CN110650103A - Lens antenna array channel estimation method for enhancing sparsity by using redundant dictionary - Google Patents

Lens antenna array channel estimation method for enhancing sparsity by using redundant dictionary Download PDF

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CN110650103A
CN110650103A CN201910879291.9A CN201910879291A CN110650103A CN 110650103 A CN110650103 A CN 110650103A CN 201910879291 A CN201910879291 A CN 201910879291A CN 110650103 A CN110650103 A CN 110650103A
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pilot
matrix
antenna array
receiving end
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CN110650103B (en
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高镇
万子维
廖安文
肖振宇
杨凯
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Beijing University of Technology
Beijing Institute of Technology BIT
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/0224—Channel estimation using sounding signals
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01Q—ANTENNAS, i.e. RADIO AERIALS
    • H01Q19/00—Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic
    • H01Q19/06—Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using refracting or diffracting devices, e.g. lens
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01Q—ANTENNAS, i.e. RADIO AERIALS
    • H01Q21/00—Antenna arrays or systems
    • H01Q21/0006—Particular feeding systems
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01Q—ANTENNAS, i.e. RADIO AERIALS
    • H01Q23/00—Antennas with active circuits or circuit elements integrated within them or attached to them
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0456—Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting
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    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
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Abstract

The invention discloses a lens antenna array MIMO channel estimation method for enhancing sparsity by utilizing a redundant dictionary. In the millimeter wave lens antenna array MIMO, the receiving and transmitting ends adopt an antenna selection network to replace a full connection-phase shift network, so that the hardware cost and the power consumption of a communication system are reduced; in the pilot frequency interaction stage, the antenna selection network obtains the channel detection result of partial angle range in each switching; through multiple times of switching, the radio frequency link traverses and connects all array elements of the lens antenna array in a plurality of continuous time slots, thereby completely observing a channel by using limited hardware resources; and the receiving end performs further sparse representation on the channel of the original sparse lens antenna array by using the redundant dictionary, and then performs channel recovery by adopting a compressed sensing theory. By utilizing the redundant dictionary, the power leakage caused by the continuous distribution of the AoDs and the AoAs of the channels can be resisted, so that the channel representation is sparser, and the performance of a subsequent sparse signal recovery algorithm is facilitated.

Description

Lens antenna array channel estimation method for enhancing sparsity by using redundant dictionary
Technical Field
The invention relates to the field of channel estimation in mobile communication, in particular to a millimeter wave lens antenna array MIMO channel estimation method for enhancing sparsity by utilizing a redundant dictionary.
Background
The millimeter wave technology is a communication technology with wide application prospect in a next generation mobile communication (5G) system. The ultra-large bandwidth and ultra-high data rate provided by millimeter waves can strongly support the rapidly growing user demand in 5G. In order to fully utilize the advantages of millimeter waves, a large-scale MIMO (Multiple-Input Multiple-Output) system based on an analog/digital hybrid architecture is widely used for a communication transceiver, but a huge phase shifter network widely used in the hybrid architecture brings hard-to-bear hardware cost and power consumption to the system.
In order to improve the cost efficiency of the communication system, a brand new antenna array form, namely a lens antenna array, is applied to the field of communication. By utilizing the angle-based energy focusing characteristics of the lens antenna array and the angular sparsity of the millimeter wave channel, the lens antenna array-based communication system can achieve data capacity optimization over a very small number of radio frequency links. However, the use of very few rf links will greatly reduce the dimension of receiving the pilot signal, making the channel estimation problem in this system more challenging to study.
For millimeter wave MIMO systems based on lens antenna arrays, various channel estimation methods have been proposed at present. Specifically, y.zeng et al, using the energy focusing property of a lens antenna array, decomposes a MIMO channel based on a lens antenna into several SISO (Single-input Single-Output) channels under the assumption that AoDs (angle of Departure) and AoAs (angle of Arrival) of the channels fall within the angle-resolving range of the lens antenna array, and estimates each SISO channel separately using a conventional LS (Least square) algorithm. Qi et al, by analyzing the MIMO channel based on the lens antenna array, finds out the "double cross" structure of the channel in the supporting set distribution, and thus proposes a supporting set detection method to estimate the channel, which has better estimation performance than the traditional OMP (Orthogonal Matching Pursuit) algorithm. Jin et al propose a deep learning model for MIMO channel estimation based on lens antenna arrays, making it possible to estimate MIMO channels for lens antenna arrays using deep learning. Wen et al treat the MIMO channel based on a lens antenna array as an image signal with compressibility and estimate the channel using well-established image restoration techniques.
However, for the above method of estimating MIMO channel based on lens antenna array, on one hand, in the process of modeling channel estimation problem, in order to ensure that the structure of the channel is not destroyed, it is often assumed that aods (aoas) of the channel are sufficiently spaced from each other at the transmitting end (receiving end). This assumption is difficult to satisfy in a practical system, and thus the aforementioned channel estimation algorithm is difficult to take advantage of in practice; on the other hand, in order to ensure that an observation matrix during channel estimation satisfies RIP (restricted isometry property) to ensure the performance of a sparse signal recovery algorithm, the foregoing methods mostly use a complex fully-connected phase shifter network to randomize a transmitted pilot signal, so as to assist MIMO channel estimation based on a lens antenna array, which is contradictory to the original design of the lens antenna array, namely space diversity is realized by antenna selection and hardware complexity is reduced. The very large number of phase shifters required for a fully connected-phase shifter network would result in unacceptable hardware cost and power consumption for the system.
Disclosure of Invention
In view of the above, the present invention provides a millimeter wave lens antenna array MIMO channel estimation method using a redundant dictionary to enhance sparsity, which solves the problems of hardware cost and power consumption caused by a huge phase shifter network in the conventional channel estimation method by using a simple antenna selection network. Meanwhile, the invention firstly proposes to use the redundant dictionary in the channel estimation of the originally sparse lens antenna array, so that the performance of the channel estimation can be effectively improved.
In order to solve the technical problem, the invention is realized as follows:
a millimeter wave lens antenna array MIMO channel estimation method for strengthening sparsity by using a redundant dictionary comprises the following steps:
in the millimeter wave lens antenna array MIMO, an antenna selection network is adopted for both a transmitting end and a receiving end to replace a full-connection-phase shift network; in the antenna selection network, the number of radio frequency links is less than that of antennas, each radio frequency link can be connected with only one antenna at the same time, and each antenna can be connected with only one radio frequency link at the same time;
in the pilot frequency interaction stage, when the antenna selection network is switched every time, the radio frequency link is connected with part of array elements to obtain a channel detection result in a part of angle range; through multiple switching, the radio frequency link is in continuous NpilotTraversing and connecting all array elements of the lens antenna array in each time slot to obtain a complete channel observation result;
thirdly, the receiving end performs further sparse representation on the channel of the originally sparse lens antenna array by utilizing the redundant dictionary; and recovering the representation of the original channel on a redundant dictionary by adopting a compressed sensing theory based on the channel observation result and the known pilot signal, and further obtaining the estimation H of the original channel.
Preferably, let NTAnd NRThe number of antennas at the transmitting end and the receiving end respectively;and
Figure BDA0002205374450000035
the number of radio frequency links equipped for a transmitting end and a receiving end respectively; using NpilotEstimating each pilot signal;
the second step comprises the following steps:
the channel estimation is converted into the following form:
Figure BDA0002205374450000031
wherein, (.)HIs a transposed symbol, vec (-) is a column-wise vectorized symbol,
Figure BDA0002205374450000032
representing the Kronecker product, diag (·) representing diagonalized symbols; total received signal of receiving end
Figure BDA0002205374450000033
Total received noise
Figure BDA0002205374450000034
ynFor the pilot data vector received at the receiver of the nth slot, N is 1,2pilot，nnIs complex white Gaussian noise, s, at the receiving end of the nth time slotnA pilot signal transmitted for an nth slot; h is a millimeter wave channel under the lens antenna array; hybrid combiner F at the nth slot transmitting endn＝FRF,nFBB,nIs an analog precoder FRF,nAnd baseband digital precoder FBB,n(iii) a cascade of; hybrid combiner W at the receiving end of the nth time slotn＝WRF,nWBB,nIs an analog combiner WRF,nAnd baseband digital combiner WBB,n(iii) a cascade of;
wherein: let FBB,n,
Figure BDA0002205374450000041
Each element in the (1) is a zero-mean complex Gaussian variable subject to independent and same distribution;
in all, N is carried outpilotSecondary switching; let GRF,nRepresents FRF,nOr WRF,nLet N represent NTOr NR，NRFRepresents
Figure BDA0002205374450000042
OrNTAnd NRThe number of antennas at the transmitting end and the receiving end respectively;
Figure BDA0002205374450000044
andthe number of radio frequency links equipped for a transmitting end and a receiving end respectively;
at the time of the nth switching, for GRF,nPerforming steps 1) and 2):
step 1) if
Figure BDA0002205374450000046
Then order
Figure BDA0002205374450000047
Represents a matrix formed by randomly rearranging an NxN unit matrix among columns, wherein a symbol mod (a, b) represents a remainder of dividing a by b; then step 2) is executed; otherwise, directly executing the step 2);
step 2) order
Figure BDA0002205374450000048
Here, the
Figure BDA0002205374450000049
Symbol [ A ]]p:qRepresenting a sub-matrix consisting of the p-th to q-th columns of matrix A, symbols
Figure BDA00022053744500000410
Representing rounding down on a.
Preferably, let NTAnd NRThe number of antennas at the transmitting end and the receiving end respectively,
Figure BDA00022053744500000411
and
Figure BDA00022053744500000412
the number of radio frequency links equipped for the transmitting end and the receiving end respectively requires that the number of antennas at the transmitting and receiving ends is integral multiple of the number of radio frequency links; definition of
Figure BDA00022053744500000413
The second step comprises the following steps:
a plurality of time slots form a time block; considering the total use of transmitting ends
Figure BDA00022053744500000414
Transmit pilot frequency in one time block and require
Figure BDA00022053744500000415
Is the number of blocks
Figure BDA00022053744500000416
Integer multiples of; is provided withThe pilot signal transmitted by the transmitting end in the mth time block is represented as:
sm＝FRF,pfBB,m
where FRF,pAnd fBB,mRespectively a radio frequency part for transmitting pilot frequency and a baseband part
Figure BDA0002205374450000051
Symbol
Figure BDA0002205374450000052
Represents rounding up on a; this indicates that, in totalIn each pilot, every continuation
Figure BDA0002205374450000054
One base band pilot frequency shares one radio frequency pilot frequency FRF,p；
At the receiving end, each time block is further decomposed into
Figure BDA0002205374450000055
A plurality of time slots with equal length, in the nth time slot,the pilot signal received by the receiving end from the mth time block can be expressed as
Figure BDA0002205374450000057
Wherein H is a millimeter wave channel under the lens antenna array; n isn,mIs complex white Gaussian noise at the receiving end, and
Figure BDA0002205374450000058
WRF,nand WBB,nAnalog combiner W in hybrid combiner for nth slot receiverRF,nAnd baseband digital combiner WBB,n；
On the receiving endPilot receiving signal for all m time blocks in continuous time slot
Figure BDA00022053744500000510
Performing combined treatment to obtain
Here:
Figure BDA00022053744500000513
Figure BDA00022053744500000514
Figure BDA00022053744500000515
pilot reception signal for all time blocksPerforming combined treatment to obtain
Here:
Figure BDA00022053744500000518
Figure BDA0002205374450000061
vectorizing the formula (I) to obtain the final product
Figure BDA0002205374450000062
This step is a two-pair simulation precoder FRFThe design is as follows: get
Figure BDA0002205374450000063
Represents NT×NTA matrix formed by randomly rearranging the unit matrix among columns
Figure BDA0002205374450000064
To analog combiner WRFThe design is as follows: get
Figure BDA0002205374450000065
Represents NR×NRA matrix formed by randomly rearranging the unit matrix among columns
Figure BDA0002205374450000066
Using designed analog precoder FRFAnd an analog combiner WRFAnd finishing the pilot interaction.
Preferably, the third step is:
step 31, setting redundant dictionary A of receiving end and transmitting endR、ATRespectively as follows:
Figure BDA0002205374450000067
Figure BDA0002205374450000068
Figure BDA0002205374450000069
wherein, aRAnd aTThe guide vectors of the lens antenna array are respectively assembled for the receiving end and the transmitting end; ghAnd GvThe angle resolution unit numbers of the redundant dictionary in the horizontal direction and the vertical direction are respectively; the T or R of the subscript represents the transmitting end or the receiving end, and the n of the subscript represents the nth element in the vector;
Figure BDA00022053744500000611
and
Figure BDA00022053744500000612
respectively, a normalized vertical aperture and a normalized horizontal aperture of the full-dimensional lens antenna array; (alphan,βn) Is the angle coordinate of each antenna distribution in the lens antenna array;
Figure BDA00022053744500000613
representing the departure angle AoD and the arrival angle AoA in the vertical direction and the horizontal direction on the ith path in the L main sparse multipath components; gamma rayRAnd gammaTRespectively normalizing factors of guide vectors of a receiving end and a transmitting end;
step 32, the channel matrix H is represented as follows
Figure BDA0002205374450000071
Wherein HaThe method is characterized in that sparse representation of a channel of a lens antenna array on a redundant dictionary is also a sparse matrix, and E is a quantization error;
step 33, substituting the channel matrix H into the channel estimation model to obtain:
Figure BDA0002205374450000072
wherein phi is an observation matrix, and phi is,
Figure BDA0002205374450000073
as a total redundant dictionary matrix, symbol (.)*Denotes the conjugation, neffΦ vec (e) + n is the equivalent noise vector, n is the total noise vector;
step 34, firstly, using the channel estimation model to HaMaking an estimate and then substituting into the formula
Figure BDA0002205374450000074
The original channel H is obtained.
Preferably, the precoder for obtaining the optimized baseband pilot signal is designed by using the principle of minimizing the total cross-correlation of the sensing matrix
Figure BDA0002205374450000075
Merging device
Figure BDA0002205374450000076
Comprises the following steps:
Figure BDA0002205374450000077
Figure BDA0002205374450000078
wherein
Figure BDA0002205374450000079
Is to satisfy the condition
Figure BDA00022053744500000710
Of arbitrary matrix, symbol INThe unitary matrix is expressed by NxN, i.e. a sub-matrix composed of some columns of an arbitrary unitary matrix is used as the unitary matrix
Figure BDA00022053744500000711
Is arbitraryIs used to generate the unitary matrix.
Preferably, when the method is applied to a millimeter wave lens antenna array MIMO-OFDM system, the steps from one to three are executed for each subchannel.
Has the advantages that:
(1) the present invention can greatly reduce the hardware cost and power consumption of a communication system by allowing for the use of a simple antenna selection network in the lens antenna array instead of a full link-phase shifter network.
(2) The invention considers the strict hardware limiting condition brought by the antenna selection network, designs two pilot frequency transmission schemes for estimating the channel, and can utilize the limited hardware resources to carry out complete observation on the channel.
(3) The invention provides that a redundant dictionary is utilized in the channel representation of the lens antenna array, so that the power leakage caused by the continuous distribution of the AoDs and the AoAs of the channels can be resisted, the channel representation is sparser, and the performance of a subsequent sparse signal recovery algorithm is facilitated.
(4) In the invention, under the theory of compressed sensing, the pilot frequency signal used in the channel estimation stage is designed by minimizing the total cross-correlation of the sensing matrix, and the designed pilot frequency can effectively improve the channel estimation performance.
Drawings
Fig. 1 is a block diagram of a millimeter wave lens antenna array MIMO system using an antenna selection network.
FIG. 2 is a schematic diagram of a channel estimation method according to the present invention
FIG. 3 is a comparison graph of NMSE (Normalized Mean Square Error) performance evaluation of different channel estimation methods as a function of channel multipath number L at signal-to-noise ratios of {0,10,20} dB.
Fig. 4 is a comparison graph of NMSE performance evaluation as a function of signal-to-noise ratio for different channel estimation methods at the same pilot overhead.
Detailed Description
The main idea of the invention is to replace the traditional fully connected-phase shifter network with a simple antenna selection network for channel estimation for MIMO systems based on lens antenna arrays. The antenna selection network can effectively reduce the hardware complexity and power consumption of the system, but it makes the system face more strict hardware limitation when algorithm design is performed.
In consideration of the hardware limitations, the invention provides two pilot frequency transmission schemes under the framework of compressed sensing, and in a plurality of set time slots, a radio frequency link traverses all array elements of a lens antenna array, so that the channel can be detected in all directions, and the pilot frequency overhead is reduced.
In order to solve the problem of power leakage caused by continuous distribution of AoDs and AoAs, the invention firstly proposes to design a redundant dictionary for the lens antenna array, improves the sparsity of channels and is beneficial to the implementation of a subsequent sparse signal reconstruction algorithm.
In addition, in a fully-connected phase shifter network, the traditional pilot design relies on the RIP theory, and adopts independent and equally-distributed random sequence sequences as pilot sequences. When the antenna selection network is applied in the MIMO system, the strict hardware limitation will make this pilot design method completely ineffective. In order to solve the problem, the invention provides a pilot signal design based on the principle of minimizing the total cross-correlation of the sensing matrix, which can obviously improve the accuracy of channel estimation.
The invention is described in detail below by way of example with reference to the accompanying drawings.
Step one, setting an antenna selection network
In order to solve the problems of hard-to-bear hardware complexity and power consumption caused by a full-connection phase shifter network in the traditional scheme, the invention adopts a simple antenna selection network to replace the full-connection phase shifter network at a transmitting end and a receiving end in the millimeter wave lens antenna array MIMO.
As shown in fig. 1. In the system, a transmitting end and a receiving end are respectively provided with NTAnd NRLens antenna array of root antenna, and
Figure BDA0002205374450000091
and
Figure BDA0002205374450000092
and a radio frequency link, wherein the number of the radio frequency links is less than that of the antennas. In consideration of the hardware limitation of the antenna selection network, each radio frequency link can be connected with only one antenna at the same time, and each antenna can be connected with only one radio frequency link at the same time. In other words, when the antenna selects the network application, all N of the transmitting end are in one slotTOnly one antenna can have
Figure BDA0002205374450000093
The subset composed of the root antenna array elements is activated for use; similarly, in a time slot, all N of the receiving endROnly one antenna can have
Figure BDA0002205374450000094
A subset of the root antenna element elements is activated for use.
And step two, pilot signals are interacted between the transmitting end and the receiving end.
As previously mentioned, in contrast to the fully connected-phase shifter networks that have been widely used previously, systems using antenna selection networks can only activate a subset of the antenna elements, but not all of them, in a time slot, and thus cannot completely probe the entire angular range of the channel in a time slot. This undoubtedly renders the pilot transmission scheme designed for conventional fully-connected-phase shifter networks ineffective in antenna selection networks. The method aims to solve the integrity problem caused by the antenna selection network set in the step one. The invention provides a pilot frequency transmission scheme, which mainly comprises the following steps: the antenna selection network performs multiple switching, and during each switching, the radio frequency link is connected with part of array elements to obtain channel detection results in part of angle ranges; then the radio frequency link is at consecutive N via multiple handoverspilotAnd traversing and connecting all array elements of the lens antenna array in each time slot to obtain a complete channel observation result. For design convenience, the number of array elements is generally designed to be just an integral multiple of the number of radio frequency links.
The embodiment of the invention provides two implementation schemes.
Scheme 1, pilot frequency transmission method based on sparse matrix
The method has the following steps: in each time slot, the baseband pilot signal of the transmitting and receiving end is assigned as a complex Gaussian distribution random variable, and different values are given to the analog pre-coding/combining part of the transmitting and receiving end. By designing the value of the analog precoding/combining part, when the pilot frequency overhead (the total time slot number used for transmitting the pilot frequency) is greater than a certain value, all array elements of the lens antenna array can be activated, so that the complete detection of a channel is ensured. Scheme 1 has less requirements on system parameters, and the selection of the number of antennas, the number of radio frequency links and the number of pilot frequency overhead is relatively free.
The implementation of scheme 1 is given below. In the system shown in FIG. 1, the transmitting end is atTransmitting a pilot signal s in the nth time slotnA pilot data vector y received at the receiving endnComprises the following steps:
here, (.)HFor conjugate transpose symbols, hybrid combiner W at the receiving endn＝WRF,nWBB,nIs formed by an analog combiner WRF,nAnd baseband digital combiner WBB,nCascaded, H is millimeter wave channel under lens antenna array, and hybrid combiner F at transmitting endn＝FRF,nFBB,nThen it can be regarded as an analog precoder FRF,nAnd baseband digital precoder FBB,nIs cascaded. snAnd nnThe pilot signal vector sent in the nth time slot and the complex white gaussian noise of the receiving end are respectively. Note that since here the analog precoder FRF,nAnd an analog combiner WRF,nAre implemented by antenna selection networks and the limitations they face will be more severe than if phase shifter networks were used. In particular, for matrix FRF,nAnd WRF,nThey can have only one element per column that is non-zero (only one rf link per antenna at a time) and only one element per row that is non-zero (only one antenna per rf link at a time). To normalize the total transmit power, assume FRF,nAnd WRF,nThe non-zero element in (1).
Consider the use of N in totalpilotThe pilot signals are estimated, i.e. N is more than or equal to 1 and less than or equal to NpilotThen, at the receiving end pair NpilotA pilot frequency receiving signal
Figure BDA0002205374450000111
Performing combined treatment to obtain
Figure BDA0002205374450000112
This (·)HIs a transposed symbolVec (·) is a vectorized symbol (by column),
Figure BDA0002205374450000113
representing the Kronecker product, diag (·) representing (block) diagonalized symbols, the total received signal
Figure BDA0002205374450000114
Total received noise
Figure BDA0002205374450000115
The equivalent observation matrix is phi, and the total equivalent noise vector is
Figure BDA0002205374450000116
The task of channel estimation is to recover the channel vec (h) from the total received signal y contaminated by noise and the known observation matrix Φ.
As mentioned above, in the application of the antenna selection network, the pilot signal needs to be designed to ensure complete detection of the channel. Pair F proposed by the inventionn＝FRF,nFBB,n，n＝1,2,...,NpilotThe design method is as follows:
1) let FBB,n,
Figure BDA0002205374450000117
Each element in the (1) is a zero-mean complex Gaussian variable subject to independent and same distribution;
2) at the time of the nth switching, for FRF,nCarrying out steps a) and b)
a) If it is
Figure BDA0002205374450000118
Then order
Figure BDA0002205374450000119
Represents NT×NTThe unit matrix is formed by random rearrangement between columns. Where the symbol mod (a, b) represents the remainder of a divided by b; then step b) is executed; otherwise, directly executing the step b);
b) order to
Figure BDA0002205374450000121
Here, the
Figure BDA0002205374450000122
Symbol [ A ]]p:qRepresenting a sub-matrix consisting of the p-th to q-th columns of matrix A, symbols
Figure BDA0002205374450000123
Representing rounding down on a.
Note that only F in the above step is requiredRF,nAnd FBB,nAre respectively replaced by WRF,nAnd WBB,n，NTAnd
Figure BDA0002205374450000124
are respectively replaced by NRAnd
Figure BDA0002205374450000125
a hybrid combiner W can be obtainedn＝WRF,nWBB,nThe design method of (1) is not described herein again.
As can be seen from the above process, in total NpilotIn each pilot, every continuation
Figure BDA0002205374450000126
All N of the transmitting terminal can be ensured by one pilot frequencyTThe antennas being activated once, and for the same reason, each time in succession without being redundant
Figure BDA0002205374450000127
The pilot frequency can ensure all N of the receiving endRThe root antenna is activated once without being overlooked. Therefore, only need to set
Figure BDA0002205374450000128
It is guaranteed that all directions of the channel are covered by pilots.
Scheme 2, pilot frequency transmission method based on time block
In this scheme, the transmitting end transmits pilots in units of time blocks (multiple slots), each time block sharing the same analog precoding part. In order to ensure the integrity of the channel detection at the transmitting end, the number of blocks during transmission needs to be equal to the number of array elements at the transmitting end divided by the number of radio frequency links. At the receiving end, each time block is reduced to a plurality of time slots. In order to ensure that all time slots can activate the array elements of the lens antenna array without leakage, the number of the time slots contained in each time block is the number of the array elements at the receiving end divided by the number of the radio frequency links. A schematic diagram of the method is given in fig. 2. As can be seen from fig. 2, the scheme 2 has a certain requirement on the relationship between the number of antennas at the transmitting and receiving ends and the number of radio frequency links, that is, the number of antennas is an integer multiple of the number of radio frequency links, and at the same time, has a certain limitation on the number of pilot overheads (total number of slots used) at the transmitting end, that is, the total number of slots used for transmitting pilots at the transmitting end should be an integer multiple of the number of time blocks. Neither of these limitations need be present in scheme 1. Nevertheless, since each time block of scheme 2 uses the same analog precoding part, the signal model is more compact than that of scheme 1, which is beneficial to design the pilot frequency by using the compressed sensing theory subsequently.
The implementation of scheme 2 is given below. In the system shown in FIG. 1, assume NTAnd NRAre respectively
Figure BDA0002205374450000131
And
Figure BDA0002205374450000132
integer multiple of and define
Figure BDA0002205374450000133
In order to effectively utilize the antenna selection network to estimate the MIMO channel of the millimeter wave lens antenna array and to facilitate the subsequent pilot frequency design under the theoretical basis of compressed sensing, the method considers the time block used for transmitting signals in the channel estimation stage. Considering the total use of transmitting ends
Figure BDA0002205374450000134
Transmitting pilot at one time block and assuming
Figure BDA0002205374450000135
Is thatInteger multiples of. In the m-th time block
Figure BDA0002205374450000137
The pilot signal transmitted by the transmitting end is represented as
sm＝FRF,pfBB,m, (3)
Where FRF,pAnd fBB,mRespectively a radio frequency part and a baseband part for transmitting a pilot,
Figure BDA0002205374450000138
symbol
Figure BDA0002205374450000139
Indicating rounding up a. Can be easily seen
Figure BDA00022053744500001310
This indicates that, in total
Figure BDA00022053744500001311
In each pilot, every continuation
Figure BDA00022053744500001312
One base band pilot frequency shares one radio frequency pilot frequency FRF,pAs shown in fig. 2.
At the receiving end, each time block is further decomposed into
Figure BDA00022053744500001313
A plurality of time slots with equal length are arranged in the nth time slot
Figure BDA00022053744500001314
The pilot signal received by the receiving end from the mth time block can be expressed as
Where n isn,mIs complex white Gaussian noise at the receiving end, and
Figure BDA00022053744500001316
WRF,nand WBB,nAnalog combiner W in hybrid combiner for nth slot receiverRF,nAnd baseband digital combiner WBB,n(ii) a On the receiving end
Figure BDA00022053744500001317
Pilot receiving signal for all m time blocks in continuous time slotPerforming combined treatment to obtain
Figure BDA00022053744500001319
Here:
Figure BDA00022053744500001320
Figure BDA00022053744500001321
Figure BDA00022053744500001322
Figure BDA0002205374450000141
it is assumed that the receiving end processes the pilot signal of each time block as above, and receives the pilot signals of all time blocks
Figure BDA0002205374450000142
Performing combined treatment to obtain
Figure BDA0002205374450000143
Here:
Figure BDA0002205374450000144
Figure BDA0002205374450000145
Figure BDA0002205374450000146
vectorizing the formula (6) to obtain
Figure BDA0002205374450000147
As with method 1, the task of channel estimation is to recover the channel vec (h) from the total received signal y contaminated by noise and the known observation matrix Φ. In order to ensure complete detection of the channel and meet the strict hardware limitation introduced by the antenna selection network, the invention provides a total analog precoder FRFAnd total analog combiner WRFThe design is as follows:
get
Figure BDA0002205374450000148
Represents NT×NTA matrix formed by randomly rearranging the unit matrix among columnsIn the same way
Figure BDA00022053744500001410
It will be readily apparent that, with this design, all
Figure BDA00022053744500001411
All antennas are activated once by each pilot without leakage, thereby ensuring the integrity of channel detection. A schematic diagram of the method is given in fig. 2. In the method 2, the observation matrix phi is more compact in form, which is beneficial to the subsequent baseband pilot signal F under the theory of compressed sensingBBAnd (5) designing.
Step three: and the receiving end carries out digital signal processing to complete channel estimation.
After the second step is completed, the receiving end obtains the channel observation (e.g. y in formula (2) and formula (7)) polluted by noise, so that the receiving end can recover the channel H to be estimated by using the channel observation and the pilot signal (e.g. Φ in formula (2) and formula (7)) known in advance through a digital signal processing algorithm, thereby completing the channel estimation process of the communication system.
Due to the sparsity of millimeter wave channels (the multipath component of the channel is far less than the number of elements of the antenna array) caused by the energy focusing characteristic of the lens antenna array and the high path loss and easy-to-shield transmission characteristic of millimeter waves, it can be seen that the channel H itself has sparsity, that is, a few elements in the channel H occupy most of the energy of the channel H. The characteristic lays a foundation for the application of advanced compressed sensing technology in channel estimation. However, since AoDs and AoAs are continuously distributed in the angular domain, they are not necessarily matched with the spatial angles corresponding to the discrete array elements of the lens antenna array, which will cause a power leakage phenomenon, so that the sparsity of the channel to be recovered is damaged, and the performance of the subsequently applied compressed sensing algorithm is impaired.
In order to solve the problem of power leakage, the invention provides that a redundant dictionary is adopted in a lens antenna array to solve the problem of power leakage, and the basic idea is that a receiving end utilizes the redundant dictionary to carry out further sparse representation on an originally sparse channel of the lens antenna array; and recovering the representation of the original channel on a redundant dictionary by adopting a compressed sensing theory based on the channel observation result and the known pilot signal, and further obtaining the estimation H of the original channel.
The specific operation is as follows:
1) setting redundant dictionaries at levelThe number of angle resolution cells (quantization number) in the direction and the vertical direction are GhAnd GvAnd satisfy GvGh＞＞max(NT,NR)；
2) The set of quantized spatial angles is set as follows:
wherein, thetagRepresents the spatial angle AoD of the g-th angle-resolving unit in the vertical direction,
Figure BDA0002205374450000161
represents the spatial angle AoA of the g-th angle-resolving element in the horizontal direction.
3) Defining the redundant dictionary matrix of the receiving end and the transmitting end as follows:
Figure BDA0002205374450000162
wherein, aRAnd aTAnd guiding vectors of the lens antenna array are respectively assembled for a receiving end and a transmitting end.
The steps are described to illustrate how the redundant dictionary in equation (9) is used to make the estimate of H.
Considering a geometric millimeter wave sparse channel model in which only L different scatterers between the receiving end and the transmitting end correspond to L main sparse multipath components, the channel matrix can be expressed as
Figure BDA0002205374450000163
For the l path, glIs the channel complex gain obeying a zero-mean complex gaussian distribution.
Figure BDA0002205374450000164
And
Figure BDA0002205374450000166
aod (aoa) in the vertical direction and horizontal direction, respectively. In the invention, the receiving end and the transmitting end both adopt the full-dimensional lens antenna array, so the transmitting end guide vector in the formula (9)
Figure BDA0002205374450000167
Can be expressed as
Figure BDA0002205374450000168
Wherein
Figure BDA0002205374450000169
And
Figure BDA00022053744500001610
respectively a normalized vertical aperture and a normalized horizontal aperture of the transmitting-end full-dimensional lens antenna array,
Figure BDA00022053744500001611
is the angle coordinate of each antenna distribution in the lens antenna array; the T or R of the subscript represents the transmitting end or the receiving end, and the n of the subscript represents the nth element in the vector; the "sinc" function is defined as
Figure BDA00022053744500001612
Gamma is a normalization factor, gamma ensures the normalization of the channel matrix energy, i.e.
Figure BDA00022053744500001613
Here | · | non-conducting phosphor2Representing the 2-norm of the vector. Receiver side steering vector
Figure BDA00022053744500001614
The form of (c) is consistent with that of formula (11), and will not be described herein.
The flow of the receiving end for channel estimation by using the redundant dictionary is as follows:
1) the channel matrix H in equation (10) is expressed as follows
Figure BDA0002205374450000171
Wherein A isRAnd ATIs a redundant dictionary matrix defined in equation (9), HaIs the representation of the channel on the redundant dictionary, and is also a sparse matrix, and E is the quantization error. When G isvAnd GhWhen large, E can be considered as slight noise and ignored.
2) Substituting the formula (12) into the channel estimation model represented by the formula (2) or (7) can obtain
WhereinAs a total redundant dictionary matrix, symbol (.)*Denotes the conjugation, neffΦ vec (e) + n is the equivalent noise vector, n is the total noise vector. Due to the use of redundant dictionaries, the phenomenon of power leakage is reduced.
3) Firstly, the formula (13) is utilized to HaThe estimation is performed, and then the original channel H is obtained by using the formula (12), so that better estimation performance can be obtained than the estimation H directly by using the formula (2) or (7).
This flow ends by this point.
For the scheme 2 described in the second step, because the observation matrix (i.e. pilot) is compact in form and convenient for mathematical processing, the invention utilizes the total cross-correlation theory of the minimized sensing matrix in compressed sensing to design the baseband pilot signal in the scheme 2, so as to further improve the channel estimation performance.
For the formula (13), according to the compressive sensing theory, the performance of the subsequent sparse signal recovery algorithm can be greatly improved by designing the value of the sensing matrix Φ Ψ. As mentioned before, the precoder F of the analog partRFAnd combiner WRFAll are considering the limits of strict hardware conditionsThe design is completed under the design, therefore, the part only considers the base band pilot signal F under the guidance of the compressed sensing theoryBBAnd WBBAnd (5) designing. In order to achieve the purpose, the invention adopts an optimization method of minimizing the total cross-correlation principle in the compressed sensing theory to optimize the sensing matrix, namely, designing the baseband pilot frequency, and modeling the process as the following optimization problem
Figure BDA0002205374450000181
Here, theRepresenting the precoder and combiner of the optimized baseband pilot signal.
Figure BDA0002205374450000183
Total cross-correlation, defined as the matrix phi psi, symbol [ A ]]nRepresenting a column vector consisting of the nth column of the matrix A, | · | | non-conducting phosphorFRepresenting the Frobenius norm of the vector. The significance of the limitation on the optimization objective is to take into account the power constraints at the transceiving end.
In scheme 2, by mathematical derivation, the closed-form solution of the optimization problem (14) can be obtained as follows
Figure BDA0002205374450000184
Figure BDA0002205374450000185
Wherein
Figure BDA0002205374450000186
Is to satisfy the condition
Figure BDA0002205374450000187
Of arbitrary matrix, symbol INThe unitary matrix is expressed by NxN, i.e. a sub-matrix composed of some columns of an arbitrary unitary matrix is used as the unitary matrix
Figure BDA0002205374450000188
Is arbitraryIs used to generate the unitary matrix.
It should be noted that the above design method is performed under the theoretical guidance of compressive sensing, and therefore, the design method is not limited to a specific compressive sensing algorithm, and the design result can be applied to a plurality of existing compressive sensing algorithms, such as OMP algorithm, BP (base Pursuit) algorithm, and the like. Theoretically, the above design method will achieve better estimation performance in these algorithms than without pilot design.
It should be noted that, although the present invention only considers the problem of channel estimation of a narrow-band flat fading channel (without considering channel delay spread), considering that in a MIMO-OFDM (MIMO-orthogonal frequency Division Multiplexing) system, a channel may be divided into several independent sub-channels, and each sub-channel may be regarded as a narrow-band flat fading channel, therefore, the scheme proposed by the present invention may be directly applied to a lens antenna array MIMO-OFDM system, that is, each sub-channel may be processed by adopting the above scheme.
In summary, the present invention is directed to a millimeter wave MIMO system based on a lens antenna array, and in order to solve the problems of power loss and hardware complexity caused by a huge phase shifter network under a traditional hybrid analog-digital precoding architecture, the present invention employs an antenna in the lens antenna array. And a network architecture is selected, so that the hardware complexity can be greatly reduced. Meanwhile, in order to solve the problem that the traditional channel estimation method in the antenna selection network is invalid, the invention provides two pilot frequency transmission schemes under the condition of considering the limitation brought by the special hardware structure of the antenna selection network. In addition, the invention also designs the baseband pilot signal of the transmitting and receiving end under the theoretical frame of compressed sensing, which can effectively improve the performance of sparse channel estimation and obviously improve the accuracy of channel estimation.
For the purpose of illustrating the inventionThe advantages of the millimeter wave lens antenna array MIMO in performance in channel estimation are illustrated in fig. 3 and 4. In the simulation, on the basis of the pilot design proposed by the present invention, the channel in formula (13) is estimated by using the classical compressed sensing algorithm — the OMP algorithm. The specific values of the design of the baseband pilot frequency are as follows:
Figure BDA0002205374450000191
get
Figure BDA0002205374450000192
DFT (Discrete Fourier transform) matrix of (1),get one
Figure BDA0002205374450000194
Before DFT matrix of
Figure BDA0002205374450000195
The columns constitute a sub-matrix.
Fig. 3 compares the NMSE (normalized mean Square Error) performance of the proposed redundancy dictionary and pilot design method under the OMP algorithm and the DC (Dual Cross) based channel estimation method under the same pilot overhead. Note that DC-based channel estimation methods require a complex fully-connected-phase shifter network as an aid. For DC-based methods, see in particular the literature "translation name: the author, english name and publication of the Channel estimation in the millimeter wave massive MIMO based on the 3-D lens are "w.ma and c.qi," Channel estimation for 3-dpen millimetre wave massive MIMO system, "IEEE com.lett., vol.21, No.9, pp.2045-2048, sep.2017," as can be seen from fig. 3, although only a simple antenna selection network is used for Channel estimation in the present invention, its NMSE performance is superior to the contrast method at each SNR (Signal-to-Noise Ratio). In particular, as the number of multipaths L increases, the performance of the DC-based method gradually deteriorates to eventually fail to function properly. This is because as the number of multipaths increases, the power leakage phenomenon becomes more serious and the structure (cross shape) of the channel will be destroyed. However, it can be seen that due to the use of the redundant dictionary, the method provided by the present invention has significant robustness to the multipath number L, and compared with the comparison scheme, the millimeter wave channel with more multipath components can be effectively estimated.
Fig. 4 compares the NMSE performance of the proposed channel estimation method as a function of signal-to-noise ratio compared to the conventional method. As can be seen from fig. 4, the performance of the basic compressed sensing channel estimation method will be significantly better than the conventional LS algorithm. For example, at a SNR of 0dB, the proposed scheme has a gain on NMSE of approximately 10dB over the LS algorithm. In addition, as can be seen from fig. 4, the design of the redundant dictionary and the design of the baseband pilot used in the present invention effectively improve the performance of channel estimation.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A millimeter wave lens antenna array MIMO channel estimation method for enhancing sparsity by using a redundant dictionary is characterized by comprising the following steps:
in the millimeter wave lens antenna array MIMO, an antenna selection network is adopted for both a transmitting end and a receiving end to replace a full-connection-phase shift network; in the antenna selection network, the number of radio frequency links is less than that of antennas, each radio frequency link can be connected with only one antenna at the same time, and each antenna can be connected with only one radio frequency link at the same time;
in the pilot frequency interaction stage, when the antenna selection network is switched every time, the radio frequency link is connected with part of array elements to obtain a channel detection result in a part of angle range; through multiple switching, the radio frequency link is in continuous NpilotTraversing and connecting all array elements of the lens antenna array in each time slot to obtain a complete channel observation result;
thirdly, the receiving end performs further sparse representation on the channel of the originally sparse lens antenna array by utilizing the redundant dictionary; and recovering the representation of the original channel on a redundant dictionary by adopting a compressed sensing theory based on the channel observation result and the known pilot signal, and further obtaining the estimation H of the original channel.
2. The method of claim 1, wherein let N beTAnd NRThe number of antennas at the transmitting end and the receiving end respectively;
Figure FDA0002205374440000011
and
Figure FDA0002205374440000012
the number of radio frequency links equipped for a transmitting end and a receiving end respectively; using NpilotEstimating each pilot signal;
the second step comprises the following steps:
the channel estimation is converted into the following form:
Figure FDA0002205374440000013
wherein, (.)HIs a transposed symbol, vec (-) is a column-wise vectorized symbol,
Figure FDA0002205374440000014
representing the Kronecker product, diag (·) representing diagonalized symbols; total received signal of receiving endTotal received noise
Figure FDA0002205374440000021
ynFor the pilot data vector received at the receiver of the nth slot, N is 1,2pilot，nnIs complex white Gaussian noise, s, at the receiving end of the nth time slotnA pilot signal transmitted for an nth slot; h is a lensA millimeter wave channel under the antenna array; hybrid combiner F at the nth slot transmitting endn＝FRF,nFBB,nIs an analog precoder FRF,nAnd baseband digital precoder FBB,n(iii) a cascade of; hybrid combiner W at the receiving end of the nth time slotn＝WRF,nWBB,nIs an analog combiner WRF,nAnd baseband digital combiner WBB,n(iii) a cascade of;
wherein: let FBB,n,
Figure FDA0002205374440000022
Each element in the (1) is a zero-mean complex Gaussian variable subject to independent and same distribution;
in all, N is carried outpilotSecondary switching; let GRF,nRepresents FRF,nOr WRF,nLet N represent NTOr NR，NRFRepresents
Figure FDA0002205374440000023
Or
Figure FDA0002205374440000024
NTAnd NRThe number of antennas at the transmitting end and the receiving end respectively;
Figure FDA0002205374440000025
and
Figure FDA0002205374440000026
the number of radio frequency links equipped for a transmitting end and a receiving end respectively;
at the time of the nth switching, for GRF,nPerforming steps 1) and 2):
step 1) if
Figure FDA0002205374440000027
Then order
Figure FDA0002205374440000028
Representing NxN unit momentsThe matrix is formed after random rearrangement between the columns, wherein the symbol mod (a, b) represents the remainder of dividing a by b; then step 2) is executed; otherwise, directly executing the step 2);
step 2) order
Figure FDA0002205374440000029
Here, the
Figure FDA00022053744400000210
Symbol [ A ]]p:qRepresenting a sub-matrix consisting of the p-th to q-th columns of matrix A, symbols
Figure FDA00022053744400000211
Representing rounding down on a.
3. The method of claim 1, wherein let N beTAnd NRThe number of antennas at the transmitting end and the receiving end respectively,
Figure FDA00022053744400000212
and
Figure FDA00022053744400000213
the number of radio frequency links equipped for the transmitting end and the receiving end respectively requires that the number of antennas at the transmitting and receiving ends is integral multiple of the number of radio frequency links; definition of
Figure FDA00022053744400000214
The second step comprises the following steps:
a plurality of time slots form a time block; considering the total use of transmitting endsTransmit pilot frequency in one time block and require
Figure FDA00022053744400000216
Is the number of blocksInteger multiples of; is provided with
Figure FDA00022053744400000218
The pilot signal transmitted by the transmitting end in the mth time block is represented as:
sm＝FRF,pfBB,m
where FRF,pAnd fBB,mRespectively a radio frequency part for transmitting pilot frequency and a baseband part
Figure FDA0002205374440000031
Symbol
Figure FDA0002205374440000032
Represents rounding up on a; this indicates that, in total
Figure FDA0002205374440000033
In each pilot, every continuation
Figure FDA0002205374440000034
One base band pilot frequency shares one radio frequency pilot frequency FRF,p；
At the receiving end, each time block is further decomposed intoA plurality of time slots with equal length, in the nth time slot,
Figure FDA0002205374440000036
the pilot signal received by the receiving end from the mth time block can be expressed as
Figure FDA0002205374440000037
Wherein H is a millimeter wave channel under the lens antenna array; n isn,mIs a receiving endComplex white gaussian noise, and
Figure FDA0002205374440000038
WRF,nand WBB,nAnalog combiner W in hybrid combiner for nth slot receiverRF,nAnd baseband digital combiner WBB,n；
On the receiving end
Figure FDA0002205374440000039
Pilot receiving signal for all m time blocks in continuous time slot
Figure FDA00022053744400000310
Performing combined treatment to obtain
Figure FDA00022053744400000311
Here:
Figure FDA00022053744400000312
Figure FDA00022053744400000313
Figure FDA00022053744400000314
pilot reception signal for all time blocks
Figure FDA00022053744400000316
Performing combined treatment to obtain
Figure FDA00022053744400000317
Here:
Figure FDA00022053744400000318
Figure FDA0002205374440000041
Figure FDA0002205374440000042
vectorizing the formula (I) to obtain the final product
Figure FDA0002205374440000043
This step is a two-pair simulation precoder FRFThe design is as follows: get
Figure FDA0002205374440000044
Represents NT×NTA matrix formed by randomly rearranging the unit matrix among columns
Figure FDA0002205374440000045
To analog combiner WRFThe design is as follows: get
Figure FDA0002205374440000046
Represents NR×NRA matrix formed by randomly rearranging the unit matrix among columns
Figure FDA0002205374440000047
Using designed analog precoder FRFAnd an analog combiner WRFAnd finishing the pilot interaction.
4. The method of claim 1, wherein step three is:
step 31, setting redundant dictionary A of receiving end and transmitting endR、ATRespectively as follows:
Figure FDA0002205374440000049
Figure FDA00022053744400000411
wherein, aRAnd aTThe guide vectors of the lens antenna array are respectively assembled for the receiving end and the transmitting end; ghAnd GvThe angle resolution unit numbers of the redundant dictionary in the horizontal direction and the vertical direction are respectively; the T or R of the subscript represents the transmitting end or the receiving end, and the n of the subscript represents the nth element in the vector;
Figure FDA00022053744400000412
and
Figure FDA00022053744400000413
respectively, a normalized vertical aperture and a normalized horizontal aperture of the full-dimensional lens antenna array; (alphan,βn) Is the angle coordinate of each antenna distribution in the lens antenna array;
Figure FDA00022053744400000414
representing the departure angle AoD and the arrival angle AoA in the vertical direction and the horizontal direction on the ith path in the L main sparse multipath components; gamma rayRAnd gammaTOf receive-side and transmit-side steering vectors, respectivelyA normalization factor;
step 32, the channel matrix H is represented as follows
Figure FDA0002205374440000051
Wherein HaThe method is characterized in that sparse representation of a channel of a lens antenna array on a redundant dictionary is also a sparse matrix, and E is a quantization error;
step 33, substituting the channel matrix H into the channel estimation model to obtain:
Figure FDA0002205374440000052
wherein phi is an observation matrix, and phi is,
Figure FDA0002205374440000053
as a total redundant dictionary matrix, symbol (.)*Denotes the conjugation, neffΦ vec (e) + n is the equivalent noise vector, n is the total noise vector;
step 34, firstly, using the channel estimation model to HaMaking an estimate and then substituting into the formula
Figure FDA0002205374440000054
The original channel H is obtained.
5. The method of claim 3, wherein the baseband pilot signal is designed using the total cross-correlation principle of minimizing the sensing matrix to obtain the optimized baseband pilot signal precoder
Figure FDA0002205374440000055
Merging device
Figure FDA0002205374440000056
Comprises the following steps:
Figure FDA0002205374440000057
Figure FDA0002205374440000058
wherein
Figure FDA0002205374440000059
Is to satisfy the condition
Figure FDA00022053744400000510
Of arbitrary matrix, symbol INThe unitary matrix is expressed by NxN, i.e. a sub-matrix composed of some columns of an arbitrary unitary matrix is used as the unitary matrix
Figure FDA00022053744400000511
Figure FDA00022053744400000512
Is arbitraryIs used to generate the unitary matrix.
6. The method of claim 1, wherein the method is applied to a millimeter wave lens antenna array MIMO-OFDM system, and wherein the steps from one to three are performed for each sub-channel.
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