T  zz [1] ,..., Ψ T  zz [ P   − 1]] T  ∈ C NP  ( L e +1) × NP  ( L e +1) (17)with Ψ zz [ m ]  P  − 1 [  R zz [ m ] P  ,  R zz [ m − 1] P  ,...,  R zz [ m − P   +1] P  ]  ∈  C N  ( L e +1) × NP  ( L e +1) . Solving (15) with respect to  f   andsubstituting the solution back into (16), one obtains:  f   =  Φ − 1 zz  ( I P   ⊗ Θ δ )[( I P   ⊗ Θ T δ  ) Φ − 1 zz  ( I P   ⊗ Θ δ )] − 1        F δ ∈ C NP  ( Le +1) × NP  ( δ +1)  γ  δ .  (18)The analysis of the channel shortening capabilities of the PTV-MMOE TEQ is omitted for brevity, since it closely follows thatreported in [6, Theorem 1]. It can be shown that a necessary condi-tion to asymptotically (as  σ 2 w  → 0 ) achieve  ideal  channel shorteningwith  L eff   ≤  L cp  is that  L h  ≤  ( N   − 1)( L e  − δ  ) , from which it isapparent that increasing values of   N >  1  allows one to shortenlonger channels.The PTV-MMOE TEQ admits an alternative form, known asFourier series representation [7], which can be obtained by substi-tuting into (3) the expression of  f  [ n ]  in terms of its DFS. One has: y [ n ] =  f  H  ζ  [ n ]  (19)where  ζ  [ n ]    e [ n ]  ⊗  z [ n ]  ∈  C NP  ( L e +1) collects frequency-shifted versions of   z [ n ] , with  e [ n ]    P  − 1 [1 ,e − j 2 π ( n/P  ) ,...,e − j 2 πn ( P  − 1) /P  ] T  ∈ C P  . By substituting (4) and (18) into (19), andassuming ideal channel shortening, one obtains y [ n ] =  y u [ n ] +  y v [ n ]  (20)where  y u [ n ]     γ  H δ  F H δ  ( I P   ⊗ H win )( e [ n ] ⊗  u win [ n ])  and  y v [ n ]    γ  H δ  F H δ  ( e [ n ] ⊗ v [ n ])  represent the useful and noise components, H win    [ h 0 , h 1 ,..., h L eff  ]  ∈  C N  ( L e +1) × ( L eff  +1) , and   u win [ n ]   [ u [ n ] ,u [ n − 1] ,...,u [ n − L eff  ]] T  ∈  C L eff  +1 . This expression al-lows one to optimize the constraint vector   γ  δ  containing the DFScoef  Þ cients of   γ  δ [ n ] , by maximizing the time-averaged shorteningsignal-to-noise ratio (TA-SSNR):TA-SSNR   <  E [ | y u [ n ] | 2 ><  E [ | y v [ n ] | 2 > =  σ 2 s  γ  H δ   F H δ  ( I P   ⊗ H win H H  win )  F δ  γ  δ σ 2 w  γ  H δ  F H δ   F δ  γ  δ =   γ  H δ   F H δ  Φ zz  F δ  γ  δ σ 2 w  γ  H δ  F H δ   F δ  γ  δ − 1 (21)whose solution is the eigenvector corresponding to the maximumeigenvalue of the matrix pencil  (  F H δ  Φ zz  F δ ,σ 2 w  F H δ   F δ ) .As regards OFDM symbol recovery, after channel shortening y [ n ]  undergoes a polyphase decomposition of order  P  , i.e.,  y p [ n ]  y [ nP   +  p ] , for  0  ≤  p  ≤  P   − 1 . Since  L eff   ≤  L cp , after removingthe CP from the vector  y [ n ]    [ y 0 [ n ] ,y 1 [ n ] ,...,y P  − 1 [ n ]] T  , oneobtains the IBI-free model  y [ n ]  R cp y [ n ] =  R cp GT cp W IDFT s [ n ] + d [ n ]  ,  (22)where  R cp  ∈  C M  × P  and  T cp  ∈  C P  × M  are the CP removal andinsertion matrices [6], respectively, W IDFT  ∈ C M  × M  represents theunitary symmetric IDFT matrix, s [ n ] ∈ C M  and d [ n ] ∈ C M  denotethe vertical stacking of the data symbols  s m [ n ]    s [ nM   +  m ]  andof the disturbance  d m [ n ]  f  H  [ nM   + m ] v [ nM   + m ] , for  0 ≤ m ≤ M   − 1 , respectively, and,  Þ nally, the channel matrix G ∈ C P  × P  is G  =  g ∗ 0 [0] 0 0  ...  0 ...  g ∗ 0 [1] 0  ...  0 g ∗ L eff  [ L eff  ]  ... ...  ... .........  ... ...  00  ... g ∗ L eff  [ P   − 1]  ... g ∗ 0 [ P   − 1]  , (23)with  g ℓ [ n ]  being the  ( ℓ  + 1) th entry of the  effective  channel vector g [ n ]    H H  f  [ n ] . Observe that, unlike conventional OFDM equal-ization, as a consequence of the time-varying TEQ Þ ltering, G losesits Toeplitz structure and, thus, matrix  R cp GT cp  is no more di-agonalized by the FFT. Nevertheless, provided that  R cp GT cp  isnonsingular, the data symbol vector  s [ n ]  can still be recovered byquantizing to the nearest information symbol block (in the Euclideanmetric) the vector  W DFT  ( R cp GT cp ) − 1  y [ n ] . It should be noted,however, that this operation entails a higher complexity comparedto conventional OFDM decoding. Such a complexity, which is typ-ical of OFDM systems operating in time-varying channels, can bereduced by resorting to low-complexity equalization schemes [11]. 4. NUMERICALRESULTS In this section, numerical results, obtained via Monte Carlo com-puter simulations, are presented. The considered OFDM system em-ploys  M   = 16  subcarriers with QPSK signaling, and a CP length L cp  = 4 . The order of the FIR channel is  L h  = 7 , and the CIRsamples are modeled as iid zero-mean complex circular Gaussianrandom variables. As a performance measure, we adopt the averagebit-error-rate (ABER), which is calculated as the arithmetic averageof the  M   subcarrier BERs.We considered two different versions of the proposed PTV-MMOE TEQ implemented with  N   = 2  and  L e  = 11 : the  Þ rstone is based on (18) and employs all the  P   = 20  DFS coef  Þ cients,which are needed to exactly represent  f  [ n ] ; the second one (labeled 3578                                                                               Fig.1 . Average BER versus SNR.with the suf  Þ x “low” in the  Þ gures) is a low-complexity version thatconsiders only the  5  dominant DFS coef  Þ cients of   R zz [ n ]  for thesynthesis of   f  [ n ] . We compared the performances of the proposedreceiver with those of its time-invariant counterpart (TI-MMOE).Allreceivers (PTV-MMOEand TI-MMOE)are implemented both intheir exact version (labeled with “exact”), i.e., by assuming perfectknowledge of the weight vectors, as well as in their data-dependentversion (labeled with “data-estimated”), i.e., by estimating the re-quired SOS on the basis of a sample-size of   K   OFDM symbols. Asa reference, we reported also the performance of the exact OFDMreceiver without channel shortening (labeled with “w/o TEQ”).In Fig. 1, the ABER is reported as a function of the signal-to-noise ratio (SNR), de Þ ned as SNR    σ 2 s /σ 2 w , with a sample-size K   = 500  for the data-estimated versions of the receivers. Theresults show that, when implemented exactly, the proposed PTV-MMOE TEQ outperforms the TI-MMOE by about 4-5 dB for al-most all values of SNR, whereas its low complexity version per-forms signi Þ cantly worse. Turning to the data-estimated versions,however, the low complexity PTV-MMOE TEQ performs compara-bly to its complete version, and both receivers outperform the data-estimated version of the TI-MMOE. Note, however, that due to Þ nitesample-size effects, all the performance curves of the data-estimatedreceivers, unlike theirexact counterparts, exhibit adistinct BER ß oorwhen the SNR increases. Finally, observe that the performances of the receiver without TEQ are completely unsatisfactory in this sce-nario.In Fig. 2, the ABER for the data-dependent receivers is reportedas a function of the sample size  K   ranging from  500  to  3000  andfor SNR = 30 dB. The results con Þ rm the advantage of the data-dependent PTV-MMOE (both versions) over the TI-MMOE one,whose performances are quite insensitive to the increase of the sam-ple size. On the contrary, the performances of the complete PTV-MMOE TEQ exhibit a marked improvement with the sample-size,while those of its low-complexity version exhibit an intermediatebehavior. In particular, the low-complexity receiver performs com-parably or better than its complete counterpart for small values of  K  , due to the reduced number of parameters to be estimated in itssynthesis, whereas, for larger values of   K  , it shows a marked perfor-mance degradation compared to its complete version.                                                              Fig.2 . Average BER versus sample-size  K  . 5. REFERENCES [1] N. Al-Dhahir and J. M. Ciof  Þ , “Optimum  Þ nite-length equal-ization for multicarrier transceivers,”  IEEE Trans. Commun. ,vol. 44, pp. 56–64, Jan. 1996.[2] R. K. Martin, J. Balakrishnan, W. A. Sethares, and C. R. John-son, “A blind adaptive TEQ for multicarrier systems,”  IEEE Signal Process. Lett. , vol. 9, , pp. 341–343, Nov. 2002.[3] J. Balakrishnan, R.K. Martin and C.R. Johnson, “Blind adap-tive channel shortening by sum-squared auto-correlation min-imization (SAM),”  IEEE Trans. Signal Process. , vol. 51, pp.3086–3093, Dec. 2003.[4] T. Miyajima and Z. Ding, “Second-order statistical approachesto channel shortening in multicarrier systems”,  IEEE Trans.Signal Process. , vol. 52, pp. 3253-3264, Nov. 2004.[5] R.K. Martin, J.M. Walsh and C.R. Johnson, “Low-complexityMIMO blind adaptive channel shortening,”  IEEE Trans. SignalProcess. , vol. 53, pp. 1324–1334, Apr. 2005.[6] D. Darsena and F. Verde, “Minimum-mean-output-energyblind adaptive channel shortening for multicarrier SIMOtransceivers,”  IEEE Trans. Signal Process. , vol. 55, pp. 5755–5771, Jan. 2007.[7] W. A. Gardner,  Introduction to Random Processes.  New York:McGraw-Hill, 1990.[8] A. Scaglione, G.B. Giannakis, and S. Barbarossa, “Redundant Þ lterbank precoders and equalizers. Part I: uni Þ cation and op-timal designs,”  IEEE Trans. Signal Process. , vol. 47, pp. 1988–2006, July 1999.[9] G.B. Giannakis, “Filterbanks for blind channel identi Þ cationand equalization,”  IEEE Signal Processing Letters , vol. 4, pp.184–187, June 1997.[10] E. Serpedin and G. B. Giannakis, “Blind channel identi Þ ca-tion and equalization with modulation induced cyclostationar-ity,”  IEEE Trans. Signal Process. , vol. 46, pp. 1930–1944, July1998.[11] P. Schniter, “Low-complexity equalization of OFDM in doublyselective channels,”  IEEE Trans. Signal Process. , vol. 52, pp.1002–1011, Apr. 2004. 3579
Search
Tags
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks
SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!

x