KR20100065078A - Method for estimating channel under environment in which interference exists in wireless communication system and apparatus thereof - Google Patents

Abstract

PURPOSE: A method for estimating channel under channel environment in which interference of neighboring base station exists in a wireless communication system and a device thereof are provide to maximally enhance existing probability of reverse matrix within channel estimation result, thereby reducing interference within receiving signal. CONSTITUTION: A unit block generating unit(11) divides PUSC(Partial Usage Sub-channel) clusters into unit blocks. A pilot matrix configuration unit(12) configures pilot matrix by each unit block. A channel estimation unit(13) estimates channel through pilot subcarrier signals. A reverse matrix determining unit determines the existence of the inverse matrix of the covariance matrix related to the channel estimation result to the pilot matrices. In case of the inverse matrix does not exist, the matrix inverse determining unit indicates the configuration of the new pilot matrix to the pilot matrix configuration unit.

Description

Method and estimating channel under environment in which interference exists in wireless communication system and apparatus according to the present invention.

The present invention relates to a channel estimation method and apparatus, and more particularly, to a method and apparatus for estimating a channel under a channel environment in which interference by neighboring base stations exist.

Recently, orthogonal frequency division multiplexing (OFDM) schemes have been actively studied and utilized in wireless communication systems. The OFDM method is a method of transmitting data using a multi-carrier, and a plurality of sub-carriers having parallel orthogonal conversion of symbol strings serially inputted and each of which are orthogonal to each other Multi Carrier Modulation (MCM) is a type of multi-carrier modulation that is modulated and transmitted.

In the wireless communication system to which the OFDM scheme is applied, the frequency domain is divided into subchannels consisting of a plurality of subcarriers, the time domain is divided into a plurality of time slots, and a fan is allocated for each user to consider both time and frequency domains. Performs allocation and transmits data in block units.

In the downlink, one subchannel may be allocated to different receivers, and in the uplink, multiple transmitters may be allocated to one or more subchannels. According to the difference in the method of allocating subcarriers, the subchannel may be a partial usage sub-channel (PUSC) subchannel, a diversity subchannel, a full usage of sub-channel (FUSC) subchannel, and an adaptive modulation and coding sub-channel (AMC). And subchannels.

There is a method of estimating a channel using a cluster of one or more PUSC subchannels in a channel environment in which interference of a neighboring base station exists. The pilot subcarrier signals included in a PUSC cluster received at a terminal may be represented as follows.

 [Equation 1]

는 단말에 수신된 신호를 각각 나타내며,

는 j-번째 기지국에서 전송되는 i-번째 파일럿 부반송파 신호,

는 i-번째 기지국에서 전송되는 신호가 겪는 무선경로 이득, 그리고

는 i-번째 수신신호에 더해지는 백색잡음을 나타낸다.From here,

Represents a signal received at the terminal,

I-th pilot subcarrier signal transmitted from the j-th base station,

Is the radio path gain experienced by the signal transmitted from the i-th base station, and

Denotes white noise added to the i-th received signal.

When the received signal expressed as in Equation 1 above is used with a maximum likelihood channel estimation method, the channel estimate may be represented as in Equation 2 below.

[Equation 2]

는

행렬의 역행렬(Inverse Matrix)을 나타낸다. Here, P ^H is the Hermitian operation (Hermitian) matrix of the matrix P, i.e., changing the rows and columns of the matrix P, respectively, it converted into the complex conjugate of matrix elements. And

Is

Represents the inverse matrix of the matrix.

In the case of estimating a channel using this method, the singularity for the matrix ( P ^H · P ) ^-1 is simple because the pattern used by the system for randomizing the pilot is simple as "1" and "-1". ) Is a problem that occurs. That is, since the pilot subcarriers used for channel estimation are modulated and transmitted according to a random pattern of "1" or "1", there is a high probability that there is no inverse matrix for the ( P ^H · P ) matrix of Equation 2 There is this.

An object of the present invention is to improve the reliability of channel estimation while reducing interference when channel estimation is performed in a channel environment in which interference of adjacent base stations exists.

According to an aspect of the present invention, a channel estimating method is a method in which a wireless communication system estimates a channel in a channel environment in which interference between neighboring base stations exists, and a plurality of subchannel clusters composed of a data subcarrier signal and a pilot subcarrier signal include a subcarrier axis. And are arranged on a radio resource having a symbol axis, and are divided into a plurality of unit blocks. In this case, the channel estimating method includes: constructing a pilot matrix using pilot subcarrier signals of at least one subchannel cluster for a predetermined unit block; Performing channel estimation using the pilot matrix; Determining whether an inverse of a covariance matrix related to the pilot matrix exists in the channel estimation result; And when the inverse matrix exists, using the channel estimation result of the pilot matrix as the channel estimation result. In the case where the inverse does not exist, the step of constructing the pilot matrix configures a new pilot matrix by extending the pilot matrix in the symbol axis direction.

A channel estimating apparatus according to another aspect of the present invention is an apparatus for estimating a channel for a channel environment in which interference between neighboring base stations exists in a wireless communication system, and is arranged in a space consisting of a subcarrier axis and a symbol axis, and a data subcarrier. A unit block dividing unit for dividing a plurality of subchannel clusters composed of a signal and a pilot subcarrier signal into a plurality of unit blocks; A pilot matrix constructing unit for constructing a pilot matrix using pilot subcarrier signals of one subchannel cluster for each unit block; A channel estimator for performing channel estimation using a pilot matrix constructed by the pilot matrix constructing unit; And an inverse matrix determiner that determines whether an inverse of a covariance matrix related to a pilot matrix exists in a channel estimation result obtained by the channel estimator. In this case, when the inverse matrix does not exist, the pilot matrix constructing unit configures a new pilot matrix by extending the pilot matrix in the direction of the symbol axis when the inverse matrix does not exist, and applies a channel estimation result for the new pilot matrix. The process of extending the pilot matrix is repeated until there is an inverse of the covariance matrix.

According to an embodiment of the present invention, it is possible to improve the reliability of the channel estimation result in a channel environment in which interference of neighboring base stations exists.

In particular, for channel estimation, it is determined whether there is an inverse matrix related to a matrix of pilot subcarriers included in at least one PUSC cluster, and accordingly, the pilot matrix is extended using pilot subcarriers included in an adjacent PUSC cluster. By performing matrix expansion and performing channel estimation again, it is possible to increase the probability of existence of the inverse of the channel estimation result.

Therefore, the probability that an inverse matrix is included in the channel estimation result increases as much as possible, thereby reducing the interference included in the received signal and increasing the performance of the system.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings so that those skilled in the art may easily implement the present invention. As those skilled in the art would realize, the described embodiments may be modified in various different ways, all without departing from the spirit or scope of the present invention. In the drawings, parts irrelevant to the description are omitted in order to clearly describe the present invention, and like reference numerals designate like parts throughout the specification.

Throughout the specification, when a part is said to "include" a certain component, it means that it can further include other components, without excluding other components unless specifically stated otherwise.

Embodiments of the present invention will now be described with reference to the accompanying drawings.

In a wireless communication system according to an embodiment of the present invention, a Partial Usage Sub-channel (PUSC) subchannel transmitted from a base station is composed of a plurality of PUSC clusters, and the PUSC cluster is composed of 14 subcarriers and two symbols.

1 is a diagram illustrating a PUSC cluster structure for transmitting a signal in a wireless communication system.

As shown in FIG. 1, one PUSC cluster includes 24 data subcarriers and 4 pilot subcarriers for use in channel estimation.

2 is a diagram illustrating signals transmitted and received between a base station and a terminal in a wireless communication system according to an exemplary embodiment of the present invention.

As shown in FIG. 2, PUSC cluster signals having a structure as shown in FIG. 1 are transmitted from the base stations 100 and 200 to the terminal 300. In an embodiment of the present invention, the pilot subcarriers used for channel estimation may be two out of four pilot subcarriers included in each PUSC cluster signal based on an OFDM symbol axis.

Next, a channel estimation method according to an embodiment of the present invention will be described.

3 is a diagram illustrating a process of classifying PUSC clusters into unit blocks constituting a matrix according to an embodiment of the present invention.

In the embodiment of the present invention, as shown in FIG. 3, PUSC clusters are arranged along a symbol axis on a radio resource including a subcarrier axis and a symbol axis, and the PUSC clusters are divided into unit blocks for estimating a channel. do. In more detail, a PUSC cluster arranged along a symbol axis may include a first subcarrier including a predetermined number (eg, two) of pilot subcarriers disposed closest to a symbol axis among pilot subcarriers included in each PUSC cluster. The second unit block B2 including the pilot subcarriers other than the pilot subcarriers included in the first unit block B1 among the pilot subcarriers included in the unit block B1 and the PUSC clusters arranged along the symbol axis. In addition, some pilot subcarriers included in the first unit block B1 and some pilot subcarriers included in the second unit block B2 are divided into a third unit block B3. Here, the third unit block B3 may include a predetermined number of pilot subcarriers (for example, one) and a second unit in the order of the furthest distance from the symbol axis among the pilot subcarriers of the first unit block B1. The pilot subcarriers of the block B2 include a predetermined number (for example, one) of pilot subcarriers in the order of being arranged closest to the symbol axis.

In the embodiment of the present invention, as described above, the PUSC clusters are divided into a plurality of unit blocks, and then channel estimation is performed on the unit blocks. In this case, channel estimation may be performed in the order of unit blocks.

If the inverse matrix does not exist in the channel estimation result according to the corresponding unit block, channel estimation is performed on the next unit block, and in particular, the channel estimation result for the first unit block B1 and the second unit block B2 If the inverse does not exist, channel estimation is performed using the third unit block B3. Hereinafter, it is assumed that the channel does not change for each unit block used for channel estimation.

A method of estimating a channel based on such unit blocks will be described.

4 is a flowchart illustrating a channel estimation method according to an exemplary embodiment of the present invention, and FIG. 5 is a diagram illustrating a process of constructing a matrix for each unit block according to an exemplary embodiment of the present invention.

First, as shown in FIG. 4, a matrix for the pilot subcarriers included in the PUSC cluster signal received by the terminal 300 is obtained. That is, a matrix is configured for a predetermined unit block composed of some pilot subcarriers of the PUSC clusters. Here, an example of constructing a matrix and estimating a channel in the first unit block B1 will be described.

Specifically, as shown in FIG. 5, first, a matrix having a first size (for example, 2 × 2) is constructed by using some pilot subcarriers among the pilot subcarriers constituting the first unit block B1. (S100). The matrix of the first size consists of a predetermined number (for example, two) of pilot subcarriers in the order arranged closest to the carrier axis among the pilot subcarriers constituting the first unit block B1. For convenience, it is called a P ₁ matrix (S100). Representation of the received signal using the P ₁ matrix is as follows.

&Quot; (3) "

는 단말기에 수신된 PUSC 클러스터 내의 i-번째 OFDM 심볼의 j-번째 수신 신호를 각각 나타내며, P는 채널의 특성을 나타내는 파일럿 행렬이며,

는 k-번째 기지국에서 전송되는 i-번째 OFDM 심볼의 j-번째 파일럿 부반송파 신호를 나타낸다.

는 i-번째 기지국에서 전송되는 신호가 겪는 무선경로 이득을 나타내며, 그리고

는 i-번째 OFDM 심볼의 j-번째 수신 신호에 더해지는 백색잡음을 나타낸다.From here,

Denotes the j-th received signal of the i-th OFDM symbol in the PUSC cluster received at the terminal, P is a pilot matrix indicating the characteristics of the channel,

Denotes a j-th pilot subcarrier signal of an i-th OFDM symbol transmitted from a k-th base station.

Denotes the radiopath gain experienced by the signal transmitted from the i-th base station, and

Denotes white noise added to the j-th received signal of the i-th OFDM symbol.

When the received signal represented by Equation 3 above is processed by using a maximum likelihood channel estimation method, a channel estimate value shown in Equation 4 can be obtained.

&Quot; (4) "

는 채널 추정값을 나타내며,

는 P₁ 행렬의 에르미트(Hermitian) 행렬, 즉, P₁ 행렬의 열과 행을 바꾸고, 행렬 원소들을 각각 켤레 복소수로 변환하는 연산이다. 그리고

는

행렬의 역행렬(Inverse Matrix)을 나타낸다.From here,

Represents the channel estimate ,

Is an Hermitian matrix of the P ₁ matrix, that is, a column and a row of the P ₁ matrix, and the matrix elements are converted into conjugate complex numbers, respectively. And

Is

Represents the inverse matrix of the matrix.

The channel estimate matrix inverse of the result, the pilot matrix, the channel estimate for the received signal in accordance with P ₁ matrix of a first size of the first unit block, that is, the inverse matrix for the (P _1, ^H · P ₁₎ matrices involved in the pilot matrix (P ^It is determined whether ^H · P ) ^-1 is present (S110). The matrix corresponding to the product of the pilot matrix obtained as a result of the channel estimation and the Hermit matrix, ie, P ^H · P , is referred to as a "covariance matrix" for convenience of explanation.

When the inverse of the covariance matrix exists in the channel estimation result for the received signal according to the P ₁ matrix of the first size block of the first unit block, the channel estimation value obtained as a result of the channel estimation based on Equation 4 above is used as the first unit. It is used as a channel estimate for the block B1 (S120).

On the other hand, if it is determined that the inverse of the covariance matrix does not exist in the channel estimation result for the received signal according to the P ₁ matrix having the first size of the first unit block, the P ₁ matrix is expanded to form a new matrix and perform channel estimation again. To perform.

In detail, when the inverse of the covariance matrix does not exist, information about the pilot included in the cluster cannot be obtained, and pilot information cannot be used for channel estimation. Accordingly, in order to increase the probability of acquiring the information on the pilot, as shown in FIG. 5, the second size extended in the symbol axis direction while including pilot subcarriers of the P ₁ matrix having the first size in the first unit block B1 ( For example, a new extension matrix having 4 × 2 is configured (S140). The second-sized matrix P2 further includes a predetermined number (eg, two) pilot subcarriers arranged in the symbol axis direction in addition to the pilot subcarriers included in the first-sized matrix P1. For convenience, hereinafter referred to as P ₂ matrix. The expanded P ₂ matrix is expressed as an equation like a received signal as follows.

 [Equation 5]

Next, in the same manner as in Equation 4 above, a channel estimate for the P ₂ matrix is obtained, and it is determined whether an inverse of ( P ₂ ^H · P ₂ ) exists in the obtained channel estimate (S150).

If that is the inverse of the channel estimate for the P ₂ matrix (P ₂ ^H · P ₂₎ exists, use the channel estimation result obtained channel estimation value for the P ₂ matrix, the channel estimation value for the first unit block (B1) (S130).

On the other hand, if there is no inverse of ( P ₂ ^H · P ₂ ) in the channel estimate for the P ₂ matrix, matrix expansion is performed again to form a new matrix and perform channel estimation.

That is, as shown in FIG. 5, the first unit block B1 includes pilot subcarriers of the first size P ₁ matrix P1 and the second size P ₂ matrix P2 and extend in the symbol axis direction. A new extension matrix P3 having 3 sizes (for example, 6 × 2) is configured (S160). The third-sized matrix P3 has a predetermined number (eg, two) arranged in the symbol axis direction in addition to the pilot subcarriers included in the first-sized matrix P1 and the second-sized matrix P3. Pilot subcarriers are further included, and for convenience of description, hereinafter, referred to as a P ₃ matrix. The extended P ₃ matrix is expressed as an equation like a received signal as follows.

 &Quot; (6) "

Next, in the same manner as in Equation 4 above, a channel estimate for the P ₃ matrix is obtained, and it is determined whether or not an inverse of ( P ₃ ^H · P ₃ ) exists in the obtained channel estimate (S170).

If that is the inverse of the channel estimate for the P ₃ matrix (P ₃ ^H · P ₃₎ exists, use the channel estimation result obtained channel estimation value for the P ₃ matrix as a channel estimation value for the first unit block (B1) (S130).

If it is determined that the inverse does not exist, it is determined that channel estimation cannot be performed with respect to the first unit block B1, and an impossible indication that channel estimation cannot be performed is performed (S180).

As described above, in an embodiment of the present invention, a covariance matrix corresponding to a product of a corresponding channel of a pilot matrix and a Hermitian matrix is obtained from a channel estimation result of a unit block consisting of predetermined pilot subcarriers of PUSC clusters for which a channel is to be estimated. If the inverse does not exist, channel estimation is performed again while extending the pilot matrix of the corresponding unit block until the inverse exists.

The channel estimation process is similarly applied to other unit blocks, that is, the second unit block B2 or the third unit block B3.

As described above, according to an embodiment of the present invention, whether a covariance matrix, that is, an inverse of the matrix ( P ^H · P ), for a matrix of pilot subcarriers included in one or more clusters as a symbol axis for channel estimation is determined. If not, the matrix can be extended and channel estimation is performed again, thereby increasing the probability of inverse matrix existence of the ( P ^H · P ) matrix and increasing the reliability of channel estimation.

The channel estimation apparatus according to the embodiment of the present invention for implementing the channel estimation method according to the embodiment of the present invention may have the following structure.

6 is a structural diagram of a channel estimating apparatus according to an embodiment of the present invention.

The channel estimating apparatus according to an embodiment of the present invention performs channel estimation on signals received in base stations mounted on a terminal.

As shown in FIG. 5, the channel estimating apparatus 10 according to an exemplary embodiment of the present invention includes a unit block generator 11, a pilot matrix constructing unit 12, a channel estimating unit 13, and an inverse matrix determination unit. (14).

The unit block generation unit 11 classifies PUSC clusters arranged on a radio resource including a symbol axis and a subcarrier axis for each unit block. As shown in FIG. 3, the PUSC clusters are divided into a first unit block B1, a second unit block B2, and a third unit block B3.

The pilot matrix constructing unit 12 configures a pilot matrix for each unit block, and specifically constructs a pilot matrix using pilot subcarrier signals included in at least one PUSC cluster in the symbol axis direction.

The channel estimator 13 estimates a channel using pilot subcarrier signals. In other words, the channel is estimated using the pilot subcarrier signals of the pilot matrix configured by the pilot matrix configuration unit 12, and the channel estimation process is performed for each unit block.

The inverse matrix determination unit 14 determines whether an inverse of the covariance matrix related to the pilot matrix exists in the channel estimation result of the channel estimating unit 13, and if not, sends a new pilot matrix to the pilot matrix constructing unit 12 again. Instruct it to configure. In this case, the pilot matrix constructing unit 12 constructs a new matrix by extending the existing constructed pilot matrix. That is, as shown in FIG. 5, a new second pilot matrix is formed by extending the size of the initially formed first pilot matrix in the symbol axis direction, and then the inverse matrix of the covariance matrix also appears in the channel estimation result of the newly constructed second pilot matrix. If not present, the size of the second pilot matrix is expanded again in the symbol axis direction to form a new third pilot matrix. It is determined to include pilot subcarriers of new PUSC clusters while including pilot subcarriers of a newly extended matrix matrix.

Meanwhile, when the inverse of the covariance matrix related to the pilot matrix exists in the channel estimation result, the channel estimator 13 uses the channel estimation result as the channel estimation result of the corresponding unit block. In addition, even though the channel is estimated by extending the pilot matrix of the unit block, the inverse matrix of the covariance matrix does not exist in the channel estimation result, indicating that channel estimation for the unit block is impossible.

The channel estimating apparatus according to the embodiment of the present invention having such a structure operates based on the channel estimating method described above, and a person skilled in the art can implement the operation of the channel estimating apparatus based on the above method. Detailed description thereof will be omitted.

An embodiment of the present invention is not implemented only through the above-described apparatus and / or method, but may be implemented through a program for realizing a function corresponding to the configuration of the embodiment of the present invention, a recording medium on which the program is recorded, and the like. Such implementations may be readily implemented by those skilled in the art from the description of the above-described embodiments.

Although the embodiments of the present invention have been described in detail above, the scope of the present invention is not limited thereto, and various modifications and improvements of those skilled in the art using the basic concepts of the present invention defined in the following claims are also provided. It belongs to the scope of rights.

1 is a diagram illustrating a PUSC cluster structure for transmitting a signal in a wireless communication system.

2 is a diagram illustrating signals transmitted and received between a base station and a terminal in a wireless communication system according to an exemplary embodiment of the present invention.

3 is a diagram illustrating a process of classifying PUSC clusters into unit blocks constituting a matrix according to an embodiment of the present invention.

4 is a flowchart illustrating a channel estimation method according to an embodiment of the present invention.

5 is a diagram illustrating a process of constructing a matrix for pilot subcarriers included in a PUSC cluster signal for each unit block according to an embodiment of the present invention.

6 is a structural diagram of a channel estimating apparatus according to an embodiment of the present invention.

Claims (11)

A method for estimating a channel in a channel environment in which a wireless communication system has interference of neighbor base stations, A plurality of subchannel clusters consisting of a data subcarrier signal and a pilot subcarrier signal are arranged on a radio resource consisting of a subcarrier axis and a symbol axis, and are divided into a plurality of unit blocks. Constructing a pilot matrix using pilot subcarrier signals of at least one subchannel cluster for a predetermined unit block; Performing channel estimation using the pilot matrix; Determining whether an inverse of a covariance matrix related to the pilot matrix exists in the channel estimation result; And Using the channel estimation result of the pilot matrix as the channel estimation result of the corresponding unit block when the inverse matrix exists Including, And if the inverse does not exist, constructing the pilot matrix expands the pilot matrix in the symbol axis direction to construct a new pilot matrix. The method of claim 1 The configuring of the pilot matrix may include repeating the process of constructing a new matrix by extending the pilot matrix in the direction of the symbol axis until there is an inverse of the covariance matrix related to the pilot matrix in the channel estimation result. . The method of claim 2 Configuring the pilot matrix, Constructing a first pilot matrix of a first magnitude using pilot subcarrier signals of at least one subchannel cluster; Constructing a second pilot matrix having a second size by extending the first pilot matrix in the symbol axis direction when no inverse matrix exists in a channel estimation result according to the first pilot matrix; And If there is no inverse matrix in the channel estimation result according to the second pilot matrix, constructing a third pilot matrix having a third size by extending the second pilot matrix in the symbol axis direction Channel estimation method comprising a. The method of claim 3, And if the inverse does not exist in the channel estimation result for the third pilot matrix, indicating that channel estimation is impossible. The method according to any one of claims 1 to 4. Subchannel clusters arranged along the symbol axis on the radio resource; A first unit block including all of a set number of pilot subcarriers in an order arranged closest to a symbol axis among pilot subcarriers included in each subchannel cluster; A second unit block including pilot subcarriers other than pilot subcarriers included in the first unit block among pilot subcarriers included in subchannel clusters arranged along the symbol axis; And A third unit block including some pilot subcarriers included in the first unit block and some pilot subcarriers included in the second unit block The channel estimation method further comprising the step of separating. The method according to any one of claims 1 to 4. The performing of channel estimation may include performing channel estimation on the pilot matrix using a maximum likelihood channel estimation scheme. The method according to any one of claims 1 to 4. And a covariance matrix related to the pilot matrix is a matrix corresponding to a product of a Hermitian matrix and the pilot matrix. An apparatus for estimating a channel for a channel environment in which interference between neighboring base stations exists in a wireless communication system, A unit block dividing unit arranged in a space consisting of a subcarrier axis and a symbol axis, and dividing a plurality of subchannel clusters composed of a data subcarrier signal and a pilot subcarrier signal into a plurality of unit blocks; A pilot matrix constructing unit for constructing a pilot matrix using pilot subcarrier signals of one subchannel cluster for each unit block; A channel estimator for performing channel estimation using a pilot matrix constructed by the pilot matrix constructing unit; And An inverse matrix determination unit that determines whether an inverse of a covariance matrix related to a pilot matrix exists in a channel estimation result obtained by the channel estimator. Including, If the inverse matrix does not exist as a result of the determination by the inverse matrix determination unit, the pilot matrix configuration unit configures a new pilot matrix by extending the pilot matrix in the symbol axis direction, and covariates the channel estimation result for the new pilot matrix. And repeating the process of extending the pilot matrix until there is an inverse of the matrix. The method of claim 7, The pilot matrix configuration unit Construct a first pilot matrix of a first magnitude using pilot subcarrier signals of at least one subchannel cluster, If the inverse matrix does not exist in the channel estimation result according to the first pilot matrix, the first pilot matrix is extended in the symbol axis direction to form a second pilot matrix having a second size, When the inverse matrix does not exist in the channel estimation result according to the second pilot matrix, the second pilot matrix is extended in the symbol axis direction to form a third pilot matrix having a third size. Channel estimation device. The method of claim 7, And the channel estimator uses the channel estimation result as a channel estimation result for the corresponding unit block when the inverse matrix exists as a result of the determination by the inverse matrix determination unit. The method according to any one of claims 7 to 10. And the covariance matrix is a matrix corresponding to a product of a Hermitian matrix and the pilot matrix of the pilot matrix.

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