CN114205195B - Cross-frequency-band MIMO space domain statistical CSI estimation method - Google Patents
Cross-frequency-band MIMO space domain statistical CSI estimation method Download PDFInfo
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Abstract
The invention relates to a cross-frequency-band MIMO space domain statistical CSI estimation method, which is characterized in that firstly, elements in a space domain covariance matrix of different frequency channels are modeled as a function only related to array element serial numbers, wavelengths and array element intervals based on similarity of different frequency channels in an angle domain and a transformation relation between the angle domain and the space domain statistical CSI in the same cross-frequency-band MIMO system, and then estimation of the space domain covariance matrix is converted into extrapolation of the function. And then, a linear extrapolation model of channel covariance elements is established under two scenes of ULA and UPA configured at the base station side, and extrapolation coefficients in the two scenes are solved. And finally, estimating the channel covariance matrix of all other frequencies under the condition that the channel covariance matrix of one frequency is known through an extrapolation model. Compared with the traditional method, the method greatly reduces the measurement overhead, has low implementation complexity and has higher estimation precision compared with the traditional interpolation method.
Description
Technical Field
The invention belongs to the field of communication, and particularly relates to a method for establishing a linear extrapolation model under the condition of knowing a channel covariance matrix on a certain frequency in a cross-frequency band communication system so as to finish conversion between channel covariance among different frequencies.
Background
With the rapid development of large-scale Multiple-Input Multiple-Output (MIMO) technology, the number of users accessing in a system is increasing, the capacity requirement of the system is increasing, and the lack of spectrum resources on the existing frequency also enables the channel resources of higher frequency to be continuously researched and utilized, so that the phenomenon that a system communicates on Multiple frequencies simultaneously is more common, for example, WLAN usually works on two frequencies of 2.4GHz and 5GHz, and the existing dual-card dual-standby communication device is also a common multi-frequency cooperative communication. In addition, most of the millimeter wave communication systems are often limited by array size and hardware, and their own channel state information (CSI, channle State Information) is difficult to directly obtain through measurement, so sub-6GHz channels are also equipped to form a multi-frequency communication system together, and the CSI on the low frequency channel is used to assist the information transmission on this frequency.
The collaborative communication of different frequencies in cross-frequency communication is completed by counting the correlation of CSI, channels on different frequencies in the same multi-frequency communication system are different in instantaneous CSI despite being influenced by the center frequency, the channels on different frequencies show similar energy distribution in an angle domain due to the fact that the positions and environments are the same, and in the scene of configuring a uniform linear array (ULA, uniform LinearArray) and a uniform area array (UPA, uniform PlanarArray) on a base station side, the relationship of Fourier transformation is satisfied between the angle spectrum and a spatial domain covariance matrix, so that the covariance matrices of the different frequency channels can be mutually converted through a certain relationship by virtue of the transformation relationship and the similarity of the angle spectrum. For ULA, the parameter estimation method and the cubic spline interpolation method are used in the existing research, but these methods have certain limitations for extrapolation scene from low-dimensional array to high-dimensional array. In addition, the mutual extrapolation of the covariance matrix of the uplink and downlink channels in the frequency division duplex (FDD, frequency DivisionDuplex) system is also an important research content, and the least square method, the fourier operator interpolation method, the projection algorithm and the like have been given in the existing research, but the transformation of the array dimension is not considered in the FDD system. Meanwhile, for UPA, as the array dimension increases, the parameters required to be estimated are increased compared with ULA, therefore, the invention aims to find a channel space domain covariance estimation method suitable for different scenes and different array arrangements, so that the mutual conversion of space domain statistical parameters can be directly carried out on the premise of not carrying out angle domain statistical parameter estimation, the measurement and calculation on all frequencies are avoided, and the system overhead is reduced.
Disclosure of Invention
Technical problems: aiming at a cross-frequency-band MIMO system, the invention provides a cross-frequency-band MIMO space domain statistical CSI estimation method, namely a space domain channel covariance estimation method applicable to ULA and UPA array forms, under the condition that a channel covariance matrix on one frequency is known, a linear method is used for extrapolating the channel covariance matrix on the other frequencies, measurement and calculation on all frequencies are avoided, relevance among different frequencies is fully utilized, and system overhead and implementation complexity are reduced.
The technical scheme is as follows: in order to achieve the above object, the present invention provides a cross-band MIMO spatial domain statistical CSI estimation method comprising the steps of:
in a cross-frequency band communication system, transmitting antenna arrays corresponding to different frequencies are arranged in parallel in a co-location way, and are positioned in the same base station, and the base station communicates with all users in a cell at the same time; the higher the frequency is, the higher the dimension of the array is configured, and the noise interference on the high-frequency channel is higher than the low frequency, so that the acquisition of the space domain covariance matrix is more complex; since the statistical CSI on different frequencies shows high similarity in an angle domain, under the condition that a certain low-frequency channel covariance matrix is known, the spatial domain covariance matrix of a high-frequency channel can be directly extrapolated and estimated on the premise of not measuring the high-frequency channel by utilizing the similarity;
step 2, analyzing a cross-band communication system with a uniform linear array ULA (Uniform LinearArray) configured at a base station side, and modeling the relationship between elements in a channel space domain covariance matrix and an angle power spectrum as one-dimensional Fourier transformation based on the characteristic of approximate mutual independence between channel complex gains at different angles;
step 3, modeling the spatial domain covariance matrix elements on different frequencies in the same cross-frequency band communication system as a function only related to array element serial number difference, wavelength and array element spacing by using Fourier transformation relation between the spatial domain covariance matrix elements and the angle spectrum, and further converting estimation of the channel spatial domain covariance matrix between different frequencies into extrapolation problem of the function;
step 4, under the condition that the space between the antenna array elements on different frequencies is half wavelength, based on an autoregressive AR (AutoRegressive) model, establishing a linear extrapolation model of the covariance matrix elements of the spatial domain of the channels on different frequencies, solving a linear extrapolation coefficient by utilizing a Levinson-Durbin algorithm, combining the linear extrapolation method with a traditional cubic spline interpolation method when the space between the antenna array elements is not half wavelength, completing conversion of the covariance matrix corresponding to the channels on different frequencies by combining extrapolation and interpolation, and estimating the covariance of the spatial domain of the channels on other frequencies;
step 5, spreading a covariance matrix conversion method in ULA to a scene of configuring a uniform area array UPA (Uniform PlanarArray) at a base station side, wherein due to the increase of UPA compared with ULA dimension, when the UPA horizontal and vertical array elements are half-wavelength at different frequencies, channel covariance matrix elements can be modeled as a two-dimensional function related to the serial number difference of the horizontal and vertical array elements, a linear extrapolation model between covariance matrices can be established based on a two-dimensional AR model, and the linear extrapolation coefficient is solved by using a method for establishing a vector space;
and 6, estimating element values in the covariance matrix of the space domain of the high-frequency channel by using a linear extrapolation model under the condition of configuring ULA and UPA at the base station side, and sequentially arranging the covariance element values obtained by estimation based on the symmetry characteristic of the covariance matrix to obtain the covariance matrix on another frequency so as to finish the conversion of the covariance matrix of the space domain of the channel from the low-frequency channel to the high-frequency channel.
Wherein:
in the cross-band communication system described in step 1, on the premise that the channel covariance matrix on a certain frequency is known, the spatial domain covariance matrix of the channels corresponding to all other frequencies can be obtained by direct linear extrapolation through the known covariance matrix, so that training and measurement on all frequencies are avoided.
In the cross-frequency communication system described in step 3, under the condition that the array element spacing is half wavelength on all frequencies, letting any frequency f i The covariance matrix of the channel between the upper base station and the user k is R k (f i )，R k (f i ) All elements of row a and column b of the list can be written as a function r with an argument of a-b k (·)，I.e.
[R k (f i )] a,b ＝r k (a-b).
In the same cross-frequency communication system, it is assumed that two frequencies in the system are f 1 And f 2 And has f 1 ＜f 2 ，f 1 And f 2 Corresponding to the ULA array dimensions of 1 XN 1 1 XN 2 Let m=a-b, then frequency f 1 Upper channel covariance matrix R k (f 1 ) From [1-N 1 ,-1]∪[0,N 1 -1]R corresponding to all integer values m in the range k (m) composition, frequency f 2 Upper channel covariance matrix R k (f 2 ) From [1-N 2 ,-1]∪[0,N 2 -1]R corresponding to all integer values m in the range k (m) composition. At a known R k (f 1 ) Under (2) to estimate R k (f 2 ) The problems of (1) can be translated into those of the known [1-N ] 1 ,-1]∪[0,N 1 -1]All integer values m in the range correspond to r k Under the condition of (m), extrapolating [1-N 2 ,-N 1 ]∪[N 1 ,N 2 -1]R corresponding to all integer values m in the range k The (m) value, i.e., the function extrapolation problem.
Said R is k (f 1 ) The linear extrapolation model described in step 4, given the known premise, is expressed as:
i.e. any unknown covariance element r k (m) the first N 1 -a linear combination representation of 1 known covariance matrix elements, σ 2 Represents the error power of the linear extrapolation, q.epsilon.1, N 1 -1]Representing forward linear extrapolation order, a 1 (q) is a linear extrapolation coefficient corresponding to the order q, and delta (·) represents an impulse function; based on the extrapolation model, m E [ N ] 1 ,N 2 -1]All r corresponding to k (m) the values can be obtained by linear combination of known covariance functions one by one; similarly for m E [1-N ] 2 ,-N 1 ]All r at the time k (m) the following backward predictive model can be built
Wherein p is E [1, N 1 -1]Representing backward linear extrapolation order, a 2 (p) is the linear extrapolation coefficient corresponding to the order p, then r k (m) post-availability N 1 -a linear combination representation of 1 known covariance matrix elements.
The step 4 of solving the linear extrapolation coefficient by using the Levinson-Durbin algorithm is to solve the linear extrapolation coefficients a in two extrapolation directions by using the Levinson-Durbin algorithm 1 (q) and a 2 (p) iteratively, in calculating a 1 In the process of (q), N is performed 1 -1 iteration, the number of iterations is indicated by a superscript, then in the ith iteration, let k i Representing the scaled value of the coefficient in the ith iteration and defining the parameter epsilon i ＝ε i-1 [1-(k i ) 2 ]Then k i The calculation mode is that
Using coefficient k i Establishing linear extrapolation coefficientsAnd the coefficients obtained in the last iteration +.>Extrapolation relationship between:
and the algorithm starts epsilon 0 ＝r k (0). In the process of N 1 -obtaining a after 1 iteration 1 The final value of (q), i.eWhile coefficients in the other extrapolation direction are calculated based on the conjugate symmetry properties of the covariance matrix, i.e. +.>
In step 4, when the antenna element spacing is not half a wavelength, the channel covariance matrix elements at different frequencies are modeled as a function related to only the element sequence number difference, the wavelength and the element spacing, for the frequency f i The array element spacing and the wavelength of the base station side array corresponding to the frequency are respectively d i And lambda (lambda) i Then the elements in the covariance matrix are used as a function r k (. Cndot.) is expressed as:
at this time from R k (f 1 ) Estimating R k (f 2 ) Solving the problem combines a linear extrapolation method with a cubic spline interpolation, and solves all independent variable values using the linear extrapolation methodCorresponding +.>While->When the number is non-integer, the linear extrapolation method is used to obtain the sum +.>R corresponding to the nearest two integer values k (. Cndot.) then interpolating between the two values using a cubic spline interpolation method to obtain +.>
In step 5, when the base station side configures UPA, the intervals between the array elements in the horizontal and vertical directions of UPA on different frequencies are half-wavelength, assuming that the two frequencies in the system are f 1 And f 2 And has f 1 ＜f 2 ，f 1 And f 2 The dimension of UPA array configured corresponding to the base station side is N respectively 1 ×N 1 N 2 ×N 2 ，f 1 And f 2 The covariance matrix of the channel between the upper base station side and the user k is respectively represented by R k (f 1 ) And R is k (f 2 ) A representation; analog ULA array, R k (f 1 ) And R is k (f 2 ) All elements in (a) can use a two-dimensional function r k (m, n) represents, according to the known R k (f 1 ) Estimating R k (f 2 ) The problem of the medium element can be converted into a linear extrapolation model as follows; in UPA, r is increased by the dimension, similar to ULA k In (m, n), two independent variables m and n respectively correspond to two extrapolation directions and coexist in four linear extrapolation directions; taking one of the extrapolated directions as an example, it is assumed that R is known k (f 1 ) Under the condition of (1) extrapolating m, N E [ N ] 1 ,N 2 -1]Time-corresponding covariance matrix element r k (m, n), the model of the linear extrapolation is expressed as:
wherein q and l each independently represent r k Extrapolation order of the argument in (m, n) in two dimensions, a 1 (q, l) represents a linear extrapolation coefficient, delta (m, n) represents a two-dimensional impulse response function, sigma 2 Representing the error power of the linear extrapolation.
The linear extrapolation coefficient in the step 5 is solved by using a method for establishing a vector space, and the specific steps include:
a. according to the known R k (f 1 ) Extracting all elements therein as known r k The (m, n) function takes the value to form a coefficient matrix, and converts the linear extrapolation model into the form of a matrix multiplied equation set, wherein the coefficient matrix of the equation set is R AR All of the solutionsSolution a 1 (q, l) forming a coefficient vector b to be solved 1 ；
b. Give two vectors v and w with respect to R AR Definition of orthogonality, using orthogonalization method, find R in complex space AR An orthogonal set of bases;
c. to be solved for coefficient vector b 1 Use of R AR An orthogonal set of baseline combinations is shown.
After the element values in the covariance matrix are solved by using the linear extrapolation model in the step 6, the covariance matrix of the ULA meets the toeplitz symmetry property, the covariance matrix of the UPA meets the massive toeplitz symmetry property, and the extrapolated covariance elements are arranged by utilizing the symmetry properties of the covariance matrix, so that all covariance matrices on the other frequencies are reconstructed.
The beneficial effects are that: compared with the prior art, the invention has the following advantages:
1. the method converts the mutual extrapolation of the covariance matrixes of the spatial domains corresponding to different frequencies into the extrapolation of the functions and the solution of interpolation problems, avoids the complicated angle spectrum estimation step in the traditional method,
2. the method considers array dimension change among different dimensions, simultaneously considers two scenes of configuring ULA and UPA at the base station side, fully utilizes the known covariance matrix, has wider application range compared with the traditional method, and has higher accuracy compared with the traditional interpolation method.
3. According to the method, the covariance matrix conversion between different frequencies is completed by using a linear model under the two array forms of ULA and UPA, and the channel covariance matrix on all other frequencies can be extrapolated only by detecting the channel covariance matrix on one frequency, so that the realization complexity is low, and the huge expenditure caused by channel measurement on high frequencies is avoided.
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FIG. 1 is a flow chart of a method according to an embodiment of the invention.
Detailed Description
In order to make the present invention better understood by those skilled in the art, the following description will make clear and complete descriptions of the technical solutions of the embodiments of the present invention with reference to the accompanying drawings in which it is apparent that the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the present invention without making any inventive effort, shall fall within the scope of the present invention.
The invention relates to a cross-frequency-band MIMO space domain statistical CSI estimation method which comprises the following steps:
in a cross-frequency band communication system, transmitting antenna arrays corresponding to different frequencies are arranged in parallel in a co-location way, and are positioned in the same base station, and the base station communicates with all users in a cell at the same time; the higher the frequency is, the higher the dimension of the array is configured, and the noise interference on the high-frequency channel is higher than the low frequency, so that the acquisition of the space domain covariance matrix is more complex; since the statistical CSI on different frequencies shows high similarity in an angle domain, under the condition that a certain low-frequency channel covariance matrix is known, the spatial domain covariance matrix of a high-frequency channel can be directly extrapolated and estimated on the premise of not measuring the high-frequency channel by utilizing the similarity;
step 2, analyzing a cross-band communication system with a uniform linear array (ULA, uniform LinearArray) configured at a base station side, and modeling the relationship between elements in a channel space domain covariance matrix and an angle power spectrum as one-dimensional Fourier transformation based on the characteristic of approximate mutual independence between channel complex gains at different angles;
step 3, modeling the spatial domain covariance matrix elements on different frequencies in the same cross-frequency band communication system as a function only related to array element serial number difference, wavelength and array element spacing by using Fourier transformation relation between the spatial domain covariance matrix elements and the angle spectrum, and further converting estimation of the channel spatial domain covariance matrix between different frequencies into extrapolation problem of the function;
step 4, under the condition that the space between the antenna array elements on different frequencies is half wavelength, based on an autoregressive (AR, autoRegressive) model, establishing a linear extrapolation model of the covariance matrix elements of the spatial domain of the channels on different frequencies, solving a linear extrapolation coefficient by utilizing a Levinson-Durbin algorithm, combining the linear extrapolation method with a traditional cubic spline interpolation method when the space between the antenna array elements is not half wavelength, completing conversion of the covariance matrix corresponding to the channels on different frequencies by combining extrapolation and interpolation, and estimating the covariance of the spatial domain of the channels on other frequencies;
step 5, spreading a covariance matrix conversion method in ULA to a scene of configuring a uniform area array (UPA, uniform PlanarArray) at a base station side, wherein due to the increase of UPA compared with ULA dimension, when the UPA horizontal and vertical array elements are half-wavelength at different frequencies, channel covariance matrix elements can be modeled as a two-dimensional function related to the serial number difference of the horizontal and vertical array elements, a linear extrapolation model between covariance matrices can be established based on a two-dimensional AR model, and the linear extrapolation coefficient is solved by using a method for establishing vector space;
and 6, estimating element values in the covariance matrix of the space domain of the high-frequency channel by using a linear extrapolation model under the condition of configuring ULA and UPA at the base station side, and sequentially arranging the covariance element values obtained by estimation based on the symmetry characteristic of the covariance matrix to obtain the covariance matrix on another frequency so as to finish the conversion of the covariance matrix of the space domain of the channel from the low-frequency channel to the high-frequency channel.
The following describes specific steps of an embodiment of the present invention with reference to specific scenarios:
1) Cross-band MIMO system model
In a multi-frequency communication system, a base station and a user side communicate through channels of different frequencies. Taking a dual-frequency ULA communication system as an example, it is assumed that the frequencies of two-channel communication are f respectively 1 And f 2 ULA arrays are arranged on both frequencies with dimensions of 1 XN respectively 1 And 1 XN 2 . Two ULA arrays in the system are arranged in parallel at the same site and are positioned at the same base station, and the two base stations communicate with all users in a cell at the same time. Although the instantaneous complex gain is different in different channels due to the influence of frequency, the two channels are located in the same environment, especially when f 1 And f 2 When the frequency difference between the two is small, the statistical parameters related to the propagation environment in the angle domain are highly consistent.And when f 1 And f 2 When the frequency difference is large, the channel angle spectrums on the two frequencies are also affected by the frequencies, but still have high correlation, and can be treated as consistent. For simplicity of representation, the angular spectrum is considered as a frequency independent quantity. And the frequency of the channel directly affects the wavelength and thus the array response vector. In order to distinguish channels on different frequencies, for a multi-frequency communication system configuring ULA, let d i And lambda (lambda) i Representing the frequency f i The array element spacing of the upper ULA and the channel wavelength (i=1, 2), then according to the spatial domain model of the channel, the total number of paths is assumed to be P, f i The covariance matrix of the channel between the base station and the kth user on the frequency can be expressed as
Wherein S is k (θ p ) Representing the angle theta p Incident power on, v (θ p ,f i ) Representing the frequency f i Upward θ p Array response vector in direction, expressed as ULA array
Wherein [] T Representing the transpose of the matrix. In summary, the influence of frequency on the channel space domain covariance matrix is mainly reflected on the array response vector, and the values of the element in the row a and the column b in the space domain covariance matrix can be expressed as follows
Therefore, in the multi-frequency system, the element values in the covariance matrix of any channel space domain can be written as the following functional form
That is, elements in the covariance matrix can all be written as argumentsIs a function r of (2) k (. Cndot.) then f 1 Channel covariance corresponding argument in frequency is +.>Function r within a range k Value of (-), f 2 Channel covariance corresponding argument in frequency is +.>Function r within a range k (. Cndot.) take the value, then R is known k (f 1 ) Under the condition of>The function value in the range can be obtained by extrapolationThe function value in the range, thereby obtaining R k (f 2 ) The element value of (a) and (b) avoiding the frequency f 2 And the channel measurement reduces the complexity of system implementation and the calculation overhead. Since in most cases, in order to reduce the aliasing of the beam domain, the spacing between the array elements is half-wavelength, the situation that the spacing between the array elements of all ULA antennas in the same system is half-wavelength is taken, namely +.>At this time, the problem is simplified to R k (f 1 ) And R is k (f 2 ) The element values in the matrix are only related to the rank number difference, namely, the elements of the channel covariance matrix on different frequencies can be written as an independent variable as a function of the rank number difference, namely
[R k (f 1 )] a,b ＝[R k (f 2 )] a,b ＝r k (a-b),
2) Covariance matrix extrapolation in base station side configuration ULA scene
Let m=a-b, let r be the element of row a and column b in the covariance matrix for simplicity of representation k (m) represents the first N of the array signal by 1 The corresponding channel impulse response values of each array element can be extrapolated according to certain coefficient linear combination 1 +1 to N 2 The corresponding channel impulse response values of the individual array elements, reflected on the covariance element, can be expressed as
I.e. the mth covariance element can be used with its top N 1 -a linear combination representation of 1 known covariance matrix elements, σ 2 Error power representing linear combination prediction, delta (·) representing impulse function, q e [1, N 1 -1]Representing forward linear extrapolation order, a 1 (q) is a linear extrapolation coefficient corresponding to the order q, and based on the extrapolation model, m E [ N ] 1 ,N 2 -1]All r corresponding to k The values of (m) can be obtained by linear combination of known covariance functions one by one. Similarly, for m E [1-N ] 2 ,-N 1 ]All r at the time k (m) the following push-back model can be built
P E [1, N ] in the above 1 -1]Representing forward linear extrapolation order, a 2 (p) is a linear extrapolation coefficient corresponding to the order p, and a is obtained according to the conjugate symmetry characteristic of the one-dimensional sequence autocorrelation function 1 (p) and a 2 (q) also satisfies the nature of conjugate symmetry, and at the same time compares the forward prediction model and the backward estimation model, the expression of the forward prediction model is consistent with the expression form of the autoregressive (AR, autoRegressive) model, so that the linear prediction coefficient in covariance conversion can be obtained by solving the AR model, thereby the method does not advanceAnd directly completing the mutual conversion of covariance matrixes on different frequencies under the condition of row angle spectrum estimation. Levinson-Durbin is a traditional algorithm for solving the AR coefficients, and is accomplished in an iterative manner. A is a 1 The calculation procedure of (q) is exemplified in the calculation of a 1 In the process of (q), N is performed 1 -1 iteration, the number of iterations is indicated by a superscript, then in the ith iteration, let k i Representing the scaled value of the coefficient in the ith iteration and defining the parameter epsilon i ＝ε i-1 [1-(k i ) 2 ]Then k i The calculation mode is that
Using coefficient k i Establishing linear extrapolation coefficientsAnd the coefficients obtained in the last iteration +.>Extrapolation relationship between:
and the algorithm starts epsilon 0 ＝r k (0). In the process of N 1 -obtaining a after 1 iteration 1 The final value of (q), i.eCoefficient a in the other extrapolation direction 2 (p) can be directly applied to a 1 And (q) obtaining the conjugate.
3) Covariance matrix extrapolation in a base station side configured UPA scene
The linear extrapolation model in the analogy ULA, under ULA conditions there are forward prediction and backward estimation of the two extrapolated directions, whereas for UPA, with increasing dimensions, the two serial number variables m and n in horizontal and vertical directions correspond to the two extrapolated directions forward and backward respectively,the extrapolation model of UPA thus corresponds to a total of four extrapolated directions. Analysis of the case with forward prediction for both m and n, i.e. by knowing R k (f 1 ) R provided in (1) k (m, N), extrapolated m, N ε [ N ] 1 ,N 2 -1]R corresponds to k (m, n), the linearly extrapolated model may be expressed as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 1 (q, l) represents the extrapolation coefficient, delta (·) represents the two-dimensional impulse function, sigma 2 Representing the error power.
Normalizing the linearly extrapolated coefficientsThe model of linear extrapolation above may be further expressed as
By usingRepresenting the space formed by all M N-dimensional complex matrices, the method of solving the linear extrapolation model in UPA can be converted into the method of solving all linear extrapolation coefficients b when it is composed 1 Coefficient vector formed by (q, l)>I.e.
b 1 ＝[b 1 (0,0) b 1 (1,1)…b 1 (N 1 -1,1) …b 1 (N 1 -1,N 1 -1) … b 1 (1,N 1 -1)] T
Thus the linear extrapolation model can be written in the form of matrix multiplication, then the vector b is solved 1 The problem of (2) can be converted into solving an equation
Wherein the coefficient matrix R AR From R k (f 1 ) Provided forA known correlation coefficient r k (m, n) and the matrix is arranged as follows
Wherein R is i,j (i,j＝0,1,…,N 1 1) a matrix block, when neither i nor j is 0,and the elements in the matrix are arranged as
And when i=0, j+.0,and the elements are arranged as
R i,j ＝[r k (-j,-1) … r k (-j,-j) … r k (-1,-j)],
Conversely, when i noteq0, j = 0,and the elements are arranged as
R i,j ＝[r k (i,1) … r k (i,i) … r k (1,i)] T
The algorithm for solving the linear extrapolation coefficient equation can be implemented by establishing a vector space, and the detailed process is as follows:
step 1: given the definition below, by allComplex value space omega, # constituted by vectors in (2)>Forming a group of substrates in the space, wherein e i A vector having an i-th element of 1 and the remainder of 0. Two vectors v and +.>With respect to R AR Is the inner product of (2)
Where v (q, l) and w (q, l) are elements in v and w. Definition of an analog inner product, defining a set of vectors in the set ΩSo that any two vectors in the set of vectors satisfy the relation R AR Orthogonalization, i.e
Step 2: according to the definition above, the linear extrapolation vector b 1 Can be regarded asBy a coefficient alpha i Linearly combined vectors, denoted +.>The matrix equation of the linear extrapolation model can be further expressed as
Based on the above definition, the two ends of the equation are respectively multiplied byAccording to the definition in step 1, then for any i ε [1, (N) 1 -1) 2 +1]Is available in the form of
The linear extrapolation vector can be expressed as
Solving the corresponding linear extrapolation model of the multi-frequency communication system consisting of UPA can be converted into solving a set of R-related spatial AR Orthogonal bases.
Step 3: according to the definition above, the Gram-Schmidt method of analogically solving unit orthorhombic basis in vector space is applied to basis in any one group ΩConstruction is related to R AR The method of orthogonal group base is that
Then for i=2, 3, …, (N) 1 -1) 2 +1, with
Converting any group of radicals in space to R by the above method AR Mutually orthogonal substrates. After a group of bases satisfying the conditions are obtained by the above method, the obtained bases are normalized and substituted into the expression in step 2The linear extrapolation coefficient in forward prediction can be obtained.
Step 4: since the UPA corresponds to four linear extrapolation directions, a similar linear extrapolation model can be built for the other three extrapolation directions. Assuming m forward prediction, n is backward estimation, then the extrapolation model is built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 2 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing the error power. Using the model, mE [ N ] can be extrapolated 1 ,N 2 -1]And N is E [1-N ] 2 ,-N 1 ]All r in the range k (m, n). For simultaneous backward estimation of m and n, the extrapolation model can be built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 3 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing the error power. Using the model, m, n.epsilon.1-N can be extrapolated 2 ,-N 1 ]All r in the range k (m, n). Whereas for the m backward estimation, n is the forward prediction, then the extrapolation model is built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 4 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing the error power. Using the model, mE [1-N ] can be extrapolated 2 ,-N 1 ]And N is E [ N ] 1 ,N 2 -1]All r in the range k (m, n). The extrapolation coefficients corresponding to all the above prediction directions can be solved by adopting the method of constructing vector space and orthogonal basis, so far, the known r can be utilized k (m, n) extrapolation of all R k (f 2 ) The elements in (2) can be used for preparing r by using the arrangement form of the massive toeplitz k (m, n) are arranged in a certain form, and R is further defined as k (f 2 ) And (5) reconstructing.
In summary, the channel covariance estimation method provided by the invention can achieve accurate covariance estimation performance under different scenes, especially when the number of paths in the scene is within the order of the linear extrapolation model, is suitable for the scene of a base station side provided with ULA and UPA arrays, greatly reduces the complexity of algorithm realization by using a linear method, and simultaneously avoids the frequency f 2 And the channel measurement reduces the measurement overhead.
Claims (4)
1. A cross-band MIMO space domain statistical CSI estimation method is characterized by comprising the following steps of: the method comprises the following steps:
in a cross-frequency band communication system, transmitting antenna arrays corresponding to different frequencies are arranged in parallel in a co-location way, and are positioned in the same base station, and the base station communicates with all users in a cell at the same time;
step 2, analyzing a cross-band communication system of a uniform linear array ULA configured at a base station side, and modeling the relationship between elements in a channel space domain covariance matrix and an angle power spectrum as one-dimensional Fourier transformation based on the characteristic of approximate mutual independence between channel complex gains at different angles;
step 3, modeling the spatial domain covariance matrix elements on different frequencies in the same cross-frequency band communication system as a function only related to array element serial number difference, wavelength and array element spacing by using Fourier transformation relation between the spatial domain covariance matrix elements and the angle spectrum, and further converting estimation of the channel spatial domain covariance matrix between different frequencies into extrapolation problem of the function;
step 4, under the condition that the space between the antenna array elements at different frequencies is half wavelength, based on an autoregressive AR model, establishing a linear extrapolation model of the covariance matrix elements of the spatial domain of the channels at different frequencies, solving a linear extrapolation coefficient by utilizing a Levinson-Durbin algorithm, combining the linear extrapolation method with a traditional cubic spline interpolation method when the space between the antenna array elements is not half wavelength, and completing conversion of the covariance matrix corresponding to the channels at different frequencies in a mode of combining extrapolation and interpolation to estimate the covariance of the spatial domain of the channels at other frequencies;
step 5, spreading a covariance matrix conversion method in ULA to a scene of configuring uniform area array UPA at a base station side, modeling a channel covariance matrix element as a two-dimensional function related to sequence number difference of horizontal and vertical array elements when the UPA is half wavelength between the array elements in the horizontal and vertical directions on different frequencies due to the increase of UPA compared with ULA dimension, establishing a linear extrapolation model between covariance matrices based on a two-dimensional AR model, and solving the linear extrapolation coefficient by using a method for establishing vector space;
step 6, estimating element values in a covariance matrix of a high-frequency channel space domain by using a linear extrapolation model under the condition of configuring ULA and UPA at a base station side, and sequentially arranging the covariance element values obtained by estimation based on the symmetry characteristic of the covariance matrix to obtain a covariance matrix on another frequency so as to finish the conversion of the covariance matrix of the channel space domain from a low-frequency channel to a high-frequency channel;
wherein,
in the cross-band communication system described in step 3, under the condition that the array element spacing is half wavelength on all frequencies, letting any frequency f i The covariance matrix of the channel between the upper base station and the user k is R k (f i )，R k (f i ) All elements of row a and column b of the list can be written as a function r with an argument of a-b k (. Cndot.) i.e
[R k (f i )] a,b ＝r k (a-b).
In the same cross-frequency communication system, two frequencies in the system are f respectively 1 And f 2 And has f 1 ＜f 2 ，f 1 And f 2 Corresponding to the ULA array dimensions of 1 XN 1 1 XN 2 Let m=a-b, then frequency f 1 Upper channel covariance matrix R k (f 1 ) From [1-N 1 ,-1]∪[0,N 1 -1]R corresponding to all integer values m in the range k (m) composition, frequency f 2 Upper channel covariance matrix R k (f 2 ) From [1-N 2 ,-1]∪[0,N 2 -1]R corresponding to all integer values m in the range k (m) composition; at a known R k (f 1 ) Under (2) to estimate R k (f 2 ) The problems of (1) can be translated into those of the known [1-N ] 1 ,-1]∪[0,N 1 -1]All integer values m in the range correspond to r k Under the condition of (m), extrapolating [1-N 2 ,-N 1 ]∪[N 1 ,N 2 -1]R corresponding to all integer values m in the range k (m) value, i.e., function extrapolation problem;
said R is k (f 1 ) The linear extrapolation model described in step 4, given the known premise, is expressed as:
i.e. any unknown covariance element r k (m) the first N 1 -a linear combination representation of 1 known covariance matrix elements, σ 2 Represents the error power of the linear extrapolation, q.epsilon.1, N 1 -1]Representing forward linear extrapolation order, a 1 (q) is a linear extrapolation coefficient corresponding to the order q, and delta (·) represents an impulse function; based on the extrapolation model, m E [ N ] 1 ,N 2 -1]All r corresponding to k (m) obtaining the values by linear combination of known covariance functions one by one; similarly for m E [1-N ] 2 ,-N 1 ]All r at the time k (m) the following backward predictive model can be built
Wherein p is E [1, N 1 -1]Representing the backward linear extrapolation order,a 2 (p) is the linear extrapolation coefficient corresponding to the order p, then r k (m) post-availability N 1 -a linear combination representation of 1 known covariance matrix elements;
in step 4, when the antenna element spacing is not half a wavelength, the channel covariance matrix elements at different frequencies are modeled as a function related to only the element sequence number difference, the wavelength and the element spacing, for the frequency f i The array element spacing and the wavelength of the base station side array corresponding to the frequency are respectively d i And lambda (lambda) i Then the elements in the covariance matrix are used as a function r k (. Cndot.) is expressed as:
at this time from R k (f 1 ) Estimating R k (f 2 ) Solving the problem combines a linear extrapolation method with a cubic spline interpolation, and solves all independent variable values using the linear extrapolation methodCorresponding +.>While->When the number is non-integer, the linear extrapolation method is used to obtain the sum +.>R corresponding to the nearest two integer values k (. Cndot.) then interpolating between the two values using a cubic spline interpolation method to obtain +.>
In step 5, when the base station side configures UPA, UPA levels at different frequenciesThe distance between the array elements in the vertical direction is half wavelength, and the two frequencies in the system are f respectively 1 And f 2 And has f 1 ＜f 2 ，f 1 And f 2 The dimension of UPA array configured corresponding to the base station side is N respectively 1 ×N 1 N 2 ×N 2 ，f 1 And f 2 The covariance matrix of the channel between the upper base station side and the user k is respectively represented by R k (f 1 ) And R is k (f 2 ) A representation; analog ULA array, R k (f 1 ) And R is k (f 2 ) All elements in (a) can use a two-dimensional function r k (m, n) represents, according to the known R k (f 1 ) Estimating R k (f 2 ) The problem of the medium element can be converted into a linear extrapolation model as follows; in UPA, r is increased by the dimension, similar to ULA k In (m, n), two independent variables m and n respectively correspond to two extrapolation directions and coexist in four linear extrapolation directions; m, n are forward predicted, then R is known k (f 1 ) Under the condition of (1) extrapolating m, N E [ N ] 1 ,N 2 -1]Time-corresponding covariance matrix element r k (m, n), the model of the linear extrapolation is expressed as:
wherein q and l each independently represent r k Extrapolation order of the argument in (m, n) in two dimensions, a 1 (q, l) represents a linear extrapolation coefficient, delta (m, n) represents a two-dimensional impulse response function, sigma 2 Representing the error power of the linear extrapolation;
m forward prediction, n is backward estimation, then extrapolation model is built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 2 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing errorsThe difference power can be extrapolated to mE [ N ] by using the model 1 ,N 2 -1]And N is E [1-N ] 2 ,-N 1 ]All r in the range k (m,n)；
For simultaneous backward estimation of m and n, the extrapolation model can be built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 3 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing error power, using the model, m, N E [1-N ] can be extrapolated 2 ,-N 1 ]All r in the range k (m,n)；
m backward estimation, n is forward prediction, then extrapolation model is built as
Where q and l represent the extrapolation order in two independent dimensions, a, respectively 4 (q, l) represents the extrapolation coefficient in this direction, delta (·) represents the two-dimensional impulse function, σ 2 Representing the error power.
2. The cross-band MIMO spatial domain statistical CSI estimation method according to claim 1, wherein: the step 4 of solving the linear extrapolation coefficient by using the Levinson-Durbin algorithm is to solve the linear extrapolation coefficients a in two extrapolation directions by using the Levinson-Durbin algorithm 1 (q) and a 2 (p) iteratively, in calculating a 1 In the process of (q), N is performed 1 -1 iteration, the number of iterations is indicated by a superscript, then in the ith iteration, let k i Representing the scaled value of the coefficient in the ith iteration and defining the parameter epsilon i ＝ε i-1 [1-(k i ) 2 ]Then k i The calculation mode is that
Using coefficient k i Establishing linear extrapolation coefficientsAnd the coefficients obtained in the last iteration +.>Extrapolation relationship between:
and the algorithm starts epsilon 0 ＝r k (0) In the process of N 1 -obtaining a after 1 iteration 1 The final value of (q), i.eWhile coefficients in the other extrapolation direction are calculated based on the conjugate symmetry properties of the covariance matrix, i.e. +.>
3. The cross-band MIMO spatial domain statistical CSI estimation method according to claim 1, wherein: the linear extrapolation coefficient in the step 5 is solved by using a method for establishing a vector space, and the specific steps include:
a. according to the known R k (f 1 ) Extracting all elements therein as known r k The (m, n) function takes the value to form a coefficient matrix, and converts the linear extrapolation model into the form of a matrix multiplied equation set, wherein the coefficient matrix of the equation set is R AR All solved a 1 (q, l) forming a coefficient vector b to be solved 1 ；
b. Give two vectors v and w with respect to R AR Definition of orthogonality, using orthogonalization method, find R in complex space AR An orthogonal set of bases;
c. to be solved for coefficient vector b 1 Use of R AR An orthogonal set of baseline combinations is shown.
4. The cross-band MIMO spatial domain statistical CSI estimation method according to claim 1, wherein: after the element values in the covariance matrix are solved by using the linear extrapolation model in the step 6, the covariance matrix of the ULA meets the toeplitz symmetry property, the covariance matrix of the UPA meets the massive toeplitz symmetry property, and the extrapolated covariance elements are arranged by utilizing the symmetry properties of the covariance matrix, so that all covariance matrices on the other frequencies are reconstructed.
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