Method, codebook and base station for precoding

Abstract

A method, a codebook, and a Base Station (BS) for precoding are provided. The precoding method includes: obtaining a total uplink power of a User Equipment (UE); if the total uplink power is greater than of a rated total transmit power of antennas, selecting a codeword from a first codebook with imbalanced power between layers; otherwise, selecting a codeword from the first codebook and a second codebook with balanced power between layers, so as for precoding data to be transmitted according to the selected codeword.

Claims

1. A method performed by a mobile device in a telecommunication network for precoding data, the method comprising: receiving, from a network access entity of the telecommunication network, a message comprising an index of a codeword, wherein the index is for identifying the codeword within a codebook stored in the mobile device, wherein the codebook consists of exactly twelve codewords, each codeword being based on a Binary Phase Shift Keying (BPSK) character set and the codewords being: $A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],$ wherein A is a multiplier; precoding data using the identified codeword; and transmitting the precoded data to the network access entity.

2. The method for precoding data of claim 1, wherein the data is uplink data.

3. The method for precoding data of claim 1, wherein the telecommunication network is a Long Term Evolution-Advanced (LTE-A) system network, and the steps of receiving and transmitting are performed according to LTE-A protocols.

4. The method for precoding data of claim 1, wherein the mobile device is user equipment.

5. The method for precoding data of claim 1, wherein A is .

6. A mobile device for communicating over a telecommunication network, comprising a receiver, a transmitter, a processor, and a memory, wherein the memory stores a codebook, and wherein the codebook consists of exactly twelve codewords, each codeword being based on a Binary Phase Shift Keying (BPSK) character set and the codewords being: $A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],$ wherein A is a multiplier; and the processor configured to: receive, through the receiver, a message from a network access entity of the telecommunication network, wherein the message comprises an index of a codeword, wherein the index is for identifying the codeword within the codebook stored in the memory; precode data using the identified codeword; and transmit, through the transmitter, the precoded data to the network access entity.

7. The mobile device of claim 6, wherein the data is uplink data.

8. The mobile device of claim 6, wherein the telecommunication network is a Long Term Evolution-Advanced (LTE-A) system network, and the operations of receiving and transmitting are performed according to LTE-A protocols.

9. The mobile device of claim 6, wherein the mobile device is user equipment.

10. The mobile device of claim 6, wherein A is .

11. A computer program product comprising computer executable instructions stored on a non-transitory computer readable medium such that when executed by a processor of a mobile device cause the mobile device to receive, from a network access entity of a telecommunication network, a message comprising an index of a codeword, wherein the index is for identifying the codeword within a codebook, and wherein the codebook consists of exactly twelve codewords, each codeword being based on a Binary Phase Shift Keying (BPSK) character set and the codewords being: $A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],A\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],$ wherein A is a multiplier; precode data using the identified codeword; and transmit the precoded data to the network access entity.

12. The computer program product of claim 11, wherein the data is uplink data.

13. The computer program product of claim 11, wherein the telecommunication network is a Long Term Evolution-Advanced (LTE-A) system network and the operations of receiving and transmitting are performed according to LTE-A protocols.

14. The computer program product of claim 11, wherein the mobile device is user equipment.

15. The computer program product of claim 11, wherein A is .

16. The computer program product of claim 11, wherein the codebook is stored in the mobile device, and the computer executable instructions further cause the mobile device to retrieve the identified codeword from the codebook stored in the mobile device.

17. The computer program product of claim 11, wherein the codebook is stored on the non-transitory computer readable medium.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) To illustrate the technical solutions according to the embodiments of the present invention more clearly, the accompanying drawings for describing the embodiments are introduced briefly in the following. Apparently, the accompanying drawings in the following description are only some embodiments of the present invention, and persons of ordinary skill in the art can derive other drawings from the accompanying drawings without creative efforts.

(2) FIG. 1 is a flow chart of an embodiment of a precoding method according to the present invention;

(3) FIG. 2 is a flow chart of another embodiment of the precoding method according to the present invention;

(4) FIG. 3 is a comparison diagram between simulation results of a codebook of the present invention and a codebook as shown in expression (1);

(5) FIG. 4 is a schematic structural view of an embodiment of a BS according to the present invention;

(6) FIG. 5 is a schematic structural view of another embodiment of the BS according to the present invention;

(7) FIG. 6 is a schematic structural view of yet another embodiment of the BS according to the present invention; and

(8) FIG. 7 is a schematic structural view of further another embodiment of the BS according to the present invention.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

(9) The technical solution of the embodiments of the present invention will be clearly and completely described in the following with reference to the accompanying drawings. It is obvious that the embodiments to be described are only a part rather than all of the embodiments of the present invention. All other embodiments obtained by persons skilled in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

(10) FIG. 1 is a flow chart of an embodiment of a precoding method according to the present invention. As shown in FIG. 1, the precoding method in this embodiment includes the following steps.

(11) In Step 101, a total uplink power of a UE is obtained.

(12) For example, a BS, for example, an eNodeB, can obtain the total uplink power of the UE, in which the total uplink power can be a total power after the uplink power control. The eNodeB can determine a transmit power that the antenna needs to transmit data to the UE according to the total uplink power of the UE, that is, select which codebook structure for precoding data to be transmitted.

(13) In Step 102, if the total uplink power is greater than of a rated total transmit power of antennas, a codeword is selected from a first codebook with imbalanced power between layers; otherwise, a codeword is selected from the first codebook and a second codebook with balanced power between layers, so the UE precodes data to be transmitted according to the selected codeword.

(14) In the prior art, a codebook used for 3-layer transfer (rank=3) of four transmit antennas mainly includes the following three types.

(15) A first codebook structure is as shown in expression (1).

(16) $\begin{array}{cc}\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ x& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ x& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ x& 0& 0\end{array}\right]& \left(1\right)\end{array}$

(17) x{+1, 1, +j, j}. The 3 columns of the codebook stand for 3-layer transfer, and 4 rows stand for 4 transmit antennas.

(18) As x has 4 values, 16 codewords in total exist in the codebook, and in the expression (1) 8 codewords are a Quadrature Phase Shift Keying (QPSK) character set, that is, codewords with x=+j or j. The rest 8 codewords are a Binary Phase Shift Keying (BPSK) character set, that is, codewords with x=+1 or 1.

(19) For a codebook structure as shown in expression (1), the transfer power of four transmit antennas corresponding to the first column is larger than the transfer power in the other two layers, and the transfer power in each layer is imbalanced. In a case of a high signal-to-noise ratio, loss occurs to the transmit performance of the transmit antenna of the codebook structure. Meanwhile, as in the codebook in expression (1), the QPSK character set occupies half of the character set, the calculation complexity during the precoding is also large.

(20) The second codebook structure is as shown in expression (2).

(21) $\begin{array}{cc}\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ -1/\sqrt{2}& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ -1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 1/\sqrt{2}j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ 1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}j& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ -1/\sqrt{2}j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ -1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}j& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}j& 0& 0\end{array}\right]& \left(2\right)\end{array}$

(22) For the codebook structures as shown in expression (2), the transmit power of the first transmit antenna and the transmit power of the third transmit antenna are only half of that of the second transmit antenna and the fourth transmit antenna, and if the transmit power of the transmit antennas are restricted, the power amplification of the four transmit antennas is unable to be completely used for full power transfer, that is, the power amplification of all antennas are imbalanced.

(23) A third codebook structure is as shown in expression (3).

(24) $\begin{array}{cc}\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ 1& 0& 1\\ j& 0& -j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ -1& 0& 1\\ -j& 0& -j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ j& 0& 1\\ 1& 0& j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ -j& 0& 1\\ -1& 0& j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right]& \left(3\right)\end{array}$

(25) A is a normalized matrix.

(26) For the codebook structure as shown in expression (3), as data of the carrier overlap exists among all antennas, the CM characteristics of the SC-FDMA are unable to be preserved.

(27) Different from the precoding data to be transmitted by adopting an existing codebook structure, the codebook structure adopted in this embodiment can include two parts and K codewords in total. M codewords belong to the first codebook with imbalanced power between layers, and other (KM) codewords belong to the second codebook with balanced power between layers.

(28) In another embodiment of the precoding method according to the present invention, the first codebook includes at least one codeword of

(29) $\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ x& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ x& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ x& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ x& 0& 0\end{array}\right],\mathrm{and}$$\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right],\mathrm{in}\mathrm{which}$$x\left\{+1,-1,+j,-j\right\}.$

(30) It can be seen from the structure of the first codebook that the transmit power in the first layer is twice as much as that in the second layer and that in the third layer, that is, the power between layers in the first codebook is imbalanced.

(31) The second codebook includes at least one codeword of

(32) $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right]H,\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ x& 0& 0\end{array}\right]H,\left[\begin{array}{ccc}1& 0& 0\\ x& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]H,\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ x& 0& 0\end{array}\right]H,\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ x& 0& 0\end{array}\right]H,\mathrm{and}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right]H,\mathrm{in}\mathrm{which}H=\left[\begin{array}{ccc}\frac{1}{\sqrt{6}}& 0& 0\\ 0& \frac{1}{\sqrt{3}}& 0\\ 0& 0& \frac{1}{\sqrt{3}}\end{array}\right],\mathrm{and}$$x\left\{+1,-1,+j,-j\right\}.$

(33) It can be seen from the structure of the second codebook that the three layers have an equal transmit power, that is, the power of different layers in the second codebook is balanced.

(34) In this embodiment, it is assumed that the rated transmit power of the antenna is 1, the rated total transmit power of the four antennas is 4, so the eNodeB can learn whether the total uplink power of the UE is greater than 4=3. If the total uplink power of the UE is greater than 3, for the second codebook, as the power of different layers is balanced, the transmit power of each layer is greater than 1. For example, the eNodeB learns that the total uplink power of the UE is 4, so each layer of the second codebook has a transmit power of 4/31.33, which exceeds the rated transmit power 1 of the each transmit antenna in each layer. For the first codebook, as the power of different layers is imbalanced, full power can be adopted in all three layers for data transmission, that is, the transmit power in the first layer is 2, and the transmit power in both the second layer and the third layer is 1. If the eNodeB learns that the total uplink power of the UE is smaller than or equal to 4=3, as the transmit power of each antenna in the first codebook and the second codebook can be smaller than the rated transmit power 1, the eNodeB can select the codebook in the first codebook, and can also select the codebook in the second codebook.

(35) Therefore, if the eNodeB learns that the total uplink power of the UE is greater than of the rated total transmit power of the antennas, the eNodeB may select a codeword from the first codebook, and if the eNodeB learns that the total uplink power is smaller than or equal to of the rated total transmit power of the antennas, the eNodeB may select a codeword from a codebook formed by the first codebook and the second codebook.

(36) For the method in which the eNodeB selects a codeword from a codebook formed by a first codebook and a second codebook for precoding data to be transmitted by using the selected codeword, any method in the prior art can be adopted, the description of which is omitted.

(37) In yet another embodiment of the precoding method according to the present invention, the first codebook and the second codebook have 16 codewords in total, in which the first codebook has 8 codewords. Therefore, in this embodiment, the codebook including the first codebook and the second codebook may be:

(38) $\begin{array}{ccccc}\mathrm{Index}0\mathrm{to}3& \frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ j& 0& 0\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -j& 0& 0\end{array}\right]\\ \mathrm{Index}4\mathrm{to}7& \frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ j& 0& 0\\ 0& 0& 1\end{array}\right]& \frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -j& 0& 0\\ 0& 0& 1\end{array}\right]\\ \mathrm{Index}8\mathrm{to}11& \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]& \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]& \left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]& \left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right]\\ \mathrm{Index}12\mathrm{to}15& \left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]& \left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]& \left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]& \left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right]\end{array}$

(39) In yet another embodiment of the precoding method according to the present invention, on the basis of the embodiment as shown in FIG. 1, the precoding method further includes the following steps. A first codebook and a second codebook are selected from the codebook so as to maximize a minimal chordal distance of the codewords of the first codebook and the second codebook, and an antenna performance corresponding to a codeword in the first codebook is different from an antenna performance corresponding to a codeword in the second codebook, so that if the total uplink power is greater than of a rated total transmit power of antennas, a codeword is selected from a codebook formed by the first codebook and the second codebook. The step and Step 101 may have no specific order.

(40) Specifically, the eNodeB can use many codewords for precoding, and the large amount of codewords form a codebook. However, in this embodiment, a first codebook and a second codebook are selected from the codebook by using two rules. It is assumed that the selected first codebook and the selected second codebook have K codewords in total, so one principle of selecting the K codewords is so as to maximize a minimal chordal distance of the K selected codewords. A chordal distance of any two codewords u.sub.i and u.sub.j is defined as:

(41) $d\left(u_i,u_j\right)=\frac{1}{\sqrt{2}}{.Math.u_i{u}_i^H-u_j{u}_j^H.Math.}_F,\mathrm{in}\mathrm{which}$${.Math.u_i.Math.}_F={.Math.u_j.Math.}_F=1.$

(42) Thus, the equation can be adopted to calculate a chordal distance of any two codewords, and a codeword is selected according to the calculated chordal distance, so as to construct the first codebook and the second codebook respectively.

(43) A principle of selecting the K codewords is that if the influences of power allocation matrix are not considered, the corresponding codewords in the first codebook and the second codebook are same. In the process of selecting the codewords in the whole codebook, this situation of same codewords needs to be avoided.

(44) For example, if

(45) 0$\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$
is selected as a codeword in the first codebook,

(46) $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\frac{1}{\sqrt{6}}& 0& 0\\ 0& \frac{1}{\sqrt{3}}& 0\\ 0& 0& \frac{1}{\sqrt{3}}\end{array}\right]$
is not allowed to be a codeword in the second codebook, and vice versa, that is, the antenna performance corresponding to the codeword in the first codebook is different from the antenna performance corresponding to the codeword in the second codebook.

(47) In the embodiments of the precoding method according to the present invention, the BS can select a codeword from a corresponding codebook for precoding data to be transmitted according to a relationship between the total uplink power reported by the UE and the maximal rated total power of the antenna of the BS. As in the method in this embodiment, the codebook structure adopts the first codebook with imbalanced power between layers and the second codebook with balanced structures between layers, if a codeword is selected from the second codebook for precoding, loss of the antenna performance at a high signal-to-noise ratio can be reduced, and if a codeword is selected from the first codebook for precoding, the loss of the power amplification of the antenna can be reduced if the transmit power of the antennas is restricted.

(48) FIG. 2 is a flow chart of another embodiment of a precoding method according to the present invention. As shown in FIG. 2, the method in this embodiment includes the following steps.

(49) In Step 201, a sub-carrier resource is allocated to a UE.

(50) For example, an eNodeB can allocate a sub-carrier resource to the UE, in which the sub-carrier resource is used for data transmission between the UE and the eNodeB. The eNodeB may allocate a continuous sub-carrier resource to the UE and may also allocate discontinuous sub-carrier resource to the UE.

(51) In Step 202, if the sub-carrier resource is continuously allocated, a CMP codebook is selected for precoding data to be transmitted; otherwise, a CMF codebook is selected for precoding data to be transmitted.

(52) The eNodeB can correspondingly select different codebooks for precoding data to be transmitted according to whether the sub-carrier resource allocated by the UE is continuously allocated or discontinuously allocated. If the eNodeB learns that the sub-carrier resource allocated to the UE is continuously allocated, transmitted data of the sub-carriers do not need overlap, so if this resource allocation mode is adopted for data transmission, the eNodeB can select a CMP codebook. If the eNodeB learns that the sub-carrier resource allocated to the UE is discontinuously allocated, transmitted data of the sub-carriers do need overlap, so if this resource allocation mode is adopted for data transmission, the eNodeB can select a CMF codebook. The CMF codebook refers to a codebook that incompletely preserves CM characteristics. For any codeword in a CMP codebook, each row only has one non-zero element. For any codeword in a CMF codebook, in some rows, more than one non-zero elements exist, but in the rows, not all elements are non-zero. For example, one row only has two non-zero elements, and others are still zero. Therefore, compared with the CMP codebook, some CMs are added in the CMF codebook. However, not many CMs are added. Therefore, compared with a codebook in which the CM characteristic design is completely not considered, the CMF codebook has better CM characteristics.

(53) In a precoding method according to an embodiment of the present invention, the CMP codebook includes at least one codeword of

(54) $\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ x& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ x& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ x& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathrm{and}$$\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ x& 0& 0\end{array}\right],$
in which x{+1, 1, +j, j}; or includes at least one codeword of

(55) $\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ -1/\sqrt{2}& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ -1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ 1/\sqrt{2}j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ 1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}j& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ 1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ 0& 1& 0\\ -1/\sqrt{2}j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 1/\sqrt{2}& 0& 0\\ 0& 0& 1\\ -1/\sqrt{2}j& 0& 0\end{array}\right],\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1/\sqrt{2}& 0& 0\\ -1\sqrt{2}j& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathrm{and}$$\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1/\sqrt{2}& 0& 0\\ -1/\sqrt{2}j& 0& 0\end{array}\right].$

(56) The CMF codebook includes at least one codeword of

(57) $\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ 1& 0& 1\\ j& 0& -j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ -1& 0& 1\\ -j& 0& -j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ j& 0& 1\\ 1& 0& j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& 0\\ -j& 0& 1\\ -1& 0& j\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],\mathrm{and}\left[\begin{array}{ccc}1& 1& 0\\ j& -j& 0\\ 1& 0& 1\\ -1& 0& 1\end{array}\right],$
in which is a normalized matrix.

(58) In the embodiment of the precoding method according to the present invention, the BS selects a codeword in different codebooks respectively for precoding data to be transmitted according to different types of sub-carrier resources allocated by the UE, such that the CM characteristics are guaranteed by selecting the codeword in the CMP codebook if the sub-carrier resource is continuously allocated.

(59) An embodiment of the present invention provides a codebook, which includes at least one codeword of

(60) $\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],\mathrm{and}$$\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right].$

(61) In this embodiment the codebook is a BPSK CMP codebook, in the codebook in this embodiment, all codewords are BPSK characters, and the number is at most 12, so compared with the codebook as shown in expression (1), for the codebook in this embodiment, the complexity of the precoding using the codebook is relatively reduced.

(62) Furthermore, in this embodiment, the codebook further includes at least one codeword of

(63) $\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ j& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -j& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ j& 0& 0\\ 0& 0& 1\end{array}\right],\mathrm{and}$$\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -j& 0& 0\\ 0& 0& 1\end{array}\right].$

(64) Therefore, the codebook in this embodiment includes at least one codeword of the following codebook structures:

(65) $\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ j& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\\ -j& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -j& 0& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right],\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],\mathrm{and}$$\frac{1}{2}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right].$

(66) Compared with the codebook as shown in expression (1), the codebook provided in this embodiment has a greater average code distance, in the codebook only 4 codewords are a QPSK character set, and the rest 12 codewords are a BPSK character set, so compared with the codebook as shown in expression (1), for the codebook in this embodiment, the complexity of precoding using the codebook is reduced to a certain degree. FIG. 3 is a comparison diagram between simulation results of a codebook of the present invention and a codebook as shown in expression (1). As shown in FIG. 3, through the link simulation, in a case of the same signal-to-noise ratio, the codebook in this embodiment (that is, the codebook as shown by curve 1 in FIG. 3) has a certain gain in terms of throughput compared with the codebook as shown in expression (1) (that is, the codebook as shown by curve 2 in FIG. 3) and the codebook as shown by curve 3 in FIG. 3, so as to transfer more data.

(67) An embodiment of the present invention further provides another codebook, which includes at least one codeword of

(68) $\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5& 1& 0\\ 0.5& 0& 1\\ 0.5j& 0& -j\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5j& 0& 1\\ -0.5j& j& 0\\ -0.5j& 0& 1\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5& 1& 0\\ 0.5j& 0& 1\\ 0.5& 0& j\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5j& 0& 1\\ 0.5& 0& j\\ -0.5& 1& 0\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5j& 0& 1\\ -0.5& 1& 0\\ -0.5& 0& j\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5j& -j& 0\\ -0.5& 0& 1\\ 0.5& 0& 1\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5j& -j& 0\\ 0.5j& 0& 1\\ 0.5j& 0& -1\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5& 0& 1\\ -0.5j& 0& -j\\ -0.5& 1& 0\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5& -1& 0\\ 0.5j& 0& 1\\ -0.5& 0& -j\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5& 0& 1\\ -0.5& 0& -1\\ -0.5j& j& 0\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5& 0& 1\\ 0.5j& -j& 0\\ -0.5& 0& 1\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5j& 0& 1\\ -0.5j& 0& -1\\ 0.5j& -j& 0\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ 0.5& 0& 1\\ 0.5& -1& 0\\ -0.5j& 0& j\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5j& j& 0\\ -0.5j& 0& 1\\ 0.5j& 0& 1\end{array}\right],\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5& 0& 1\\ 0.5& 0& 1\\ 0.5j& -j& 0\end{array}\right],\mathrm{and}$$\frac{1}{\sqrt{5}}\left[\begin{array}{ccc}0.5& 1& 0\\ -0.5j& j& 0\\ -0.5& 0& 1\\ -0.5& 0& -1\end{array}\right].$