Method and system for digital signal transmission

[00002] This application claims priority under 35 U.S.C. Section 119 based on Finnish Application Serial Number 20000406 filed on Feb. 22, 2000, which has been filed as co-pending U.S. application Ser. No. 10/225,457 and Finnish Application Serial Number 20001944 filed on Sep. 4, 2000, which has been filed as co-pending U.S. application Ser. No. 10/378,068. This application is also related to and claims priority to U.S. Provisional Application No. 60/193,402 filed on Mar. 29, 2000, entitled “CLASS OF SPACE-TIME BLOCK CODES FOR MORE THAN TWO ANTENNAS” all assigned to assignee of present application and all incorporated herein by reference.

[00003] This invention relates generally to methods and systems for achieving transmit diversity in a telecommunication system. More particularly, the present invention relates to an apparatus and associated method for using a space-time block code to reduce bit error rates of a wireless communication in a spread spectrum receiver.

[heading-00004] Mathematical Foundations

[00005] The review of mathematical tools, nomenclature, and notations follows which is intended to be used for an understanding of the present invention. Multiple notations may be used in same parts of the application to better describe the invention. Other mathematical techniques, nomenclature, and notations may be used to describe the present invention without departing from the spirit and scope thereof.

[00006] A vector may be represented in Dirac notation proposed by P.A.M. Dirac:

• • [00007] Bra a| which may be represented by a row matrix or α_i=[α₁,α₂,α₃]
  • [00008] Ket |b which may be represented by a column matrix ${\beta }_i=\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ {\beta }_3\end{array}\right];$
    and
  • [00010] C is defined as $=\sum _{i}l❘i\rangle \langle i$

[00011] Thus BraCKet notation is an analogy with the dot product $A·B=\sum _i{A}_i{B}_i.$

[00012] Definition: The inner product of two vectors in a vector space V is a complex number, such that
a|b=b|a*
a|(β|b+γ|c)=βa|b+γa|c
a|a≧0, and a|a=0iff|a=0.

[00016] Therefore, the inner product may be defined in summation notation as $\langle ab\rangle \equiv {\alpha }_1^{*}{\beta }_1+{\alpha }_2^{*}{\beta }_2+\dots +{\alpha }_n^{*}{\beta }_n=\sum _{i=1}^n{\alpha }_i^{*}{\beta }_i.$

[00017] A state vector may be represented as a linear combination with suitable coefficients of a set of base vectors I, j: $\langle ia\rangle =C_1\iff a\rangle =\sum _ii\rangle C_i\langle ib\rangle =D_1\iff b\rangle =\sum _ii\rangle D_i$ $\langle ai\rangle =\langle i{a\rangle }^{*}\langle b=\sum _j{D}_j^{*}\langle j\langle ba\rangle =\sum D_j^{*}\langle ji\rangle C_i$

• • [00018] but $\langle ij\rangle ={\delta }_{\mathrm{ij}}\equiv \left\{\begin{array}{cc}1& \mathrm{if}i=j\\ 0& \mathrm{if}i\ne j\end{array}\right\}$
    (the Kronecker delta δ_ij) is a symbol that is defined to be 0 for i≠j and to be 1 when i=j and can be represented by the unit matrix $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\hspace{1em}$
    which may also be referred to as the identity matrix I₂ or for an N dimension basis I_N. The vectors i and j are normalized and form an orthonormal basis. Sometimes {|e_i}_i=1^N is used as a basis in a N-dimensional vector space.

[00021] Vectors |a,|bεV are orthogonal if a|b=0.

[00022] Operators can be represented as $A_{\mathrm{ij}}\equiv \langle iAj\rangle =\left[\begin{array}{ccccc}{A}_{11}& A_{12}& \dots & \dots & A_{1j}\\ A_{21}& A_{22}& \dots & \dots & A_{2j}\\ ⋮& ⋮& ⋰& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋰& ⋮\\ A_{\mathrm{i1}}& A_{\mathrm{i2}}& \dots & \dots & A_{\mathrm{ij}}\end{array}\right]$

• • [00023] |b=A|a operator A performs operation on vector state |a to produce vector state |b, $\langle ib\rangle =\sum _j\langle iAj\rangle \langle ja\rangle .$
  • [00024] b|A|a*=a|A^H|b where A^H is an operator whose matrix elements are A_ij^H=(A_ji) and is called Hermitian when A^H=A.

[00025] A system may be represented in terms of the unit matrix and a set of matrices: $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=a\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)+b\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)+c\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)+d\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$

[00026] An example of such a set of matrices is called Pauli-Spin matrices: ${\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),{\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\mathrm{and}{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right).$  σ_x²=1,σ_y²=1,σ_z²=1. σ_xσ_y=−σ_yσ_x=iσ_z, σ_yσ_z=−σ_zσ_y=iσ_x, and σ_zσ_x=−σ_xσ_z=iσ_y.

[00027] The present application will also use the mathematical methods of group theory and algebras. Simply, a group is a set G of elements g together with a binary operation called multiplication with the properties:
∃ε∈G ε*g=g*ε=g and ∀g∈G,∃g⁻¹g*g⁻¹=g⁻¹*g=ε

• • [00029] which reads in other words: there exist an unique element ε of the group G such that ε operating on g equals g operating on ε equals g (the element ε is referred to as the identity); and for every element g belonging to group G, there exists an element g⁻¹ (called the inverse) such that g operating on g-inverse equals g-inverse operating on g equals the identity ε.

[00030] A set of linearly independent vectors may be represented as group elements, a vector space may be generated by taking a linear combination of the elements of the set. The product already defined on the basis set can be extended to all the elements of the vector space by linearity defining a group algebra. The Pauli-Spin matrices may be thought of as group elements.

[00031] As wireless communication systems evolve, wireless system design has become increasingly demanding in relation to equipment and performance requirements. Future wireless systems, which will be third generation (3G) systems and fourth generation systems compared to the first generation analog and second generation digital systems currently in use, will be required to provide high quality high transmission rate data services in addition to high quality voice services. Concurrent with the system service performance requirements will be equipment design constraints, which will strongly impact the design of mobile terminals. The third and fourth generation wireless mobile terminals will be required to be smaller, lighter, more power-efficient units that are also capable of providing the sophisticated voice and data services required of these future wireless systems.

[00032] Time-varying multi-path fading is an effect in wireless systems whereby a transmitted signal propagates along multiple paths to a receiver causing fading of the received signal due to the constructive and destructive summing of the signals at the receiver. This occurs regardless of the physical form of the transmission path, (i.e. whether the transmission path is a radio link, an optical fiber or a cable). Several methods are known for overcoming the effects of multi-path fading, such as time interleaving with error correction coding, implementing frequency diversity by utilizing spread spectrum techniques, transmitter power control techniques, and the like. Each of these techniques, however, has drawbacks in regard to use for third and fourth generation wireless systems. Time interleaving may introduce unnecessary delay, spread spectrum techniques may require large bandwidth allocation to overcome a large coherence bandwidth, and power control techniques may require higher transmitter power than is desirable for sophisticated receiver-to-transmitter feedback techniques that increase mobile station complexity. All of these drawbacks have negative impact on achieving the desired characteristics for third and fourth generation mobile terminals. Diversity is another way to overcome the effects of multi-path fading.

[00033] Antenna diversity is one type of diversity used in wireless systems. In antenna diversity, two or more physically separated antennas are used to receive a signal, which is then processed through combining and switching to generate a received signal. A drawback of diversity reception is that the physical separation required between antennas may make diversity reception impractical for use on the forward link in the new wireless systems where small mobile station size is desired.

[00034] A second technique for implementing antenna diversity is transmit (Tx) diversity. In transmit (Tx) diversity, a signal is transmitted from two or more antennas and then processed at the receiver by using maximum likelihood sequence estimator (MLSE) or minimum mean square error (MMSE) techniques. Transmit diversity has more practical application to the forward link in wireless systems in that it is easier to implement multiple antennas in the base station than in the mobile terminal. FIG. 1 is an illustration showing an example of a transmit diversity system. Transmit (Tx) diversity system 100 comprises multiple transmit antennas (Tx₁. . . Tx_n; n=1 to N) 110, multiple channels (h₁ . . . h_K) 120 where h_k=α_ke^jθk∀k=1 to K, and receiving antenna Rx 130. Channel interference may be created by structures or other features 140 among various channel paths.

[00035] Use of Space-Time Block Code (STBC) aka Space-Time Code (STC) may be considered as diversity-creating. In space-time block code design, the essential design criteria are the achieved diversity, the rate of the code, and the delay. Diversity is characterized by the number of independently decodable channels. For full diversity, this equals the number of transmit antennas. The rate of the code is the ratio of the space-time coded transmission rate to the rate of an one-antenna transmission (i.e. the ratio of the transmission rate of the block code to transmission rate of the uncoded scheme). The delay is the length of the space-time block code frame. Depending on the underlying modulation scheme, space-time block codes may be divided into real and complex codes. Rate in the context of space-time block code may be understood as inefficiency in use of the antenna resources, leading to dilution of maximal bit-rates as compared to the inherent capacity of the underlying wireless system specifications. This inefficient use of antenna resources may give rise to fluctuating of transmit powers. Such fluctuation of transmit powers is referred to as the power-unbalance problem. Thus, the aims in the design of space-time block codes is to achieve unit rate (R=1) in order to use antenna resources as efficiently as possible. The aim of diversity is to achieve maximal diversity.

[00036] Transmit diversity for the case of two antennas (N=2) is well studied. Both open-loop and closed-loop transmit diversity methods have been under consideration for 3G Wide-band Code Division Multiple Access (WCDMA) system.

[00037] An example of an open-loop concept is provided by Alamouti, who proposed a method of transmit diversity employing two antennas that offers second order diversity for complex valued signals. (S. M. Alamouti, “A Simple Transmit Diversity Technique for Wireless Communications,” IEEE Journal on Selected Areas of Communications, vol. 16, no. 8, pp. 1451-1458, October 1998 and publication WO 99/14871 of ALAMOUTI et al. entitled “Transmitter Diversity Technique for Wireless Communication”) The Alamouti method employs two transmit antennas, has a rate R=1 and a maximum-ratio diversity-combining detector.

[00038] The method achieves effective communication by encoding symbols that comprise negations and conjugation of symbols (i.e. negation of imaginary parts) and simultaneously transmitting two signals (K=2) from two antennas (N=2) to a receiving antenna (Rx) during a symbol period (T) also referred to as a time epoch. During one symbol period (T₁=t₀+T), the signal transmitted from a first antenna (Tx₁) is denoted by s₁ and the signal transmitted from the second antenna (Tx₂) is denoted by s₂.where s₁ and s₂ are complex numbers. During the next symbol period (T₂=T₁+T), the signal−s₂*(negation/conjugate) is transmitted from the first antenna (Tx₁) and the signal s₁* (conjugate) is transmitted from the second antenna (Tx₂), where * is the complex conjugate operator.

[00002]

TABLE 1
Period	Trans. Ant. 1 (Tx1)	Trans Ant. 2 (Tx2)	Receive Ant. (Rx)
T1	s1	s2	r1
T2	−s2*	s1*	r2

[00039] The baseband signals during a first interval can be written as:
r₁=h₁s₁+h₂s₂+n₁  (1)
r₂=−h₂s₂*+h₁s₁*+n₂*  (2)

• • [00042] where n₁ and n₂ are noise factors and let us assume impulse response coefficients, e.g. flat-fading case h_i, where i=1 to K associated with the two antennas (K=2) are constant over the two-symbol time interval.

[00043] We can map the table into matrix form with the columns correspond to antennas and the row to time epochs, where the column index represents or is associates with the antenna index and the row index represents or is associated with the time index. Thus, the Alamouti Space-Time Block code (STBC) $\begin{array}{cc}{C}_{\mathrm{Ala}}=\begin{array}{c}T\\ i\\ m\\ e\\ \downarrow \end{array}\stackrel{\mathrm{Antenna}->}{\left[\begin{array}{cc}{s}_1& s_2\\ -s_2^{*}& s_1^{*}\end{array}\right]}& \left(3\right)\end{array}$
is optimal with complex signal constellations. It reaches diversity 2, with a linear decoding scheme which yields estimates for both symbols with the two channels maximal ratio combined.

[00045] The impulse response coefficients can be represented by $h=h\rangle =\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]$
and the received signals by $r=r\rangle =\left[\begin{array}{c}{r}_1\\ r_2\end{array}\right]$
and the noise by $n=n\rangle =\left[\begin{array}{c}{n}_1\\ n_2\end{array}\right].$

[00048] The signal received at Rx 130 is given by: $\begin{array}{cc}\left[\begin{array}{c}{r}_1\\ r_2\end{array}\right]=C_{\mathrm{Ala}}\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]+n\iff r\rangle =\langle h_iCh_j\rangle +n\rangle & \left(4\right)\end{array}$

• • [00049] or, equivalently, $\left[\begin{array}{c}{r}_1\\ r_2^{*}\end{array}\right]=\left[\begin{array}{cc}{h}_1& h_2\\ h_2^{*}& -h_1^{*}\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\end{array}\right]+\left[\begin{array}{c}{n}_1\\ n_2^{*}\end{array}\right]\iff r=H_{12}s+n$

[00050] There is interest to derive space-time codes (STC) for more than two antennas. However, extension of the Alamouti method to more than two antennas is not straightforward. Note that the channel matrix H₁₂ in equation (5) is orthogonal, thus to decode we use:
s=H₁₂^Hr  (6)

[00052] The detection is:
ŝ=sign (H₁₂^Hr)=sign[(||h₁||²+||h₂||²)I₂s+H₁₂^Hn]  (7)

[00054] The C_Ala matrix may be recognized as proportional to a general unitary unimodular matrix which is commonly written in the art as: $\begin{array}{cc}U\left(a,b\right)=\left[\begin{array}{cc}a& b\\ -b^{*}& a^{*}\end{array}\right]& \left(8\right)\end{array}$
where a and b are complex numbers which satisfy the unimodular condition |a|²+|b|²=1. The orthogonality of the space-time block codes may thus be expressed as $\begin{array}{cc}{C}^{H}C=\sum _i{z_i}^{2}I& \left(9\right)\end{array}$

[00056] C is the code matrix, I is the identity matrix of the same dimension, and the superscript H is the hermitean operator (complex conjugate transpose). The maximum-ratio-combining property of the code is a direct consequence of the appearance of the sum of the symbol powers on the main diagonal of the hermitean square of the code matrix.

[00057] In general, the Alamounti STBC is the Radon-Hurwitz submatrix form and is an unitary unimodular matrix given in general form by: $\begin{array}{cc}{C}_{RH}=\left[\begin{array}{cc}{c}_n& c_{n+1}\\ -c_{n+1}^{*}& c_n^{*}\end{array}\right]& \left(10\right)\end{array}$

[00058] The mathematical work of Radon and Hurwitz in the 1920's [proved that an orthogonal design of size N exists if and only if N=2, 4, and 8.] is cited in Calderbank et al. in U.S. Pat. No. 6,088,408 issued to Calderbank et al. on Jul. 11, 2000 and in (V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space-Time Block Codes from Orthogonal Designs,” IEEE Transactions on Information Theory, pp. 1456-1467, July 1999), both incorporated herein by reference, showed the Hurwitz-Radon proof and showed that for more than two antennas complex orthogonal designs that achieve R=1 do not exist. Space-Time Block Codes. Calderbank et al. proposed a method using rate=½, and ¾ Space-Time Block codes for transmitting on three and four antennas using complex signal constellations.

[00059] As an example, the code rate ¾ is given by: $\begin{array}{cc}{C}_{\frac{3}{4}}(s_1,s_2,s_3)=\hspace{1em}[\begin{array}{cccc}{s}_1& s_2& \frac{s_3}{\sqrt{2}}& \frac{s_3}{\sqrt{2}}\\ -s_2^{*}& s_1^{*}& \frac{s_3}{\sqrt{2}}& -\frac{s_3}{\sqrt{2}}\\ \frac{s_3^{*}}{\sqrt{2}}& \frac{s_3^{*}}{\sqrt{2}}& \frac{\left(s_2-s_2^{*}-s_1-s_1^{*}\right)}{2}& \frac{\left(s_1-s_1^{*}-s_2-s_2^{*}\right)}{2}\\ \frac{s_3}{\sqrt{2}}& -\frac{s_3^{*}}{\sqrt{2}}& \frac{\left(s_1-s_1^{*}+s_2+s_2^{*}\right)}{2}& \frac{\left(s_1+s_1^{*}+\mathrm{s2}-s_2^{*}\right)}{2}\end{array}]& \left(11\right)\end{array}$

[00060] This method has a disadvantage in a loss in transmission rate and the fact that the multi-level nature of the ST coded symbols increases the peak-to-average ratio requirement of the transmitted signal and imposes stringent requirements on the linear power amplifier design. Other methods proposed include a rate (R=1), orthogonal transmit diversity (OTD)+space-time transmit diversity scheme (STTD) four antenna method.

[00061] When complex modulation is used, full diversity codes with a code rate 1 are only described in connection with two antennas in the publication WO 99/14871 Published on Mar. 25, 1999 and entitled TRANSMITTER DIVERSITY TECHNIQUE FOR WIRELESS COMMUNICATIONS and the publication Tarokh, V., Jafarkhani, H., Calderbank, A. R.: Space-Time Block Coding for Wireless Communications: Performance Results, IEEE Journal on Selected Areas In Communication, Vol. 17 pp. 451-460, March 1999, both incorporated herein by reference present a rate ½ code which is constructed from the full rate real code by setting the complex signals on top of the same, but conjugated signals. This way rate ½ codes for two to eight antennas are obtained. In the following, an example of a code for three antennas is given: $\begin{array}{cc}{C}_{1/2}\left(s_1,s_2,s_3,s_4\right)->\left[\begin{array}{ccc}{s}_1& s_2& s_3\\ -s_2& s_1& -s_4\\ -s_3& s_4& s_1\\ -s_4& -s_3& s_2\\ s_1^{*}& s_2^{*}& s_3^{*}\\ -s_2^{*}& s_1^{*}& -s_1^{*}\\ -s_3^{*}& s_4^{*}& s_1^{*}\\ -s_4^{*}& -s_3^{*}& s_2^{*}\end{array}\right]& \left(12\right)\end{array}$

• • [00062] where a star (*) refers to a complex conjugate. These codes are not delay-optimal.

[00063] So far, all complex space-time block codes have belonged to two categories: a group based on real codes, halving the code rate, such as the above example, or a group based on square unitary matrices.

[00064] It is desirable that ‘Open-loop diversity’ should have these four properties:

• • [00065] 1. Full diversity in regard to the number of antennas.
  • [00066] 2. Only linear processing is required in a transmitter and a receiver.
  • [00067] 3. Transmission power is divided equally between the antennas.
  • [00068] 4. The code rate efficiency is as high as possible.

[00069] A drawback of the above solutions is that only the requirements 1 and 2 can be fulfilled. For example, the transmission power of different antennas is divided unequally, (i.e. different antennas transmit at different powers). This causes problems in the planning of output amplifiers. Furthermore, the code rate is not optimal.

[00070] For 3 and 4 antennas, this maximal rate is ¾. Because of the inefficiency of codes with rates less than one (R<1) transmit power of a given antenna fluctuates in time; thus, presenting a power-unbalanced problem. Therefore, there is a need to provide a power-balance full-rate code. Since a decrease in rate may not acceptable, some other features of space-time block codes have to be relaxed.

[00071] For example, uncoded diversity gain may be sacrificed and rely on coding to exploit the diversity provided by additional antennas. Motorola introduced Orthogonal Transmit Diversity+Space-Time Transmit Diversity (OTD+STTD) scheme, (L. Jalloul, K. Rohani, K. Kuchi, and J. Chen, “Performance Analysis of CDMA Transmit Diversity Methods,” Proceedings of IEEE Vehicular Technology Conference, vol. 3, pp. 1326-1330 Fall 1999, hereinafter referred to as Jalloul). Jalloul states that extension of OTD to more than two antennas is straightforward; but “STTD is not directly extendable to more than two antennas, since rate one S-T block codes (STC) are non-existent for greater that 2 antennas.” They extended the two antenna STTD scheme to four transmit antennas by combining the STTD with two-branch OTD, $\begin{array}{cc}{C}_{\mathrm{STOTD}}\left(s_1,s_2,s_3,s_4\right)=\left[\begin{array}{cccc}{s}_1& s_2& s_1& s_2\\ -s_2^{*}& s_1^{*}& -s_2^{*}& s_1^{*}\\ s_3& s_4& s_3& s_4\\ -s_4^{*}& s_3^{*}& -s_4^{*}& s_3^{*}\end{array}\right]& \left(13\right)\end{array}$

[00072] The STOTD scheme is completely balanced and is also orthogonal, so that linear decoding gives maximal likelihood results. However, the diversity order achieved is only two, which is the same as the Alamouti STTD scheme.

[heading-00073] Space-Time Code Design Criteria

[00074] There are three design criteria for space-time block codes, which are all formulated in terms of the codeword difference matrix D_ce=C_c−C_e, where C_c and C_e are the code matrices corresponding to two distinct sets of information c and e. Minimizing the pair-wise error probability of deciding in favor of C_e when transmitting C_c leads to the following design criteria:

• • [00075] 1. The rank criterion: The diversity gained by a multiple transmitter scheme is:
    diversity=min_e≠cRank[D_ce]<=min[T;N]  (14)

[00077] To achieve maximal diversity, D_ce should have full rank for all distinct code words c and e.

• • [00078] 2. The determinant criterion: To optimize performance in a Rayleigh fading environment, Code (C) should be designed to maximize
    min_e≠sdet′[D_ce^HD_ce].  (15)

[00080] Where the prime in the determinant indicates that zero eigenvalues should be left out from the product of eigenvalues when computing the determinant.

• • [00081] 3. The trace criterion: To optimize performance in flat fading channels, Code (C) should be designed to maximize the Euclidean distance
    Tr[D_ce^HD_ce].  (16)

[00083] Moreover, to optimize performance in fading channels, the eigenvalues of D_ce^H D_ce should be as close to each other as possible.

[00084] From linearity, it follows that the codeword difference matrix D_ce inherits the unitarity property of the code matrix C: $\begin{array}{cc}{D}_{\mathrm{ce}}^H{D}_{\mathrm{ce}}=\sum _k{z_{k,c}-z_{k,e}}^2{I}_N.& \left(17\right)\end{array}$

[00085] Thus, all design criteria are fulfilled:

• • [00086] Rank criterion: As an unitary matrix, D_ce is full rank for all distinct code word pairs. Thus, all space-time block codes give full diversity, equaling the number of Tx antennas.

[00087] Determinant criterion: As D_ce is unitary, $\begin{array}{cc}\det D_{\mathrm{ce}}^H{D}_{\mathrm{ce}}=\sum _k{z_{k,c}-z_{k,e}}^{2N}& \left(18\right)\end{array}$

[00088] This is the maximum given a fixed transmit power.

[00089] Trace criterion: As D_ce is unitary, $\mathrm{Tr}D_{\mathrm{ce}}^H{D}_{\mathrm{ce}}=N\sum _k{z_{k,c}-z_{k,e}}^2.$
This is the maximum given a fixed transmit power. Moreover, all eigenvalues are the same.

[00091] Thus, there is a need for a design which provides full, diversity, full rate and power-balanced space-time codes.

[00092] It is thus an object of the invention to implement a method and a system by which optimal diversity is achieved with different numbers of antennas. This is achieved by a method of transmitting a digital signal consisting of symbols, which method comprises the steps of coding complex symbols to channel symbols in blocks having the length of a given K and transmitting the channel symbols via several different channels and two or more antennas. In the method of the invention, coding is performed such that the coding is defined by a code matrix, which can be expressed as a sum of 2K elements, in which each element is a product of a symbol or symbol complex conjugate to be transmitted and a N×N representation matrix of a complexified anti-commutator algebra, extended by a unit element, and in which each matrix is used at most once in the formation of the code matrix.

[00093] Further, in the method of the invention the coding is performed such that the coding is defined by a code matrix which is formed by freely selecting 2K−1 unitary, anti-hermitean N×N matrices anti-commuting with each other, forming K−1 pairs of said matrices, whereby the remaining matrix forms a pair with an N-dimensional unit matrix, forming two matrices of each pair such that the second matrix of the pair, multiplied by the imaginary unit, is added to and subtracted from the first matrix of the pair, and in which each matrix formed in the above manner defines the dependence of the code matrix on one symbol or symbol complex conjugate to be coded.

[00094] The invention also relates to an arrangement for transmitting a digital signal consisting of symbols, which arrangement comprises a coder for coding complex symbols to channel symbols in blocks having the length of a given K, means for transmitting the channel symbols via several different channels and two or more antennas. In the arrangement of the invention, the coder is arranged to code the symbols using a code matrix, which can be expressed as a sum of 2K elements, in which each element is a product of a symbol or symbol complex conjugate to be transmitted and a a N×N representation matrix of a complexified anti-commutator algebra, extended by a unit element, and in which each matrix is used at most once in the formation of the code matrix.

[00095] Furthermore, in the arrangement of the invention the coder is arranged to code the symbols using a code matrix which is formed by freely selecting 2K-1 unitary, anti-hermitean N×N matrices anti-commuting with each other, forming K−1 pairs of said matrices, whereby the remaining matrix forms a pair with an N-dimensional unit matrix, forming two matrices of each pair such that the second matrix of the pair, multiplied by the imaginary unit, is added to and subtracted from the first matrix of the pair, and in which each matrix formed in the above manner defines the dependence of the code matrix on one symbol or symbol complex conjugate to be coded.

[00096] The solution of the invention can provide a system in which any number of transmit and receive antennas can be used and a full diversity gain can be achieved by space-time block coding. In a preferred embodiment, the maximal code rate and the optimal delay are achieved by square codes having a dimension that is a power of two.

[00097] The solution of the invention employs complex block codes. In a preferred embodiment codes are used, which are based on matrices whose all elements have the form of ±s_k, ±s*_k or 0. The prior art solutions reveal no codes in whose elements the term 0 appears. First, square codes are given, from which non-square codes are obtained by eliminating columns (antennas). In these codes known as basic codes the elements depend only on one symbol, or on the real part of a symbol and the imaginary part of another symbol. In another preferred embodiment, full diversity codes which do not have the above restriction can be used.

[00098] Also provided is a method and arrangement in which a code matrix is formed by matrices of a portion of the symbols placed on the diagonal of the code matrix and by matrices of a second portion of symbols along the anti-diagonal of the code matrix.

[00099] A sub-optimal solution is also provided by an embodiment of the invention using a rate ⅓ convolution code.

[00100] An equal distribution of transmission power between different antennas is also achieved by means of the solution of the invention. The solution of the invention preferably also provides coding in which the ratio of the maximum power to the average power or the ratio of the average power to the minimum power can be minimized.

[00101] At the receiver, the transmitted symbols may be recovered using a maximum likelihood sequence estimator (MLSE) decoder implemented with the Viterbi algorithm with a decoding trellis according to the transmitter.

[00110] The invention may be used in radio systems which allow the transmission of at least a part of a signal by using at least three or more transmit antennas or three or more beams that are accomplished by any number of transmit antennas. A transmission channel may be formed by using a time division, frequency division or code division multiple access method. Also systems that employ combinations of different multiple access methods are in accordance with the invention. The examples describe the use of the invention in a universal mobile communication system utilizing a broadband code division multiple access method implemented with a direct sequential technique, yet without restricting the invention thereto.

[00111] Referring to FIG. 2A, a structure of a mobile communication system is described by way of example. The main parts of the mobile communication system are core network CN, UMTS terrestrial radio access network UTRAN and user equipment UE. The interface between the CN and the UTRAN is called lu and the air interface between the UTRAN and the UE is called Uu.

[00112] The UTRAN comprises radio network subsystems RNS. The interface between the RNSs is called lur. The RNS comprises a radio network controller RNC and one or more nodes B. The interface between the RNC and B is called lub. The coverage area, or cell, of the node B is marked with C in the figure.

[00113] The description of FIG. 2A is relatively general, and it is clarified with a more specific example of a cellular radio system in FIG. 2B. FIG. 2B includes only the most essential blocks, but it is obvious to a person skilled in the art that the conventional cellular radio system also includes other functions and structures, which need not be further explained herein. It is also to be noted that FIG. 2B only shows one exemplified structure. In systems according to the invention, details can be different from what is shown in FIG. 2B, but as to the invention, these differences are not relevant.

[00114] A cellular radio network typically comprises a fixed network infrastructure, i.e. a network part 200, and user equipment 202, which may be fixedly located, vehicle-mounted or portable terminals. The network part 200 comprises base stations 204, a base station corresponding to a B-node shown in the previous figure. A plurality of base stations 204 are, in turn, controlled in a centralized manner by a radio network controller 206 communicating with them. The base station 204 comprises transceivers 208 and a multiplexer 212.

[00115] The base station 204 further comprises a control unit 210 which controls the operation of the transceivers 208 and the multiplexer 212. The multiplexer 212 arranges the traffic and control channels used by several transceivers 208 to a single transmission connection 214, which forms an interface lub.

[00116] The transceivers 208 of the base station 204 are connected to an antenna unit 218 which is used for implementing a bi-directional radio connection 216 to the user equipment 202. The structure of the frames to be transmitted in the bi-directional radio connection 216 is defined separately in each system, the connection being referred to as an air interface Uu.

[00117] The radio network controller 206 comprises a group switching field 220 and a control unit 222. The group switching field 220 is used for connecting speech and data and for combining signalling circuits. The base station 204 and the radio network controller 206 form a radio network subsystem 224 which further comprises a transcoder 226. The transcoder 226 is usually located as close to a mobile services switching center 228 as possible, because speech can then be transferred in a cellular radio network form between the transcoder 226 and the radio network controller 206, which saves transmission capacity.

[00118] The transcoder 226 converts different digital speech coding forms used between a public switched telephone network and a radio network to make them compatible, for instance from a fixed network form to another cellular radio network form, and vice versa. The control unit 222 performs call control, mobility management, collection of statistical data and signalling.

[00119] FIG. 2B further shows the mobile services switching center 228 and a gateway mobile services switching center 230 which controls the connections from the mobile communications system to the outside world, in this case to a public switched telephone network 232.

[00120] In accordance with the present invention there is provided a sub-optimal class of Space-Time Codes (STC) based on the Radon-Hurwitz sub-matrix. The invention can thus be applied particularly to a system in which signal transmission is carried out by using complex space-time block coding in which the complex symbols to be transmitted are coded to channel symbols in blocks having the length of a given K in order to be transmitted via several different channels and two or more antennas. These several different channels can be formed of different time slots. As a result of the coding, the symbol block forms into a code matrix in which the number of columns corresponds to the number of antennas used for the transmission and the number of rows corresponds to the number of different channels, which, in case of space-time coding, is the number of time slots to be used. Correspondingly, the invention can be applied to a system in which different frequencies or different spreading codes are used instead of time slots. In this case it does not naturally deal with space-time coding but rather with space-frequency coding or space-code-division coding. The space-frequency coding could be used in an OFDM (orthogonal frequency division multiplexing) system, for example.

[00121] The codes of the present invention have rate R=1 and are easily decodable. For the 4-antenna case, a diversity of order 3 is achieved. The present invention minimizes the inherent non-orthogonality and have simple linear decoding, which may be iterated. The present invention is also backward compatible with 3GPP release 99 open-loop diversity mode.

[00122] Let us first examine the forming of a freely selected square complex space-time block code. Assuming the number of transmit antennas is N=2^K−1, where K is an integer and bigger than two. By means of the obtained code, K complex number modulated symbols can be transmitted during N symbol periods. These symbols can be marked with s_k, k=1, . . . K.

[00123] A square complex space-time block code is based on a unitary N×N matrix, whose elements depend on linearly transmitted symbols s_k and their complex conjugates. A unitary matrix is a square matrix whose inverse matrix is proportional to its hermitean conjugate. On the other hand, the hermitean conjugate is the complex conjugate of the matrix transpose. In addition, the proportional coefficient between the product of the code matrix and its hermitean conjugate, and the unit matrix is a linear combination of the absolute value squares of the symbols to be transmitted. This linear combination can be called unitarity coefficient. By interpreting the symbols to be transmitted appropriately, this linear combination can always be seen as a sum of the absolute value squares of the symbols to be transmitted.

[00124] The unitarity (complex-orthogonality) of the space-time block codes may thus be expressed with the inner products proportional to the sum of the squared amplitudes of the symbols: $\begin{array}{cc}{C}^{H}C=\sum _i{z_i}^{2}I& \left(19\right)\end{array}$

[00125] C is the code matrix, I is the identity matrix of the same dimension, and H is the hermitean conjugate (complex conjugate transpose operator). The maximum-ratio-combining property of the code is a direct consequence of the appearance of the sum of the symbol powers on the diagonal of the hermitean square of the code matrix.

[00126] The linearity of the symbols z₁, z₂ allows the code matrix to be expanded thus $\begin{array}{cc}C=\sum _{k=1}^K\left(z_k{\beta }_k^{-}+z_k^{*}{\beta }_k^{+}\right)& \left(20\right)\\ =\sum _{k=1}^K\left(x_k{\beta }_{2k-2}+y_k{\beta }_{2k-1}\right).& \left(21\right)\end{array}$

[00127] {β}_k=0^2K−1 is a set of 2K constant T×N matrices with complex entries. The real variable x_k and y_k are the real and imaginary part of the complex variable z_k=x_k+jy_k, and the matrices β_k^± are linear combination of β_k $\begin{array}{cc}{\beta }_k^{\pm }=\frac{1}{2}\left({\beta }_{2k-2}\pm j{\beta }_{2k-1}\right).& \left(22\right)\end{array}$

[00128] Combining the above based on unitarity and linearity criteria the following is a restriction on the β_k matrices.
β_k^Hβ_j+β_j^Hβ_k=2δ_jkI_N.  (23)

[00130] For square matrices (T=N), it can be seen the code matrix C(z) and the coefficient matrices β_k are unitary which lends itself to the use of group algebras. Moreover, the matrices belong to the unitary group U(N).

[00131] As stated above in regards to Pauli-Spin matrices, taking a set of linearly independent vectors as group elements, a vector space may be generated by taking a linear combination of the elements of the set. The product already defined on the basis set can be extended to all the elements of the vector space by linearity. Thus, defining a group algebra. Due to this, the well developed tools of the representation theory of continuous groups lend themselves to the analysis of the problem.

[00132] Redefine
γ_k=β_k^Hβ_k;k=0, . . . ,2K−1,  (24)

[00134] γ₀=I_N, and the algebra β_k^Hβ_j+β_j^Hβ_k=2δ_jkI_N from above remains unchanged for the γ:s. From the relation between γ₀ and the other γ:s, we then see that these should be anti-hermitean
γ_k^H=−γ_k,k=1, . . . ,2K−1,  (25)

[00136] The algebra of these remaining γ:s is now
γ_kγ_j+γ_jγ_k=−2δ_jkI_N,j,k=1, . . . ,2K−1.  (26)
γ_kγ_j

[00139] This is the defining relation of generators of the Clifford algebra. There is a coincidence that the investigation of the Clifford algebra originates in the work of theoretical physicist P.A.M. Dirac on fermionic particles in space-time. From a group theoretical point of view, the γ-matrices generate the so-called spinor representation of the special orthogonal subgroup SO(2K−1), and here we are dealing with the problem of embedding spinorial SO(2K−1) representation in representations of unitary groups.

[00140] We thus get the following generic prescription for finding a complex modulation space-time block code:

• • [00141] 1. Find a N-dimensional representation of the Clifford algebra γ_kγ_j+γ_jγ_k=−2δ_jkI_N,j,k=1 . . . ,2K −1 for anti-hermitean matrices γ_k, k=1, . . . ,2K−1.
  • [00142] 2. Take an unitary matrix β₀εU(N).
  • [00143] 3. Define β_k=β₀γ_k,k=1, . . . ,2K−1.
  • [00144] 4. Use the matrices β_k,k=0, . . . ,2K−1 to create a code matrix C(z) according $\mathrm{to}C=\sum _{k=1}^K\left(x_k{\beta }_{2k-2}+y_k{\beta }_{2k-1}\right).$

[00145] By construction, this prescription yields all possible complex linear space-time codes with N antennas and time epochs, full diversity, and rate K/N. The codes with minimal dimensions for a given rate are thus delay optimal codes.

[00146] Similar to what was shown as a result of the Hurwitz-Radon proof, the representation theory of Clifford algebas give very stringent conditions on the existence of block codes with arbitrary rate K/N. For an orthogonal complex modulation space-time block code employing N antennas, the maximal achievable rate is: $\begin{array}{cc}\frac{\left[{\log }_{2}N\right]+1}{2^{\left[{\log }_{2}N\right]}}& \left(27\right)\end{array}$

[00147] However, one may not always need optimal codes. An embodiment of the invention provides a sub-optimal solution as will be presented below.

[heading-00148] Representation Theory of Clifford Algebras

[00149] Let us look at an example using Clifford algebra representation. Suppose you have K objects that fulfil the defining relations of the generators of a Clifford algebra γ_kγ_j+γ_jγ_k=−2δ_jkI_N,j,k=1, . . . ,2K−1.That is, they are fourth roots of the identity matrix, (i.e. gεG g²=−I), and they anti-commute. We want to represent these objects as anti-hermitean square matrices, and we are interested in the minimal dimension of irreducible representations. (An irreducible representation is one that cannot be decomposed into a direct product of two lower-dimensional ones). For anti-hermitean matrices, idempotency translates to unitarity. The enveloping algebra of these objects is the algebra Clifford_K. We shall use this terminology loosely, and refer with “Clifford_K” to the algebra of the generators of Clifford_K.

[00150] In constructing representations for Clifford algebras, we shall use the following matrices: $\begin{array}{cc}{\sigma }_1=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{\sigma }_2=\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right),\mathrm{and}{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)& \left(28\right)\end{array}$

[00151] Matrices (28) are proportional to Pauli matrices. For later convenience, these have been chosen so that σ₁ and or σ₂ are anti-hermitean, and σ₃ is hermitean. The matrices that σ₁, σ₂, and I σ₃ constitute a basis for the algebra Su₂, the algebra of anti-hermitean 2×2 matrices with vanishing trace, which is a real form of Sl₂(C)=A1 in Cartan's classification of Lie algebras. The fundamental representation (the minimal dimension faithful representation), is generated by these 2×2 matrices.

[heading-00152] Minimal Dimensions for Irreducible Representations

[00153] First consider the case K=2. Clearly, the objects γ₁, γ₂ cannot be represented as complex numbers. The minimal dimension to represent two anti-commuting objects is 2. It is easy to find two anti-commuting anti-hermitean unitary 2×2 matrices, take e.g.
γ₁=σ₁; γ₂=σ₂.  (29)

[00155] Now consider a Clifford-algebra generated by three elements γ₁;γ₂; γ₃. From the anti-commutation relations, it follows that the product γ₁γ₂;γ₃ commutes with all generators of the algebra. As an hermitean/Casimir operator, its matrix representation can be taken to be proportional to the unit matrix I₂. Thus, γ₃ can be represented in the same matrix dimension as γ₁ and γ₂, and is simply proportional to their product. Corresponding to γ₁,=σ₁; γ₂=σ₂. we have e.g.
γ₃=γ₁ γ₂=Iσ₃.  (30)
Here we see that Clifford₃ coincides with the algebra su₂. Also, Clifford₂ is a two-dimensional sub-algebra of su₂.

[00158] Taking the matrices β in $C=\sum _{k=1}^K\left(x_k{\beta }_{2k-2}+y_k{\beta }_{2k-1}\right).$
to be the above representation of the three generators of Clifford₃, together with the two-dimensional unit matrix, we get exactly the 2×2 Alamouti code. Next add a fourth generator to the Clifford-algebra. Denoting {tilde over (γ)}₃=−jγ₁γ₂γ₃, {tilde over (γ)}₄=−jγ₁γ₂γ₄, we see from the anti-commutation relations that the sub-algebras generated by γ₁, γ₂ and {tilde over (γ)}₃, {tilde over (γ)}₄ commute. The two sub-algebras, on the other hand, are isomorphic to Clifford₂. Thus each of these subalgebras can be represented using the matrices in Equation (A.27). The two commuting sub-algebras, however, have to be represented using the tensor product. Thus, we get the following matrix representation for the generators of Clifford₄:
γ₁=I₂{circle around (×)}σ₁;γ2=I₂{circle around (×)}σ₂ {tilde over (γ)}₃=σ₁{circle around (×)}I₂;{tilde over (γ)}₄=σ₂{circle around (×)}I₂.  (31)

[00161] Adding a fifth generator, we again note that jγ₁ γ₂ γ₃ γ₄ γ₅ is an hermitean/Casimir operator, and its matrix representation can be taken proportional to the unit matrix. Thus, we can define e.g. γ₅=−j γ₁γ₂γ₃γ₄.

[00162] We have constructed the following representation of Clifford₅: $\begin{array}{cc}{\gamma }_1=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right],{\gamma }_2=\left[\begin{array}{cccc}0& j& 0& 0\\ j& 0& 0& 0\\ 0& 0& 0& j\\ 0& 0& j& 0\end{array}\right],{\gamma }_3=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& -1\\ -1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],{\gamma }_4=\left[\begin{array}{cccc}0& 0& j& 0\\ 0& 0& 0& -j\\ j& 0& 0& 0\\ 0& -j& 0& 0\end{array}\right],{\gamma }_5=\left[\begin{array}{cccc}j& 0& 0& 0\\ 0& -j& 0& 0\\ 0& 0& -j& 0\\ 0& 0& 0& j\end{array}\right],& \left(32\right)\end{array}$

[00163] To get the form $\begin{array}{cc}{C}_{3/4}\left(s_1,s_2,s_3\right)=\left[\begin{array}{cccc}{s}_1& s_2& s_3& 0\\ -s_2^{*}& s_1^{*}& 0& -s_3\\ -s_3^{*}& 0& s_1^{*}& s_2\\ 0& s_3^{*}& -s_2^{*}& s_1\end{array}\right]& \left(33\right)\end{array}$

• • [00164] using prescription $C=\sum _{k=1}^K\left(x_k{\beta }_{2k-2}+y_k{\beta }_{2k-1}\right).,$
    one can use the matrices above with the indices cyclically permuted.

[00166] The same construction can be applied to find inductively the irreducible representations of a Clifford algebra generated by an arbitrary number of elements K. There are two different sets of induction steps, from even K−1 to odd K, and from even K−2 to even K. For even K, the induction steps are the following:
{tilde over (γ)}_K=(−j)^K/2−1(Π_i=1^K−2γ_j)γ_K,

[00168] Define
{tilde over (γ)}_K−1=(−j)^K/2−1(Π_i=1^K−2γ_j)γ_K−1.  (34)

[00170] Note that {γ_j}_j=1^K−2 and {{tilde over (γ)}_K−1,{tilde over (γ)}_K}generate commuting Clifford sub-algebras isomorphic to Clifford_K−2 and Clifford₂.

[00171] Use a tensor product of existing irreducible representations R_K−2 of Clifford_K−2 and R₂ of Clifford₂ to represent these sub-algebras:
R_K(γ_j)=I₂{circle around (×)}R_K−2(γ_j)j=1, . . . K−2
R_K({tilde over (γ)}_K−1)=R₂(γ₁){circle around (×)}I_dimR_K−2
R_K({tilde over (γ)}_K)=R₂(γ₂){circle around (×)}I_dimR_K−2  (35)

[00175] Finally, invert the definitions of the {tilde over (γ)}:s, $\begin{array}{cc}{R}_K\left({\gamma }_{\left\{K-1,K\right\}}\right)=j^{K/2-1}{R}_K\left(\prod _{j=1}^{K-2}{\gamma }_j{\tilde{\gamma }}_{\left\{K-1,K\right\}}\right)& \left(36\right)\end{array}$

[00176] The induction steps from even K−1 to odd K are following $\begin{array}{cc}\mathrm{Define}{\Lambda }_K={\left(-j\right)}^{\left(K+1\right)/2}\prod _{j=1}^K{\gamma }_j& \left(37\right)\end{array}$

[00177] Note that Λ_K is a Casimir operator; it commutes with all {γ_j}_j=1^K.

[00178] Represent it by the unit operator in the representation R_K−1.

[00179] Take

[00180]
R_K(γ_j)=R_K−1(γ_j),j=1, . . . K−1  (38) $\begin{array}{cc}{R}_K\left({\gamma }_K\right)=j^{\left(K+1\right)/2}{R}_{K-1}\left(\prod _{j=1}^{K-1}{\gamma }_j\right)& \left(39\right)\end{array}$

[00181] The induction steps above can be used to prove the following theorem: The minimal dimension to represent a Clifford algebra generated by K elements is 2^[K/2]. The minimal representation matrices are elements in the space $\begin{array}{cc}\underset{\underset{\left[K/2\right]\mathrm{elements}}{︸}}{R_K^{\mathrm{minimal}}={\tilde{R}}_2\left({\mathrm{su}}_2\right)\otimes {\tilde{R}}_2\left({\mathrm{su}}_2\right)\otimes \dots \otimes {\tilde{R}}_2\left({\mathrm{su}}_2\right)}.& \left(40\right)\end{array}$

[00182] Here, R₂(su₂) is the fundamental two-dimensional representation of su₂, the algebra of anti-hermitean 2×2 matrices with vanishing trace. {tilde over (R)} denotes this representation space enlarged by matrices proportional to the two-dimensional unit matrix.

[00183] A minimal (i.e. 2^K−1) dimensional representation of Clifford_K constructed by the induction described above, is $\begin{array}{cc}\begin{array}{c}{\gamma }_2=\underset{\underset{K-2\mathrm{times}}{︸}}{I_2\otimes I_2\otimes \dots \otimes I_2\otimes I_2}\otimes {\sigma }_1\\ {\gamma }_3=\underset{\underset{K-3\mathrm{times}}{︸}}{I_2\otimes I_2\otimes \dots \otimes I_2\otimes I_2}\otimes {\sigma }_2\\ {\gamma }_4=\underset{\underset{K-3\mathrm{times}}{︸}}{I_2\otimes I_2\otimes \dots \otimes I_2\otimes I_2}\otimes {\sigma }_1\otimes {\sigma }_3\\ {\gamma }_5=\underset{\underset{K-3\mathrm{times}}{︸}}{I_2\otimes I_2\otimes \dots \otimes I_2\otimes I_2}\otimes {\sigma }_2\otimes {\sigma }_3\\ {\gamma }_6=\underset{\underset{K-4\mathrm{times}}{︸}}{I_2\otimes \dots \otimes I_2}\otimes {\sigma }_1\otimes {\sigma }_3\otimes {\sigma }_3\\ {\gamma }_7=\underset{\underset{K-4\mathrm{times}}{︸}}{I_2\otimes \dots \otimes I_2}\otimes {\sigma }_2\otimes {\sigma }_3\otimes {\sigma }_3\\ {\gamma }_{2k}=\underset{\underset{K-1-k\mathrm{times}}{︸}}{I_2\otimes \dots \otimes I_2}\otimes {\sigma }_1\otimes \underset{\underset{K-1\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}\\ {\gamma }_{2k+1}=\underset{\underset{K-1-k\mathrm{times}}{︸}}{I_2\otimes \dots \otimes I_2}\otimes {\sigma }_2\otimes \underset{\underset{K-1\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}\\ ⋮\\ {\gamma }_{2K-2}={\sigma }_1\otimes \underset{\underset{K-2\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}\\ {\gamma }_{2K-1}={\sigma }_2\otimes \underset{\underset{K-2\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}\\ {\gamma }_1=j\underset{\underset{K-1\mathrm{times}}{︸}}{{\sigma }_3\otimes {\sigma }_3\otimes \dots \otimes {\sigma }_3}\end{array}& \left(41\right)\end{array}$

[00184] To classify all possible representations, one has to assess the free parameters in this construction.

[00185] First, note that the anti-commutation relations γ_kγ_j+γ_jγ_k=−2δ_jkI_N,j,k=1, . . . ,2K−1 have the symmetry
γ_j→V^Hγ_jV,j=1 . . . ,L  (42)
where V is an unitary dimR_K×dimR_K matrix. This symmetry is large enough to accommodate any choice of basis in Clifford₂ sub-algebras.

[00188] Second, it is easy to see that representation in higher than the minimal dimensions can be found by taking any three higher dimensional anti-commuting matrices instead of σ₁, σ₂, and σ₃ as a basis in any of the elements in the tensor-product above. Anti-commuting fourth roots of the identify matrix, however, are hard to come by.

[00189] A generic representation space would thus be $\begin{array}{cc}{R}_K^{\left(\mathrm{d1},\dots ,d_{\left[K/2\right]}\right)}=\left({\oplus }_{k=1}^{d_1/2}{\tilde{R}}_2\left({\mathrm{su}}_2\right)\right)\otimes \left({\oplus }_{k=1}^{d_2/2}{\tilde{R}}_2\left({\mathrm{su}}_2\right)\right)\otimes \dots \otimes \left({\oplus }_{k=1}^{d_{\left[K/2\right]}/2}{\tilde{R}}_2\left({\mathrm{su}}_2\right)\right)& \left(43\right)\end{array}$

• • [00190] where the dimensions of the commuting sub-algebra representation (d₁, . . . ,d_[K/2]) are even integers ≧2.

[00191] The above can be placed in a less rigorous mathematical narrative. By multiplying by a unitary N×N matrix, a square space-time block code which is freely selected from the left can be brought to a form in which the real part of a symbol to be transmitted appears only on the diagonal of the code matrix. If the symbols to be transmitted are interpreted in the above manner, said real part appears in every diagonal element, multiplied by the same real number. In this case, the dependence of the code matrix on the real part of the symbol is proportional to an N-dimensional unit matrix.

[00192] Let us next examine a method in which a unitary N×N matrix is formed, the elements of which depend linearly on symbols s_k, the unitarity coefficient of which is proportional to the sum of the absolute value squares of the symbols s_k and the dependence of which on the real part of a symbol so is proportional to an N-dimensional unit matrix.

[00193] Let us take a freely selected 2K−1 quantity of N×N matrices, which are all antihermitean and unitary and which all anti-commute with each other. An anti-hermitean matrix refers to a matrix, the hermitean conjugate of which is the matrix itself multiplied by −1. Anti-commutation means that when two matrices can be multiplied by each other in two orders, then if one product is −1 times the other product, the matrices anti-commute. The above family, to which 2K−1 matrices belong, can be called an N-dimensional anti-commutator algebra presentation of 2K−1 elements.

[00194] Let us form K−1 pairs of these 2K−1 matrices. Since there is an uneven number of matrices, an N-dimensional unit matrix is used to form a pair with the remaining matrix. Two matrices are formed of each matrix pair such that the second matrix of the pair, multiplied by the imaginary unit, is added to and subtracted from the first matrix of the pair. The unit matrix is interpreted in its own pair as the first matrix. This way, 2K matrices are formed. These matrices form a complexified anti-commutator algebra extended by a unit element. In short they are called complex anti-commutator matrices.

[00195] A code matrix is formed such that each of the matrices formed as above defines the dependence of the code matrix on one and only one s_k or the complex conjugate of s_k. Thus, the code matrix is the sum of 2K elements and each element is the product of some s_k or s_k complex conjugate and an N×N complex anti-commutator matrix, such that each symbol, complex conjugate, and matrix only appears once in the expression.

[00196] Let us examine a method of forming an N-dimensional anti-commutator algebra presentation of 2K−1 elements.

[00197] First, three 2×2 matrices are freely selected, the matrices fulfilling the following conditions:

• • [00198] matrices are anti-hermitean and unitary; and
  • [00199] matrices anti-commute with each other.

[00200] So, the matrices form an anti-commutator algebra presentation of the freely selected 3 elements. Two matrices are selected from the above defined matrices, and they can be called an elementary pair. The remaining matrix is multiplied by the imaginary unit, and the result is called a third elementary matrix. In addition, a matrix proportional to a two-dimensional unit matrix is used as a fourth elementary matrix. This matrix can be called an elementary unit matrix.

[00201] K-I pairs of N×N matrices are formed of these matrices by formulating tensor products of K−1 elementary matrices for example in the following manner:

• • [00202] The first matrix pair is established as a tensor product of K−2 elementary unit matrices and members of the elementary pair. Each member of the elementary pair appears as separately tensored with the unit matrices. This gives two matrices, (i.e. a matrix pair).

[00203] The second matrix pair is obtained by tensoring K−3 elementary matrices, one member of the elementary pair and the third elementary matrix, in this order.

[00204] The Ith matrix pair is obtained by tensoring K-I−1 elementary unit matrices, one member of the elementary pair and I−1 third elementary matrices, in this order.

[00205] K-1th pair is obtained by tensoring one member of the elementary pair and K−2 third elementary matrices.

[00206] The tensor product of two matrices can be understood as a block form by considering a matrix with as many blocks as the first matrix to be tensored has elements, each block being as big as the second matrix to be tensored. A block of the tensor product is the corresponding element of the first matrix times the second matrix.

[00207] In the above manner we arrive at 2K−2 N-dimensional complex anti-commutator matrices, where N=2^K−1. The 2K−1th anti-commutator matrix is obtained by tensoring K−1 third elementary matrices and by multiplying by the imaginary unit.

[00208] Let us next examine an example of the above method. The following anti-hermitean unitary 2×2 matrices anti-commuting with each other are selected: $\begin{array}{cc}{\sigma }_1=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right);{\sigma }_2=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right);\tau =\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right).& \left(44\right)\end{array}$

[00209] Here the imaginary unit is marked with the letter i. Let us call the pair σ₁, σ₂ an elementary pair and the matrix σ₃=it as a third elementary matrix. As a fourth elementary matrix, a 2-dimensional unit matrix $\begin{array}{cc}{1}_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)& \left(45\right)\end{array}$

• • [00210] is used, which is called an elementary unit matrix.

[00211] N=2^K−1-dimensional complex anti-commutator matrices are formed as tensor products of the elementary matrices: $\begin{array}{cc}{\gamma }_2=\underset{\underset{K-2\mathrm{times}}{︸}}{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}\otimes {\sigma }_1{\gamma }_3=\underset{K-2\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_2{\gamma }_4=\underset{K-3\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_1\otimes {\sigma }_3{\gamma }_5=\underset{K-3\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_2\otimes {\sigma }_3{\gamma }_6=\underset{K-4\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_1\otimes {\sigma }_3\otimes {\sigma }_3{\gamma }_7=\underset{K-4\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_2\otimes {\sigma }_3\otimes {\sigma }_3{\gamma }_{2K}=\underset{K-1-k\mathrm{times}}{\underbrace{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}}\otimes {\sigma }_1\otimes \underset{\underset{k-1\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}{\gamma }_{2K+1}=\underset{\underset{K-1-k\mathrm{times}}{︸}}{1_2\otimes 1_2\otimes 1_2\otimes \dots \otimes 1_2}\otimes {\sigma }_2\otimes \underset{\underset{K-1-k\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}{\gamma }_{2K-2}={\sigma }_1\otimes \underset{\underset{K-2\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}{\gamma }_{2K-1}={\sigma }_2\otimes \underset{\underset{K-2\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}{\gamma }_1=|{\sigma }_3\otimes \underset{\underset{K-1\mathrm{times}}{︸}}{{\sigma }_3\otimes \dots \otimes {\sigma }_3}& \left(46\right)\end{array}$

[00212] The formed matrices are an example of a 2K−1 quantity of N=2^K−1-dimensional antihermitean unitary matrices anticommuting with each other.

[00213] From these matrices, 2K complex anticommutator matrices {γ_k+, γ_k−}k=1^K are formed in the following way: $\begin{array}{cc}{\gamma }_{k\pm }=\frac{\left({\gamma }_{2k-2}\pm i{\gamma }_{2k-1}\right)}{2},k=1,\dots ,K.& \left(47\right)\end{array}$

[00214] Matrices γ_k defined above are used herein. In addition, the 2^K−1-dimensional unit matrix is marked with γ₀. The matrices have also been normalized by dividing by two. The code matrix can now be formed for example as follows: $\begin{array}{cc}C=\sum _{k=1}^K\left(s_k{\gamma }_{k-}+s_k^{*}{\gamma }_{k+}\right)& \left(48\right)\end{array}$

[00215] The obtained code is a delay optimal basic block code. All possible basic block codes of a given code rate can be created simply by interchanging the places of rows and/or columns in all γ matrices simultaneously, or by multiplying the γ matrices by any combination of terms, or changing the numbering of the γ matrices, or by multiplying all γ matrices from right and/or left by a unitary matrix which has four elements diverging from zero, the elements being an arbitrary combination of the numbers ±1, ±i.

[00216] For example, the basic rate ¾ code for four transmit antennas as formed in the above manner has the form $\begin{array}{cc}\left(s_1,s_2,s_3\right)\to \left(\begin{array}{cccc}{s}_1& s_2& s_3& 0\\ -s_2^{*}& s_1^{*}& 0& -s_3\\ -s_3^{*}& 0& s_1^{*}& s_2\\ 0& s_3^{*}& -s_2^{*}& s_1\end{array}\right)& \left(49\right)\end{array}$

[00217] Here N=4 and K=3. Further, the rate 1/2 code for eight antennas, for example, is $\begin{array}{cc}\left(s_1,s_2,s_3\right)\to \left(\begin{array}{cccccccc}{s}_1& s_2& s_3& 0& s_4& 0& 0& 0\\ -s_2^{*}& s_1^{*}& 0& -s_3& 0& -s_4& 0& 0\\ -s_3^{*}& 0& s_1^{*}& s_2& 0& 0& -s_4& 0\\ 0& s_3^{*}& -s_2^{*}& s_1& 0& 0& 0& s_4\\ -s_4^{*}& 0& 0& 0& s_1^{*}& s_2& s_3& 0\\ 0& s_4^{*}& 0& 0& -s_2^{*}& s_1& 0& -s_3\\ 0& 0& s_4^{*}& 0& -s_3^{*}& 0& s_1& s_2\\ 0& 0& 0& -s_4^{*}& 0& s_3^{*}& -s_2^{*}& s_1^{*}\end{array}\right)& \left(50\right)\end{array}$

[00218] Here the rate ¾ code is in the upper left corner and the corresponding inverted complex conjugate in the lower right corner.

[00219] By the above manner, ‘basic codes’ are obtained, in which the elements only depend on one signal, or the real part of one signal and the imaginary part of another. The combination of any N′<N code matrix column gives a full diversity non-square code for N′ antennas. Using these codes, full diversity codes, which do not have the above restriction, can be constructed in the solution of the invention. In a solution according to a preferred embodiment of the invention the elements are allowed to be linear combinations. This way, provided that full diversity is provided, block codes that are unitarily converted are obtained, having the form
{overscore (C)}=UC(s)V,  (51)

• • [00221] where C(s) is a basic block code, such as above. It is an N×N′ matrix, where N is the number of time slots and N′ is the number of antennas. U and V are N×N and N′×N′ unitary matrices. The phase shifts caused by U and V are irrelevant. U and V can be assumed to be unitary matrices with determinant 1.

[00222] This construction gives a family of block codes with N²+N′²−2 continuous parameters. The square codes obtained this way comprise delay optimal maximal rate block codes when the number of antennas is proportional to a power of two.

[00223] Consider, for example, the rate ¾ code for four antennas which were described above (49). A generic unitary 4×4 matrix with a unit determinant can be written, for example, as $\begin{array}{cc}V=\exp \frac{i}{2}\left(\begin{array}{cccc}{\Phi }_{13}+\frac{1}{\sqrt{3}}{\Phi }_{14}+\frac{1}{\sqrt{6}}{\Phi }_{15}& w_1& w_2& w_3\\ w_1^{*}& -{\Phi }_{13}+\frac{1}{\sqrt{3}}{\Phi }_{14}+\frac{1}{\sqrt{6}}{\Phi }_{15}& w_4& w_5\\ w_2^{*}& w_4^{*}& \frac{-2}{\sqrt{3}}{\Phi }_{14}+\frac{1}{\sqrt{6}}{\Phi }_{15}& w_6\\ w_3^{*}& w_5^{*}& w_6^{*}& \frac{-3}{\sqrt{6}}{\Phi }_{15}\end{array}\right)& \left(52\right)\end{array}$

• • [00224] where the exp operation is a matrix exponential, the six parameters ω_k are complex and the three parameters Φ_k are real. U is of the same form. All in all this makes 30 free real parameters. All possible generalizations of said ¾ code (49) can now be constructed by applying the transformation (51) and using the above described U and V.

[00225] It would be desirable that transmission power is distributed equally between different antennas. However, when for example the prior art ¾ code (the ¾ code mentioned in the introduction of the application, for instance) is used, some antennas transmit using only half of their power at certain times. If the code (49) according to a preferred embodiment of the invention is used, the ratio of the peak power to the average power can be made lower. In addition, this construction allows that instead of transmitting zeros of the code matrix, a signal that is orthogonalized in some other way (for example by a different spreading code), a pilot signal for instance, may be transmitted. This way a fully power-uniformized transmission can be provided.

[00226] In a system with several users, especially in a code division and frequency division system, users may be provided with different versions (for example, a version with a permutated antenna order) of the block code, and thus the transmission powers can be uniformized.

[00227] Sometimes, in a time division system with several users, for example, it is preferable to balance the transmission of one user directly without using the above mentioned ways. In a solution according to a preferred embodiment of the invention, an unequal distribution of transmission power between different antennas may be avoided and the above described unitary transformation (51) is applied. The power spectrum of different antennas cannot necessarily be uniformized in respect to each other as a function of time, but by selecting the unitary transformation preferable, the average transmission powers of the antennas are uniformized and the ratio of the peak power to the average power and the ratio of the minimum power to the average power can be minimized.

[00228] Let us examine this embodiment by means of an example. Consider the above described ¾ code (49) for four antennas. Depending on which parameter needs to be improved, the matrices U and Vare selected in an appropriate manner. If the minimum-to-average power needs to be optimized, V is selected as the unit matrix and U as the 4×4 Hadamard matrix: $\begin{array}{cc}U=\frac{1}{2}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 1& -1& -1\\ 1& -1& -1& 1\end{array}\right)& \left(53\right)\end{array}$

[00229] Now by applying the transformation (51) to the code (49) with the above mentioned matrices U and V, the power-uniformized code is $\begin{array}{cc}\stackrel{\_}{C}=\mathrm{UC}\left(z\right)=\left(\begin{array}{cccc}{s}_1-s_2^{*}-s_3^{*}& s_1^{*}+s_2^{*}+s_3^{*}& s_1^{*}-s_2^{*}+s_3^{*}& s_1+s_2-s_3\\ s_1+s_2^{*}-s_3^{*}& -s_1^{*}+s_2^{*}-s_3^{*}& s_1^{*}+s_2^{*}+s_3^{*}& -s_1+s_2+s_3\\ s_1-s_2^{*}+s_3^{*}& s_1^{*}+s_2^{*}-s_3^{*}& -s_1^{*}+s_2^{*}+s_3^{*}& -s_1-s_2-s_3\\ s_1+s_2^{*}+s_3^{*}& -s_1^{*}+s_2^{*}+s_3^{*}& -s_1^{*}-s_2^{*}+s_3^{*}& s_1-s_2+s_3\end{array}\right).& \left(54\right)\end{array}$

[00230] On the other hand, if the peak-to-average power needs to be minimized, U and V can be selected for example as follows: (It is assumed herein that the signal constellation is 8-PSK.) $\begin{array}{cc}U=\frac{1}{2}\left(\begin{array}{cccc}1& e^{-i\frac{\pi }{8}}& e^{-i\frac{\pi }{4}}& e^{-i\frac{3\pi }{8}}\\ 1& -e^{-i\frac{\pi }{8}}& e^{-i\frac{\pi }{4}}& -e^{-i\frac{3\pi }{8}}\\ 1& e^{-i\frac{\pi }{8}}& -e^{-i\frac{\pi }{4}}& -e^{-i\frac{3\pi }{8}}\\ 1& -e^{-i\frac{\pi }{8}}& -e^{-i\frac{\pi }{4}}& e^{-i\frac{3\pi }{8}}\end{array}\right)\mathrm{and}& \left(55\right)\\ V=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& e^{i\frac{\pi }{8}}& 0& 0\\ 0& 0& e^{-i\frac{\pi }{4}}& 0\\ 0& 0& 0& e^{i\frac{3\pi }{8}}\end{array}\right)& \left(56\right)\end{array}$

[00231] Applying now the transformation (51) to the code (49) with the above matrices U and V, a power-uniformized code is achieved, which has a minimal peak-to-average power $\begin{array}{cc}\stackrel{\_}{C}=\mathrm{UC}\left(s\right)V=\frac{1}{2}\left(\begin{array}{cccc}{z}_1-e^{-i\frac{\pi }{8}}{z}_2^{*}-e^{-i\frac{\pi }{4}}{z}_3^{*}& z_1^{*}+e^{i\frac{\pi }{8}}{z}_2+e^{-i\frac{\pi }{4}}{z}_3^{*}& z_1^{*}-e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{i\frac{\pi }{4}}{z}_3& z_1+e^{i\frac{\pi }{8}}{z}_2-e^{i\frac{\pi }{4}}{z}_3\\ z_1+e^{-i\frac{\pi }{8}}{z}_2^{*}-e^{-i\frac{\pi }{4}}{z}_3^{*}& -z_1^{*}+e^{i\frac{\pi }{8}}{z}_2-e^{-i\frac{\pi }{4}}{z}_3^{*}& z_1^{*}+e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{i\frac{\pi }{4}}{z}_3& -z_1+e^{i\frac{\pi }{8}}{z}_2+e^{i\frac{\pi }{4}}{z}_3\\ z_1-e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{-i\frac{\pi }{4}}{z}_3^{*}& z_1^{*}+e^{i\frac{\pi }{8}}{z}_2-e^{-i\frac{\pi }{4}}{z}_3^{*}& -z_1^{*}+e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{i\frac{\pi }{4}}{z}_3& -z_1-e^{i\frac{\pi }{8}}{z}_2+e^{i\frac{\pi }{4}}{z}_3\\ z_1+e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{-i\frac{\pi }{4}}{z}_3^{*}& -z_1^{*}+e^{i\frac{\pi }{8}}{z}_2+e^{-i\frac{\pi }{4}}{z}_3^{*}& -z_1^{*}-e^{-i\frac{\pi }{8}}{z}_2^{*}+e^{i\frac{\pi }{4}}{z}_3& z_1-e^{i\frac{\pi }{8}}{z}_2+e^{i\frac{\pi }{4}}{z}_3\end{array}\right).& \left(57\right)\end{array}$

[00232] Let us next examine a decoding method which can be applied to the reception of the signals that are coded in the above manners. Let us assume that a receiver has M antennas. Let us further assume that N′ antennas are used for transmission in a transmitter and that the block code uses N time slots. The channel between the nth transmit antenna and the mth receive antenna is denoted by the term α_nm. The channels can be assumed to be static over the frame N. The channel terms are collected into the N′×M matrix $\begin{array}{cc}\alpha =\left(\begin{array}{cccc}{\alpha }_{11}& {\alpha }_{12}& \dots & {\alpha }_{1M}\\ {\alpha }_{21}& {\alpha }_{22}& \dots & {\alpha }_{2M}\\ ⋮& ⋮& ⋰& ⋮\\ {\alpha }_{\mathrm{N1}}& {\alpha }_{\mathrm{N2}}& \dots & {\alpha }_{\mathrm{NM}}\end{array}\right)& \left(58\right)\end{array}$

[00233] Correspondingly, the signal received by the antenna m at the time slot t is denoted by r_tm. The N×M matrix of these signals is obtained from the formula
R={overscore (C)}(s)α+n,  (59)

• • [00235] where n is an N×M matrix of additive complex Gaussian noise. The block code {overscore (C)} is constructed as above ((1), (2) and (4)), possibly by restricting the number of antennas. Now denote
    {overscore (γ)}_k±=U_γk±V, k=1 . . . ,K  (60)

[00237] Using these markings, the maximum likelihood detection metric for the kth transmitted symbol s_k is
M_k=|Tr({overscore (γ)}_k+αR^H+Rα^H{overscore (γ)}^H_k−)−s_k|²+(Tr(αα^H)−1)s_k|²  (61)

• • [00239] where Tr refers to a matrix trace, (i.e. the sum of diagonal elements, and H refers to the complex conjugate transpose). Thus, the aim is to minimize the metric, (i.e. it is used as a criterion for deciding which symbol s_k comprises).

[00240] FIG. 3 is an illustration of an arrangement according to an embodiment of the invention—to be use as exemplar only. The figure shows a situation where channel-coded symbols are transmitted via three antennas at different frequencies, at different time slots or by using a different spreading code. Firstly, the figure shows a transmitter 300, which is in connection with a receiver 302. The transmitter comprises a modulator 304 which receives as input a signal 306 to be transmitted, which consists of bits in a solution according to a preferred embodiment of the invention. The bits are modulated to symbols in the modulator. The symbols to be transmitted are grouped into blocks having the length of a given K. It is assumed in this example that the length of the block is three symbols and that the symbols are s₁, s₂ and s₃. The symbols are conveyed to a coder 308. In the coder each block is coded to N×N′ channel symbols. The channel symbols 310 are conveyed in this example via radio frequency parts 312 to three antennas 314 to 318 from where they are to be transmitted.

[00241] In the present example, the block comprises the symbols s₁, s₂ and s₃. The coder performs coding, the defining code matrix of which is formed of 2K elements, in which each element is a product of a symbol or symbol complex conjugate to be transmitted and a complex N×N anticommutator matrix, and in which each matrix is used at most once in the formation of the code matrix.

[00242] A code matrix can for example be the matrix (54) described above, which means that the coder performs the coding $\begin{array}{cc}\left(s_1,s_2,s_3\right)\to \left(\begin{array}{cccc}{s}_1-s_2^{*}-s_3^{*}& s_1^{*}+s_2^{*}+s_3^{*}& s_1^{*}-s_2^{*}+s_3^{*}& s_1+s_2-s_3\\ s_1+s_2^{*}-s_3^{*}& -s_1^{*}+s_2^{*}-s_3^{*}& s_1^{*}+s_2^{*}+s_3^{*}& -s_1+s_2+s_3\\ s_1-s_2^{*}+s_3^{*}& s_1^{*}+s_2^{*}-s_3^{*}& -s_1^{*}+s_2^{*}+s_3^{*}& -s_1-s_2-s_3\\ s_1+s_2^{*}+s_3^{*}& -s_1^{*}+s_2^{*}+s_3^{*}& -s_1^{*}-s_2^{*}+s_3^{*}& s_1-s_2+s_3\end{array}\right).& \left(62\right)\end{array}$