Alamouti Model With Multiple Receiving Antennas

𝑥 = arg min

(∑ 𝑛 𝑅|ℎ | 2+ |ℎ | 2− 1)|𝑥 | 2+ 𝑑 2 (𝑥 , 𝑥

) (3.13)

1 𝑥 1 ∈𝑆

𝑗=1 1 2 1 1 1

𝑛 𝑅

𝑥 2= arg min ( |ℎ 1 | 2+ |ℎ 2 | 2− 1) |𝑥 2 | 2+ 𝑑 2 (𝑥 2 , 𝑥 2 )

𝑥 2 ∈𝑆

𝑗=1

For M-PSK modulated signals, all signals in the constellation have equal energy. The maximum correlation decoding law is equivalent to the case of a single receiving antenna.

3.2.2. STBC space-time code

3.2.2.1. Block space-time coding

Figure 3.7 illustrates the coding structure for a block space-time code. In general, a transmit matrix 𝑋 of size 𝑛 𝑇 × 𝑝 describes a block space-time code. Where 𝑛 𝑇 is the number of transmit antennas, 𝑝 is the number of time intervals for transmitting a block of the coded symbol.

Suppose that the coding signal constellation is 2 𝑚 points. At each coding algorithm, a block of 𝑘𝑚 information bits is mapped into the signal constellation to select 𝑘 modulated signals 𝑥 1 , 𝑥 2 , … , 𝑥 𝑘 , where each group of 𝑚 bits selects a signal in the constellation. The modulated signal k is encoded by a block space-time encoder to transmit

𝑛 𝑇 parallel signal sequences of length 𝑝 pass through the transmit matrix 𝑋 . These sequences are transmitted on the transmit antenna 𝑛 𝑇 continuously for a period of time 𝑝 .

In block space-time coding, the number of coded symbols is 𝑘 . The number of transmission intervals required to transmit coded symbols across multiple transmit antennas is 𝑝 . On the other hand, there are 𝑝 space-time symbols transmitted from each antenna for each block of 𝑘 symbols. The rate of block space-time coding is the ratio of the number of symbols fed into the encoder to the number of coded space-time symbols transmitted on each antenna given by:

𝑅 = 𝑘⁄𝑝

Information Source

Modulator

Space-time Block Encoder

X

Tx

𝑥 1

𝑇𝑥𝑛 𝑇

𝑥

𝑛𝑇

(3.14)

Figure 3.7. Encoding for STBC

The spectral coefficient of a block space-time code is given by the formula:

𝜂 =𝑟 𝑏=𝑟 𝑠 𝑚𝑅= 𝑘𝑚

bits/s/Hz (3.15)

𝐵 𝑟 𝑠 𝑝

where 𝑟 𝑏 and 𝑟 𝑠 are the bit rate and symbol rate respectively, and 𝐵 is the bandwidth. The input to the transmit matrix 𝑋 is a linear combination of the 𝑘 modulated symbols

𝑥 1 , 𝑥 2 , … , 𝑥 𝑘 and its complex conjugate 𝑥 ∗ , 𝑥 ∗ , ⋯ , 𝑥 ∗ . To obtain full transmit diversity

1 2 𝑘

of 𝑛 𝑇 , the generation matrix 𝑋 must be constructed based on the following orthogonal design:

𝑇

𝑇

𝑋. 𝑋 𝐻 = 𝑐(|𝑥 1 | 2 + |𝑥 2 | 2 + ⋯ + |𝑥 𝑘 | 2 )𝐼 𝑛 (3.16) Where 𝑐 is a constant, 𝑋 𝐻 is the Hermitian of 𝑋 and 𝐼 𝑛 is the unit matrix of size

The block space-time code rate with full transmit diversity is less than or equal to one, 𝑅  1 . A code with rate 𝑅 = 1 requires no spreading, while a code with rate 𝑅 < 1 requires spreading 1/𝑅 . A block space-time code with 𝑛 𝑇 transmit antennas, symbol transfer matrix 𝑋 𝑛 𝑇 . This code is called a block space-time code with size 𝑛 𝑇 .

Note the orthogonal design applied to the generation of the space-time code.

block. The rows of the transmit matrix 𝑋 𝑛 𝑇 are orthogonal to each other. That is, in each block, the signal sequences from the two transmit antennas are orthogonal. For example, if we assume, 𝑥 𝑖 = 𝑥 𝑖,1 , 𝑥 𝑖,2 , ⋯ , 𝑥 𝑖,𝑝 is the transmit sequence from the 𝑖 = 1, 2, … , 𝑛 𝑇- th antenna , we have:

𝑥 . 𝑥

= ∑ 𝑝

𝑥 . 𝑥 ∗ = 0 , 𝑖 𝑗, 𝑖, 𝑗  {1, 2, … , 𝑛

} (3.17)

𝑖 𝑗

𝑡=1

𝑖,𝑡 𝑗,𝑡 𝑇

Where, 𝑥 𝑖 . 𝑥 𝑗 is the series product of 𝑥 𝑖 and 𝑥 𝑗 . Orthogonality can achieve full transmit diversity with respect to the number of transmit antennas. Furthermore, it allows the receiver to receive two signals transmitted from different and consecutive antennas, making decoding simple.

3.2.2.2. STBC for real signal constellation

Based on the shape of the signal constellation, block space-time codes can be classified into block space-time codes with real signals and complex signals.

In general, if a real generating matrix 𝑋 𝑛 𝑇 of size 𝑛 𝑇 × 𝑝 with variable

𝑥 1 , 𝑥 2 , … , 𝑥 𝑘 satisfy:

𝑋 𝑛

. 𝑋 𝑇 = 𝑐(|𝑥 | 2 + |𝑥 | 2 + ⋯ + |𝑥 | 2 )𝐼

(3.18)

𝑇 𝑛 𝑇 1 2 𝑘 𝑛 𝑇

Block space-time coding can provide full transmit diversity 𝑛 𝑇 at code rate 𝑘/𝑝 .

More simply we can consider a block space-time code with a square transmit matrix 𝑋 𝑛 𝑇. For any real signal constellation, modulated by M-ASK, a block space-time code with a square matrix 𝑋 𝑛 𝑇 of size 𝑛 𝑇 × 𝑛 𝑇 exists if and only if the number of transmit antennas 𝑛 𝑇 = 2, 4, or 8. The transmit matrix for 𝑛 𝑇 = 2 is as follows:

𝑋 2

𝑥 1 −𝑥 2

= [ 𝑥 2 𝑥 1 ] (3.19)

] (3.20)

𝑥 1 −𝑥 2

𝑥 2 𝑥 1

𝑥 3 −𝑥 4

𝑥 4 𝑥 3

𝑋 8 =𝑥 5 𝑥 6

𝑥 6 −𝑥 5

𝑥 7 −𝑥 8

[𝑥 8 𝑥 7

−𝑥 3 −𝑥 4

𝑥 4 −𝑥 3

𝑥 1 𝑥 2

−𝑥 2 𝑥 1

𝑥 7 𝑥 8

𝑥 8 −𝑥 7

−𝑥 5 𝑥 6

−𝑥 6 −𝑥 5

−𝑥 5 −𝑥 6

𝑥 6 𝑥 5

𝑥 7 −𝑥 8

𝑥 8 𝑥 7

𝑥 1 −𝑥 2

𝑥 2 𝑥 1

𝑥 3 −𝑥 4

𝑥 4 𝑥 3

−𝑥 7 −𝑥 8

𝑥 8 −𝑥 7

𝑥 5 𝑥 6

−𝑥 6 𝑥 5

−𝑥 3 −𝑥 4

𝑥 4 −𝑥 3

𝑥 1 𝑥 2

−𝑥 2 𝑥 1 ]

(3.21)

The square matrix has rows orthogonal to the input ±𝑥 1 , ±𝑥 2 , … , ±𝑥 𝑘 . From the matrices, observing a block of modulated symbols 𝑘 , both the number of transmit antennas 𝑛 𝑇 and the number of time periods p are required to be equal to the block length 𝑘 . For example, with four transmit antennas, the encoder inputs 𝑘 = 4 real-number modulated symbols 𝑥 1 , 𝑥 2 , 𝑥 3 , and 𝑥 4 . At time 𝑡 = 1 , the signal

𝑥 1 , 𝑥 2 , 𝑥 3 , and 𝑥 4 transmit on antennas 1 to 4 respectively. At time 𝑡 = 2 , the signal

The transmitted signals on the corresponding antennas are −𝑥 2 , 𝑥 1 , 𝑥 2 , −𝑥 4 , and 𝑥 3 , and so on. This example requires four antennas and four time slots to transmit four symbols. Therefore, this code does not require bandwidth spreading.

Construct a model with full rate 𝑅 = 1 for any number of transmitting antennas. In general, for 𝑛 𝑇 transmitting antennas, the transmission period p has a minimum value given by the following formula:

min(2 4𝑐+𝑑 ) (3.22)

In there:

𝑐, 𝑑|0 ≤ 𝑐, 0 ≤ 𝑑 ≤ 4, 𝑎𝑛𝑑 8𝑐 + 2 𝑑 ≥ 𝑛 𝑇 (3.23)

With 𝑛 𝑇  8 , the minimum value of p is as follows:

𝑛 𝑇 = 2, 𝑝 = 2

𝑛 𝑇 = 3, 𝑝 = 4

𝑛 𝑇 = 4, 𝑝 = 4

𝑛 𝑇 = 5, 𝑝 = 8

𝑛 𝑇 = 6, 𝑝 = 8

𝑛 𝑇 = 7, 𝑝 = 8

𝑛 𝑇 = 8, 𝑝 = 8 (3.24)

Using these values, we can generate non-square matrices 𝑋 3 , 𝑋 5 , 𝑋 6 , and 𝑋 7 based on real orthogonal design with dimensions 3, 5, 6, and 7 respectively. These matrices are as follows:

𝑥 1 −𝑥 2 −𝑥 3 −𝑥 4

𝑋 3 = [ 𝑥 2

𝑥 3

𝑥 1

−𝑥 4

𝑥 4

𝑥 1

−𝑥 3 ] (3.25)

𝑥 2

𝑋 5 =

𝑥 1 −𝑥 2

𝑥 2 𝑥 1

𝑥 3 −𝑥 4

𝑥 4 𝑥 3

−𝑥 3 −𝑥 4

𝑥 4 −𝑥 3

𝑥 1 𝑥 2

−𝑥 2 𝑥 1

−𝑥 5 −𝑥 6

𝑥 6 𝑥 5

𝑥 7 −𝑥 8

𝑥 8 𝑥 7

−𝑥 7 −𝑥 8

𝑥 8 −𝑥 7

𝑥 5 𝑥 6

−𝑥 6 𝑥 5

[𝑥 5 𝑥 6 𝑥 7 𝑥 8 𝑥 1 −𝑥 2 −𝑥 3 −𝑥 4 ]

𝑋 6 =

𝑥 1 −𝑥 2

𝑥 2 𝑥 1

𝑥 3 −𝑥 4

𝑥 4 𝑥 3

−𝑥 3 −𝑥 4

𝑥 4 −𝑥 3

𝑥 1 𝑥 2

−𝑥 2 𝑥 1

−𝑥 5 −𝑥 6

𝑥 6 𝑥 5

𝑥 7 −𝑥 8

𝑥 8 𝑥 7

−𝑥 7 −𝑥 8

𝑥 8 −𝑥 7

𝑥 5 𝑥 6

−𝑥 6 𝑥 5

𝑥 5 𝑥 6 𝑥 7 𝑥 8 [𝑥 6 −𝑥 5 𝑥 8 −𝑥 7

𝑥 1 −𝑥 2 −𝑥 3 −𝑥 4

𝑥 2 𝑥 1 𝑥 4 −𝑥 3 ]

𝑥 1 −𝑥 2

𝑥 2 𝑥 1

𝑥 3 −𝑥 4

𝑋 =𝑥 4 𝑥 3

−𝑥 3 −𝑥 4

𝑥 4 −𝑥 3

𝑥 1 𝑥 2

−𝑥 2 𝑥 1

−𝑥 5 −𝑥 6

𝑥 6 𝑥 5

𝑥 7 −𝑥 8

𝑥 8 𝑥 7

−𝑥 7 −𝑥 8

𝑥 8 −𝑥 7

𝑥 5 𝑥 6

−𝑥 6 𝑥 5

7𝑥 𝑥 𝑥

𝑥 𝑥

−𝑥

−𝑥

−𝑥

5 6 7

8 1 2 3 4

𝑥 6 −𝑥 5 𝑥 8 −𝑥 7 [𝑥 7 −𝑥 8 −𝑥 5 𝑥 6

𝑥 2 𝑥 1 𝑥 4 −𝑥 3

𝑥 3 −𝑥 4 𝑥 1 𝑥 2 ]

Explain 𝑋 6 , a block space-time code matrix with six transmit antennas. The input to the block space-time encoder is a block of eight symbols 𝑥 1 , 𝑥 2 , …, 𝑥 8 from the modulation constellation. After encoding, the coded symbols are transmitted on six antennas in eight time intervals, the third antenna, signals 𝑥 3 , 𝑥 4 , 𝑥 1 , −𝑥 2 , −𝑥 7 , −𝑥 8 , 𝑥 5 and 𝑥 6 are transmitted on antennas one through six.

3.3. STTC grid space-time coding

3.3.1. Introduction

STTC allows full diversity and high code rate. STTC is a convolutional code extended for MIMO. The convolutional code structure is particularly suitable for communication

space and satellite, by using only simple encoding but achieving high efficiency thanks to complex decoding method.

3.3.2 STTC encoding structure

In STTC, the encoder maps binary data into signals into modulation symbols, with the modulation function described by a mesh diagram.

Consider the M-PSK modulated grid space-time encoder with 𝑛 𝑇 transmitting antennas as shown in Figure 3.8. The input current, denoted by 𝑐 , is given as follows:

𝑐 = (𝑐 0 , 𝑐 1 , 𝑐 2 , ⋯ , 𝑐 𝑡 , ⋯ ) (3.26) Where 𝑐 𝑡 is the group of 𝑚 = log 2 𝑀 bits of time information 𝑡

𝑐 𝑡 = (𝑐 1 , 𝑐 1 , ⋯ , 𝑐 𝑚 ) (3.27)

𝑡 𝑡 𝑡

The encoder maps the input signal sequence into an M-PSK modulated signal sequence.

𝑥 = (𝑥 0 , 𝑥 1 , 𝑥 2 , ⋯ , 𝑥 𝑡 , ⋯ ) (3.28) Where 𝑥 𝑡 is the space-time symbol at time 𝑡

( 1 1

𝑛 𝑇 𝑇

𝑥 𝑡 =

𝑥 𝑡 , 𝑥 𝑡 , ⋯ , 𝑥 𝑡 )

(3.29)

The modulated signal, 𝑥 1 , 𝑥 1 , ⋯ , 𝑥 𝑛 𝑇 , is transmitted through 𝑛

transmitting antenna

𝑡 𝑡 𝑇

(𝑔 1 … … 𝑔 1 )

0.1

0.𝑛𝑇

(𝑔 1 … … 𝑔 1 )

1.1

1,𝑛𝑇

(𝑔 1 … … 𝑔 1

)

𝑐 1

𝑣 1 ,1

𝑣 1 ,𝑛𝑇

𝑥 1 , 𝑥 2 , … . . 𝑥 𝑛𝑇

𝑐 𝑚

(𝑔 𝑚 … … 𝑔 𝑚

𝑣 𝑚 ,1

𝑣 𝑚 ,𝑛𝑇

)

(𝑔 𝑚 … … 𝑔 𝑚

1.1

1,𝑛𝑇

(𝑔 𝑚 … … 𝑔 𝑚 )

0.1

0.𝑛𝑇

Figure 3.8 STTC encoder

The encoder is shown in Figure 3.8, a string of 𝑚 binary bits is input into the encoder, consisting of 𝑚 shift registers. The input string 𝑐 𝑘 = (𝑐 𝑘 , 𝑐 𝑘 , 𝑐 𝑘 , ⋯ , 𝑐 𝑘 , ⋯ ) , 𝑘 = 1, 2, … 𝑚 , to the shift register

0 1 2 𝑡

shift register 𝑘 and multiplied by the set of encoding coefficients. The output of the multiplier from all the shift registers and added modulo 𝑀 , the encoder output is 𝑥 = (𝑥 1 , 𝑥 2 , ⋯ , 𝑥 𝑛 𝑇 ) . The shift register and adder components modulo 𝑀 are described by the sets of multipliers 𝑚 as follows:

𝑔 1 = [(𝑔 1 , 𝑔 1 , ⋯ , 𝑔 1 ), (𝑔 1 , 𝑔 1 , ⋯ , 𝑔 1 ), ⋯ , (𝑔 1 , 𝑔 1 , ⋯ , 𝑔 1 )] (3.30)

0.1

0.2

0.𝑛 𝑇

1.1

1.2

1,𝑛 𝑇

𝑣 1 ,1

𝑣 1 ,2

𝑣 1 ,𝑛 𝑇

𝑔 2 = [(𝑔 2 , 𝑔 2 , ⋯ , 𝑔 2 ), (𝑔 2 , 𝑔 2 , ⋯ , 𝑔 2 ), ⋯ , (𝑔 2 , 𝑔 2 , ⋯ , 𝑔 2 )]

0.1

⋮

0.2

0.𝑛 𝑇

1.1

1.2

1,𝑛 𝑇

𝑣 2 .1

𝑣 2 .2

𝑣 2 ,𝑛 𝑇

𝑔 𝑚 = [(𝑔 𝑚 , 𝑔 𝑚 , ⋯ , 𝑔 𝑚 ), (𝑔 𝑚 , 𝑔 𝑚 , ⋯ , 𝑔 𝑚 ), ⋯ , (𝑔 𝑚 , 𝑔 𝑚 , ⋯ , 𝑔 𝑚 )]

0.1

0.2

0.𝑛 𝑇

1.1

1.2

1,𝑛 𝑇

𝑣 𝑚 ,1

𝑣 𝑚 ,2

𝑣 𝑚 ,𝑛 𝑇

𝑗,𝑖

𝑡

The output of the encoder of the 𝑖th transmitting antenna at time 𝑡 , denoted by 𝑥 𝑖, is calculated as

after:

𝑥 𝑖 = ∑ 𝑚

∑ 𝑣 𝑘

𝑔 𝑘

𝑐 𝑘 mod 𝑀, 𝑖 = 1,2, ⋯ , 𝑛

(3.31)

𝑡 𝑘=1

𝑗=0

𝑗,𝑖

𝑡−𝑗 𝑇

These output values ​​are the components of the M-PSK signal set. The spatio-temporal modulated signal transmitted at time 𝑡

( 1 1

𝑛 𝑇 𝑇

𝑥 𝑡 =

𝑥 𝑡 , 𝑥 𝑡 , ⋯ , 𝑥 𝑡 )

(3.32)

The M-PSK grid space-time code has a bandwidth factor of 𝑚 bits/s/Hz. The total memory of the encoder 𝑣

𝑣 = ∑

𝑚

𝑘=1

𝑣 𝑘

(3.33)

𝑣 𝑘

= ⌊ 𝑣+𝑘−1 ⌋ (3.34)

log 2 𝑀

The total number of states of the grid encoder is 2 𝑣 . The set of multipliers 𝑚 is called the generator sequence.

Consider a QPSK grid space-time code with two transmit antennas. The encoder consists of two shift registers. The encoder structure for the model with memory 𝑣 is shown in Figure 3.9

Two binary strings 𝑐 1 = (𝑐 1 , 𝑐 1 , 𝑐 1 , ⋯ , 𝑐 1 , ⋯ ) and 𝑐 2 = (𝑐 2 , 𝑐 2 , 𝑐 2 , ⋯ , 𝑐 2 , ⋯ ) give

0 1 2 𝑡 0 1 2 𝑡

into the upper and lower encoding shift registers. The memories of the upper and lower shift registers are v 1 and v 2 respectively , with v = v 1 + v 2 . These two input strings are multiplied by the factors:

𝑔 1 = [(𝑔 1 , 𝑔 1 ), (𝑔 1 , 𝑔 1 ), ⋯ , (𝑔 1 , 𝑔 1 )] (3.35)

0.1

0.2

1.1

1.2

𝑣 1 ,1

𝑣 1 ,2

𝑔 2 = [(𝑔 2 , 𝑔 2 ), (𝑔 2 , 𝑔 2 ), ⋯ , (𝑔 2 , 𝑔 2 )]

0.1

0.2

1.1

1.2

𝑣 2 .1

𝑣 2 .2

𝑗,𝑖

In which 𝑔 𝑘 ∈ {0,1,2,3} , 𝑘 = 1, 2 ; 𝑖 = 1, 2 ; 𝑗 = 0, 1, … , 𝑣 𝑘 . The output of the multiplier continues to be added modulo 4 together

𝑥 𝑖 = ∑ 2

∑ 𝑣 𝑘

𝑔 𝑘

𝑐 𝑘 mod 4, 𝑖 = 1.2 (3.36)

𝑡 𝑘=1

𝑗=0

𝑗,𝑖

𝑡−𝑗

The output of the adder is 𝑥 1 and 𝑥 2 which are points in the QPSK constellation. Then

𝑡 𝑡

These points are on the first and second antennas respectively.

(𝑔 1 , 𝑔 1 )

0.1 0.2

(𝑔 1 , 𝑔 1 )

1.1 1.2

𝑐 1

(𝑔 1 ,1 , 𝑔 1 ,2 )

𝑣 1𝑣 1

𝑡

(𝑥 1 , 𝑥 2 )

𝑡 𝑡

𝑐 2

𝑡

(𝑔 , 𝑔 )

2 2

𝑣 ,1 𝑣 ,2

2 2

(𝑔 2 , 𝑔 2 )

1.1 1.2

(𝑔 2 … … 𝑔 2 )

0.1

0.2

The signal received at the receiver will be decoded by the Space-Time Maximum Likelihood Decoder (STMLD). The STMLD will be implemented into a Viterbi vector algorithm, the code line with the smallest cumulative metric will be selected as the decoded data sequence. The complexity of the decoder increases exponentially with the number of states on the constellation diagram and the number of grid states. An STTC code with a diversity level of D transmits data at a rate of 𝑅 bps, then the complexity of the decoder is proportional to the system 2 𝑅(𝐷−1) .

Figure 3.9 STTC encoder with two transmit antennas

STTC provides much better code gain than STBC. The code gain of STTC increases as the number of states of the code grid increases. However, the complexity of STBC is much lower than that of STTC. Because STBC is encoded and decoded simply by linear processing algorithms, STBC is more suitable for practical applications in MIMO systems.

3.4. BLAST class space-time encoding

After discovering that when the scattering paths are large enough, multipath radio channels can provide quite large capacities thanks to suitable processing architectures. In 1996, G.J. Foshchisi of Bell Laboratories proposed the D-BLAST (Diagonal-Bell Laboratories Layered Space-Time) architecture using multiple transmit and receive antennas with diagonal layered coding, each data block will be transmitted diagonally. In Rayleigh scattering environments, this architecture can provide capacity that increases linearly with the number of transmit and receive antennas (assuming that the number of transmit and receive antennas is equal) and can reach nearly 90% of the Shannon capacity. However