Indoor Terminal Angle of Arrival Regression Estimation Method Based on Small-Sample CSI Fingerprint

Abstract

The present invention discloses an indoor terminal angle-of-arrival regression estimation method based on small-sample CSI fingerprints. In a multiple-input multiple-output (MIMO) two-terminal communication system, the characteristics of the small-sample CSI fingerprint library are adjusted as follows: taking the characteristics of the CSI phase difference between antennas at different angles under a certain path planning as the data medium, and establishing a sample set by establishing a weight label and fingerprints according to the actual angle; building a Multilayer Perceptron (MLP) regressor for regression analysis to extract high-dimensional feature mappings, and jointly performing linear regression prediction with the angle label to output the angle-of-arrival of unknown samples, so as to reduce the workload of data pre-acquisition and resource consumption, and have high estimation accuracy and low positioning cost.

Description

Technical Field

[0001] The present invention belongs to the technical field of indoor positioning, and particularly relates to a method for indoor terminal angle-of-arrival regression estimation based on small-sample CSI fingerprints. Background Art

[0002] Indoor positioning technology has become one of the research hotspots in recent years. One reason is the growing demand of the public for location-based services (LBS), with the main service requirements such as tracking the precise location of users. Another reason is that although the global positioning system (GPS) for outdoor positioning has been widely used, the building walls block or reflect the satellite signals, making it difficult to accurately locate indoors. Therefore, it is necessary to study other indoor positioning techniques. WiFi, Bluetooth, etc. have been widely deployed in indoor environments, and the deployment and cost are low. Indoor positioning techniques based on the analysis of such transmission signals emerge in an endless stream.

[0003] Traditional indoor positioning techniques based on channel state information (CSI) fingerprints are a general term for a class of indoor positioning methods that use pre-collected CSI as a data medium and save it as a data fingerprint library, and perform matching through signal modeling or machine learning algorithms. However, it usually requires building a large number of fingerprint libraries to ensure high matching accuracy. In addition, the complex indoor environment will also interfere with the captured signal samples, making the development of this technology full of challenges. Regarding the need to build a large number of fingerprint libraries, the vast majority of algorithms default the correctness of this acquisition method and ignore the problem that there may be a large amount of redundancy in the samples themselves. However, on the other hand, if the pre-collection workload is to be reduced, the number of collection points or the sample volume at each collection point has to be reduced, which will turn a class of indoor positioning techniques based on machine learning algorithms into a few-shot learning problem, and the positioning accuracy will inevitably decline. There are also many scenarios in daily life where only a small amount of data can be collected. Therefore, to solve this problem, there are a large number of studies on this problem in the prior art. Some scholars deal with the sample quantity problem through data augmentation, but it is more suitable for data such as pictures. Some scholars study the similarity between samples by researching models based on transfer learning, but the models are too complex. Currently, there is a lack of a truly effective method to reduce the high cost of pre-collection work while forming small CSI samples. Summary of the Invention

[0004] To solve the above problems, the present invention provides an indoor terminal angle of arrival regression estimation method based on small-sample CSI fingerprints, aiming to adjust the CSI small-sample fingerprint library features reference in a multiple-input multiple-output (MIMO) two-terminal communication system, and use the regression characteristics of a multilayer perceptron (MLP) regressor to replace the fingerprint classification work for prediction output, so as to reduce the workload and resource consumption of data pre-acquisition, and have high estimation accuracy and low positioning cost.

[0005] The indoor terminal angle of arrival regression estimation method based on small-sample CSI fingerprints according to the present invention comprises the following steps:

[0006] Step 1: Configure the router as the mobile terminal and the receiving terminal respectively. According to the angle of arrival principle, obtain the CSI data of all points in the area to be located according to the fixed route rule, and form a CSI small-sample set.

[0007] Step 2: Extract the frequency-domain raw phase information in the CSI small-sample set, perform unwrapping processing, splice all the antenna array phase information in the dimension direction as the original CSI phase set, and select any one antenna as the reference antenna, and calculate the phase difference between the remaining antennas and the reference antenna to form a CSI small-sample fingerprint library.

[0008] Step 3: Mark the labels according to the prior geographical location information, and normalize the angle numerical label set to the range of [0, 1], and convert it into a weight numerical label set y.

[0009] Step 4: Build an MLP regressor and perform offline training on the CSI small-sample fingerprint library.

[0010] Step 5: In the online prediction stage, the mobile terminal sends a location service request to the nearby receiving terminal for communication. The receiving terminal receives a number of Wi-Fi data packets through a one-dimensional antenna array, extracts the CSI data, and also obtains the CSI test set of the data to be located according to Steps 1 and 2. Use the built and trained MLP regressor for azimuth regression prediction.

[0011] Further, Step 1 is specifically:

[0012] The mobile terminal collects set b on a tangent line that is 45 degrees offset from the normal direction of the receiving terminal and more than 2 m away from the fixed end, and moves several times at the edge of the area to be located to collect set c. The mobile terminal continuously sends a number of Wi-Fi data packets to the one-dimensional antenna array of the receiving terminal through the antenna, and repeats this step until the receiving terminal has collected all positions to form a CSI small-sample set.

[0013] A CSI small-sample set of the positioning area is collected by this method, which can be expressed as a set

[0014] Among them, N is the total number of data packets, represents the set of integers; for a set of subcarrier transmission paths with a total of m antennas in the antenna array, the phase data in the independently extractable CSI obtained at a certain moment t is used as the original CSI phase sample..

[0015] Furthermore, in step 2, the specific steps to obtain the CSI small sample fingerprint library are as follows:

[0016] Taking each path k as a unit, its channel impulse response is calculated as:

[0017]

[0018] a p , θ p , τ p respectively represent the amplitude attenuation, phase shift, and time delay of the p-th path, n is the total number of propagation paths, and δ(τ) is the Dirac impulse function;

[0019] Mapping from the time domain to the frequency domain, the frequency domain information of the CSI of the i-th subcarrier collected is calculated as:

[0020]

[0021] H(f r ) represents the CSI of the subcarrier with a center frequency of f i , where ||H(f r )|| and ∠H(f r ) respectively represent the amplitude and phase of the r-th subcarrier;

[0022] Taking out the phase information alone, the CSI phase of any one of the m antennas is simply denoted as Then the CSI phase sample of any data packet of the one-dimensional antenna array is calculated as:

[0023]

[0024] When using the s-th antenna as the reference, the difference is taken to obtain the CSI single phase difference vector sample:

[0025]

[0026] When taking the phase difference, any phase difference needs to be calculated according to unwrapping, and the calculation method is:

[0027]

[0028] Further, in step 4, the specific steps for constructing the MLP regressor and performing offline training are as follows:

[0029] Step 4-1: Establish the network structure of the MLP regressor in a one-way stacked manner, where each neuron in each layer is independent of each other, and the layers are connected to each other. Each neuron in the next layer processes the outputs of all neurons in the previous layer, without cross-layer connections.

[0030] Step 4-2: The MLP regressor takes the CSI small sample fingerprint library and the label set y as inputs, updates the network data state until the model converges and saves it.

[0031] Step 4-3: The MLP regressor uses an iterator to update the network parameters, selects the Huber_loss as the regression loss function for gradient update until the set epoch stops.

[0032] Further, in step 4-1, the constructed MLP regressor is specifically as follows: The MLP regressor includes an input layer, a hidden layer, and an output layer; the first layer is the input layer, and the input is the CSI small sample fingerprint library, and the input dimension size is (m - 1)×56, where m refers to the number of antennas; the second and third layers are the hidden layers, and the hidden layers are connected to each other in a fully connected manner, and there is no connection between the neurons within the hidden layer, and the hidden layer dimension is greater than (m - 1)×56; the last layer is the output layer, and the predicted value of the high-level feature weight is output through a linear linear regression neuron, and the network output is the angle of arrival regression estimation result, and finally the orientation of the mobile terminal is output in online prediction.

[0033] Further, in step 4-2, the specific steps for updating the network data state until the model converges and saves it are as follows:

[0034] First, the CSI small sample fingerprint library and the label set y are briefly recorded as the data set D, then the quantization calculation is as follows:

[0035]

[0036] Among them, the input sample csi (i)T represents a small sample CSI fingerprint set with a total of N data packets and d dimensional attributes, where d = (m - 1)×56; the real-valued vector y (i)T is the label set, which is a one-dimensional vector; their corresponding matrices are respectively represented as CSI N×d and Y N ; there is a weight matrix at a certain moment t between the input layer and the hidden layer:

[0037]

[0038] Among them, q is the number of neurons in the hidden layer, q > d, such that the input vector of the neuron with serial number h of a certain sample in the first hidden layer The state matrix under the sample input at a certain moment t is:

[0039]

[0040] Where represents the transposed matrix of, and the hyperbolic tangent function tanh is selected as the activation function for the hidden layer. Therefore, the output state of the neuron with serial number h in the first layer of the hidden layer According to the automatic neuron threshold γ h is calculated as:

[0041]

[0042] Finally, in the same way, the state matrix of the second hidden layer at a certain moment t is obtained Until the model converges and is saved, the high-dimensional output weight matrix W obtained from the CSI small sample fingerprint library mapped by the MLP hidden layer is obtained q And the input value matrix C of the output layer N , the input vector e (i)T , and the output vector g is output by the linear regression neuron of the output layer according to the automatic neuron threshold δ (i)T :

[0043] C N = B N× q·(W T ) q

[0044] g (i)T = Linear(e (i)T - δ)

[0045] Among them, (W T ) q is the transposed matrix of W q .

[0046] The beneficial effects of the present invention are as follows: By means of MLP regression, a large amount of fingerprint information at the positions to be collected is avoided or reduced, thereby reducing the workload, and azimuth prediction is carried out by learning the characteristic relationship of samples changing with angles between regions. Compared with the original fingerprint positioning method, the pre-collection workload is greatly reduced; while reducing the number of samples required for training, since the sample volume is no longer huge, there is no need to build a very complex learning model for prediction estimation, solving the problem of creating a high-cost fingerprint library, and the model construction is relatively simple and easy to use. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is the overall flowchart of the present invention;

[0048] Figure 2 It is a schematic diagram of an embodiment of the angle-of-arrival acquisition point of the present invention. Among them, the small figure (a) represents the point diagram of the candidate acquisition area, and the small figure (b) represents the acquisition scheme for forming small samples of the present invention;

[0049] Figure 3 It is a schematic diagram of the CSI phase difference processing flow in the present invention. Among them, the small figure (a) represents the different distribution schematic diagrams of CSI phase information at different points, the curve in the small figure (b) represents the phase value performance of three data packets on 56 subcarriers of each of the three antennas, and the small figure (c) represents the CSI small sample phase difference overlap formed after the three data packets are processed by the present invention, and the values have good stability;

[0050] Figure 4 It is the model structure diagram of the MLP regressor of the present invention. Specific embodiments

[0051] In order to make the content of the present invention easier to be clearly understood, the present invention will be further described in detail below according to specific embodiments and in conjunction with the accompanying drawings.

[0052] The present invention provides an indoor terminal angle-of-arrival regression estimation method based on small-sample CSI fingerprints, and the specific embodiment process is as Figure 1 shown:

[0053] Step 1: Configure the router as the mobile end and the receiving end respectively. According to the angle-of-arrival principle, obtain the CSI data of all points in the area to be located according to a fixed route rule, and form a CSI small sample set;

[0054] Step 2: Extract the frequency-domain raw phase information in the CSI small sample set, perform unwrapping processing, splice all antenna array phase information in the dimension direction as the original CSI phase set, and select any one antenna as the reference antenna, calculate the phase difference between the remaining antennas and the reference antenna, and form a CSI small sample fingerprint library;

[0055] Step 3: Mark labels according to the prior geographical location information, and normalize the angle numerical label set to the range of [0, 1], and convert it into a weight numerical label set y;

[0056] Step 4: Build an MLP regressor and perform offline training on the CSI small sample fingerprint library;

[0057] Step 5: In the online prediction stage, the mobile device sends a location service request to a nearby receiver for communication. The receiver receives several Wi-Fi data packets through a one-dimensional antenna array, extracts CSI data, and also obtains the CSI test set of the data to be located according to Steps 1 and 2. Then, the established and trained MLP regressor is used for azimuth regression prediction.

[0058] In this embodiment, the configuration environment and model parameters are as follows: The acquisition devices are a Lenovo laptop g480 and two three-day routers using Atheros series wdr4310 network cards. The experimental system is a Windows 11-compatible desktop computer, and the experimental environment is an Intel(R) Core(TM) i5-9400F CPU@2.90GHz six-core processor. The programming software involves Matlab and Python, and Keras is used as the model framework.

[0059] As Figure 2 shown, Figure 2 the small figure (a) represents the candidate acquisition area point map, Figure 2 the small figure (b) represents the acquisition scheme for forming small samples of the present invention; The sample set acquisition and its configuration are as follows: Set up two three-day routers with Atheros series network cards that can communicate in an indoor room. One is fixed in position, and the other is used as a mobile device to move along a ray that is 45 degrees offset from the normal direction of the fixed end and more than 2 m away from the fixed end, and also move several times along the edge of the area to be located for acquisition. Use the CSI_tool tool command line to set a high packet sending rate, set the network frequency band to 5G, and use the packet sending and receiving commands to continuously send several Wi-Fi data packets to the receiving terminal. Repeat this step until the receiver has collected 10,000 packets at each position, and randomly select 1,000 packets at each position to form the original CSI small sample set.

[0060] As Figure 3 shown, where Figure 3 the small figure (a) represents the schematic diagram of different distributions of CSI phase information at different points, Figure 3 the curve in the small figure (b) represents the phase value performance of three data packets on 56 subcarriers of three antennas, Figure 3 the small figure (c) shows the overlap of the CSI small sample phase differences formed after the processing of the present invention for three data packets, and the values have good stability. The steps of data preprocessing include:

[0061] Enter Matlab and use the CSI extraction tool code command to extract CSI, and then enter Python to write a data splicing algorithm to splice the original CSI small sample sets received by the antenna array in the dimension direction to obtain phase samples;

[0062] The CSI phase sample calculation for a single data packet of a three - antenna one - dimensional antenna array is as follows:

[0063] represents the numbers of the three antennas from left to right, and \(T\) represents vector transpose;

[0064] Based on the phase samples, unwrapping and differencing operations are performed on the antennas to obtain the phase - difference samples corresponding to each sample;

[0065] When taking the first antenna as the reference, the difference is taken to obtain the CSI single - phase - difference vector sample:

[0066]

[0067] Considering orthogonality, the mobile terminal is offset by 45 degrees in the direction perpendicular to the normal of the fixed end; to avoid the near - distance unstable area, set \(b\) is collected on a tangent line more than 2m away from the fixed end, and set \(c\) is collected by moving several times at the edge of the area to be located. Thus, a sample matrix of 1000 data packets for a single collection point can be obtained

[0068]

[0069] When calculating the phase difference, if there is or the case, \(2\pi\) needs to be added for correction.

[0070] Among them, any phase difference has a dimension of 56, and after processing by the three antennas, it gets \(2\times26 = 112\) dimensions.

[0071] Taking the collection set of a point in this embodiment as an example, its sample set \(X\) b The corresponding file name is loc1_c.csv, and its 112 - dimension sample form is:

[0072] 2.329089,2.374091,2.411158,…,5.834802,5.802786,5.806453.

[0073] It should be noted that the selection method of the reference antenna is not unique.

[0074] Weight label: In this embodiment, the angle range \([0,90]\) is selected as the experimental object, and the actual angle measurements and records are made for all collection points, and then normalized from \([0,90]\) to the range \([0,1]\) according to the actual angle values. Taking loc1_c.csv as an example again, its 1 - dimension label form is: 0.2.

[0075] MLP Regressor Construction: The network structure is established in a unidirectional cascading manner. Each neuron in each layer is independent of each other, and the connections between layers establish relationships. Each neuron in the next layer processes the outputs of all neurons in the previous layer, without cross-layer connections. The MLP regressor includes an input layer, hidden layers, and an output layer; the first layer is the input layer, and the input is a small sample set of CSI phase differences at all acquisition points, with an input dimension size of 112; the second and third layers are hidden layers, and the hidden layers are connected to each other in a fully connected manner, and there are no connections between neurons within the hidden layers. The tanh activation function is selected, and the number of neurons required for each layer is specified; the last layer is the output layer, and the predicted value of the high-level feature weight is output through a linear regression neuron. The relevant parameter settings of the MLP regressor are as follows: the parameter optimizer in the model compiler is adam, the loss function is huber_loss, and the specific parameters remain default, and the learning rate is adjusted according to the specific situation. The parameter epoch in the model fitter is 30, batch_size is 32, and shuffle is True. In this embodiment, 160 hidden units are set for each layer, with a total of 2 hidden layers, and the network structure is as shown in Figure 4 shown. The input sample csi (i)T contains 112-dimensional attributes, and the real-valued vector y (i)T is a one-dimensional vector, and the corresponding matrix representation is CSI N×112 , Y N ; There is a weight matrix at a certain moment t between the input layer and the hidden layer:

[0076]

[0077] Among them, the input vector of the neuron with the serial number h of a certain sample in the first hidden layer makes the state matrix under the sample input at a certain moment t be:

[0078]

[0079] where represents the transpose matrix of; the tanh is selected as the activation function for the hidden layer, so the output state of the neuron with the serial number h in the first layer can be calculated according to the automatic neuron threshold γ h corresponding to the algorithm as:

[0080]

[0081] Finally, similarly, the state matrix of the second hidden layer at a certain moment t can be obtained.Until the model converges and is saved, obtaining the high-dimensional output weight matrix W obtained by mapping the CSI phase difference small sample set by the MLP hidden layer 160 And the input value matrix C of the output layer N , input vector e (i)T , the linear regression neuron of the output layer outputs vector g according to the automatic neuron threshold δ (i)T :

[0082] C N = B N×160 ·(W T ) 160

[0083] g (i)T = Linear(e (i)T - δ)

[0084] Among them, (W T ) q is the transpose matrix of W q .

[0085] Online prediction: Use the trained MLP regressor to predict non-training points and give the azimuth prediction result.

[0086] The present invention aims at the problem that the small sample CSI fingerprint learning data volume is insufficient and difficult to train, builds an MLP regressor to learn the high-dimensional features of the samples for angle of arrival estimation; compared with other learning models, the present invention re-plans the acquisition scheme and re-considers the limitations on model selection in terms of the pre-acquired data volume and the number of categories; by adjusting the CSI sample acquisition method, builds an MLP regressor for non-linear regression training and prediction; through linear regression output of the high-dimensional features and angle mapping extracted from the CSI small sample fingerprint library by the regressor to predict the angle of arrival of unknown samples, simplifies the model building work to a certain extent, and reduces the huge workload and resource consumption of the pre-acquisition work.

[0087] The above is only the preferred solution of the present invention and is not used as a further limitation of the present invention. All equivalent changes made using the content of the specification and drawings of the present invention are within the protection scope of the present invention.

Claims

1. An indoor terminal angle of arrival regression estimation method based on small - sample CSI fingerprints, characterized in that, The method steps are as follows: Step 1: Configure the router as the mobile end and the receiving end respectively. According to the angle-of-arrival principle, obtain the CSI data of all points in the area to be located according to the fixed route rule, and form a CSI small sample set. Step 2: Extract the original phase information in the frequency domain from the CSI small sample set, perform unwrapping processing, splice all the antenna array phase information in the dimension direction as the original CSI phase set, and select any one antenna as the reference antenna, calculate the phase difference between the remaining antennas and the reference antenna, and form a CSI small sample fingerprint library. Step 3: Mark the tags according to the prior geographical location information, and normalize the angle numerical tag set to the range of [0, 1], and convert it into a weight numerical tag set y. Step 4: Build an MLP regressor and perform offline training on the CSI small sample fingerprint library; specifically: Step 4-1: Establish the network structure of the MLP regressor in a one-way stacked manner, where each neuron in each layer is independent of each other, and the layers are connected to generate connections. Each neuron in the next layer processes the outputs of all neurons in the previous layer, and no cross-layer connections are made. Step 4-2: The MLP regressor takes the CSI small sample fingerprint library and the tag set y as inputs, updates the network data state until the model converges and saves it. Step 4-3: The MLP regressor uses an iterator to update the network parameters, selects Huber_loss as the regression loss function for gradient update until the set epoch stops. Step 5: In the online prediction stage, the mobile end sends a location service request to the nearby receiving end for communication; the receiving end receives several Wi-Fi data packets through a one-dimensional antenna array, extracts the CSI data, and also obtains the CSI test set of the data to be located according to Steps 1 and 2, and uses the built and trained MLP regressor for azimuth regression prediction.

2. The indoor terminal angle of arrival regression estimation method based on small - sample CSI fingerprints according to claim 1, characterized in that, Step 1 is specifically as follows: The mobile end collects set b on a tangent line that is 45 degrees offset from the normal direction of the receiving end and more than 2 m away from the fixed end, and moves several times at the edge of the area to be located to collect set c. The mobile end continuously sends several Wi-Fi data packets to the one-dimensional antenna array of the receiving end through the antenna, and repeats this step until the receiving end has collected all positions to form a CSI small sample set. A CSI small sample set of the positioning area is acquired by this method, denoted as the set where N is the total number of data packets, represents the set of integers; for a set of subcarrier transmission paths with a total of m antennas in the antenna array, the phase data in the independently extractable CSI obtained at a certain moment t is used as the original CSI phase sample.

3. The indoor terminal angle of arrival regression estimation method based on small - sample CSI fingerprints according to claim 2, characterized in that, In Step 2, the specific steps to obtain the CSI small sample fingerprint library are as follows: Taking each path k as a unit, its channel impulse response is calculated as: a p 、 θ p 、 τ p respectively represent the amplitude attenuation, phase shift, and time delay of the p-th path, n is the total number of propagation paths, and δ(τ) is the Dirac impulse function; Mapping from the time domain to the frequency domain, the frequency domain information of the CSI of the i-th subcarrier collected is calculated as: H(f r ) represents the CSI of the sub - carrier with the central frequency of f i , where, ∥H(f r )∥ and ∠H(f r ) represent the amplitude and phase of the r - th sub - carrier respectively; The phase information is taken out separately, and the CSI phase of any one of the total m antennas is simply denoted as Then the CSI phase sample of any data packet of the one-dimensional antenna array is calculated as: When taking the s-th antenna as the reference, the difference is obtained to get a single CSI phase difference vector sample: When calculating the phase difference, any phase difference needs to be calculated based on unwrapping, and the calculation method is as follows:

4. The indoor terminal angle of arrival regression estimation method based on small - sample CSI fingerprints according to claim 3, characterized in that, In step 4-1, the constructed MLP regressor is specifically as follows: The MLP regressor includes an input layer, a hidden layer, and an output layer. The first layer serves as the input layer, and the input is the CSI small-sample fingerprint database, with the input dimension size being (m - 1)×56, where m refers to the number of antennas. The second and third layers serve as the hidden layers, and the hidden layers are connected to each other in a fully connected manner. There is no connection between the neurons within the hidden layer, and the dimension of the hidden layer is greater than (m - 1)×56. The last layer serves as the output layer, and the predicted value of the high-level feature weight is output through the linear regression neuron. The network output is the angle-of-arrival regression estimation result, and finally, the orientation of the mobile terminal is predicted and output online.

5. The indoor terminal angle of arrival regression estimation method based on small - sample CSI fingerprints according to claim 4, characterized in that, In step 4-2, the specific steps to update the network data status until the model converges and saves are as follows: First, the CSI small-sample fingerprint database and the label set y are briefly denoted as the dataset D, then the quantization calculation is as follows: Among them, the input sample csi (i)T represents a small sample CSI fingerprint set of a total of N data packets and d dimensional attributes, where d = (m - 1) × 56; the real-valued vector y (i)T is the label set, which is a one-dimensional vector; their corresponding matrices are respectively denoted as CSI N×d , Y N ; there is a weight matrix at a certain moment t between the input layer and the hidden layer: where q is the number of neurons in the hidden layer, q > d, and the input vector of the neuron with serial number h of a certain sample in the first hidden layer is the state matrix under the sample input at a certain moment t is: wherein denotes the transposed matrix; the hidden layer selects tanh as the activation function, so the output state of a neuron with a certain serial number h in the first layer of the hidden layer is calculated according to the automatic neuron threshold γ corresponding to the algorithm h as follows: Finally, in the same way, the state matrix of the second hidden layer at a certain moment t is obtained Until the model converges and is saved, the high-dimensional output weight matrix W obtained by mapping the CSI small sample fingerprint library by the MLP hidden layer is obtained q And the input value matrix C of the output layer N , the input vector e (i)T , and the output vector g is output by the linear regression neuron of the output layer according to the automatic neuron threshold δ (i)T : C N = B N×q ·(W T ) q g (i)T = Linear(e (i)T -δ) where (W T ) q is the transpose matrix of W q .

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