Source: https://patents.google.com/patent/US20110268207A1/en
Timestamp: 2020-02-20 05:30:53
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Matched Legal Cases: ['§119', 'Application No. 61', 'Application No. 61', 'Application No. 61', 'Application No. 61', 'Application No. 61', 'Application No. 61', '§119', 'Application No. 10']

US20110268207A1 - Multiple input multiple output communication system using codebook corresponding to each reporting mode - Google Patents
Multiple input multiple output communication system using codebook corresponding to each reporting mode Download PDF
US20110268207A1
US20110268207A1 US13/097,719 US201113097719A US2011268207A1 US 20110268207 A1 US20110268207 A1 US 20110268207A1 US 201113097719 A US201113097719 A US 201113097719A US 2011268207 A1 US2011268207 A1 US 2011268207A1
US13/097,719
US9876553B2 (en
2010-08-16 Priority to US37394210P priority
2010-10-01 Priority to US38873610P priority
2010-12-30 Priority to US201061428348P priority
2011-04-05 Priority to KR1020110031200A priority patent/KR101817724B1/en
2011-04-05 Priority to KR10-2011-0031200 priority
2011-04-29 Priority to US13/097,719 priority patent/US9876553B2/en
2011-04-29 Assigned to SAMSUNG ELECTRONICS CO., LTD. reassignment SAMSUNG ELECTRONICS CO., LTD. ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: CHOI, JOON-IL, CLERCKX, BRUNO, KIM, KI IL
2011-04-29 Application filed by Samsung Electronics Co Ltd filed Critical Samsung Electronics Co Ltd
2011-11-03 Publication of US20110268207A1 publication Critical patent/US20110268207A1/en
2018-01-23 Publication of US9876553B2 publication Critical patent/US9876553B2/en
230000000875 corresponding Effects 0 claims description title 29
239000011159 matrix materials Substances 0 abstract claims description 246
239000003138 indicator Substances 0 description 119
A multiple input multiple output (MIMO) communication system using a first codebook and a second codebook is provided. The first codebook and the second codebook may independently exist, or may exist in a form of an overall codebook in which the first codebook and the second codebook are integrated with each other. A receiver may extract a first precoding matrix indicator from the first codebook, and may extract a second precoding matrix indicator from the second codebook. The receiver may also extract the first precoding matrix indicator and the second precoding matrix indicator from the overall codebook. The first precoding matrix indicator and the second precoding matrix indicator may be fed back to a transmitter. The transmitter may determine a precoding matrix based on the first precoding matrix indicator and the second precoding matrix indicator.
This application claims the benefit under 35 U.S.C. §119(e) of U.S. Provisional Application No. 61/329,634, filed on Apr. 30, 2010, U.S. Provisional Application No. 61/355,681, filed on Jun. 17, 2010, U.S. Provisional Application No. 61/356,768, filed on Jun. 21, 2010, U.S. Provisional Application No. 61/373,942, filed on Aug. 16, 2010, U.S. Provisional Application No. 61/388,736, filed on Oct. 1, 2010, and U.S. Provisional Application No. 61/428,348, filed on Dec. 30, 2010, all of which were filed in the United States Patent and Trademark Office, and claims the benefit under 35 U.S.C. §119(a) of Korean Patent Application No. 10-2011-0031200, filed on Apr. 5, 2011, in the Korean Intellectual Property Office, the entire disclosures of which are incorporated herein by reference for all purposes.
The following description relates to a multiple input multiple output (MIMO) communication system using a codebook, and more particularly, to codebooks corresponding to respective reporting modes used by a transmitter and a receiver included in a MIMO communication system.
A multiple input multiple output (MIMO) communication system may include a transmitter and at least one receiver. For example, the MIMO communication system may include a base station and at least one terminal. In a downlink, the base station may perform a functionality as the transmitter, and each of the at least one terminal may perform a functionality as the receiver.
The transmitter or the receiver operating in the MIMO communication system may include a plurality of antennas, and may transmit and receive data using the plurality of antennas. A wireless channel may be formed between each transmit antenna of the transmitter and each receive antenna of the receiver. The transmitter and the receiver may share information associated with the wireless channel, thereby achieving a high data rate.
In a closed-loop MIMO communication system, feedback information to be shared between the transmitter and the receiver may include a rank indicator indicating a preferred rank of the receiver, a precoding matrix indicator indicating a preferred precoding matrix, channel quality information indicating a quality of a wireless channel, and the like. The receiver may select one of matrices or vectors included in a codebook using a predefined codebook, and may feed back an index of the selected matrix or vector as the precoding matrix indicator.
In one general aspect, there is provided a communication method of a receiver in a multiple input multiple output (MIMO) communication system including a transmitter having eight transmit antennas and the receiver, the communication method including extracting a first precoding matrix indicator corresponding to a first codeword included in a first codebook and a second precoding matrix indicator corresponding to a second codeword included in a second codebook, and transmitting, to the transmitter, the first precoding matrix indicator and the second precoding matrix indicator.
In another general aspect, there is provided a communication method of a receiver in a MIMO communication system including a transmitter having eight transmit antennas and the receiver, the communication method including feeding back, to the transmitter, a first precoding matrix indicator corresponding to a first codeword included in a first codebook in order to indicate a recommended precoding matrix at a first reporting point in time, and feeding back, to the transmitter, a second precoding matrix indicator corresponding to a second codeword included in a second codebook in order to indicate a recommended precoding matrix at a second reporting point in time.
In still another general aspect, there is provided a communication method of a transmitter in a MIMO communication system including the transmitter having eight transmit antennas and a receiver, the communication method including receiving, from the receiver, a first precoding matrix indicator corresponding to a first codeword included in a first codebook and a second precoding matrix indicator corresponding to a second codeword included in a second codebook, accessing a memory that stores the first codebook and the second codebook, and generating a precoding matrix using the first precoding matrix indicator and the second precoding matrix indicator.
In yet another general aspect, there is provided a communication method of a transmitter in a MIMO communication system including the transmitter having eight transmit antennas and a receiver, the communication method including receiving, from the receiver, a first precoding matrix indicator corresponding to a first codeword included in a first codebook, the first precoding matrix indicator indicating a recommended precoding matrix at a first reporting point in time, receiving, from the receiver, a second precoding matrix indicator corresponding to a second codeword included in a second codebook, the second precoding matrix indicator indicating a recommended precoding matrix at a second reporting point in time, accessing a memory that stores the second codebook, and generating the recommended precoding matrix at the second reporting point in time using the second precoding matrix indicator received at the second reporting point in time.
FIG. 1 is a diagram illustrating an example of a multiple input multiple output (MIMO) communication system.
FIG. 2 is a diagram illustrating an example of a communication method of a receiver and a transmitter that share channel information using a single codebook.
FIG. 3 is a diagram illustrating an example of a relationship between two codebooks and a precoding matrix.
FIG. 4 is a diagram illustrating an example of a communication method of a receiver and a transmitter that share channel information using two codebooks.
FIG. 5 is a diagram illustrating an example of a communication method of a receiver and a transmitter that operate in a physical uplink control channel (PUCCH) 1-1 sub-mode 2.
FIG. 6 is a diagram illustrating an example of a communication method of a receiver and a transmitter that operate in PUCCH 2-1 sub-modes 1 and 2.
FIG. 1 illustrates an example of a multiple input multiple output (MIMO) communication system.
Referring to FIG. 1, the MIMO communication system may include a transmitter 110 and a plurality of receivers 120, 130, and 140.
Nt transmit antennas may be installed in the transmitter 110. The transmitter 110 may function as a base station in a downlink, and may function as a terminal in an uplink. Nr receive antennas may be installed in the receivers 120, 130, and 140. Each of the receivers 120, 130, and 140 may function as a terminal in the downlink, and may function as a base station in the uplink. Hereinafter, embodiments will be described based on an operation of the transmitter 110 and the receivers 120, 130, and 140 in the downlink. The embodiments may be applicable to the uplink.
Channels may be formed between the transmitter 110 and the receivers 120, 130, and 140. Data may be transmitted from the transmitter 110 to the receivers 120, 130, and 140 via the channels. The transmitter 110 may precode at least one data stream using a precoding matrix, enhancing a performance of the MIMO communication system. A data stream may also be referred to as data.
The transmitter 110 may generate or determine a more accurate precoding matrix by verifying information associated with channel direction and information associated with channel quality. Information associated with the channel direction and information associated with the channel quality may be one example of channel information. Information associated with the channel direction may include a precoding matrix indicator.
For example, the transmitter 110 and the receivers 120, 130, and 140 may share the precoding matrix indicator using a codebook. The codebook may include a plurality of codewords. Each of the plurality of codewords may correspond to a vector or a matrix. A size of the codebook may correspond to a number of codewords. For example, a 3-bit codebook may include eight codewords, and a 4-bit codebook may include 16 codewords.
Each of the receivers 120, 130, and 140 may select a single codeword from the plurality of codewords, and may generate an indicator of the selected codeword as a precoding matrix indicator. The precoding matrix indicator may be fed back to the transmitter 110. The transmitter 110 may verify a codeword indicated by the precoding matrix indicator, using the codebook. The transmitter 110 may generate or determine an optimal precoding matrix based on the codeword corresponding to the precoding matrix indicator.
A dimension of a precoding matrix may be dependent on a rank of the transmitter 110. The rank of the transmitter 110 may correspond to a number of data streams desired to be transmitted or a number of layers of the transmitter 110.
FIG. 2 illustrates an example of a communication method of a receiver and a transmitter that share channel information using a single codebook.
Referring to FIG. 2, at 210, the transmitter may transmit a well-known signal to the receiver. The well-known signal may be a pilot signal.
At 220, the receiver may estimate a channel formed from the transmitter to the receiver based on the well-known signal.
At 230, the receiver may select, from a codebook, a codeword suitable for the estimated channel and generate a precoding matrix indicator including an index of the selected codeword. In this example, the same codebook may be stored in both the transmitter and the receiver.
At 240, the receiver may feed back a precoding matrix indicator to the transmitter. The receiver may also feed back channel quality information and a rank indicator.
At 250, the transmitter may generate or determine an optimal precoding matrix based on the fed back precoding matrix indicator. At 260, the transmitter may transmit data using the precoding matrix.
The communication method of the transmitter and the receiver when the transmitter and the receiver use the same single codebook is described above with reference to FIG. 2. According to embodiments, two codebooks may be used for the receiver and the transmitter to share two precoding matrix indicators.
Hereinafter, it is assumed that a first codebook C1 and a second codebook C2 are present, and two codebooks are stored in the receiver and the transmitter, respectively. It is also assumed that a precoding matrix W is finally recommended by the receiver and is used by the transmitter.
FIG. 3 illustrates an example of a relationship between two codebooks and a precoding matrix.
Referring to FIG. 3, both a transmitter and a receiver may store a first codebook C1 310 and a second codebook C2 320. The receiver may select a preferred first codeword W1 from the first codebook C1 310, and may select a preferred second codeword W2 from the second codebook C2 320. A first precoding matrix indicator may be fed back to the transmitter as an index of the preferred first codeword W1, and a second precoding matrix indicator may be fed back to the transmitter as an index of the preferred second codeword W2.
Based on the first precoding matrix indicator and the second precoding matrix indicator, the transmitter may find the preferred first codeword W1 from the first codebook C1 310, and may find the preferred second codeword W2 from the second codebook C2 320. The transmitter may determine a precoding matrix W=f(W1, W2) based on the preferred first codeword W1 and the preferred second codeword W2.
In W=f(W1, W2), a function f may be variously defined. For example, W=f(W1, W2)=W2W1 or W=f(W1, W2)=W1W2 may be defined.
W1 corresponds to the preferred first codeword of the receiver corresponding to the first precoding matrix indicator selected by the receiver from the first codebook C1. W2 corresponds to the preferred second codeword of the receiver corresponding to the second precoding matrix indicator of the receiver selected from the second codebook C2. The first codebook C1 or the first precoding matrix indicator may be used to indicate a property of a channel in a wideband including a plurality of subbands, or to indicate a long-term property of the channel. The second codebook C2 or the second precoding matrix indicator may be used to indicate a property of a channel in a subband or to indicate a short-term property of the channel.
In W=f(W1, W2)=W2W1, W may have a dimension of Nt×R and W1 may have a dimension of Nt×R. W2 may have a dimension of Nt×Nt. In W=f(W1, W2)=W1W2, W may have a dimension of Nt×R and W1 and W2 may have a variety of dimensions based on R. Here, R corresponds to a rank and indicates a number of data streams or a number of layers.
Hereinafter, the first codebook C1 including candidates of W1 and the second codebook C2 including candidates of W2 when the transmitter includes eight transmit antennas will be defined with respect to each of various ranks. Since W1 is indicated by a combination of W1 and W2, to define the candidates of W1 and the candidates of W2 may be equivalent to define candidates of W. In addition to the first codebook C1 and the second codebook C2, the candidates of W may also be defined.
Design of rank 1 codebook when the transmitter includes eight transmit antennas:
In dual polarized channels, a precoding matrix in one subband may be expressed by,
W = 2 2  [ 2 -  α  2  A α   B ]
A and B may correspond to unit norm vectors having a dimension of Nt/2×1 and may independently perform beamforming in each polarization. Each polarization may appear as an effectively single antenna after beamforming is performed in each polarization using A and B. To design codebooks with respect to A and B may be dependent on statistical properties of a channel in each polarization. Without further assumption with respect to properties, A and B may account for subband/short-term information and wideband/long-term information.
Beamforming of polarizations may be performed by vector
[ 2 -  α  2 α ] .
Here, α corresponds to a complex scalar and may account for a phase difference and a magnitude difference. The phase difference between the polarizations may typically correspond to a short-term property and the magnitude difference may correspond to a function of the subband/short-term property and wideband/long-term property. A cross-polarization discrimination factor is generally referred to as XPD of a channel. XPD indicates a wideband/long-term property of a dual polarization channel and a mean value with respect to α may vary.
In general, A and B may be selected to be different from each other. However, when an interval between antennas is relatively close and each angle spread is relatively low, a beamforming vector with respect to a first polarization and a beamforming vector with respect to a second polarization may be regarded to be identical to each other. Since beamforming is invariant over a phase shift, B=ejφA may be established. Here, a selection of φ may not affect the performance of the dual polarization channel. When the interval between antennas is close, A, B, and φ may be associated with wideband/long-term properties of a channel. Accordingly, a precoding matrix in a subband may be expressed by,
W = 2 2  [ 2 -  α  2  A α    j   φ  A ] = 2 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A  j   φ  A ]
For an appropriate design of A, discrete Fourier transformation (DFT) vectors may be used. In the above equation, a last equal mark may remind a structure of W2W1. A subband/short-term matrix may be expressed by
W 2 = [ 2 -  α  2  I n t / 2 α   I n t / 2 ]
A wideband/long-term matrix may be expressed by
W 1 = 2 2  [ A  j   φ  A ]
In a special case where ejφ=1:
W  = ( a )   2 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A ] = ( b )   2 2  [ 2 -  α  2 α ] ⊗ A = ( c )   2 2  [ A A ]  [ 2 -  α  2 α ]
As shown in the above equation, in the special case where ejφ=1, many equivalent methods may be used to express the same precoding matrix. For example, in the above equation, (a) corresponds to a method of using the structure of W2W1, (b) corresponds to a method of using Kronecker product, and (c) corresponds to a method of using a structure of W1W2.
When the interval between antennas is close, the precoding matrix may be expressed using the aforementioned equations in a single polarization channel. In this example, α=1, a value of φ may be A-dependent and be selected to obtain DFT vectors for eight transmit antennas. For example, W2 may correspond to an identity matrix and W1 may provide a wideband precoding matrix of DFT vectors. Contrast to dual polarization channels, the selection of φ may affect the performance of single polarization channels.
According to the structure of W2W1 shown in
W = 2 2  [ 2 -  α  2  A α    j   φ  A ] = 2 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A  j   φ  A ] ,
the wideband/long-term matrix
[ A  j   φ  A ]
may have a significantly robust physical meaning That is, in its given Nt×1 dimension, the wideband/long-term matrix may be equivalent to a rank and thus, may provide a direct insight to a rank 1 wideband PMI structure. Also, in the aforementioned W2W1 structure, a structure
may not be associated with the rank and may not provide any information associated with a wideband PMI structure.
A full utilization of power amplifiers may be used as an important design criterion. When only a phase shift keying (PSK) is used to decrease the complexity of PMI search, there is a need to constrain a precoding matrix. It may be assumed that the precoding matrix becomes constant modulus and |α|=1. In this scenario, α may use a subband/long-term property with respect to a phase shift between polarizations.
Design of rank 2 codebook when the transmitter includes eight transmit antennas:
A rank 2 precoding matrix may include two orthogonal columns, which may be expressed by
W ( 1 ) = 2 2  [ 2 -  α 1  2  I n t / 2 α 1  I n t / 2 ]  [ A 1 B 1 ] W ( 2 ) = 2 2  [ 2 -  α 2  2  I n t / 2 α 2  I n t / 2 ]  [ A 2 B 2 ]
The full utilization of power in each antenna may force |α1|2+|α2|2=2, and may establish α2=√{square root over (2−|α|2)}ejδ with α1=α. In this example, the following equations may be expressed.
W ( 1 ) = 2 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A 1 B 1 ] W ( 2 ) = 2 2  [  α   I n t / 2 2 -  α  2   j   δ  I n t / 2 ]  [ A 2 B 2 ]
To obtain mutually orthogonal columns, A1 HA2=0 and B1 HB2=0 may be sufficient. A1, A2, B1, and B2 may be approximated by two dominant eigenvectors of Nt×Nt covariance matrix. Many combinations may be used for design of the precoding matrix, which may cause great overheard. In a scenario with a narrow interval between antennas, A1=A, A2=A, B1=ejφ 1 A, and B2=ejφ 2 A. A cross-polarized setup may help achievement of rank 2 transmission in a configuration where the interval between antennas is narrow.
Parameters φ1 and φ2 may be selected to guarantee so that W(1) and W(2) may be orthogonal with respect to each other. In this example, φ1=φ, and φ2=φ+π. The rank 2 precoding matrix may be expressed by
W = 1 2  [ W ( 1 ) W ( 2 ) ] = 1 2  [ 2 -  α  2  A  α   A α    j   φ  A - 2 -  α  2   j   δ   j   φ  A ]
The precoding matrix may be expressed using the W2W1 structure, as follows:
W = 1 2  [ W ( 1 ) W ( 2 ) ] = 1 2  [ 2 -  α  2  A  α   A α    j   φ  A - 2 -  α  2   j   δ   j   φ  A ] = 1 2  [ A  j   φ  A ]  [ 2 -  α  2  α  α - 2 -  α  2   j   δ ]
W 1 = [ A  j   φ  A ] and W 2 = 1 2  [ 2 -  α  2  α  α - 2 -  α  2   j   δ ] .
The precoding matrix may be expressed using a variety of methods. For example, the precoding matrix may be expressed by
W = 1 2  [ W ( 1 ) W ( 2 ) ] = 1 2  [ 2 -  α  2  A  α   A α    j   φ  A - 2 -  α  2   j   δ   j   φ  A ] = [ 2 -  α  2  α  α - 2 -  α  2   j   δ ] ∘ [ 1 2  [ A  A   j   φ  A -  j   φ  A ] ]
In this equation, ∘ corresponds to Hardmard product, and
W 1 = 1 2  [ A  A   j   φ  A -  j   φ  A ] ,  and W 2 = 1 2  [ 2 -  α  2  α  α - 2 -  α  2   j   δ ] .
When |α|=1 is assumed to maintain the precoding matrix as constant modulus, and to maintain a PSK alphabet, the rank 2 precoding matrix may include two orthogonal columns W(1) and W(2). Each column may satisfy the structure of the rank 1 precoding matrix, for example, as follows:
W ( 1 ) = 2 2  [ 2 -  α 1  2  I n t / 2 α   I n t / 2 ]  [ A  j   φ 1  A ] W ( 2 ) = 2 2  [ 2 -  α 2  2  I n t / 2 αI n t / 2 ]  [ A  j   φ 2  A ]
Two rank 1 precoding matrices may be differentiated using only the parameter φ. The parameters φ1 and φ2 may be selected to guarantee that W(1) and W(2) are orthogonal to each other. When φ1=φ and φ2=φ+π, the rank 2 precoding matrix may be expressed by
W = 1 2  [ W ( 1 ) W ( 2 ) ] = 1 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A  A   j   φ  A -  j   φ  A ] .
Wideband/long-term matrix W1 may correspond to a wideband precoding matrix and may be given as
W 1 = 1 2  [ A  A   j   φ  A -  j   φ  A ] .
A subband matrix W2 may be expressed by
W 2 = [ 2 -  α  2  I n t / 2 α   I n t / 2 ] .
The selection of φ may not affect the performance of the wideband precoding matrix W1 in dual polarization channels, however, may have a strong influence in single polarization channels. The parameter φ may be selected so that W1 may have excellent performance even in single polarization channels.
W  = ( a )   1 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A  A  A - A ] = ( b )   1 2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  U rot  [ A  0 0 A ] = ( c )   1 2  [ 2 -  α  2  2 -  α  2 α  - α  ] ⊗ A = ( d )   1 2  [ A A ]  [ 2 -  α  2  2 -  α  2 α  - α  ] .
In the special case where ejφ=1, many equivalent methods may be used to express the same precoding matrix. For example, in the above equation, (a) corresponds to a method of using the structure of W2W1, (b) corresponds to a method of using a rotated block diagonal structure, (c) corresponds to a method of using Kronecker product, and (d) corresponds to a method of using the structure of W1W2.
Design of Rank 3 Codebook When the Transmitter Includes Eight Transmit Antennas:
A rank 3 precoding matrix may be obtained by simply extending a structure induced with respect to the rank 1 precoding matrix and the rank 2 precoding matrix. By adding, to the rank 2 precoding matrix, a column orthogonal to the rank 2 precoding matrix, the rank 3 precoding matrix may be obtained as follows:
W = 1 3  [ W ( 1 ) W ( 2 ) W ( 3 ) ] = 1 3  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A B  j   ϕ  A -  j   ϕ  A  j   ϕ  B ] or W = 1 3  [ W ( 1 ) W ( 2 ) W ( 3 ) ] = 1 3  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A B  j   ϕ  A -  j   ϕ  A -  j   ϕ  B ]
In this example, A and B may be orthogonal to each other.
Design of Rank 4 Codebook When the Transmitter Includes Eight Transmit Antennas:
Similarly with respect to rank 4, a rank 4 precoding matrix may be expressed using two rank 2 precoding matrices as follows:
W =  1 4  [ W ( 1 ) W ( 2 ) W ( 3 ) W ( 4 ) ] =  1 4  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A B B  j   ϕ  A -  j   ϕ  A  j   ϕ  B -  j   ϕ  B ]
Design of Rank r Codebook When the Transmitter Includes Eight Transmit Antennas:
With respect to rank r codebook, the precoding matrix may be expressed as follows:
When r is an odd number,
W =  1 r  [ W ( 1 ) W ( 2 ) … W ( r ) ] =  1 r  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A … C  j   ϕ  A -  j   ϕ  A …  j   ϕ  C ] or W =  1 r  [ W ( 1 ) W ( 2 ) … W ( r ) ] =  1 r  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A … C  j   ϕ  A -  j   ϕ  A … -  j   ϕ  C ]
When r is an even number,
W =  1 r  [ W ( 1 ) W ( 2 ) … W ( r - 1 ) W ( r ) ] =  1 r  2  [ 2 -  α  2  I n t / 2 α   I n t / 2 ]  [ A A … C C  j   ϕ  A -  j   ϕ …  j   ϕ  C -  j   ϕ   C ]
In this example, A, B, . . . , C may be orthogonal to each other.
The following collusion may be made. That is, the minimum requirement for achieving the excellent performance of a recommended precoding matrix may follow as:
W=W 2 W 1
Here, an outer matrix W1 corresponds to a unitary precoding matrix that is an element of a first codebook C1 and has a dimension of Nt×R. For each rank, W1 may be expressed as follows:
Rank   1  :   W 1 = 2 2  [ A  j   ϕ  A ] Rank   2  :   W 1 = 1 2  [ A A  j   ϕ  A -  j   φ  A ]
Rank r:
when r is an odd number:
W 1 = 1 r   2  [ A A … C  j   φ  A -  j   φ  A  …  j   φ  C ] or W 1 = 1 r  2   [ A A … C  j   φ  A -  j   φ  A … -  j   φ  C ]
when r is an even number:
W 1 = 1 r  2  [ A A … C C  j   φ  A -  j   φ  A …  j   φ  C -  j   φ  C ]
A, B, . . . , C may be orthogonal to each other, or may be DFT vectors.
An inner matrix W2 may correspond to a diagonal matrix that is an element of a second codebook C2 and has a dimension of Nt×Nt. For example,
W 2 = [ 2 -  α  2  I n t / 2 α   I n t / 2 ]   with    α  = 1.
In the aforementioned observation, highly correlated channels may be assumed. Feedback overhead required for reporting W2 and W1 with a sufficient accuracy may not be used. To provide some design flexibilities, and to provide balanced feedback overheard and high feedback accuracy with respect to W2 and W1, a previous observation may be extended as follows:
In this example, an outer matrix W1 corresponds to a unitary precoding matrix that is an element of a first codebook C1 and has a dimension of Nt×R. For each rank, W1 may be expressed as follows:
Rank   1  :   W 1 = 2 2  [ A  j   φ  A ] Rank   2  :   W 1 = 1 2  [ A A  jφ  A -  j   φ  A ]
W 1 = 1 r  2  [ A A … C  j   φ  A -  j   φ  A …  j   φ  C ] or W 1 = 1 r  2  [ A A … C  j   φ  A -  j   φ  A … -  j   φ  C ]
W 1 = 1 r  2  [ A A … C C  j   φ  A -  jφ  A …  j   φ  C -  j   φ  C ]
W 2 = [ 2 -  α  2  Θ 0 4 × 4 0 4 × 4 α   Θ ]   with    α  = 1.
In W2, Θ corresponds to a 4×4 matrix, and may be defined as Θ=diag{1,ejπθ,ej2πθ,ej3πθ}. diag(a, b, c, d) corresponds to a diagonal matrix that includes a, b, c, and d as diagonal elements. Θ enables tracking of a spatial correlation structure, for example, a DFT structure in a subband level above antennas 0 through 3, and above antennas 4 through 7. In this example, in a dual polarization case, the antennas 0 through 3 may generate one polarization, and the antennas 4 through 7 may generate another polarization. In a single polarization case, all the antennas may generate the same polarization.
α corresponds to a complex scalar and may process dual polarization or single polarization based on a small antennal interval. α may be selected within a subband level, for example, within a set of 1, j, ej4πθ. For example, in a single polarization case, W2 may have a structure of W2=diag{1,ejπθ,ej2πθ,ej3πθ,ej4πθ,ej5πθ,ej6πθ,ej7πθ}. In a dual polarization case, α may be selected as 1 or j.
Codebook Suggestions
Prior to suggesting codebooks, 4×r DFT matrices may be defined as follows:
DFT 1 = 1 2  [ 1 1 1 1 1 j - 1 - j 1 - 1 1 - 1 1 - j - 1 j ] ,  DFT 2 = diag  { 1 ,  jπ / 4 , j ,  j   3  π / 4 }  DFT 1 ,  DFT 3 = diag  { 1 ,  j   π  / 8 ,  j   2  π  / 8 ,  j   3  π  / 8 }  DFT 1 ,  DFT 4 = diag  { 1 ,  j   3  π  / 8 ,  j   6  π  / 8 ,  j9π / 8 }  DFT 1 ,
Suggestion 1: 4-Bit Codebook for Each Rank for W1
In suggestion 1, the first codebook C1 for rank r where r=1, . . . , 6 may include 16 4-bit elements or codewords. The first codebook C1 for rank r where r=7, 8 may include four elements.
Codebook C1
The first codebook C1 for rank r may be expressed as C1,r.
A first codebook C1,1 for rank 1 may be obtained by employing columns 1 through 16 of the following matrix:
V 1 = 2 2  [ DFT 1 DFT 2 DFT 3 DFT 4 DFT 1 - DFT 2 j   DFT 3 - j   DFT 4 ]
The 16 column vectors may correspond to DFT vectors for eight transmit antennas.
A first codebook C1,2 for rank 2 may include the following 16 matrices:
C 1 , 2 = { 1 2  [ D 1 , k D 1 , k D 1 , k - D 1 , k ] , 1 2  [ D 2 , k D 2 , k D 2 , k - D 2 , k ] , 1 2  [ D 3 , k D 3 , k j   D 3 , k - j   D 3 , k ] , 1 2  [ D 4 , k D 4 , k j   D 4 , k - j   D 4 , k ] , k = 1 , …  , 4 }
In this example, Dm,k corresponds to a kth column of DFTm. For example, D1,k corresponds to a kth column of DFT1, D2,k corresponds to a kth column of DFT2, D3,k corresponds to a kth column of DFT3, and D4,k corresponds to a kth column of DFT4.
The first codebook C1,2 may be obtained by using a first codebook for rank 1 and by adding up orthogonal columns based on
W 1 = 1 2  [ A A  j   φ  A -  j   φ  A ] .
A first codebook C1,3 for rank 3 may include the following 16 matrices:
C 1 , 3 = { 1 3  2  [ D 1 , k  D 1 , k D 1 , m D 1 , k - D 1 , k D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k  - D 2 , k D 2 , m ] , 1 3  2  [ D 3 , k D 3 , k D 3 , m j   D 3 , k - j   D 3 , k j   D 3 , m ] , 1 3  2  [ D 4 , k D 4 , k D 4 , m j   D 4 , k - j   D 4 , k j   D 4 , m ] }
In this example, k=1, . . . 4 and m=k mod 4+1.
C 1 , 3 = { 1 3  2  [ D 1 , k D 1 , k D 1 , m D 1 , k - D 1 , k - D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k - D 2 , k - D 2 , m  ] , 1 3  2  [ D 3 , k D 3 , k D 3 , m j   D 3 , k - j   D 3 , k - j   D 3 , m ] , 1 3  2  [ D 4 , k D 4 , k D 4 , m j   D 4 , k - j   D 4 , k - j   D 4 , m ] }
Other examples may also be used. For example, m may be given to be different from above, and k may also be given to be different from above. For example, various combinations of k and m may be given as (k,m)={(1,2),(1,3),(1,4),(2,3)}.
A first codebook C1,4 for rank 4 may include the following 16 matrices:
C 1 , 4 = { 1 4  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , k - D 1 , k D 1 , m - D 1 , m ] , 1 4  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , k - D 2 , k D 2 , m - D 2 , m ] , 1 4  2  [ D 3 , k D 3 , k D 3 , m D 3 , m j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m ] , 1 4  2  [ D 4 , k D 4 , k D 4 , m D 4 , m j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m ] }
Example 2) m may be given to be different from above, and k may also be given to be different from above. For example, various combinations of k and m may be given as (k,m)={(1,2),(1,3),(1,4),(2,3)}. Other examples may also be used.
A first codebook C1,5 for rank 5 may include the following 16 matrices:
C 1 , 5 = { 1 5  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n ] , 1 5  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n ] , 1 5  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n ] , 1 5  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n ] }
A combination of k,m, and n may be selected from {(1,2,3),(1,2,4),(1,3,4),(2,3,4)}.
C 1 , 5 = { 1 5  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m - D 1 , n ] , 1 5  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , k - D 2 , k D 2 , m - D 2 , m - D 2 , n ] , 1 5  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m - j   D 3 , n ] , 1 5  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m - j   D 4 , n ] }
A first codebook C1,6 for rank 6 may include the following 16 matrices:
C 1 , 6 = { 1 6  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n ] , 1 6  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n ] , 1 6  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n D 3 , n j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n - j   D 3 , n ] , 1 6  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n D 4 , n j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n - j   D 4 , n ] }
A first codebook C1,7 for rank 7 may include the following four matrices:
C 1 , 7 = { 1 7  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , p D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n D 1 , p ] , 1 7  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , p D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n D 2 , p ] , 1 7  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n D 3 , n D 3 , p j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n - j   D 3 , n j   D 3 , p ] , 1 7  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n D 4 , n D 4 , p j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n - j   D 4 , n j   D 4 , p ] } ( k , m , n , p ) = ( 1 , 2 , 3 , 4 ) .
C 1 , 7 = { 1 7  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , p D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n - D 1 , p ] , 1 7  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , p D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n - D 2 , p ] , 1 7  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n D 3 , n D 3 , p j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n - j   D 3 , n - j   D 3 , p ] , 1 7  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n D 4 , n D 4 , p j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n - j   D 4 , n - j   D 4 , p ] } ( k , m , n , p ) = ( 1 , 2 , 3 , 4 ) .
A first codebook C1,8 for rank 8 may include the following four matrices:
C 1 , 8 = { 1 8  2  [ D 1 D 1 D 1 - D 1 ] , 1 8  2  [ D 2 D 2 D 2 - D 2 ] , 1 8  2  [ D 3 D 3 j   D 3 - j   D 3 ] , 1 8  2  [ D 4 D 4 j   D 4 - j   D 4 ] }
Codebook C2
A number of codewords to be assigned to Θ and α may need to be carefully investigated.
Example 1) For example, when a single bit is assigned to Θ and α, the second codebook C2 may be expressed as follows:
For rank 2:
With respect to α ∈ {ej4πθ i } and Θi where i=1,2, when a second codebook for rank 1 including a first codeword and a second codeword is assumed as C2,1 . . . 2,
C 2 , 1   …   2 = { [ Θ 1 0 4 × 4 0 4 × 4  j4πθ 1  Θ 1 ] , [ Θ 2 0 4 × 4 0 4 × 4  j4πθ 2  Θ 2 ] } .
θ 1 = 1 16 , θ 2 = - 1 16 .
With respect to α ∈ {1,−1} and Θ=I, when the second codebook for rank 1 including a third codeword and a fourth codeword is assumed as C2,3 . . . 4,
C 2 , 3   …   4 = { [ I 4 0 4 × 4 0 4 × 4 I 4 ] , [ I 4 0 4 × 4 0 4 × 4 - I 4 ] } .
For ranks 2, 3, and 4:
With respect to α ∈ {1} and Θi where i=1, 2, when a second codebook for ranks 2, 3, and 4 including a first codeword and a second codeword is assumed as C2,1 . . . 2,
C 2 , 1   …   2 = { [ Θ 1 0 4 × 4 0 4 × 4 Θ 1 ] , [ Θ 2 0 4 × 4 0 4 × 4 Θ 2 ] } .
Example 2) A size of the second codebook may be extended to three bits by extending the aforementioned example 1).
For rank 1:
With respect to α ∈ {1,ej4πθ i } and Θi where i=1,2, when the second codebook for rank 1 including four codewords is assumed as
 C 2 , 1   …   4 ,  C 2 , 1   …   4 = { [ Θ 1 0 4 × 4 0 4 × 4  j4πθ 1  Θ 1 ] , [ Θ 2 0 4 × 4 0 4 × 4  j4πθ 2  Θ 2 ] , [ Θ 1 0 4 × 4 0 4 × 4 Θ 1 ] , [ Θ 2 0 4 × 4 0 4 × 4 Θ 2 ] } .
With respect to α ∈ {1} and Θi where i=1,2, when the second codebook for ranks 2, 3, and 4 including first through fourth codewords is assumed as
 C 2 , 1   …   4 ,  C 2 , 1   …   4 = { [ Θ 1 0 4 × 4 0 4 × 4 Θ 1 ] , [ Θ 2 0 4 × 4 0 4 × 4 Θ 2 ] , [ Θ 3 0 4 × 4 0 4 × 4 Θ 3 ] , [ Θ 4 0 4 × 4 0 4 × 4 Θ 4 ] } .
θ 1 = 1 16 , θ 2 = - 1 16 , θ 3  1 8 , θ 4 = - 1 8 .
With respect to α ∈ {1,j} and Θ=I, when the second codebook for ranks 2, 3, and 4 including fifth through sixth codewords is assumed as C2,5 . . . 6,
C 2 , 5   …   6 = { [ I 0 4 × 4 0 4 × 4 I ] , [ I 0 4 × 4 0 4 × 4 j   I ] } .
θ 3 = 1 8 , θ 4 = - 1 8
with α ∈ {j}, when the second codebook for ranks 2, 3, and 4 including seventh through eighth codewords is assumed as
C 2 , 7   …   8 , C 2 , 7   …   8 = { [ Θ 3 0 4 × 4 0 4 × 4 jΘ 3 ] , [ Θ 4 0 4 × 4 0 4 × 4 jΘ 4 ] } .
Suggestion 2: Maximum 4-Bit Codebook for Each Rank for W1
In suggestion 2, the first codebook for rank r where r=1, . . . 2 may include 16 elements, the first codebook for rank r where r=3, 4 may include eight elements, and the first codebook for rank r where r=5, 6, 7, 8 may include four elements.
The above 64 entries may be divided into four subsets each including 16 entries. To indicate one of the subsets, two bits may be used. The two bits may indicate a rank corresponding to the selected subset among rank 1, rank 2, rank 3-4, and rank 5-8.
A first codebook C1 for rank r may be indicated as C1,r.
A rank 1 first codebook C1,1 may be obtained by employing columns 1 through 16 of the following matrix:
V 1 = 2 2  [ D   F   T 1 D   F   T 2 D   F   T 3 D   F   T 4 D   F   T 1 - D   F   T 2 j   D   F   T 3 - j   D   F   T 4 ]
The column vectors 1 through 16 may correspond to DFT vectors for eight transmit antennas.
A rank 2 first codebook C1,2 may include the following 16 matrices:
The rank 2 first codebook C1,2 may be obtained by using the rank 1 first codebook and adding orthogonal columns based on
W 1 = 1 2  [ A A  jφ -  jφ  A ] .
A rank 3 first codebook C1,3 may include the following eight matrices:
C 1 , 3 = { 1 3  2  [ D 1 , k D 1 , k D 1 , m D 1 , k - D 1 , k D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k - D 2 , k D 2 , m ] , 1 3  2  [ D 3 , k D 3 , k D 3 , m j   D 3 , k - j   D 3 , k j   D 3 , m ] , 1 3  2  [ D 4 , k D 4 , k D 4 , m j   D 4 , k - j   D 4 , k j   D 4 , m ] }
In this example, k=1, 2 and m=k+2.
C 1 , 3 = { 1 3  2  [ D 1 , k D 1 , k D 1 , m D 1 , k - D 1 , k - D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k - D 2 , k - D 2 , m ] , 1 3  2  [ D 3 , k D 3 , k D 3 , m j   D 3 , k - j   D 3 , k - j   D 3 , m ] , 1 3  2  [ D 4 , k D 4 , k D 4 , m j   D 4 , k - j   D 4 , k - j   D 4 , m ] }
C 1 , 3 = { 1 3  2  [ D 1 , k D 1 , k D 1 , m D 1 , k - D 1 , k D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k - D 2 , k D 2 , m ] }  or C 1 , 3 = { 1 3  2  [ D 1 , k D 1 , k D 1 , m D 1 , k - D 1 , k - D 1 , m ] , 1 3  2  [ D 2 , k D 2 , k D 2 , m D 2 , k - D 2 , k - D 2 , m ] }
In this example, k=1, . . . , 4 and m=k mod 4+1.
In this example, (k,m)={(1,2),(1,3),(1,4),(2,3)}.
In addition to examples 1) through 4), other examples may also be employed.
A rank 4 first codebook C1,4 may include the following eight matrices:
C 1 , 4 = { 1 4  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , k - D 1 , k D 1 , m - D 1 , m ] , 1 4  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , m - D 2 , k D 2 , m - D 2 , m ] , 1 4  2  [ D 3 , k D 3 , k D 3 , m D 3 , m j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m ] , 1 4  2  [ D 4 , k D 4 , k D 4 , m D 4 , m j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m ] }
In this example, (k,m)={(1,2),(1,3)}.
C 1 , 4 = { 1 4  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , k - D 1 , k D 1 , m - D 1 , m ] , 1 4  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , k - D 2 , k D 2 , m - D 2 , m ] , }
The rank 5 first codebook C1,5 may include the following four matrices:
C 1 , 5 = { 1 5  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n ] }
In this example, (k,m,n)={(1,2,3),(1,2,4),(1,3,4),(2,3,4)}.
C 1 , 5 = { 1 5  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n ] 1 5  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n ] }
In this example, (k,m,n)={(1,2,3),(1,2,4)}.
In this example, (k,m,n)={(1,2,3)}.
A rank 6 first codebook C1,6 may include the following four matrices:
C 1 , 6 = { 1 6  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n ] }
C 1 , 6 = { 1 6  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n ] , 1 6  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n ] }
A rank 7 first codebook C1,7 may include the following four matrices:
C 1 , 7 = { 1 7  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , p D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n D 1 , p ] , 1 7  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , p D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n D 2 , p ] , 1 7  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n D 3 , n D 3 , p j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n - j   D 3 , n j   D 3 , p ] , 1 7  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n D 4 , n D 4 , p j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n - j   D 4 , n j   D 4 , p ] }
In this example, (k,m,n, p)={(1,2,3,4)}.
C 1 , 7 = { 1 7  2  [ D 1 , k D 1 , k D 1 , m D 1 , m D 1 , n D 1 , n D 1 , p D 1 , k - D 1 , k D 1 , m - D 1 , m D 1 , n - D 1 , n - D 1 , p ] , 1 7  2  [ D 2 , k D 2 , k D 2 , m D 2 , m D 2 , n D 2 , n D 2 , p D 2 , k - D 2 , k D 2 , m - D 2 , m D 2 , n - D 2 , n - D 2 , p ] , 1 7  2  [ D 3 , k D 3 , k D 3 , m D 3 , m D 3 , n D 3 , n D 3 , p j   D 3 , k - j   D 3 , k j   D 3 , m - j   D 3 , m j   D 3 , n - j   D 3 , n - j   D 3 , p ] , 1 7  2  [ D 4 , k D 4 , k D 4 , m D 4 , m D 4 , n D 4 , n D 4 , p j   D 4 , k - j   D 4 , k j   D 4 , m - j   D 4 , m j   D 4 , n - j   D 4 , n - j   D 4 , p ] }
In this example, (k,m,n,p)=(1,2,3,4).
A rank 8 first codebook C1,8 may include the following four matrices:
The second codebook C2 may be the same as in suggestion 1.
Suggestion 3: Maximum 4-Bit Codebook for Each Rank for W1
Suggestion 3 relates to the structure of W1W2. In suggestion 3, the first codebook C1 for rank r where r=1, 2 may include 16 elements, the first codebook C1 for rank r where r=3, 4 may include eight elements, and the first codebook C1 for rank r where r=5, 6, 7, 8 may include four elements.
The first codebook C1 for rank r may be indicated as C1,r.
A first codebook C1,(1,2) for ranks 1 and 2 may be obtained by the following matrices:
B = [ b 0 b 1 … b 31 ] ,  [ B ] 1 + m , 1 + n =  j  2  π   mn 32 ,  m = 0 , 1 , 2 , 3 , n = 0 , 1 , …   31 X ( k ) ∈ { 1 2  [ b 2   k   mod   32 b ( 2   k + 1 )   mod   32 b ( 2   k  + 2 )  mod   32 b ( 2   k  + 3 )  mod   32 ] : k = 0 , 1 , …  , 15 } W 1 ( k ) = [ X ( k ) 0 0 X ( k ) ] C 1 , ( 1 , 2 ) = { W 1 ( 0 ) , W 1 ( 1 ) , W 1 ( 2 ) , …  , W 1 ( 15 ) }
In this example, [B]1+m,1+n indicates an element present in an (1+m)th row and an (1+n)th column among elements belonging to B, and bz(z=0, 1, 2, . . . , 31) corresponds to a zth column vector of the matrix B, and a mod b denotes a remainder when a is divided by b.
A first codebook C1,(3,4) for ranks 3 and 4 may be obtained by the following matrices:
B = [ b 0 b 1 … b 31 ] ,  [ B ] 1 + m , 1 + n =  j  2  π   mn 32 ,  m = 0 , 1 , 2 , 3 , n = 0 , 1 , …   31 X ( k ) ∈ { 1 2  [ b 4   k   mod   32 b ( 4   k + 1 )   mod   32 … b ( 4   k  + 7 )  mod   32 ] : k = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } W 1 ( k ) = [ X ( k ) 0 0 X ( k ) ] C 1 , ( 3 , 4 ) = { W 1 ( 0 ) , W 1 ( 1 ) , W 1 ( 2 ) , …  , W 1 ( 7 ) }
A first codebook C1,(5,6,7,8) for ranks 5, 6,7, and 8 may be obtained by the following matrices:
X ( 0 ) = 1 2 × [ 1 1 1 1 1 j - 1 - j 1 - 1 1 - 1 1 - j - 1 j ] ,  X ( 1 ) = diag  { 1 ,  jπ / 4 , j ,  j   3  π / 4 }  X ( 0 ) ,  X ( 2 ) = diag  { 1 ,  jπ / 8 ,  j   2  π / 8 ,  j   3  π / 8 }  X ( 0 ) ,  X ( 3 ) = diag  { 1 ,  j3π / 8 ,  j   6  π / 8 ,  j9π / 8 }  X ( 0 ) W 1 ( k ) = { [ X ( k ) 0 0 X ( k ) ] } , k = 0 , 1 , 2 , 3 C 1 , ( 5 , 6 , 7 , 8 ) = { W 1 ( 0 ) , W 1 ( 1 ) , W 1 ( 2 ) , …  , W 1 ( 3 ) }
The second codebook C2 for rank r may be indicated as C2,r.
A second codebook C2,1 for rank 1 may be expressed by:
C 2 , 1 = { 1 2  [ Y Y ] , 1 2  [ Y j   Y ] , 1 2  [ Y - Y ] , 1 2  [ Y - j   Y ] } Y ∈ { e ~ 1 , e ~ 2 , e ~ 3 , e ~ 4 }
A second codebook C2,2 for rank 2 may be expressed by:
C 2 , 2 = { 1 2  2  [ Y 1 Y 2 Y 1 - Y 2 ] , 1 2  2  [ Y 1 Y 2 j   Y 1 - j   Y 2 ] } ( Y 1 , Y 2 ) ∈ { ( e ~ 1 , e ~ 1 ) , ( e ~ 2 , e ~ 2 ) , ( e ~ 3 , e ~ 3 ) , ( e ~ 4 , e ~ 4 ) , ( e ~ 1 , e ~ 2 ) , ( e ~ 2 , e ~ 3 ) , ( e ~ 1 , e ~ 4 ) , ( e ~ 2 , e ~ 4 ) }
In this example, {tilde over (e)}n corresponds to a selection vectors. An nth element of {tilde over (e)}n may have a value of 1 with respect to ranks 1 and 2 and all of remaining elements may have a value of zero.
A second codebook C2,3 for rank 3 may be expressed by
C 2 , 3 = { 1 3  2  [ Y 1 Y 2 Y 1 - Y 2 ] , 1 3  2  [ Y 1 Y 2 j   Y 1 - j   Y 2 ] } ( Y 1 , Y 2 ) ∈ { ( e 1 , [ e 1 e 5 ] ) , ( e 2 , [ e 2 e 6 ] ) , ( e 3 , [ e 3 e 7 ] ) , ( e 4 , [ e 4 e 8 ] ) , ( e 5 , [ e 1 e 5 ] ) , ( e 6 , [ e 2 e 6 ] ) , ( e 7 , [ e 3 e 7 ] ) , ( e 8 , [ e 4 e 8 ] ) }
A second codebook C2,4 for rank 4 may be expressed by
C 2 , 4 = { 1 4  2  [ Y Y Y - Y ] , 1 4  2  [ Y Y j   Y - j   Y ] } Y ∈ { [ e 1 e 5 ] , [ e 2 e 6 ] , [