The invention generally relates to digital communication and in particular to methods, systems, and computer program products for decoding a received data signal.
In recent years, wireless communication systems have grown at an accelerated pace, thereby becoming central components of modern modes of communications.
Different wireless communication systems are available today, such as the cellular and wireless ad-hoc networks accommodating single or multiple transmitters/receivers using single or multiple antennas, such as MIMO (Multiple INPUT Multiple OUTPUT) systems. A wireless MIMO communication system refers to radio links with multiple antennas at the transmitter side and at the receiver side.
The significant development of MIMO systems over scattering-rich wireless channels is due to their ability to meet the increasing needs in terms of communication reliability and data rate on wireless networks.
Many decoders have been proposed to retrieve signal streams sent over such wireless communication systems with an improved performance in terms of data rate and reliability. However, a major challenge of such decoders is the complexity cost. In order to warrant the deployment for real-time and high-throughput applications, it is desirable that the coding operations and decoding algorithms satisfy the prescribed computational complexity which is fixed for a given device and application.
For example, Maximum Likelihood (ML) decoders, such as the sphere decoder (E. Viterbo and E. Biglieri. A universal decoding algorithm for lattice codes. In Quatorzieme colloque GRETSI, 1993) or the Schnorr Euchner decoder (C. P. Schnorr and M. Euchner. Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. In Math. Programming, pages 181-191, 1993) are optimal decoders which require an exponential complexity in the number of antennas (B. Hassibi and H. Vikalo. On the sphere-decoding algorithm i. expected complexity. Signal processing, IEEE Transactions on, 53(8):2806-2818, August 2005).
These decoders have been adapted to reduce their complexity at a possible cost of performance degradation in terms of a tradeoff between complexity and performance, according to two different approaches. In a first approach, the decoder is based on a node pruning-based tree search algorithm which is configured to discard some nodes (and their children) in each layer if they are associated with a low likelihood function to lead to the optimal solution. Exemplary decoders operating according to this first approach comprise for example:
The second approach relates to iterative decoders. An iterative decoder is based on the equivalent channel matrix form induced from the code structure to determine partitioned signal sets to be decoded iteratively. Such iterative approach reduces the decoding complexity while not maintaining a desired error performance and full diversity.
For example, in the article “Won-Joon Choi, R. Negi, and J. M. Cioffi. Combined ML and DFE decoding for the v-blast system. In Communications, 2000. ICC 2000. 2000 IEEE International Conference on, volume 3, pages 1243-1248 vol. 3, 2000”, the channel matrix is divided into two blocks, the first block having a size q. An ML decoding scheme is performed on the first block of size q, while a decision feedback equalizer (also referred to as ZF-DFE) is applied to the remaining system given the output of the ML decoding performed on the first block (i.e. the ML output is subtracted from the received signal). Even if such solution increases the performance, the decoding is sub-optimal while not ensuring a target diversity order.
Another solution, proposed for space-time coded systems which are compatible with sphere decoders, consists in splitting the received signal into a number L (L≥2) of subsets each of a given cardinality λ. A conditional maximization of a likelihood function with respect to one of set of signal points given another is performed. This comprises:
The choice of the signal set to be detected (also referred to hereinafter as ‘decoded’) conditioned on the values of the remaining signal sub-sets has an impact on the performance of the algorithm. Inspired from the work disclosed in S. D. Howard, S. Sirianunpiboon, and A. R. Calderbank. Low Complexity Essentially Maximum Likelihood Decoding of Perfect Space-Time Block Codes. In Acoustics, Speech and Signal Processing, 2009. ICASSP 2009. IEEE International, a selection criterion for n×n space-time coded systems using the Perfect codes has been proposed in L. P. Natarajan and B. S. Rajan. An Adaptive Conditional Zero-Forcing Decoder with Full-diversity, Least Complexity and Essentially-ML Performance for STBCs. In Information Theory and its Applications (ISITA), 2012 International Symposium on, pages 235-239, October 2012, particularly for a 2×2 MIMO system using the Golden code, 3×3 and 4×4 configurations
Accordingly, the signal set selected in second step described above is the sub-set corresponding to the divided sub-matrix of the equivalent channel matrix of maximum determinant of covariance matrix overall divided sub-matrices. Moreover, sufficient conditions on the characteristics of the sub-matrices of the equivalent channel matrix involving characteristics of the used Space-Time Block Code have been disclosed in L. P. Natarajan and B. S. Rajan. An Adaptive Conditional Zero-Forcing Decoder with Full-diversity, Least Complexity and Essentially-ML Performance for STBCs. In Information Theory and its Applications (ISITA), 2012 International Symposium on, pages 235-239, October 2012. One of these sufficient conditions imposes that, in order to achieve a full diversity order under ACZF or ACZF-SIC decoding, at least one of the L sub-matrices should be full rank.
Although existing sub-detection methods offer better performance than sub-optimal linear and non-linear joint decoding schemes, they do not allow to control the diversity order while achieving a reduced complexity.
To address these and other problems, there is provided a decoder for decoding a signal received through a transmission channel in a communication system, the signal carrying information symbols selected from a given set of values and being associated with a signal vector, the transmission channel being represented by a channel matrix. The decoder comprises:
In one embodiment, the decoder may be configured to previously determine an orthogonal matrix Q and an upper triangular matrix R by performing a QR decomposition from the channel matrix, and the sub-block division unit is configured to divide the upper triangular matrix R into a number of upper triangular sub-matrices and a number of rectangular matrices, the number of upper triangular sub-matrices being superior or equal to two, the sub-block division unit being configured to divide the received signal vector into a set of sub-vectors such that each sub-vector of the received signal vector corresponds to one of the upper triangular sub-matrices.
The decoder may be then provided to determine the received signal vector by multiplying the transpose matrix of the orthogonal matrix with the received signal.
In particular, the decoder may be further arranged to determine a set of permuted matrices from the channel matrix by permuting rows or columns of the channel matrix, and to perform a QR decomposition of each permuted matrix to determine intermediary upper triangular matrices, the decoder being configured to select one of the permuted matrices from a criterion related to the components of the intermediary upper triangular matrix obtained for each permuted matrix, the sub-block division unit being configured to divide the upper triangular matrix R corresponding to the intermediary upper triangular matrix associated with the selected permuted matrix.
In certain embodiments, the decoding algorithm may be a lattice decoding algorithm.
Particularly, the decoding algorithm may be configured to solve a condition on the cumulated metric of each block with respect to a threshold.
The decoding algorithm may be a sequential decoding algorithm, and the threshold is the cumulated metric threshold of the sequential decoding algorithm.
Alternatively, the decoding algorithm may be a sphere constrained decoding algorithm, and the threshold is the initial radius of the sphere of the sphere constrained decoding algorithm.
In one embodiment, the threshold may be determined from a target quality of service indicator.
The target quality of service indicator may be the target diversity order.
In particular, the decoder may further comprise a threshold estimation unit configured to determine a threshold for the decoding of each sub-block.
The threshold estimation unit may be configured to determine the threshold from the target diversity order and from at least one further parameter chosen among a group consisting of an estimate of the signal to noise ratio, the dimension of the received signal, and the dimension of said sub-block.
The threshold estimation unit may comprise a lookup table storing a value of the threshold for each tuple of values comprising the value of the target diversity order and the at least one further parameter.
In one embodiment, the number of sub-blocks may be equal to two and the threshold values of the look-up table may be predetermined from the probability that a selected path corresponding to the first sub-block be not visited during a tree search implemented by the decoding algorithm applied for the first sub-vector to determine the candidate estimates for the first sub-block and the signal-to-noise ratio.
The threshold estimation unit may be configured to update the lookup table depending on statistical data related to the decoding of at least one other signal. In certain embodiments, each set of candidate estimates may be a data structure ordered by increasing value of the cumulated metric obtained for each estimate.
The candidate set estimation unit may be further configured to further reduce the number of candidates determined in the current set of candidate estimates for the at least one sub-block depending on a target number of candidate estimates associated with the sub-block.
The target number of candidate estimates associated with the sub-block may be determined from the number of candidate estimates of the previously processed sub-block.
The target number of candidate estimates associated with the sub-block may be a multiplicative function of the number of candidate estimates of the previously processed sub-block, the multiplicative function having a slope coefficient inferior to one.
The candidate set estimation unit may further comprise determining one estimate for the last processed sub-block by applying an optimal or sub-optimal decoding algorithm selected according to a predefined criteria.
The optimal or sub-optimal decoding algorithm may be chosen among a group consisting of a ML decoding algorithm (optimal decoding algorithm), a ZF-DFE decoding algorithm, and a MMSE decoding algorithm (the ZF-DFE decoding algorithm and the MMSE decoding algorithms being sub-optimal decoding algorithms). The ML decoding algorithm may be in particular any optimal lattice decoding algorithm).
In certain embodiments, the signal estimation unit may be configured to determine the tuple of estimates that minimizes the global metric, each tuple estimate comprising one candidate estimate from each one of the sets of candidate estimates obtained for the sub-blocks of information symbols.
In one application of the invention, the communication system may be a coded system using a space-time block code to encode the data signals transmitted over the transmission channel, the decoder being configured to vectorize the signal vector using an equivalent channel matrix, and divide the equivalent channel matrix into a number of rectangular equivalent channel sub-matrices, each rectangular sub-matrix being a function of the Linear Dispersion Matrix representing the signal sent over the transmission channel corresponding to the received signal, the sub-block division unit comprising dividing the received signal vector into a set of sub-vectors in correspondence with the division of the equivalent channel matrix.
The decoder may be configured to reorder the rectangular sub-matrices depending on the value of the determinant of the product of the Hermitian transposition of each equivalent channel sub-matrix with the equivalent channel sub-matrix.
The invention also provides a receiver for receiving and decoding an encoded signal, the receiver comprises a decoder according to any of the preceding claim for decoding the signal.
A mobile device capable of transmitting and receiving data in a wireless communication network, the mobile device comprises such receiver for receiving a signal is also provided.
There is also provided a method of decoding a signal received through a transmission channel in a communication system, the signal carrying information symbols selected from a given set of values and being associated with a signal vector, the transmission channel being represented by a channel matrix, the method comprising:
recursively determining candidate estimates of sub-blocks of the transmitted signal corresponding to the sub-vectors, each estimate of a given sub-block being determined from at least one candidate estimate of the previously processed sub-blocks,
The method may further comprise:
There is also provided a computer program product for decoding a signal received through a transmission channel in a communication system, the signal carrying information symbols selected from a given set of values and being associated with a signal vector, the transmission channel being represented by a channel matrix, the computer program product comprising:
The various embodiments of the invention make it possible to avoid the conventional exhaustive search, while offering a flexibility on desired diversity order and a reduced complexity.
Further advantages of the present invention will become clear to the skilled person upon examination of the drawings and detailed description. It is intended that any additional advantages be incorporated herein.
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate various embodiments of the invention and, together with the general description of the invention given above, and the detailed description of the embodiments given below, serve to explain the embodiments of the invention.
Embodiments of the invention provide a sub-block decoding method for decoding a received signal according to a semi-exhaustive and recursive approach. The sub-block decoding method and device according to the embodiments of the invention divide the information symbols contained in a received signal y in accordance with the division of a matrix related to the channel matrix Hc which provides N groups of information symbols s(k) (also referred to hereinafter as “blocks of information symbols”), k ranging from N to 1. The signal carries information symbols selected from a given set of values. The set of values may be a finite and discrete set of values such as an alphabet, or alternatively an infinite set of values such as infinite lattices Zn of dimension n.
In one embodiment, the division of the information symbols contained in the received signal y is made in correspondence with a division of the upper triangular matrix obtained from a QR decomposition of the channel matrix H into
of sub-blocks, N being at least equal to two. The following description of certain embodiments will be made with reference to a generation of sub-blocks of information symbols from a division of the upper triangular matrix R, for illustration purposes.
The sub-block decoding method recursively decodes each sub-block of information symbols s(k), the decoding of at least one sub-block of information symbols s(k) comprising applying at least one iteration of a decoding algorithm Dk using the sets of candidate estimates Γk+1, . . . , ΓN determined for the previously processed sub-blocks s(k+1), . . . , s(N) to determine a set of candidate estimate Γk for the current sub-block of information symbols s(k), the number of candidate estimates k of the current candidate set Γk being strictly inferior to the cardinal of the set of values from which are selected the information symbols (for example cardinal of the Alphabet). Each candidate estimate stored in a candidate set Γk represents a vector of information symbols. The following description of certain embodiments will be made with reference to information symbols selected from a finite and integer set of values, such as an alphabet.
In particular, the decoding algorithm Dk may be a lattice decoding algorithm. In one embodiment, the lattice decoding algorithm may be applied to solve a threshold condition on the cumulated metric m(s(k)) corresponding to the sub-block s(k), each iteration of the lattice decoding algorithm corresponding to a candidate estimate.
In one embodiment, the threshold condition may be related to a threshold Rth
The estimate of the signal ŝ may be then constructed from the estimates stored in the candidate sets Γ1, . . . , ΓN, for example by selecting the tuple of values that minimizes the global metric Σk=1Nm(s(k)), each value of the tuple corresponding to a value of one candidate set Γi.
By convention, in the following description of certain embodiments of the invention, the index of the first processed sub-block will be referred to as k=N and the index of the last processed sub-block will be will be referred to as k=1. According to this convention, the sub-block of information symbols s(k) are thus recursively processed from k=N to k=1 with k being decrementing for the processing of the next sub-block s(k−1) of information symbols.
In one embodiment of the invention, the decoding of the last sub-block of information s(1) may comprise determining only one estimate for the candidate set Γ1 by applying a sub-optimal decoding algorithm D1 depending on a predefined criterion (for example ML decoding criterion), such as for example a ZF-DFE decoding algorithm.
This provides a semi-exhaustive recursive block decoding of a received signal implementable in different types of systems, such as in wireless or optical communication systems, signal processing systems, cryptographic systems, and positioning systems, etc.
In one application of the invention to wireless communication systems, the communication system may comprise at least a transmitter for transmitting simultaneously a plurality of information symbols through a communication channel, and at least a receiver for receiving one or more of the symbols transmitted by the transmitter in the form of independent signals. The communication channel may be any linear AWGN (Additive White Gaussian Noise) channel or a multipath channel using single carrier or multi-carrier modulation types such as OFDM (Orthogonal Frequency-Division Multiplexing).
The MIMO system may present a centralized configuration where the transmit antennas are collocated at a same user. Alternatively, the MIMO system may be a distributed MIMO system (or multi-user MIMO) where the transmit antennas are distributed in the communication network and are located at different users. Such multi-user MIMO configurations may be used for example in mobile networks in the uplink communications applied for example in cellular 3G, 4G and LTE standards or in cooperative communications applied for example in ad-hoc networks (wireless sensor networks, machine-to-machine communications, internet of things . . . ). In such multi-user configurations, the communication system may further use, alone or in combination, any multiple access technique such as Time Division Multiple Access (TDMA), Frequency Division Multiple Access (FDMA), Code Division Multiple Access (CDMA), and Space-Division Multiple Access (SDMA).
The communication system may be alternatively a single-antenna multicarrier communication system using multicarrier communication techniques to combat frequency-selective channels and manage interference and delays such as OFDM modulations adopted in wireless standards like IEEE 802.11 (WiFi) and Filter Bank Multi-Carrier (FBMC) modulations.
In other applications of the invention, the communication system may be an optical fiber-based communication system such as a Polarization Division Multiplexing-OFDM (PDM-OFDM) system used for example as a telecommunication medium in access networks, metropolitan networks, or in computer networks in order to generate, the optical communication channel output admitting a linear (lattice) representation. In such embodiments, the information symbols conveyed by an optical transmitter device may be carried by optical signals polarized according to the different polarization states of the fiber. The optical signals may propagate along the fiber-based transmission channel according to one or more propagation modes until reaching the optical receiver device.
In some embodiments corresponding to optical communications, the optical signal carrying the information symbols may be generated using a single wavelength lasers.
In other embodiments, wavelength division multiplexing (WDM) techniques may be used at the optical transmitter devices to enable generating optical signals using a plurality of independent wavelengths.
In another application of the invention to optical communications using in particular multi-mode fibers, space division multiplexing techniques may be further used to multiplex the information symbols according to the various propagation modes.
Further, a multiple access technique such as WDMA (Wavelength Division Multiple Access) may be used in applications to optical communication systems.
The wireless network environment may comprise a plurality of wireless or optical devices capable of operating in a wireless or optical environment, such as for example base stations, user equipment, terminals, each wireless or optical device including a transmitter and/or a receiver including one or more antennas, each wireless or optical device communicating with other wireless or optical devices through a wireless or optical connection.
When applied to MIMO decoding, for a single user or multiple users detection, the dimension of the received signal or channel output depends on the dimension of the signal space at the transmitter, on the number (nt) of the Transmit (Tx) antennas and/or on the number (nr) of Receive (Rx) antennas.
Referring to
The MIMO configuration may be symmetric, in which case it includes a same number (nt) of Transmit antennas as the number (nr) of Receive antennas. Alternatively, the MIMO configuration may be asymmetric, in which case the number (nt) of Transmit antennas is different from the number (nr) of Receive antennas (in particular the number nr, at the receive side, is higher than nt, at the transmit side to avoid a rank deficiency).
The transmitter 2 can transmit a signal to a receiver 3 by means of a noisy MIMO channel. The data transmitter 2 can in particular be integrated in the base stations. The transmitter 2 may comprise for example:
The transmitter 2 codes a binary signal received as input using a convolutional code provided by the channel coder 101. The signal may be modulated by the modulator 102 according to a modulation scheme (for example, a quadrature amplitude modulation nQAM). The modulator 102 can also implement a modulation scheme generating complex symbols sc, each complex symbol belonging to a group of symbols si. The modulated symbols thus obtained may be then coded by the space-time coder 104 to form a code word STBC, such as the Golden Code (“The Golden Code: A 2×2 Full-Rate Space-Time Code with Non-Vanishing Determinants”, J.-C. Belfiore, G. Rekaya, E. Viterbo, IEEE Transactions on Information Theory, vol. 51, no. 4, pages 1432-1436, April 2005). The STBC code may be based on a complex matrix of dimension nt*T, in which nt designates the number of transmission antennas and T is the time length of the STBC code, or on a spatial multiplexing (the modulated symbols are directly sent to the transmission antennas).
The code word thus generated is converted from the time domain to the frequency domain (noted X in
The receiver 3 can be also integrated in the base stations. The receiver 3 may be configured to receive a signal Yc transmitted by the transmitter 2 in a wireless channel. The channel may be noisy (for example channel with Additive White Gaussian Noise (AWGN) subjected to fading). The signal transmitted by the transmitter 2 may be further affected by echoes due to the multiple paths and/or the Doppler effect due to the transmitter and receiver having a non-zero relative velocity.
In one exemplary embodiment, the receiver 3 may comprise:
It should be noted that the receiver 3 implements a reverse processing of the processing implemented in transmission. Accordingly, if a single-carrier modulation is implemented in transmission instead of a multi-carrier modulation, the nr OFDM demodulators are replaced by corresponding single-carrier demodulators.
The skilled person will readily understand that the various embodiments of the invention are not limited to specific applications. Exemplary applications of this new decoder include, with no limitation, multi-user communication systems, MIMO decoding in configurations implementable in wireless standards such as the WiFi (IEEE 802.11n), the cellular WiMax (IEEE 802.16e), the cooperative WiMax (IEEE 802.16j), the Long Term Evolution (LTE), the LTE-advanced, the 5G ongoing standardization, and optical communications.
Further the semi-exhaustive decoding method and device according to the various embodiments of the invention may be applied to both coded and uncoded systems. A coded communication system uses a space-time block code (STBC) to encode the digital data sequence at the transmitter side, the signal sent over the transmission channel comprising a set of independent symbols, the space-time block code being represented by a generator matrix G. In an uncoded communication system, the generator matrix is equal to the Identity matrix.
In one application of the invention to a Rayleigh fading wireless multi-antenna system to decode a signal received by the multi-antenna system (MIMO), with nt transmit and nr receive antennas using spatial multiplexing, the data signal yc received as a complex-valued vector, according to a complex-valued representation of the channel output is given by:
yc=Hcsc+wc (1)
In Equation (1), Hc∈n
In equation (2), (.) and (.) denote respectively the real and imaginary parts of a complex-valued input (vector or matrix).
The equivalent channel output can then be written as:
y=Hs+w (3)
In embodiments where a length-T Space-Time code is used, the channel output can be written in the same form of equation (3) with the equivalent channel matrix Heq given by:
Heq=HcΦ (4)
In equation (4), Φ ∈ n
According to the equivalent system obtained in (3), the received signal can be viewed as a point of the lattice generated by H and perturbed by the noise vector w.
When optimal detection is required, the receiver implements an ML decoder that attempts to determine, given the channel output and the channel matrix, an estimate ŝ of the originally transmitted symbols vector from the given data in H and y, according to the minimization of the error probability such that:
ŝ=argmiPr(ŝ≠s) (5)
In Equation (5), the finite subset represents the alphabet to which belong the real and imaginary parts of the information symbols. For example, using an 2M-ary QAM constellation to construct the complex information symbols, the alphabet is the integer sub-set given by =[−(M−1), (M−1)] ( may represent for example a M-ary QAM constellation to which belongs the complex information symbol). The minimization of the error probability under ML detection is equivalent to the minimization problem given by:
{circumflex over (s)}=argmi∥y−Hs∥2 (6)
Assuming coherent system where H is perfectly known or estimated at the receiver using estimation techniques such as least square estimators, optimal ML detection reduces to solve a closest vector problem in the n-dimensional lattice generated by H to seek the nearest lattice point to the equivalent received signal y according to the minimization problem of Equation 6.
Thus the ML detector (equivalently ML decoder) chooses the symbols vector s which yields the smallest Euclidean distance between the received vector y, and hypothesized message Hs. The ML detector represents a discrete optimization problem over candidate vectors s within the chosen alphabet. In the case of high constellations size and high dimension of the system (number of antennas), the search for the ML solution in an exhaustive way generally requires a very high complexity.
Sequential decoders implementing a tree search strategy searches the closest lattice point to the received vector using a decoding tree structure. Before transmission of the signal to such a sequential decoder, a predecoding may be performed using a QR decomposition of the channel matrix such that H=QR where Q designates an orthogonal matrix and R designates an upper triangular matrix. Given the orthogonality of Q, equation (3) can be rewritten in the following form:
{tilde over (y)}=Qty=Rs+Qtw (7),
By denoting {tilde over (w)}=Qtw, equation (7) can be rewritten:
{tilde over (y)}=Rs+{tilde over (w)} (8)
The ML decoding problem then amounts to solving the equivalent system given by:
{circumflex over (s)}=argmi∥{tilde over (y)}−Rs∥2 (9)
The triangular structure of R thus reduces the search of the closest point to a sequential tree-search. Nodes in the tree represent the different possible values of the symbols si.
The decoder 110 may comprise a complex-to-real converter 201 to transform the received signal yc into a real-valued representation, and a QR decomposition unit 206 for performing a QR decomposition of the channel matrix such that H=QR where Q designates the orthogonal matrix and R designates the upper triangular matrix. The decoder 110 may further comprise a modifier unit 209 to rewrite the received signal vector y into an equivalent received signal vector {tilde over (y)}=Rs+Qtw and a sub-block detector (also referred to as sub-block decoder) 210 configured to recursively detect in blocks the received signal from a division of the equivalent received signal vector {tilde over (y)} performed from a corresponding division of the upper triangular matrix R. In certain embodiments, the decoder 110 may also comprise a channel matrix permutation unit 204 to permute columns or lines of the channel matrix H prior to the QR decomposition.
The Space-Time decoder 110 may further comprise a real-to-complex convertor 211 configured to deliver an estimate of the complex-valued transmitted signal by converting the real-valued vector ŝ into a complex-valued vector ŝc. The conversion operation is the invert of the processing performed at the complex-to-real converter 201.
Referring to
The sub-block decoder 210 may comprise a sub-blocks decomposition unit 301 (also referred to as a “sub-block division unit”) configured to:
In particular, the upper triangular matrix R may be divided into N upper triangular sub-matrices R(k), k=1, . . . , N and
rectangular sub-matrices B(kj), k=1, . . . , N; j=k+1, . . . , N.
Accordingly, the vector {tilde over (y)} is divided into N sub-vectors {tilde over (y)}(k), k=1, . . . , N of lengths lk such that
and Σk=1Nlk=n. The same vector division is applied to the vector of symbols s to obtain N sub-vectors s(k), k=1, . . . , N of lengths lk such that
The upper triangular matrix R is thus divided into
matrices composed of N upper triangular sub-matrices R(k), k=1, . . . , N and
rectangular sub-matrices B(jk), k=1, . . . , N; j=k+1, . . . , N such that:
Each upper triangular sub-matrix R(k), k=1, . . . , N represents a square matrix of dimension lk×lk and corresponds to the sub-vector {tilde over (y)}(k), k=1, . . . , N. Further, each sub-matrix B(jk), k=1, . . . , N; j=k+1, . . . , N represents a rectangular matrix of dimension lk×lj and corresponds to a feedback matrix from block j to block k.
Accordingly, (8) can be rewritten as:
The ML decoding problem of equation (9) can thus be rewritten as:
ŝ=∥{tilde over (y)}−Rs∥2=∥Σk=1N{tilde over (y)}(k)−(R(k)s(k)+Σj=k+1NB(kj)s(j))∥2 (12)
For example, considering a number of blocks N=2, the R matrix can be divided as follows:
In the above example of a two block division of the upper triangular matrix R:
Accordingly, the corresponding symbol vectors are s(2) and s(1) of size p and n−p respectively:
Equation (12) applied to such exemplary two-block sub-decoding can be written:
This problem may be solved using the following approximation:
ŝ=(∥{tilde over (y)}(1)−R(1)s(1)−Bs(2)∥2+∥{tilde over (y)}(2)−R(2)s(2)∥2) (15)
It should be noted that the division corresponding to equation (15) may generate sub-optimalities in the decoding results such that the resolution of equation (15) may not correspond to the ML global solution.
The sub-blocks detector 210 may further comprise at least one candidate set estimation unit 305 for determining the set of candidate estimates Γk for each block s(k) and a symbol estimation unit 306 for determining an estimate ŝ of the equivalent transmitted signal from the data sets Γ1, . . . , ΓN.
Referring now to
In step 401, the channel matrix Hc and the received signal Yc may be converted into real-valued matrix H and real-valued received signal y.
In step 402, the QR decomposition of the channel matrix Hc is performed to determine the orthogonal matrix Q and upper triangular matrix R. In certain embodiments, the channel matrix Hc may be permuted prior to performing the QR decomposition using any permutation technique comprising a multiplication of the channel matrix Hc with a permutation matrix. Alternatively, the upper triangular matrix R obtained from the QR decomposition may be sorted according to other ordering methods.
In step 403, the equivalent received signal {tilde over (y)} is determined by multiplying the received real-valued signal y by the transpose matrix Qt.
In step 404, the matrix R is divided into N upper triangular matrices R(k), k=1, . . . , N and
rectangular matrices B(kj), k=1, . . . , N; j=k, . . . , N and the equivalent signal vector {tilde over (y)} is divided into N sub-vectors {tilde over (y)}(k). The same vector division is applied to the vector of symbols s to obtain N sub-vectors s(k), k=1, . . . , N of lengths lk such that
The decoding method proceeds a number of iterations of the steps 407 to 408 depending on the number of sub-blocks s(k) starting from k=N, N corresponding to the number of sub-blocks (step 405) for determining each candidate set Γk associated with each sub-block s(k).
More specifically, in step 407, if K#N, the previously estimated candidate sets ΓN, . . . , Γk+1 are retrieved from memory.
In step 408, the candidate estimates for Γk are then determined using a number of iterations of a given decoding algorithm Dk solving a condition related to the cumulated metric m(s(k))) for the k-th block s(k) the condition further depending on the previously estimated candidate sets ΓN, . . . , Γk+1 if k#N. The candidate set Γk thus comprises Ak values {ŝ1(k), ŝ2(k) . . . , ŝA
For example, the candidate estimates may be determined using a set of iterations of a lattice decoding algorithm Dk enumerating the ML solution (corresponding to the point minimizing the cumulated metric m(s(k))), and a set of neighbors satisfying a selection criterion Ck, which provides the remaining candidate estimates. Each candidate set Γk may be a list ordered by increasing value of the cumulated metric the ML solution corresponds to ŝ1(k), and the neighbors correspond to {ŝ2(k) . . . , ŝl
As used herein, the cumulated metric m(s(k)) for a k-th block s(k) is defined as:
m(s(k))=∥ŷ(k)−(R(k)s(k)+Σj=k+1NB(kj)ŝ(j))∥2 (16)
The terms B(kj)ŝ(j) correspond to inter-symbol interference, the terms B(kj)ŝ(j) for j=k+1 to N being known from previous estimations of Γ1, . . . , ΓN if k#N.
By setting
m(s(k))=∥
In certain embodiments, the lattice decoding algorithm is configured to provide a solution to the following condition related to the cumulated metric, at each iteration:
m(s(k))≤Rthk (18)
In condition (18), Rthk designates a threshold defined for each block s(k).
A new iteration of steps 407 and 408 may be performed for k=k−1 if k#1, to determine candidate estimates for Γk−1 similarly.
If k=1, one or more candidate estimates may be determined for the last candidate set Γ1 using a selected decoding algorithm D1 using the previously estimated candidate sets ΓN, . . . , Γ2. In one embodiment, only one candidate estimate may be determined for the last candidate set Γ1 using a selected decoding algorithm D1 such as an ML or ZF-DFE or MMSE and a selection criterion C1. The candidate set Γ1 thus comprises one value {ŝ1(1)}. The selection criterion C1 may be related to the metric minimization and may be used to select the decoding algorithm D1. For example, if the selection criterion C1 consists in selecting the point that minimizes the metric, an ML decoding algorithm may be applied. Otherwise, if the selection criterion C1 consists in selecting a neighbor point of the ML point, a sub-optimal decoding such as ZF-DFE or MMSE may be applied at this late iteration.
In certain embodiments, the selection criterion C1 may depend on the zero-structure of the upper triangular matrix R or the orthogonality of the upper triangular matrix R. For example, if the upper triangular matrix R is orthogonal, a ZF (Zero-Forcing) decoding is sufficient to generate the ML solution.
In step 410, an estimation ŝ of equivalent transmitted signal is constructed from the candidate sets ΓN, . . . , Γ2, Γ1. In one embodiment, an estimation ŝ is constructed by determining the tuple {ŝi
In step 411, the real-valued vector ŝ may be converted into a complex-valued vector ŝc to deliver an estimate of the complex-valued transmitted signal.
The lattice decoding algorithm Dk used to estimate each candidate set Γk in step 408 may be any type of lattice decoding algorithm capable of solving the condition (10) such as:
Particularly, if the decoding algorithm is based on a sphere constrained decoding algorithm, the threshold Rthk corresponds to the initial radius of the sphere. If the decoding algorithm is a stack algorithm, the threshold Rthk may correspond to the limiting threshold of cumulated weight associated to nodes stored in the stack, or to the size of the second stack used to store the number of candidate points if the stack decoder uses a second stack. If the lattice decoding algorithm is an SB-stack decoder, the threshold Rthk may alternatively correspond to the sphere radius.
For example, for N=2, the decoding method may comprise only one iteration of step 408 using one or more iterations of a sphere constrained decoding algorithm, the threshold Rth
In one exemplary application of the invention to a division of the received signal into two blocks s(2) and s(1), of length p and n−p respectively, step 408 may comprise applying a stack decoding algorithm D2 to determine the candidate set Γ2. The stack decoder originally generates the ML solution of the ML decoding given the system y2=R(2)s(2)+w(2) and stores the first point in a second stack to store candidate lattice points. The stack decoding algorithm proceeds recursively with the search for its (A2−1) metric-wise neighbors and save them in the second stack. The second stack thus provides the potential candidates for the candidate set Γ2. The stack decoding algorithm is described for example in A. Salah, G. Othman, R. Ouertani, and S. Guillouard. New soft stack decoder for mimo channel. In Signals, Systems and Computers, 2008 42nd Asilomar Conference on, pages 1754-1758, October 2008.
In another exemplary application of the invention to a division of the received signal into two blocks s(2) and s(1), of length p and n−p respectively, step 408 may comprise applying a SB-stack decoding algorithm D2 to determine the candidate set Γ2. The SB-stack decoder is a reduced-complexity version of stack decoder. The threshold Rth
In equation 19, vol(Λ)=det(R2) and Vp designates the volume of a unit radius sphere in the real space p,
This radius guarantees to find Np lattice points only for high values of Np.
Alternatively, by using an effective number Ne of points inside the sphere derived based on the shape of the used constellation and a multiplicative constellation-dependent factor (for example, α4QAM=3/2, α16QAM=4 and α64QAM=3/8), the effective number of lattice points in the list is related to the radius of the sphere as follows:
In equation 20, β designates an additional factor ensuring we find the needed Np lattice points inside the sphere.
In still another exemplary application of the invention to a division of the received signal into two blocks s(2) and s(1), of length p and n−p respectively, step 408 may comprise applying a sphere constrained decoding algorithm D2 to determine the candidate set Γ2. To get the A2 lattice points, the algorithm, centered on the received vector runs first time to find the ML vector within the sphere radius r. Then, centered on this solution, the algorithm restarts to find its A2−1 neighbors.
Alternatively, step 408 may comprise further reducing the number of candidates to be maintained in each candidate set Γk based on the empirical and or statistical data. In embodiments where each candidate set Γk is ordered by increasing value of the cumulated metric, such statistical data may be determined from the occurrence of each candidate value rank over previous decoding of data streams.
Alternatively, the threshold per block Rth
In another embodiment, step 408 may comprise for each sub-block s(k):
In certain embodiments, the threshold Rthk used at step 408 to enumerate candidate points of the candidate set Γk may be a function of a target quality of service indicator Qtarget (also referred to hereinafter as “QoS indicator”). In particular, the target quality of service indicator Qtarget may be the target diversity order that is to be reached dtarget. By limiting the number of candidate points to enumerate at each iteration of step 408 by a threshold Rthk depending on a target quality of service indicator Qtarget, this ensures that the target quality of service indicator is effectively reached or approached. The threshold Rthk may be a threshold defined per block k or a threshold defined for a set of blocks or for all the blocks s(k).
As used herein, the expression “target quality of service indicator” refers to a parameter related to the target QoS that is to be achieved, such as the target diversity order, the target complexity of the decoder, the target error probability, etc. The following description of certain embodiments of the invention will refer mainly to a target QoS indicator represented by the target diversity order, for illustration purpose only.
With reference to
The SNR may be estimated by an SNR estimation unit 504 provided in the candidate set estimation unit 305 or more generally in the decoder 110. The threshold estimation unit 500 may further comprise a look-up table manager 501 for updating the look-up table based on statistical data collected during the decoding of a previous data stream.
In embodiments where the decoder 110 receives a target QoS indicator Qtarget corresponding to the target diversity order dtarget and applies a sub-block decoding of the received signal into a number N=2 of sub-blocks, it has been found the threshold values of the look-up table may be predetermined from the probability that a selected path corresponding to the first sub-block be not visited during a tree search implemented by the decoding algorithm applied for the first sub-vector to determine the candidate estimates for the first sub-block and the signal-to-noise ratio, and in particular by solving the following equation:
In equation 21, σ2 designates the noise variance, p designates the size of the second block, ρ the SNR and r is an unknown variable corresponding to the threshold Rthk to be determined. Γ(x, y) designates the Gamma Function
with f being the probability density function of χ2(p) (chi-squared distribution) and
is the normalized upper Gamma function).
The first term
corresponds to the probability that a selected path corresponding to the first sub-block be not visited during a tree search implemented by the decoding algorithm applied for said first sub-vector to determine the candidate estimates for the first sub-block. Equation 21 is equivalent to determining threshold values such that the probability that the normalized noise
related to the first sub-block is larger than a normalized threshold
the normalized noise
being distributed according to the χ2(p).
For example, if the two blocks are respectively of length p and n−p (N=2), and an ML decoding algorithm D1 for the processing of block s(1), each Rth
In equation 22, c1 and c2 designate positive value constants, σ2 designates the noise variance, p designates the size of bloc, ρ the SNR and r is an unknown variable corresponding to the threshold Rthk to be determined. Γ(x, y) designates the Gamma Function.
The first term of equation 22
corresponds to the calculated Frame Error Probability (probability of having error per transmitted frame) when applying the ML decoding for the processing of block s(1) while the second term of equation 22
corresponds to target Frame Error Probability.
The first term of equation 22 from the probability of visiting a path that is outside the sphere of radius r using the ML decoding:
with w(1) designating the noise corresponding to block s(1).
In the first term of equation 22, the term that controls the achievable diversity order dtarget with the applied decoding scheme is
while the term c1ρ−n+c2ρ−n indicates a full diversity. Accordingly, each Rth
In another example, if the decoder 110 applies a sub-optimal decoding of the received signal into two blocks s(2) and s(1) respectively of length p and n−p (N=2), using a ZF-DFE decoding algorithm D1 for the processing of block s(1), each Rth
In equation 24, Ne designates the average number of the nearest neighbors in the constellation, using a nearest neighbor union bound, σ2 designates the noise variance, p designates the size of block s(2), ρ the SNR and r is an unknown variable corresponding to the threshold Rthk to be determined. Γ(x, y) designates the Gamma Function. Further ∈ is given by the following formula
with dmin representing the minimal distance of the constellation. The first term of equation 24
has been obtained from the fact that the probability Pr(Ep+1) of decoding the wrong symbol sp+1 satisfies Pr(Ep+1)≤Ne∈.
In the first term of equation 24, the term that controls the achievable diversity order dtarget with the applied decoding scheme is
while the additive term
represents the degradation of performance caused by the use of ZF-DFE in the second decoding stage corresponding to the processing of block s(2). Accordingly, each Rth
In some embodiments, the decoder 110 may receive a target QoS indicator Qtarget corresponding to the target diversity order dtarget and apply a sub-block decoding of the received signal into a number N≥2 of sub-blocks of sizes p1, . . . , pN. In such embodiments, the threshold values of the look-up table may be predetermined from an analysis of the error probability, the analysis enabling the derivation of a threshold value ri,th in correspondence with each sub-block s(i) of length p1.
The analysis of the error probability may be derived considering equal or different sub-block sizes. Moreover, the error probability may depend on the decoding algorithm used in the last stage. For example, when an ML decoder is implemented for the processing of block s(1), the frame error probability may be upper bounded according to:
In equation (26):
Accordingly, each value ri,th for i=1, . . . , N of the look-up table may be predetermined according to the satisfaction of the following inequality:
In inequality (27), δ designates a positive constant that enables to control the signal-to-noise ratio gain. Inequality (27) may be solved numerically in simulations with a margin of error as small as possible.
Given the determined thresholds, the decoder 110 may be configured to perform sub-block decoding of the N≥2 sub-blocks of sizes p1, . . . , pN according to the following steps:
The recursive sub-block decoding method and device according to the embodiments of the invention thus allows to efficiently control the target diversity order while ensuring a reduced complexity. The complexity reduction can be optimized depending on the number of candidate estimates selected in the candidate set Γk associated with each sub-block s(k), the type of the lattice decoding algorithm Dk selected for the decoding of each sub-block s(k), and/or the order according to which the sub-blocks s(k) are processed.
The diagrams of
Although the invention has been described in relation with a division of the received signal derived from a division of the upper triangular matrix R derived from the real channel matrix, the sub-block semi-exhaustive decoding method may be performed alternatively from the complex vectorized channel output. Accordingly, the division into sub-blocks may be performed from the complex channel matrix instead of using the real channel matrix.
Further, while the invention has been described in relation with certain examples of division into two blocks, and certain examples of the upper triangular matrix R, the invention is not limited to a particular number of blocks or configuration of the upper triangular matrix.
More generally, even if the semi-exhaustive sub-decoding method and device of the invention have particular advantages when the division of the received signal is derived from a division of the upper triangular matrix R, semi-exhaustive sub-decoding method and device may use alternatively another matrix related to the channel matrix H to derive the division of the received signal therefrom. For example, in one application of the invention to a coded system using a space-time block code (STBC) to encode the digital data sequence at the transmitter side, an encoded signal is sent over the transmission channel and comprises q sets of independent symbols. The signal sent over the transmission channel is denoted by a codeword matrix X, with X ∈ n
X=Σi−1q((si)A2i-1ℑ(si)A2i) (28)
In Equation (26), (si) and ℑ(si) correspond respectively to the real and imaginary parts of the si complex information symbols and matrices Al represent the linear dispersion matrices (also referred to as LD matrices).
The received signal at the receiver side can be written as:
Y=HX+W (29)
The received signal may be vectorized as follows:
vec(Y)=Heqs+vec(W) (30)
The operator vec(.) is defined as the operator that stacks the m columns of an n×m complex-valued matrix into an mn complex column vector.
In equation (28), Heq designates the equivalent channel matrix of dimension nr×q given by:
Heq=(ITH)G (31)
The vector s is obtained by vectorizing the codeword matrix X according to:
vec(X)=Gs, with G representing a generator matrix of the used linear STBC.
Heq can be rewritten as a function of the LD matrices as follows:
Heq=[vec(A1H)|vec(A2H)| . . . |vec(A2kH)] (31)
The vectorized complex system can be rewritten as:
y=Heqs+w (32)
In such coded embodiment of the invention, the sub-block decomposition unit 301 may be configured to divide the equivalent received vector y into N sub-vectors y(k), and to decompose the equivalent channel matrix Heq in correspondence with the division of the received vector. The equivalent channel matrix Heq is divided into N rectangular sub-matrices Heq(k), with k=1, . . . , N, each sub-matrix Heq(k) representing a rectangular matrix of dimension nr×lk and composed of lk column vectors of the equivalent channel matrix Heq:
Heq=[Heq(1)| . . . |Heq(k)| . . . |Heq(N)]
The received signal vector y is divided into N sub-vectors y(k), k=1, . . . , N of lengths lk such that
and Σk=1Nlk=n. The same vector division is applied to the vector of symbols s to obtain N sub-vectors s(k), k=1, . . . , N of lengths lk such that
In some embodiments, the equivalent channel sub-matrices may be reordered depending on the determinant of the product (Heq(k))H.Heq(k) (Superscripts “.H” denotes Hermitian transposition):det((Heq(k))H.Heq(k)), such that the sub-vector s(k) that corresponds to the sub-matrix of maximum determinant of the product (Heq(k))H.Heq(k) is placed in the first position (k=N). The reordered equivalent channel sub-matrices are denoted: [Heq(1′)| . . . |Heq(k′)| . . . |Heq(N′)]. The same reordering is applied to the vector of symbols s in order to obtain an equivalent system.
Accordingly, equation (32) can be rewritten as:
In equation (33), s(N) and thus Heq(N′) correspond to the sub-matrix having the maximum determinant of the product (Heq(k))H.Heq(k).
The candidate set estimation unit 305 may then determine a set of candidate estimates Γk for each block s(k) as described above, the candidate sets Γ1, . . . , ΓN being used to determine an estimates of the transmitted signal, according to step 405 to 410 of
The non-volatile ROM memory 62 may include for example:
The algorithms for implementing the method according to this embodiment of the invention can be stored in the program 620. The CPU processor 41 may be configured to download the program 620 to the RAM memory and runs the corresponding instructions. Specifically, the CPU comprises instructions that, when executed by the CPU, cause the CPU to:
The CPU is caused to determine a set of candidate estimates Γk for at least one sub-block s(k) of the transmitted signal by applying at least one iteration of a decoding algorithm Dk using the estimates Γk+1, . . . , ΓN determined for the previously processed sub-blocks, the number of candidate estimates determined for said sub-block being strictly inferior to the cardinal of the set of values from which the information symbols are selected. The CPU is further caused to calculate an estimate of the transmitted signal from the candidate estimates Γ1, . . . , ΓN determined for the sub-blocks.
The RAM memory 63 may include:
More generally, the decoding techniques described herein may be implemented by various means. For example, these techniques may be implemented in hardware, software, or a combination thereof. For a hardware implementation, the processing elements of decoder can be implemented for example according to a hardware-only configuration (for example, in one or more FPGA, ASIC or VLSI integrated circuits with the corresponding memory) or according to a configuration using both VLSI and DSP.
While the invention has been described in relation with a wireless communication system, it should be noted that the invention is not limited to such applications. For example, the decoding device and method may be integrated in a signal processing apparatus, for example electronic filters of finite impulse response (FIR) used in audio applications such as audio crossovers and audio mastering, to decode an output sequence from a given input sequence. Given an input sequence of data, the output sequence of a FIR filter of order M is a weighted sum of the recent input values observed in a sliding window of size M. Given the lattice representation in the model of the output sequence, certain embodiments of the invention may be accordingly integrated to generate an estimate of the input sequence.
In another application, methods, devices and computer program products according to some embodiments of the invention may be implemented in a Global Navigation Satellite System (GNSS), such as IRNSS, Beidou, GLONASS, Galileo; GPS comprising for instance at least a GPS receiver for estimating positioning parameters using for example carrier phase measurements.
Further, methods, devices and computer program products according to some embodiments of the invention may be implemented in cryptographic systems for determining estimates on private secret values used in a cryptographic algorithm for encrypting/decrypting data or messages during their storage, processing or communication. In lattice-based cryptography applications, data/messages are encrypted in the form of lattice points. The decryption of such encrypted data may be advantageously performed according to some embodiments of the invention, enabling for a high probability of success recovery of secret values with a reduced complexity.
More generally, while embodiments of the invention have been illustrated by a description of various examples, and while these embodiments have been described in considerable detail, it is not the intent of the applicant to restrict or in any way limit the scope of the appended claims to such detail. Particularly, the invention is not limited to particular types of lattice decoder. More generally, any type of lattice decoder may be used in step 408 such as any sequential decoder using a best-first tree-search to search for candidate lattice vectors like the Stack decoders, the Fano decoders, the decoders implementing the M-algorithm, the SB-Stack and the Zigzag Stack decoder as described in patent application EP No 14306517.5.
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8737540 | Shi | May 2014 | B1 |
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20170141788 A1 | May 2017 | US |