The field of the invention is that of digital communications. More specifically, in a system of digital communications, the invention pertains to a technique for the iterative reception of a signal which has undergone channel encoding at transmission. The term “channel encoding” is understood here and throughout the document to mean a technique for the protection of information to be transmitted by insertion of redundant information.
The invention can be applied to wire-based digital communications systems such as the ADSL or “Asymmetric Digital Subscriber Line” systems as well as to radio-frequency systems using one or more transmit and/or receive antennas.
The invention also relates more particularly to reception technique adapted to multi-carrier system such as the OFDM (“Orthogonal Frequency Division Multiplexing” or MC-CDMA (“Multi-Carrier Code Division Multiplex Access”).
There are many techniques for the reception of multi-carrier signals that have undergone channel encoding at transmission, designed to introduce redundancy into the signal.
Among these techniques, some (and this is the case with the present invention) rely on an iterative reception of signals enabling the improvement, as and when the iterations are performed, of the quality of estimation of the signal transmitted as a function of the signal received.
The following may be highlighted among these iterative techniques:
exhaustive-type receivers which implement a maximum likelihood (ML) type algorithm and have the drawback of inducing high complexity of processing, especially for MIMO type multiple-antenna transmission for example. Indeed, the complexity of these algorithms increases exponentially according to the number of antennas and the number of states of the modulation.
the receivers implementing linear decoding algorithms which have the advantage, as compared with maximum likelihood receivers of being far more complex.
The invention falls within the narrower field of linear iterative receivers for multi-carriers signals that have undergone channel encoding at transmission.
In “Low Complexity Iterative Receiver for Non-Orthogonal Space-Time Block Code with Channel Coding”, Proceedings of VTC, Los Angeles, USA, September 2004), P-J. Bouvet, M. Hélard and V. Le Nir have proposed a technique of iterative and linear reception for non-orthogonal block space-time codes.
In this article, the authors consider a MIMO channel model comprising a set of frequency-flat Rayleigh fading orthogonal channels. Indeed, a frequency-flat fading channel is equivalent to a multiple-path channel (with frequency fading) for which an OFDM modulation of the signal is performed.
As illustrated in
The iterative receiver proposed derives advantage from the channel decoding to improve the space-time block decoding by eliminating interference terms.
More specifically, the proposed receiver (see
Thus, a signal r is received on NR receive antennas referenced 251 to 25N
Since the method is iterative, this estimated binary signal {circumflex over (d)} is subjected to a new interleaving II 222 and a new mapping M 232, in order to obtain an estimated M-ary signal ŝ, that can be reinjected into the MMSE space-time decoding block 20 for a following iteration for improving the estimation of received signal.
At the first iteration, the receiver carries out a classic MMSE type equalization of the received signal. During the following iterations, however, the previously estimated symbols are used by the space-time decoder to cancel a residual inter-element interference (IEI).
According to this technique, the co-antenna interference coming from the space-time encoding is therefore cancelled iteratively by means of a static coefficient interference canceller.
One drawback of this prior art technique is that it cannot be used to cancel every type of interference affecting the signal. While it enables the cancellation of co-antenna interference linked to a MIMO type transmission context, it is on the other hand not suited to multi-user type interference cancellation (in an MC-CDMA or “Multi Carrier Code Division Multiple Access” type context) or to the cancellation of interference due to a linear pre-encoding of the signal before transmission.
Another drawback of this prior-art technique is that its performance is not the optimum and does not permit an estimation of adequate quality of the transmitted signal. In particular, an iterative reception method of this kind does not converge with sufficient speed or sufficient reliability towards the optimum limits.
It is a goal of the invention especially to overcome these drawbacks of the prior art.
More specifically, it a goal of the invention to provide a technique of iterative reception with interference cancellation, of a multi-carrier signal that has undergone channel encoding prior to transmission, of improved performance as compared with the prior art techniques.
It is also a goal of the invention to provide a technique of this kind that is adapted to signals that may or may not have undergone linear pre-encoding at transmission in any type of transmission context with one or more receive and/or transmit antennas (SISO (“Single Input Single Output”), MIMO, etc.), whether or not in a multi-user (MC-CDMA) context. It may be recalled that linear pre-encoding consists of a multidimensional rotation of the transmission constellation.
It is another goal of the invention to propose a technique of this kind that can be used to cancel any type of interference affecting the signal, whether this interference is due to the system of transmission of the signal or to the transmission channel. In particular, the invention is aimed at proposing a technique of this kind that can be used to cancel co-antenna interference, multi-user interference or again interference due to linear pre-encoding, if any, of the signal.
It is yet another goal of the invention to provide a technique of this kind that is less complex than the prior-art techniques based on maximum likelihood.
These goals as well as others that shall appear here below are achieved by means of a method of reception of a multiple-carrier data signal, called a multi-carrier signal, having undergone channel encoding at transmission, said method employing an estimation of the received signal, delivering an estimated signal.
According to the invention, a reception method of this kind comprises at least one iteration of improvement of said estimated signal as a function of said received signal and of a preceding estimated signal, and said iteration comprises, for each of said carriers of said received signal, a step of equalization of said carrier by cancellation of an interference affecting said received carrier, as a function of at least one statistical parameter of said previous estimated signal.
Thus, the invention relies on a fully novel and inventive approach to the iterative techniques of reception of multi-carrier signals with interference cancellation. Indeed, the method of the invention can be used for the iterative decoding of an encoded multi-carrier signal (and possibly linearly pre-encoded and/or space-time encoded) by the use of an interference-cancellation type equalizer whose coefficients are corrected in the course of the iterations. This gives the signal estimation process fast and reliable convergence towards the optimum convergence boundaries. The dynamic correction of the coefficients of the interference canceller is done as a function of statistical parameters of the signal estimated during the previous iteration, and does not take account solely of the transmission channel or of the noise (which would result in static coefficients).
The performance of such a reception method is therefore improved by the use of a multi-carrier signal, for example of the OFDM (orthogonal frequency division multiplexing) type. Indeed, while the intrinsic structure of a multi-carrier signal of this kind is already designed to reduce interference, the invention proposes the use of an interference cancellation structure at reception of such a signal, which could seem to be unnecessary and redundant to those skilled in the art.
Finally, the processing proposed by the invention is a block processing of the received signal, performed carrier by carrier.
Advantageously, said iteration comprises the following steps for each of said carriers:
filtering of said received carrier, delivering a filtered carrier;
determining an interference that affects said received carrier, generated at transmission of said multi-carrier signal and/or due to the transmission channel of said multi-carrier signal, said determining step delivering an estimated interference;
subtraction of said estimated interference from said filtered carrier so as to obtain an improved carrier;
equalization of said improved carrier, delivering an equalized carrier;
estimation, from at least said equalized carrier, of the sent carrier, delivering an estimated carrier,
and said estimated signal takes account of each of said estimated carriers.
Thus, the method of the invention has the characteristic feature of carrying out a processing of the signal, carrier by carrier, and enables the processing of any type of interference, whether it is of a co-antenna, multi-user type or again whether it is due to a linear pre-encoding of the signal. The estimation of the carrier sent may take account of one or more equalized carriers of the signal, as a function of the channel encoding.
Preferably, said filtering step applies a multiplication of a vector representing said carrier received by a filtering matrix taking account of at least one representative matrix representing said transmission channel, of a signal-to-noise ratio and of said statistical parameter.
According to one advantageous characteristic, said step for determining interference employs a multiplication of a vector representing said previous estimated carrier by an interference matrix taking account of at least said matrix representing said transmission channel and said statistical parameter.
Preferably, said statistical parameter is the variance of said preceding estimation of said received signal. The interference cancellation of the invention takes account, via the variance, of the energy of the estimate of the transmitted signal.
Advantageously, said preceding estimation is delivered by the preceding iteration for all the iterations except the first one, and is delivered by a step of preliminary estimation for the first iteration.
In an advantageous variant of the invention, said multi-carrier signal undergoes a linear pre-encoding at transmission by multiplication by a pre-encoding matrix, and said filtering and interference matrices also take account of said pre-encoding matrix.
In another advantageous variant of the invention, said multi-carrier signal undergoes space-time encoding at transmission and is transmitted by and/or received on at least two antennas. The context therefore is that of MIMO type transmission.
These two alternative embodiments can of course be combined to obtain MIMO type transmission of a linearly pre-encoded multi-carrier signal.
The invention also relates to a receiver of a multiple-carrier data signal, called a multi-carrier signal, having undergone channel encoding at transmission, said receiver comprising means of estimation of the received signal, delivering an estimated signal. According to the invention, a receiver of this kind comprises means of improvement of said estimated signal, as a function of said received signal and of a preceding estimated signal, implementing at least once, in the form of a iteration, and for each of said carriers of said received signal, means of equalization of said carrier by cancellation of an interference affecting said received carrier, as a function of at least one statistical parameter of said previous estimated signal.
The invention also relates to a computer program product comprising program code instructions for the execution of the steps of the method of reception of a multiple-carrier data signal described previously when said program is executed in or by a microprocessor.
Other features and advantages of the invention shall appear more clearly from the following description of a preferred embodiment given as a simple and non-restrictive illustration and from the appended drawings, of which:
a and 13b illustrate the principle of OFDM modulation before transmission and OFDM demodulation at reception, by frequency distribution followed by time distribution of the symbol in the context of SISO transmission with linear pre-encoding of
a and 14b pertain to an embodiment that is an alternative to those of
a and 15b present this same principle of OFDM modulation/demodulation in the context of MIMO transmission of
a and 16b finally illustrate this principle of OFDM modulation/demodulation in the context of MIMO transmission with linear pre-encoding as described in
The general principle of the invention relies on the iterative reception of multi-carrier signals that have undergone channel encoding at transmission. This iterative reception employs an interference cancellation whose coefficients are dynamically modified in the course of the iterations as a function of statistical parameters (such as energy or variance) of the previously estimated signal.
It can be implemented in a single-antenna or multi-antenna transmission context, in a MC-CDMA or non-MC-CDMA type multi-user system, with or without linear pre-encoding at transmission.
The operating principle of the diagram of
More specifically, an equalization is carried out for each of the carriers of the received signal, according to the following formula:
{circumflex over (x)}(p)(k)=PHr(k)−QH{circumflex over (x)}(p-1)(k), where:
{circumflex over (x)}(p)(k) designates the vector representing the signal equalized at the iteration p for the carrier k or the group of carriers indexed k.
{circumflex over (x)}(p-1)(k) designates the vector representing the signal estimated at the iteration (p-1) for the carrier k or the group of carriers indexed k,
r(k) designates the signal received after OFDM demodulation on the carrier k or the group of carriers indexed k
and where PH and QH are two matrix filters having the following form:
with C as a matrix representing the transmission channel (and comprising, if necessary, linear pre-encoding if such pre-encoding is applied at transmission), I the identity matrix, σx2 the variance of the signal x sent, σ{circumflex over (x)}
the inverse of the mean signal-to-noise ratio. It will be noted that the operator (.)H designates the conjugate transposed operator and that the operator ddiag(.) associates a matrix ddiag(A) with a matrix A, all the terms of this matrix ddiag(A) being identical with those of the matrix A except for the diagonal terms which are equal to zero.
For the sake of simplification, the description of
The variance computation block 31 computes the variance σ{circumflex over (x)}
where N is a sufficiently great integer (for example N=10000). The N samples correspond to N temporal symbols on one or more carriers as a function of the size of the system considered. A computation of time-related variance is therefore performed on the OFDM spectrum.
The equalization block 30, working by interference cancellation, receives at input the estimated M-ary signal {circumflex over (x)}(p-1) coming from the preceding iteration and the received signal r, for the carrier k, or the group of carriers indexed k. It performs the following operations:
matched filtering 302 of the received signal r by application of the filtering matrix PH delivering a filtered signal. This filtering takes account of the variance of the preceding estimated M-ary signal, the channel matrix and of the signal-to-noise ratio
as indicated in the formula proposed here above for PH. σx2 is often fixed as being equal to 1 in transmission, and σn2 is the estimated noise in reception, for example by means of pilot sequences. Thus, at output of the filtering block 302, a filtered signal is obtained for the carrier k or the group of carriers indexed k;
creation 301 of the interferences from a preceding estimated M-ary signal {circumflex over (x)}(p-1) by left multiplication of this preceding estimated M-ary signal by an interference matrix QH, which takes account of the channel matrix and of the variance of the preceding estimated M-ary signal σ{circumflex over (x)}
subtraction of the estimated interference signal obtained at output of the block referenced 301 from the filtered signal obtained at output of the filtering block 302 to obtain an equalized M-ary signal {circumflex over (x)}(p) for the carrier k or the group of carriers indexed k.
The equalized M-ary signal {circumflex over (x)}(p) is then fed into the estimation block 32 which does the estimation of:
the binary signal sent, called an estimated binary signal {circumflex over (d)}(p);
the M-ary signal sent, called an estimated M-ary signal {circumflex over (x)}(p) for the carrier k or the group of carriers indexed k.
or of only one of these signals.
The estimation block 32 can carry out especially certain of the following operations:
<<mapping>>(i.e. the conversion of binary elements into complex symbols)
de-interleaving;
channel decoding;
re-interleaving;
soft <<mapping>>.
Indeed, the modulation used may be, for example, a trellis-encoded modulation (or TCM).
The signal r received at the NR receive antennas of the MIMO system considered is delivered to the first elementary module Ite1 of the receiver, illustrated in
first of all, a classic equalization 50 of the received signal r is performed by multiplication by an equalization matrix taking account of the matrix representing the transmission channel, the linear pre-encoding if it has been implemented at transmission, and the signal-to-noise ratio
This equalization 50 delivers an equalized M-ary signal {circumflex over (x)}(1). Such an equalization is, for example, of an MMSE type;
the equalized M-ary signal {circumflex over (x)}(1) is then fed into an estimation block 51 for estimation of the binary signal and of the M-ary signal which, from the equalized M-ary signal, delivers an estimated binary signal {circumflex over (d)}(1) (which possibly is not exploited and, therefore, is not necessarily available at output) and an estimated M-ary signal {circumflex over (x)}(1).
The estimated M-ary signal {circumflex over (x)}(1) is then injected into the next elementary module Ite2.
Certain of the elementary modules Ite1 to Itep performing the estimation of the received signal deliver firstly a binary estimation a of this received signal and, secondly, a weighted estimation {circumflex over (x)} of this received signal. It is this soft estimation that is used for the following iteration if this iteration exists.
The process of improvement of the estimated signal can be stopped at any time, at the end of any one of the iterations Ite1 to Itep. Generally, the iterations are stopped when {circumflex over (x)}(p)≈{circumflex over (x)}(p+1), or more specifically when the difference between the estimated signals coming from two successive iterations is below a predetermined quality threshold. This threshold may be fixed as a function of the needs of the application in view.
Referring to
SISO Transmission with Linear Pre-Encoding
In a first embodiment, the multi-carrier signal undergoes a linear pre-encoding before transmission so as to introduce diversity of space. To this end, the vectors representing the signal to be sent are multiplied by a pre-encoding matrix sized L:
More specifically, the M-ary signal to be sent undergoes an OFDM modulation before it is sent. If we consider, for example, a linear pre-encoding LP 84 sized Lp=4, the M-ary signal x to be sent can be represented in the form of the following pre-encoded symbol vector: [s1 s2 s3 s4]T. The four pre-encoded symbols can be distributed in different ways in the time-frequency plane as illustrated by
In a first classic technique illustrated by
The OFDM modulation 130 of the pre-encoded signal therefore comprises the successive blocks presented in
a series/parallel conversion S/P 131 enabling time-frequency plane distribution of the symbols s1 to s4;
an inverse fast Fourier transform IFFT 132
a guard interval insertion block +Δ 133;
a parallel/series conversion block P/S 134.
Symmetrically, at reception, an OFDM demodulation 135 is performed on the signal received at the receive antenna 72 (
series/parallel conversion S/P 131 of the received signal;
elimination −Δ 136 of the guard interval;
forward fast Fourier transform FFT 137;
parallel/series conversion P/S 134 enabling retrieval of the received symbols r1 to r4 distributed in the time-frequency plane.
At the end of the OFDM demodulation 135, the received signal
is fed into the iterative receiver 138 of the invention as illustrated for example in
a and 14b illustrate a second OFDM technique for modulation of the signal to be sent and distribution of the symbols s1 to s4 in the time-frequency plane. In this second technique, the pre-encoded symbols are mapped arbitrarily in time and frequency as illustrated by the distribution of the symbols s1 to s4 in the time-frequency plane t-f in
The transmission scheme (
Similarly, the reception scheme of
The following is the equivalent channel matrix:
In this particular embodiment, the iterative receiver of the invention uses the matrix C defined such that:
C=H·ΘL
as well as the vector r defined such that:
It is assumed that, at each iteration, an estimate of the transmitted signal is available (given by the previous iteration). It is also assumed that the variance of the noise as well the matrix C is known to the receiver (the matrix H representing the transmission channel is for example determined according to a classic channel estimation technique implemented when the receiver is initialized, by means of reference symbols or pilot carriers).
Referring to
where N is a sufficiently great integer (for example N=10000).
A filtering (302) is then performed, carrier by carrier (or carrier group by carrier group), on the received signal r using a matrix PH taking account of at least the channel matrix, the pre-encoding matrix, the variance of the estimated signal σ{circumflex over (x)}
to obtain a filtered signal.
Furthermore, the signal estimated at the preceding iteration {circumflex over (x)}(P-1) is multiplied (301), carrier by carrier (or carrier group by carrier group) by a matrix QH representing the interferences taking account at least of the channel matrix of the pre-encoding matrix and of the variance of the estimated signal. Thus an estimated interference signal is obtained.
Then, the estimated interference signal is subtracted from the filtered signal, giving an equalized signal {circumflex over (x)}(p).
In the estimation module 32, the following are then performed:
the estimation, from the equalized signal, of the binary signal sent, called an estimated binary signal {circumflex over (d)}(p);
the estimation, from the equalized signal, of the M-ary signal called an estimated M-ary signal {circumflex over (x)}(p).
The term “estimation” covers mapping operations as well as de-interleaving, channel decoding, re-interleaving and soft mapping.
For each carrier k, or each group of carriers indexed k, of the multi-carrier signal, the equalized signal is therefore expressed in the form:
{circumflex over (x)}(p)=PHr−QH{circumflex over (x)}(p-1)
where the two matrix filters P and Q have the form:
For the first iteration implemented by the receiver, for which there is not yet any previous estimated signal available, a classic equalization is carried out, followed by a symbol estimation:
More specifically, the following operations are performed:
multiplication of the vector r representing the carrier k of the signal received by a matrix taking account at least of the channel, the linear pre-encoding and the signal-to-noise ratio
Thus an equalized signal {circumflex over (x)}(1) is obtained,
estimation, from the equalized signal, of the binary signal sent, called an estimated binary signal {circumflex over (d)}(1);
estimation, from the equalized signal, of the M-ary signal sent called an estimated M-ary signal {circumflex over (x)}(1).
It is noted that the equalized signal {circumflex over (x)}(1) can be obtained with the interference cancellation structure of
MIMO Transmission without Linear Pre-Encoding
In this second particular exemplary embodiment, a transmission without linear pre-encoding is assumed. The description is situated in the context of MIMO type multi-antenna transmission in which the multi-carrier signal undergoes space-time encoding or ST mapping before transmission.
The matrix QH representing interferences, by which the estimated signal {circumflex over (x)}(p-1) is multiplied takes account at least of the channel matrix and of the variance of the estimated signal. At the end of this multiplication, an estimated interference signal is obtained.
Furthermore, during the first iteration of the receiver, before an estimated signal is available for the first time, each vector representing a carrier k of the received signal r is multiplied by a matrix taking account at least of the channel and of the signal-to-noise ratio
This multiplication delivers an equalized signal {circumflex over (x)}(1).
Finally, the matrix C and the vector r used are defined according to the space-time encoding used in the following paragraphs 7.2.1 and 7.2.2.
Spatial Multiplexing
We assume a MIMO channel with four transmit antennas 91 to 94 (or Tx 1 to Tx 4) and four receive antennas 95 to 98, as illustrated in
The encoding scheme adopted is of a spatial multiplexing type. For a block of K=4 data symbols s1, s2, s3 and s4, the space-time mapper ST 80 multiplexes the data on a time interval IT1 according to the following scheme:
a provides a more precise illustration of this space multiplexing technique applied to the M-ary signal constituted by the symbols s1, s2, s3 and s4. The M-ary signal undergoes a series/parallel conversion S/P 150 which enables the distribution of each of the symbols s1 to s4 to a distinct processing channel. Each of these processing channels is then identical to the scheme of
The equivalent channel matrix is the following:
The iterative receiver uses the matrix C defined such that:
C=H
and the received vector is:
b illustrates the reception scheme corresponding to the transmission scheme of
Space-Time Encoding
We assume a MISO channel with four transmit antennas 91 to 94 (or Tx 1 to Tx 4) and one receive antenna 100, as illustrated in
Let hij be the channel coefficient between the antenna i and the antenna j. It is assumed that this coefficient remains constant on all four time intervals IT1 to IT4. rjk is the symbol received at the antenna j during the time interval k.
At reception, the iterative system of the invention uses the vector r defined such that:
As well as the following matrix C:
MIMO Transmission with Linear Pre-Encoding
Finally, we present a last example of an embodiment of the invention, in a context of MIMO type transmission, in which the multi-carrier signal undergoes linear pre-encoding at transmission.
The processing performed at reception on each of the carriers of the signal is identical to the processing described here above with reference to
A MIMO transmission system as described in paragraph 7.2 (with spatial multiplexing or space-time encoding of the multi-carrier signal at transmission) is considered. In the case of transmission with linear pre-encoding of the signal, the iterative receiver uses the following matrix C:
where each matrix Ck is the matrix C defined in the preceding section 7.2. The vector r is defined such that:
where the values rk are as defined in the preceding section 7.2.
a and 16b illustrate the principle of OFDM modulation/demodulation at transmission and reception in this context. These figures consider the case of a MIMO scheme with two transmit antennas 91, 92 and two receive antennas 95, 96 in the context of use of a space multiplexing technique. The size of the linear pre-encoding is Lp=4.
In
At reception, the signals received at each of the two antennas 95, 96 undergo processing identical to that of
where the symbols r1, 1 and r1, 2 come from the processing channel coming from the first receive antenna 95 and where the symbols r2, 1 and r2, 2 come from the processing channel coming from the second receive antenna 96.
As described here above in the case of the SISO, it is also possible at transmission to arbitrarily distribute the pre-encoding symbols in time and in frequency (“map F/T”). The dual operation (“demap F/T”) must then be performed at reception on each antenna 95, 96 upstream from the iterative receiver 138.
Appendix 1 here below, which forms an integral part of the present description, proposes a detailed demonstration of the computation of the coefficients of the interference canceller of the receiver of the invention.
The iterative receiver of the invention has wholly satisfactory performance and can be used especially for efficient estimation of the signal sent in the presence of major interference, whether this interference results from extensive linear pre-encoding, from the use of a BLAST system (Bell Laboratories Layered Space-Time, also called a MIMO system) with several antennas, or again from non orthogonal space-time encoding.
APPENDIX 1: Demonstration of the Computation of the Coefficients of the Interference Canceller
We may recall the expression of the signal at input of the receiver:
r=C(ekxk+ak)+n (1)
Where ek is a null vector whose kith element is equal to 1 and âk=x−xkek. At output of the interference canceller we get
{circumflex over (x)}k=PkHr−qkHâk (2)
Where p and q are two filtering vectors. The following problem has to be resolved:
The optimum vectors pkH and qkH are determined in two steps. First of all, we develop the expression of the signal-to-interference noise ratio (SNIR). Let us rewrite the expression {circumflex over (x)}k as a function of xk.
The SNIR is equal to:
We propose to determine the optimum filters which maximize SNIR at output of the equalizer. Since the SNIR maximization criterion is a generalization of the MMSE, the vectors obtained truly meet the MMSE criterion. In looking at the expression of the SNIR it can be seen that only the numerator depends on qk. Thus, the optimum vector qkopt can be determined by minimizing the σNI2. In deriving relative to qk we obtain:
On the one hand, we consider a perfect decorrelation between the symbols and the noise and, secondly, it is assumed that
In setting
we obtain the optimal first vector:
qkopt=IkCHpk (7)
Where Ik=I−ekekHεCL×L and I is the identity matrix sized L×L. Let us now rewrite the expression of the equalized signal:
The new expression of the mean square-root error (MSE) is equal to:
Let us derive this expression relative to pk:
In setting
we obtain:
pkopt=R−1P (11)
In considering a perfect decorrelation between two symbols, we have:
Where V k=diag[σr2−σr2 , . . . , σx2, . . . , σx2, . . . , σx2−σx2]εCL×L (which is actually the matrix of covariance of x-x in which the kith element of the diagonal has been
is the variance of the estimated symbols. Furthermore we have:
(13)
Thus the final expression of the optimal vector is:
It is noted that a matrix inversion
sized is necessary for each equalized symbol.
Let us rewrite the expression of
p
k
opt=σr2(C[(σx2−σx2)I+σx2ekekH]CH+σn2I)−1Ckek (15)
Using the Sherman-Morrison-Woodbury identity (A+uvH)−1=
the last equation becomes:
{tilde over (p)}
k
opt=σx2(CCH(σx2−σx2)+σn2I)−1Cek (17)
Only a matrix inversion (always sized
is necessary for a block of L symbols. Now is {tilde over (p)}lopt simply proportional to pkopt. It is therefore possible to use the expression of {tilde over (p)}kopt. In matrix notation, we finally obtain.
Where the operator stores n ddding (A) coefficients of A in a matrix of equivalent size.
Number | Date | Country | Kind |
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0502319 | Mar 2005 | FR | national |
Filing Document | Filing Date | Country | Kind | 371c Date |
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PCT/EP06/50515 | 1/30/2006 | WO | 00 | 5/13/2008 |