CN106911443B - Pilot tone optimum design method in compressed sensing based M2M communication system - Google Patents
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Abstract
The present invention provides pilot tone optimum design methods in a kind of compressed sensing based M2M communication system.This method comprises: the constellation point sets of the modulation scheme according to used by M2M communication system, determine the initial pilot sequence p of user nodek, by the initial pilot combined sequence of all user nodes at an initial pilot matrix P；Complex field is carried out to the transformation of real number field to initial pilot matrix P, matrix Q is obtained, matrix Q is used and is optimized based on the SVD optimization algorithm decomposed.The present invention is by will be after initial pilot matrix conversion to real number field, it uses and is optimized based on the SVD optimization algorithm decomposed again, under conditions of different signal-to-noise ratio, compared with random pilot matrix, the bit error rate is substantially reduced the pilot frequency sequence after method optimization of the invention；Alternatively, the length of required pilot frequency sequence is shorter in the identical situation of the bit error rate, so as to realize the accuracy for improving multiple access detection and channel Combined estimator in M2M communication system.
Description
Technical Field
The invention relates to the technical field of M2M (Machine to Machine), in particular to a pilot frequency optimization design method in M2M communication based on compressed sensing.
Background
The M2M communication is a main expression form of the current internet of things, and with the explosive development of information technology, people are no longer limited to using machines to complete work in daily life, and it is more desirable that machine devices can communicate through network interconnection without human intervention to complete corresponding tasks. For example, temperature and humidity sensors in the farmland collect data by the temperature and humidity of the land and return the data to a server of a management center, and the server analyzes the data and manages the ecological environment of the farmland by adjusting temperature and humidity controllers. The process does not require human activity to participate, and the devices communicate with each other to complete the whole task.
LTE-Advanced has now been used in M2M communication, but unlike voice call systems, user nodes in M2M communication systems are typically characterized by low activity and low data rates. Compared with the traditional communication system, the number of the user nodes in the M2M communication system communicating at the same time is not large, and the data packets transmitted are small, so the M2M communication system is a sparse communication system.
CS (Compressive Sensing) is a new theory emerging in recent years, and its core is to project a sparse or compressible high-dimensional signal onto a low-latitude space through a specific matrix transformation, and when performing signal reconstruction, reconstruct an original signal by using a linear or nonlinear recovery algorithm using the sparsity of the sparse signal or the compressed signal. In the pilot method channel estimation of the M2M communication system, because the activity of the user node is low, the channel impulse response corresponding to the inactive user node is regarded as a zero value, and the channel impulse response of the active user node is regarded as a non-zero value, that is, the user access has a sparse characteristic, the multi-user access detection and the channel state information estimation can be performed through a corresponding compressed sensing signal reconstruction algorithm.
At present, the existing multi-user access detection technology needs to allocate mutually orthogonal pilot sequences to each user, and non-orthogonal pilot sequences are used in the multi-user access and channel joint estimation based on compressed sensing, so that pilot resources can be saved. Currently, commonly used compressed sensing signal recovery algorithms include Matching Pursuit (MP), Orthogonal Matching Pursuit (OMP), compressed sampling Matching Pursuit (CoSaMP), and the like.
In multi-user access and channel joint estimation based on compressed sensing, pilot sequences of multiple users are generally randomly generated. However, the performance of multi-user access detection and channel joint estimation using randomly generated pilot sequences is not optimal. Therefore, it is desirable to design a new pilot sequence optimization design method, and the pilot sequence generated by using the method can further improve the accuracy of multi-user access detection and channel joint estimation.
Disclosure of Invention
The embodiment of the invention provides a pilot frequency optimization design method in an M2M communication system based on compressed sensing, so as to improve the accuracy of multi-user access detection and channel joint estimation in the M2M communication system.
In order to achieve the purpose, the invention adopts the following technical scheme.
A pilot optimization design method in an M2M communication system based on compressed sensing further comprises the following steps:
determining an initial pilot sequence p of a user node according to a constellation point set of a modulation scheme adopted by an M2M communication systemkCombining the initial pilot sequences of all user nodes into an initial pilot matrix P;
performing complex number domain to real number domain conversion on the initial pilot frequency matrix P to obtain a matrix Q, and optimizing the matrix Q by adopting an optimization algorithm based on SVD decomposition to obtain an optimized pilot frequency matrix Popti；
The converting the initial pilot matrix P from a complex number domain to a real number domain to obtain a matrix Q includes:
defining a function f (-) for converting a matrix from a complex number domain to a real number domain, and performing complex number domain to real number domain conversion on the initial pilot frequency matrix P by using the function f (-) to obtain a 2 Nx 2K dimensional matrix Q;
wherein P isrRepresenting the real part of the matrix P, PiRepresents the imaginary part of the matrix P;
the matrix Q is optimized by adopting an optimization algorithm based on SVD decomposition to obtain an optimized pilot matrix PopliThe method comprises the following steps:
firstly, carrying out column normalization on the 2 Nx 2K dimensional matrix Q to obtainWherein,the square of each element in the corresponding column in the 2 Nx 2K dimensional matrix Q is divided by the square sum of all elements in the column to which the element belongs;
the initial set omega is an empty set; vector w of initial length K, jth of vector wThe element isSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; the array Γ is the initial length K, the elements of the array Γ are sequentially the serial numbers of the elements of the vector w arranged from large to small, and j is equal to Γ (1), wherein Γ (1) is the 1 st element of the array; wherein,n is repeatedly and randomly selecting N elements;
③ adding element j into set omega, using 2 Nx (2K-2) dimensional matrixTo representIn (1) removingAndthe matrix left over later, by pairThe SVD is decomposed into a transposed matrix of the product of the matrix U, the matrix S and the unitary matrix V to obtainThe last column V in the unitary matrix V is selectedendUpdatingLet ρ represent the average energy of the modulation symbol set, then updateIs composed of
Updating the pilot frequency vector pjThe value of the real part isFirst N elements of (d), update pilot vector pjThe value of the imaginary part isThe last N elements of (1); updating pjEach element of (1) is a constellation point nearest to the element, and is updated according to the steps of (i) and (ii)
Defining vector with length of KThe jth element of the vector isSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; defining an array of length KThe array elements being in turn vectorsThe sequence numbers of the elements arranged from large to small;
sixthly, making t equal to 1; wherein t is the number of iterations;
seventhly, ifLet j equal t, letJumping to the step (IV); if it is notAnd isLet j equal t, letJumping to the step (IV); if it is notAnd Γ (t) ∈ Ω, let t ═ t + 1;
eighthly, if t is not larger than K, repeating the step (seventhly); if t is more than K, stopping iteration to obtain the optimal pilot frequency matrix Popli。
Further, the initial pilot sequence p of the user node is determined according to the constellation point set of the modulation scheme adopted by the M2M communication systemkCombining the initial pilot sequences of all user nodes into an initial pilot matrix P, including:
determining a constellation point set Lambda corresponding to a modulation scheme adopted by an M2M communication system, and repeatedly and randomly selecting N elements from the constellation point set Lambda to form an initial pilot sequence p of a kth userk∈ΛNSequentially generating initial pilot sequences of all user nodes, and combining the initial pilot sequences of all user nodes into an initial pilot matrix P ═ P1，p2，...，pK]∈ΛN×KAnd K is the total number of users.
Further, the method further comprises the following steps:
using the optimized pilot matrix PoptiAnd detecting multi-user access in the M2M communication system, performing channel joint estimation, and recovering information sent by a user by using a channel estimation result.
Further, the pilot matrix P after optimization is utilizedoptiDetecting and summing multi-user access in M2M communication systemChannel joint estimation, comprising:
the channel impulse response corresponding to the user node i isWherein L ishThe total number of tap time delays of the discrete channel is represented, and the observation result of the pilot frequency sequence received at the receiving end is as follows:
whereinPilot frequency of user node i after representing optimization represents convolution, n represents additive Gaussian white
Noise;
transforming the formula (1) according to a matrix convolution transformation
WhereinRepresenting a vectorConvolution moment of
And (5) arraying. Thus, the compound of formula (2) can further obtain
yp＝Aph+n (6)
WhereinRepresents a set of pilot convolution matrices, andrepresenting a set of channel impulse responses of all user nodes, wherein K is the total number of users;
and solving the sparse vector h by using a compressed sensing reconstruction signal algorithm.
According to the technical scheme provided by the embodiment of the invention, the initial pilot frequency matrix is converted into a real number domain, then the optimization algorithm based on SVD decomposition is adopted for optimization, and under the condition of different signal-to-noise ratios, the pilot frequency sequence optimized by the pilot frequency sequence optimization design method of the embodiment of the invention is compared with the random pilot frequency matrix, so that the error rate is obviously reduced; or, the length of the required pilot sequence is shorter under the condition of the same error rate, so that the accuracy of multi-user access detection and channel joint estimation in the M2M communication system can be improved.
Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive labor.
Fig. 1 is a schematic diagram of a sparse M2M communication system according to an embodiment of the present invention;
fig. 2 is a diagram illustrating an optimal pilot matrix P obtained by performing pilot optimization design on a matrix Q according to an embodiment of the present inventionoptiA flow chart of (1);
fig. 3 is a schematic diagram illustrating a principle of pilot sequence detection according to an embodiment of the present invention;
FIG. 4 is a schematic diagram illustrating a comparison between a signal-to-noise ratio and a bit error rate according to an embodiment of the present invention (P)a＝0.03，PL＝48)；
Fig. 5 is a schematic diagram illustrating a comparison between pilot length and bit error rate (SNR-8, O)a＝0.04)；
FIG. 6 shows a pilot length and bit error rate according to an embodiment of the present inventionComparative schematic (SNR 16, P)a＝0.04)。
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
As used herein, the singular forms "a", "an", "the" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. It will be understood that when an element is referred to as being "connected" or "coupled" to another element, it can be directly connected or coupled to the other element or intervening elements may also be present. Further, "connected" or "coupled" as used herein may include wirelessly connected or coupled. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items.
It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the prior art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.
For the convenience of understanding the embodiments of the present invention, the following description will be further explained by taking several specific embodiments as examples in conjunction with the drawings, and the embodiments are not to be construed as limiting the embodiments of the present invention.
The embodiment of the invention provides a pilot sequence optimization design method for multi-user access detection and channel joint estimation applied to an M2M communication system, which can reduce the overhead of pilot resources and improve the accuracy of the multi-user access detection and the channel joint estimation.
Fig. 1 shows a schematic diagram of a sparse M2M communication system according to an embodiment of the present invention, where the sparse M2M communication system has K user nodes in total, and only a users in the system need to send data to a base station at the same time, that is, the active probabilities of the users are the sameThe active users send respective pilot frequencies, and the base station performs multi-user access detection and channel joint estimation through a compressed sensing algorithm. The base station then uses the estimated channel state information to estimate the data subsequently transmitted by the user.
The invention provides a pilot frequency optimization design method for multi-user access detection and channel joint estimation in an M2M communication system based on compressed sensing, which comprises the following steps:
step 1, according to the modulation scheme adopted by the system, determining the initial pilot frequency sequence p of the user kk。
1. Determining a constellation point set Λ corresponding to a modulation scheme adopted by the system, for example, Λ of a BPSK (Binary Phase Shift Keying) modulation scheme is [ +1, -1], Λ of a QPSK (Quadrature Phase Shift Keying) modulation scheme is [1+ i, 1-i, -1+ i, -1-i ], or adopting other modulation schemes.
2. Repeatedly and randomly selecting N elements from the constellation point set Lambda to form an initial pilot sequence p of a k userk∈ΛNAnd sequentially generating initial pilot sequences of all user nodes.
3. Combining the initial pilot sequences of all user nodes into an initial pilot matrix P ═ P1，p2，...，pK]∈ΛN×KAnd K is the total number of users.
And 2, converting the initial pilot frequency matrix P from a complex number domain to a real number domain to obtain a matrix Q. Optimizing the matrix Q by adopting an optimization algorithm based on SVD decomposition to obtain an optimized pilot matrix Popli。
Since the elements in the constellation point set Λ are complex numbers, the elements in the matrix P are also complex numbers, and thus the invention defines a function f (-) to convert the matrix from a complex domain to a real domain, withWherein P isrRepresenting the real part of the matrix P, PiRepresenting the imaginary part of the matrix P. The matrix Q is optimized by adopting the pilot frequency sequence optimization method based on Singular Value Decomposition (SVD) provided by the invention to obtain the optimal pilot frequency matrix Popti。
Fig. 2 is a diagram illustrating an optimal pilot matrix P obtained by performing pilot optimization design on a matrix Q according to an embodiment of the present inventionopliThe specific processing procedure comprises the following steps:
firstly, the initial pilot frequency matrix P is converted from a complex number domain to a real number domain to obtain a 2 Nx 2K dimensional matrix
② carrying out column normalization on 2 Nx 2K dimensional matrix Q to obtainWherein, the square of each element in the corresponding column in the 2 Nx 2K dimensional matrix Q is divided by the square sum of all elements in the column to which the element belongs; .
The initial set omega is an empty set; a vector w of initial length K, the jth element of the vector w beingSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; the initial length is K, and the elements of the array are sequentially the serial numbers of the elements of the vector w arranged from large to small. Let j equal Γ (1), where Γ (1) is the 1 st element of the array.
Adding element j into set omega, using 2N x (2K-2) dimensional matrixTo representIn (1) removingAndthe matrix left over later, by pairSVD is carried out to obtain the matrix U and the matrix V in a mode of being decomposed into a transposed matrix of the product of the matrix U and the unitary matrix VThe last column V in the unitary matrix V is selectedendUpdatingLet ρ represent the average energy of the modulation symbol set, then updateIs composed of
Update the pilot frequency vector pjThe value of the real part isFirst N elements of (d), update pilot vector pjThe value of the imaginary part isThe last N elements of (1); updating pjEach element of (1) is a constellation point nearest to the element, and is updated according to the steps of (i) and (ii)
Defining a vector of length KThe jth element of the vector isSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; defining an array of length KThe array elements being in turn vectorsThe sequence numbers of the elements are arranged from large to small.
And (c) making t be 1, wherein t is the number of iterations.
IfLet j equal t, letJumping to the step (IV); if it is notAnd isLet j equal t, letJumping to the step (IV); if it is notAnd Γ (t) ∈ Ω, let t ═ t + 1;
ninthly, if t is less than or equal to K, repeating the step (b); if t is more than K, stopping iteration to obtain the optimal pilot frequency matrix Popti。
And step 3: using the optimized pilot matrix PoptiAnd detecting multi-user access in the M2M communication system, performing channel joint estimation, and recovering information sent by a user by using a channel estimation result.
A schematic diagram of a pilot sequence detection principle provided in an embodiment of the present invention is shown in fig. 3, and includes the following processing procedures;
the channel impact response corresponding to the user node i isWherein L ishRepresenting the total number of tap delays for the discrete channel. Then the observation of the pilot sequence received at the receiving end is
WhereinPilot frequency of the user node i after representing optimization represents convolution, and n represents additive white Gaussian noise;
② according to the matrix convolution transformation, the formula (4) is transformed to obtain
WhereinRepresenting a vectorConvolution moment of
And (5) arraying. Thus, the compound of formula (5) can further provide
yp＝Aph+n (9)
WhereinRepresents a set of pilot convolution matrices, where K is the total number of users andrepresenting the set of channel impulse responses for all user nodes.
Thirdly, the channel information of the inactive users is regarded as zero element, and the activity rate P of the usersa1, so h is a sparse vector, and thus equation (6) can be solved using a compressed perceptual reconstruction signal algorithm, such as Group Orthogonal Matching Pursuit (GOMP) or Group minimum absolute shrinkage and selection (Group Lasso).
In the process of each round of iterative solution, the result obtained by the SVD-based pilot frequency sequence optimization algorithm is not a pilot frequency sequence but a real number vector which does not meet the constraint of a modulation scheme constellation point set, the vector is converted into a complex number domain, and then the constellation point closest to the vector is selected as an element of a new pilot frequency sequence according to the distance.
In summary, the embodiment of the present invention provides a design method for optimizing a multi-user pilot sequence composed of constellation point elements, and as can be seen from fig. 4, under the condition of different signal-to-noise ratios, the error rate is significantly reduced when the pilot sequence optimized by the pilot sequence optimization design method of the present invention is compared with a random pilot matrix. As can be seen from fig. 5 and fig. 6, in the case of low signal-to-noise ratio (8dB) and high signal-to-noise ratio (16dB), after using the pilot sequence optimization design method of the present invention, the error rate is significantly reduced in the case of the same pilot length, or the length of the required pilot sequence is shorter in the case of the same error rate. Therefore, the accuracy of multi-user access detection and channel joint estimation in the M2M communication system can be improved.
In general, pilot sequences in channel estimation are randomly generated sequences, and the invention combines a compressed sensing technology and considers elements of modulated signal set elements of the pilot sequences, optimizes a pilot matrix consisting of the randomly generated pilot sequences in an M2M communication system based on SVD decomposition, so that the optimized pilot matrix can improve the accuracy of multi-user access detection and channel joint estimation under different conditions.
Those of ordinary skill in the art will understand that: the figures are merely schematic representations of one embodiment, and the blocks or flow diagrams in the figures are not necessarily required to practice the present invention.
From the above description of the embodiments, it is clear to those skilled in the art that the present invention can be implemented by software plus necessary general hardware platform. Based on such understanding, the technical solutions of the present invention may be embodied in the form of a software product, which may be stored in a storage medium, such as ROM/RAM, magnetic disk, optical disk, etc., and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the method according to the embodiments or some parts of the embodiments.
The embodiments in the present specification are described in a progressive manner, and the same and similar parts among the embodiments are referred to each other, and each embodiment focuses on the differences from the other embodiments. In particular, for apparatus or system embodiments, since they are substantially similar to method embodiments, they are described in relative terms, as long as they are described in partial descriptions of method embodiments. The above-described embodiments of the apparatus and system are merely illustrative, and the units described as separate parts may or may not be physically separate, and the parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of the present embodiment. One of ordinary skill in the art can understand and implement it without inventive effort.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (4)
1. A pilot optimization design method in an M2M communication system based on compressed sensing is characterized by comprising the following steps:
determining an initial pilot sequence p of a user node according to a constellation point set of a modulation scheme adopted by an M2M communication systemkCombining the initial pilot sequences of all user nodes into an initial pilot matrix P;
performing complex number domain to real number domain conversion on the initial pilot frequency matrix P to obtain a matrix Q, and optimizing the matrix Q by adopting an optimization algorithm based on SVD decomposition to obtain an optimized pilot frequency matrix Popti；
The converting the initial pilot matrix P from a complex number domain to a real number domain to obtain a matrix Q includes:
defining a function f (-) for converting a matrix from a complex number domain to a real number domain, and performing complex number domain to real number domain conversion on the initial pilot frequency matrix P by using the function f (-) to obtain a 2 Nx 2K dimensional matrix Q;
wherein P isrRepresenting the real part of the matrix P, PiRepresents the imaginary part of the matrix P;
the matrix Q is optimized by adopting an optimization algorithm based on SVD decomposition to obtain an optimized pilot matrix PoptiThe method comprises the following steps:
firstly, carrying out column normalization on the 2 Nx 2K dimensional matrix Q to obtainWherein,the square of each element in the corresponding column in the 2 Nx 2K dimensional matrix Q is divided by the square sum of all elements in the column to which the element belongs;
the initial set omega is an empty set; a vector w of initial length K, the jth element of the vector w beingSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; the array Γ is the initial length K, the elements of the array Γ are sequentially the serial numbers of the elements of the vector w arranged from large to small, and j is equal to Γ (1), wherein Γ (1) is the 1 st element of the array; wherein,n is repeatedly and randomly selecting N elements;
③ adding element j into set omega, using 2 Nx (2K-2) dimensional matrixTo representIn (1) removingAndthe matrix left over later, by pairThe SVD is decomposed into a transposed matrix of the product of the matrix U, the matrix S and the unitary matrix V to obtainThe last column V in the unitary matrix V is selectedendUpdateLet ρ represent the average energy of the modulation symbol set, then updateIs composed of
Updating the pilot frequency vector pjThe value of the real part isFirst N elements of (d), update pilot vector pjThe value of the imaginary part isThe last N elements of (1); updating pjEach element of (1) is a constellation point nearest to the element, and is updated according to the steps of (i) and (ii)
Defining vector with length of KThe jth element of the vector isSum of squares of correlation with other columns plusThe sum of squared correlation with other columns; defining an array of length KThe array elements being in turn vectorsThe sequence numbers of the elements arranged from large to small;
sixthly, making t equal to 1; wherein t is the number of iterations;
seventhly, ifLet j equal t, letJumping to the step (IV); if it is notAnd isLet j equal t, letJumping to the step (IV); if it is notAnd Γ (t) ∈ Ω, let t ═ t + 1;
eighthly, if t is not larger than K, repeating the step (seventhly); if t is more than K, stopping iteration to obtain the optimal pilot frequency matrix Popti。
2. The method of claim 1, wherein the initial pilot sequence p of the user node is determined according to a set of constellation points of a modulation scheme adopted by the M2M communication systemkCombining the initial pilot sequences of all user nodes into an initial pilot matrix P, including:
determining a constellation point set Lambda corresponding to a modulation scheme adopted by an M2M communication system, and repeatedly and randomly selecting N elements from the constellation point set Lambda to form an initial pilot sequence p of a kth userk∈ΛNSequentially generating initial pilot sequences of all user nodes, and combining the initial pilot sequences of all user nodes into an initial pilot matrix P ═ P1，p2，...，Pk]∈ΛN×KAnd K is the total number of users.
3. The method of claim 1, further comprising:
using the optimized pilot matrix PoptiAnd detecting multi-user access in the M2M communication system, performing channel joint estimation, and recovering information sent by a user by using a channel estimation result.
4. The method of claim 3 wherein the optimized pilot matrix P is utilizedoptiDetecting and channel joint estimation for multi-user access in an M2M communication system, comprising:
the channel impulse response corresponding to the user node i isWherein L ishThe total number of tap time delays of the discrete channel is represented, and the observation result of the pilot frequency sequence received at the receiving end is as follows:
juquePilot frequency of the user node i after representing optimization represents convolution, and n represents additive white Gaussian noise;
transforming the formula (1) according to a matrix convolution transformation
WhereinRepresenting a vectorThe convolution matrix of (a); thus, the compound of formula (2) can further obtain
yp＝Aph+n (3)
WhereinRepresents a set of pilot convolution matrices, K being the total number of users, anda set of channel impulse responses representing all user nodes;
and solving the sparse vector h by using a compressed sensing reconstruction signal algorithm.
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