Space-Time Block Codes

Space-time block codes operate on a block of input symbols producing a matrix output whose columns represent time and rows represent antennas. Their key feature is the provision of full diversity with extremely low encoder/decoder complexity under frequency-nonselective channels. The space-time block code system is shown in figure 2.4. Assuming n_T antennas are used, and p symbols per antenna are used to convey k uncoded symbols. The code rate is given by

R k

= p (2.1)

To describe the space-time block codes, we use the transmission matrix X which is a n_T×p matrix. The element of X in the ith row and jth column,

, , 1, , , 1, ,

i j T

x i= … n j= … p represents the signals transmitted from the antenna i at time

j.

The key property of this system is the orthogonality between the sequences generated by the different transmit antennas. This feature was generalized in [8] to an arbitrary number of transmit antennas by applying the theory of orthogonal designs.

It is also shown in [8] that to achieve full transmit diversity, the code rate of a space-time block code must be less than or equal to one, R≤1 which requires an bandwidth expansion of ¹

R.

Each element of the transmission matrix X is a linear combinations of the k

Chapter 2 Space-Time Trellis Codes

modulated symbols x₁,…,x_k and their conjugates x₁^*,…,x_k^* . To achieve the orthogonality, X must satisfy the following equation:

(

^{12 2}

)

T matrix. Through the orthogonal designing, the signal sequence from any two transmit antennas are orthogonal. This property enables the receiver to decouple the signals transmitted from different antennas and consequently, a simple maximum likelihood decoding, based only on linear processing of the received signals.

Antenna 1

Figure 2.5 Alamouti space-time block encoder

The Alamouti code [9], was the first and the most famous space-time block code.

It is achieve a full diversity gain using two transmit antennas and a simple maximum-likelihood decoding algorithm. The transmission matrix is given by

*

Figure 2.5 shows an encoder structure for Alamouti code. It is easy to see that the

transmit sequence from antennas one and two are orthogonal,

1 2 * *

1 2 2 1 0

x x x x

⋅ = − =

x x (2.4)

The decoding of space-time block codes was based on maximum likelihood algorithm. Assuming one receive antenna and perfect channel state information (CSI) is know at the receiver. Using Alamouti code as an example, the maximum

Chapter 2 Space-Time Trellis Codes

likelihood decoder choose a pair of signals ( ,x xˆ ˆ_{1 2}) from the signal modulation constellation to minimize the distance metric

( ) ( )

2 2 *

1, 1 1ˆ 2ˆ2 2, 1 2ˆ 2 1ˆ *

d r h x +h x +d r h x +h x (2.5)

With some simple transformation, the decision rule can be derived as

( )

The decision rules derivation can be extended to other cases with other number of receive antennas, and they can be found in [8].

Space-time block codes can achieve a maximum possible diversity advantage with a simple decoding algorithm. It is very attractive because of its simplicity.

However, no coding gain can be provided by space-time block codes, and also, non-full rate space-time block codes can introduce bandwidth expansion.

2.2.2 Space-Time Trellis Codes

Space-time trellis codes operate on one input symbol at a time producing a sequence of vector symbols whose length represents antenna number. Like traditional trellis coded modulation (TCM) for the single-antenna channel, space-time trellis codes provide coding gain. Since they also provide full diversity gain, their key advantage over space-time block codes is the provision of coding gain.

Space-time trellis codes are nowadays widely discussed as it can simultaneously offer a substantial coding gain, spectral efficiency, and diversity improvement on flat fading channels. The case for frequency-selective channels are studied in [13][18], and the conclusion of [13] is that the frequency-selective channel doesn’t affect the diversity

Chapter 2 Space-Time Trellis Codes

provided by space-time trellis codes. However, compared with space-time block codes, this code is far more difficult to design, and also requires a computationally intensive encoder and decoder. The key development was done by Tarokh, Seshadri and Calderbank in 1998 [1], and some other improved development was done in

[10][11][12]. In next session, we will talk about it in detail.

2.3 Space-Time Trellis Codes 2.3.1 Trellis description

The space-time trellis codes are described by trellis structures. Considering a space-time trellis coded M-PSK modulation with n_T transmit antennas. The encoder takes a group of m=log₂M information bits at time t given by

(

^{1, , m}

)

t = at at

a … (2.8)

and produces a group of mn_T coded bits

( ) ( )

(

^{,11, , , 1 , , ,1nT, , , nT}

)

t = ct ct m ct ct m

c … … … (2.9)

Each

(

^{,1, , ,}T

)

^{, 1, ,}

i i

t t n T

c … c i= … n are mapped into symbol x_tⁱ, and thus c_t is mapped into a group of n_T symbols given by

(

^{1, , nT}

)

t = xt xt

x … (2.10)

where each x_tⁱ ,i= …1, ,n_T is an M-PSK modulated signal. At each time t , depending on the state of the encoder and the input bits, a transition branch is chosen.

On the trellis, each branch transition is labeled of

(

xt^1,…^,xt^nT

)

which means the transmit antenna i is used to send symbolx_tⁱ, and all these transmissions are simultaneous.

Chapter 2 Space-Time Trellis Codes

Figure 2.6 Trellis description for a 4-state space-time trellis codes with 2 transmit antennas and 4-PSK signal constellation

0 1

2

3

(1,0) (0,1)

(-1,0)

(0,-1)

Figure 2.7 4-PSK signal constellation

Figure 2.6 shows an example of trellis description for a 4 states space-time trellis

code with 2 transmit antennas and 4-PSK signal constellation. 4-PSK signal constellation is given in figure 2.7. The number pairs a a_t¹ _t²/x x_t¹ _t² in front of each state are the labels of each branch transition starting from that state. The left-most number pair is corresponding to the top-most branch transition of the state. As an illustration, if the encoder is in the second state at time t, the input at this time is 11, then the encoder chooses the branch transition from the second state to the fourth state.

This branch transition is labeled 11/13 which means antennas (1,2) will transmit the symbol (1,3) respectively.

The trellis is a representation which can fully describe the space-time trellis codes.

However, it is common to describe the space-time trellis codes as generator descriptions in the encoders.

Chapter 2 Space-Time Trellis Codes

Figure 2.8 Encoder structure of space-time trellis codes

2.3.2 Generator Description

The encoder structure of space-time trellis code is shown in figure 2.8. The k-th

input sequence a^k=

(

a0^k,…,a_t^k,…

)

, k=1,…,m is fed into the k-th shift register and

Chapter 2 Space-Time Trellis Codes

As an example, let us consider a scheme of 4-state space-time trellis coded QPSK system with 2 transmit antennas and the generator sequences are

( ) ( )

The resulting trellis structure is shown in figure 2.6, which is the same as the one we use as an example in the previous session.

2.3.3 Design Criteria

For a given encoder structure, a set of encoder coefficients is determined by minimizing the error probability. It is shown in [1] that the error rate for slow fading channels, the upper bound is depend on the value of r which is the rank of codeword distance matrix ^{A X X}

( )

^,ˆ . It can be obtained by utilizing the codeword difference matrix ^{B X X}

( )

^,ˆ

( ) ( ) ( )

^{,ˆ = ,ˆ ⋅ H ,ˆ}

A X X B X X B X X (2.16)

where X^ˆ is the erroneous decision mad by decoder when the transmitted sequence was in fact X. Therefore, in order to minimize the error probability, we have two different criteria:

z Rank & determinant criteria: If rn_R <4, the minimum rank r over all pairs of distinct codewords should be maximized. Also, the determinant of ^{A X X}

( )

^,ˆ

along the pairs of distinct codewords with the minimum rank should maximized, too.

z Trace criteria: If rn_R ≥4, the minimum trace of ^{A X X}

( )

^,ˆ among all pairs of distinct codewords should be maximized.

Chapter 2 Space-Time Trellis Codes

These criteria were derived and summarized in [1] along with the criteria for fast fading channels. The above criteria was referred to as the Tarokh/Seshadri/Calderbank (TSC) codes. An improved criterion was proposed and referred to as the Baro/Bauch/Hansmann (BBH) codes [14].

2.4 Decoding Algorithm

2.4.1 Maximum A posterior Probability (MAP) Decoder

To recover the information bits at the receiver, it is natural to choose a receiver that achieves the minimum probability of error P a( _k ≠aˆ_k) where aˆ_k is the decision

The algorithms that achieve this task are commonly referred to as maximum a posteriori probability (MAP) algorithms.

When probabilities are concerned, it is often convenient to work with log-likelihood ratios (LLRs) rather than actual probabilities. The LLR for a variable

ak is defined as is on the contrary. Therefore, by using this property of LLRs, the decision rules (2.17) can be further written as

1, ( | ) 0

Chapter 2 Space-Time Trellis Codes

The main problem of the MAP approach is that the APPs calculation is computation-intensive. To show this, we use Bayes’ rule and the theorem of total probability [15] on P a( _k =a^ˆ_k | )y , and obtain

The computation loading in this equation is too heavy. Therefore we introduce algorithms which require less computation complexity than (2.20).

2.4.2 BCJR Algorithm

This algorithm was proposed in1974 by Bahl, Cocke, Jelinek, and Raviv [16], and is now known as the name “BCJR algorithm” composed of the initials of four authors.

This is a algorithm to efficiently compute the APPs of interestP a( _k =aˆ_k| )y . chosen trellis. For example, the trellis in figure 2.9 has the set

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Chapter 2 Space-Time Trellis Codes

is the memory order of the encoder.

We begin with the computation of the probability that the transmitted sequence path in the trellis contained the branch transition from state r_i to state r_j when the

-th

The left hand side of this equation can be written as

( )

1 1 1 1 1

( ,_{k k} , ) _k, _k , ( , _k ), _k, ( _k , _N)

P s s₊ y =P s s ₊ y …y₋ y y₊ …y (2.23)

Applying the chain rules again on (2.23), we obtain the key decomposition

( ) ( ) ( )

γ _{− −} , and therefore it can be extended to a recursive computation:

( ) ( ) ( )

has also a recursive computation formula:

( ) ( ) ( )

From (2.25) and (2.26), we know that if we have the knowledge of all

(

, 1

)

, 0, , 1

k s sk k k N

γ ₊ = … − , then we can compute all P s s

(

_k, _k+1,y

)

, k=0,…,N−1. The

Chapter 2 Space-Time Trellis Codes

term γ_k

(

s s_k, _k+1

)

can be further decomposed into

(

, 1

) (

1|

) (

| , 1

)

k s sk k P sk sk P yk s sk k

γ ₊ = ₊ ⋅ ₊ (2.28)

With specific state numbers, (2.28) can be written as

( ) ( ) ( ) ( )

where a_{i j,} represent the set of information bits corresponding to branch transition from state r_i to state r_j . Usually, the information bits are assumed to be independent identically distributed (i.i.d.), which makes ( ) ¹

2 system with only one decoder and AWGN channel, this term is simply the pdf. of the noise distribution, i.e.

2 system, and this term will be provided by the equalizer.

Our goal is to compute the APPs P a

(

k =a^{ˆ |}k y

)

. Since we have derived the

Finally, the LLRs of the APPs are

Chapter 2 Space-Time Trellis Codes this is done similarly as follows,

( )

These LLRs will be used in the turbo equalizer system described in Chapter 3.

Chapter 3 Turbo Equalization

CHAPTER

3

Turbo Equalization

For wideband systems, because the bandwidth of the transmitted signals is large and often exceeds the coherent bandwidth of the channel, the channels usually have frequency-selective characteristics. The selectivity in frequency is aroused by the multipath fading channel in which transmission paths have different delays and fadings. This channel can be charactered by a tapped-delay line filter. The objective of an equalizer is to eliminate the distortion induced by the frequency-selective channel. Structures of a variety of equalizers are surveyed in this chapter. The algorithms to derive the optimal filter coefficients are also briefly introduced here. At the end of this chapter, a promising equalization scheme incorporated with the turbo principle is introduced.

3.1 Frequency-Selective Channel

Figure 3.1 shows an equivalent discrete baseband system including transmitter

and receiver. Here we focus on channel and equalizer only. The system does not include any coding layer which is commonly found in practical communication systems. To add coding layers into this system is straightforward, that is to replace

Chapter 3 Turbo Equalization

the information source with the coded bits. Information bits a in figure 3.1 is mapped into modulated symbols

x according to a chosen modulation. A transmit

filter and a receiver filter are used to meet the spectrum requirement and to mitigate the effect of frequency-selective channel. The objective of the equalizer at the receiver is to recover the symbols

x from the output of the receive filter y . The

and the receive filter. It is the channel that equalizers have to cope with, and can be charactered by the tapped-delay line model as shown in figure 3.2. Thus, channels can be described by the impulse response

( h

T

[ ]

^{0 ,}…^,

h L

T

[ ] )

where L is the

Figure 3.1 Equivalent discrete baseband system

∑

[ ]

¹

h ^h

[ ]

^{2 h L}

[ ]

[ ]

⁰

h

Figure 3.2 Tapped delay line model of the channel

Chapter 3 Turbo Equalization

3.2 Equalizer Overview

3.2.1 Trellis-Based

Due to the tapped delay line model of the channel, we can derive a trellis description for the channel. With this trellis description, the BCJR algorithm we presented in 2.4.2 can also be applied readily to detect the symbols. To apply BCJR algorithm in the equalizer, the initial values of α0

( )

s0 and βN

( )

sN need to be modified according to the property of channels. Usually, the channel starts at the zero state and ends at arbitrary state, which leads to

0

Or, if the channel was preoccupied by the previous transmission, the value α0

( )

s0

will be α0

( )

r_i =1 for all i.

This approach of equalization requires the knowledge of channel state information (CSI). This requires another effort in channel estimating, and the accuracy of the estimate will affect the detection performance in a certain level.

However, the problem of an trellis-based equalizer is the complexity. The state number of the equivalent trellis is dependent on the channel order L , the signal constellation M-PSK, and transmit antenna

n

_T in a MIMO system. It can be calculated by

( )

^nT

state number = M ^L (3.5)

For example, if the transmitter uses 2 transmit antennas and 4-PSK modulation, and the channel order is 5, then we will have a equivalent trellis of state number

( )

^{42 5 =1048576}. This is a huge number for a decoder to build, not to mention the one with more antennas or with higher constellation. Therefore, this approach is

Chapter 3 Turbo Equalization

only practical when M, L, and

n are very small.

_T

3.2.2 Filter-Based

Due to the high complexity required by trellis-based equalizer, we turn to a much simpler approach, filter-based equalization. The trellis-based approach does not recover the signals, but it calculates the APPs and then chooses the one with max APPs to be the estimate signals. The filter-based does not calculate the APPs, but instead, it tries to recover the signals of interest and makes decisions on them whether in soft or hard decisions. The structures of filters can be categorized into linear filters or decision-feedback filters.

∑

[ ]

¹

f ^f

[ ]

^{2 f L}

[ ]

[ ]

⁰

f

Figure 3.3 Linear euqalizer

3.2.2.1 Linear Equalizer (LE)

To compensate for the channel distortion, we may employ a linear filter with adjustable coefficients f . The filter coefficients are adjusted on the basis of measurements of the channel characteristics. There are different criteria to derive the filter coefficients and will be addressed in 3.2.3. A linear equalizer is shown in

figure 3.3. Assuming the filter order is L , the linear operation of the equalization

_f

can be expressed as

ˆ_k ^H _k

x

=

f

⋅

y (3.6)

Chapter 3 Turbo Equalization symbol interval, in which case the FIR equalizer is called a symbol-space equalizer.

In this case the input to the equalizer is the sampled received sequence at a sampling rate equal to 1

T

s . On the other hand, when the time delay τ between adjacent taps is selected such that 1 1

T

s

τ

^> , the channel equalizer is said to have fractionally spaced taps and it is called a fractionally spaced equalizer. The advantages of a fractionally spaced equalizer are that it provides the function of match filtering and it is less sensitive to symbol sampling timing. The disadvantage is the computation load increase.

Figure 3.4 Decision-feedback equalizer

3.2.2.2 Decision-Feedback Equalizer (DFE)

The linear filter equalizers described above are very effective on channels where the ISI is not severe. A decision-feedback equalizer is a nonlinear equalizer that employs previous decisions to eliminate the ISI caused by previously detected symbols on the current symbol to be detected, i.e., to eliminate the post-cursor part of ISI. The DFE consists of two filters as shown in figure 3.4. The first filter is called a feedforward filter and is generally a fractionally spaced FIR filter. This filter is identical in structure to the LE described above. The second filter is a feedback filter.

Chapter 3 Turbo Equalization

It is also implemented as an FIR filter with symbol-spaced taps b. Its input is the set of previously detected symbols. In a simple system where equalizer process one symbol only once, the input of the feedback filter is usually the hard decision of previous equalized symbols. In the turbo system we proposed in chapter 4, the input to the feedback filter are the estimate symbols produced from the decoder in the previous iteration.

Assuming the feedback filter order is

L , and the feedforward filter order is

_b

L ,

_f the output of a DFE can be expressed as

ˆ_k ^H _k ^H _k it has received considerable attention from many researchers due to its improved performance over the linear equalizer and reduced implementation complexity as compared to the optimal trellis-based equalizer we mentioned in 3.2.1. However, due to the feedback of previously detected symbols, a DFE suffers from error propagation. Especially when SNR is low, the previously detected symbols are erroneous, and thus the DFE under this condition may not outperform the LE.

3.2.3 Filter Design Algorithm

Filtering theorem is well-established and has a rich history. Therefore, there are lots of algorithms and criteria to determine the filter coefficients. The zero-forcing criterion and the minimum mean square error criterion are the most famous two.

3.2.3.1 Zero-Forcing (ZF)

Chapter 3 Turbo Equalization

The ideal of zero-forcing equalizer is simple and straightforward. It is to compensate for the channel distortion and ignore all other interference including noise.

The optimal coefficients for an infinite length LE are the samples of the inverse filter of channel. The output of a linear equalizer can be expressed as

[ ] [ ] [ ]

the z-transform of the filter coefficient:

( )

T¹

( )

response (FIR) filter structure. The long division formula shown below requires little computation for small n:

[ ] ( [ ] [ ] )

The problem of zero-forcing criterion is the noise enhancement. From equation

(3.10), if the frequency response of the channel is small or even null at some frequency, then the zero-forcing equalizer compensates it by placing a large gain at that frequency. Consequently, the noise at that frequency is greatly enhanced, too.

3.2.3.2 Minimum Mean Square Error (MMSE)

The MMSE criterion is to minimize the mean-square-error (MSE) between the

Chapter 3 Turbo Equalization

actual equalizer output ˆ

x and the desired value

_k

x , i.e.,

_k

2

MSE= ⎣ ⎦E e⎡ k ⎤ (3.12)

where

e

_k =

x

_k − . To minimize the mean-square-error, we take the gradient of

x

ˆ_k MSE and find its root. Assuming a linear filter is applied, which means

*

The MMSE solution is the solution to the Winer-Hopf equation (3.15), and the filter coefficients are

−1

=

f R p (3.16)

R and p are called the autocorrelation matrix of y and the crosscorrelation

_k matrix between

y and

_k

x respectively. For decision-feedback filters where the

_k output of the equalizer can be expressed as

ˆ

Equation (3.16) can be applied readily with some modifications

Chapter 3 Turbo Equalization

The minimum-mean-square-error criterion is considered to be superior to the zero-forcing criterion. However, it requires the knowledge of the statistics

R and p which are normally unknown. Also, even the statistics are know, the direct

computation of (3.16) is heavy, too.

3.3 Adaptive Equalizer

The Wiener-Hopf solution can be found by a recursive method known as the method of steepest descent. Under the appropriate conditions, the solution obtained by the method of steepest descent will converges to the Wiener solution without the need to invert the correlation matrix of the input vector. However, it still requires the knowledge of the statistics. Using instantaneous estimates of these statistics, we obtain a simple but effective algorithm to approach the Wiener solution. This algorithm is called “least mean square algorithm”.

3.3.1 Least Mean Square Algorithm

Define a cost function ^{J f}

( )

to be the mean square error E e⎡ _k ²⎤

⎣ ⎦ as a function of filter coefficients

f . The ideal of the method of steep decent is to adjust the

coefficient

f in the direction of steepest decent, that is, in a direction opposite to the

gradient vector of the cost function ^{J f}

( )

, which is denoted by ^{∇ fJ}

( )

^.

Accordingly, the steepest decent algorithm is formally described by

(

¹

) ( )

¹

( ( ) )

n

+ =

n

−2

µ

⋅∇

J n

f f f

(3.19)

Where n denotes the iteration,

µ

is a positive constant called the step-size, and the

Chapter 3 Turbo Equalization

Accordingly, (3.19) is now calculated as

(

^{n+ =1}

) ( )

^{n + ⋅ −µ}

( ( )

ⁿ

)

f f p Rf (3.21)

To compute f by using (3.21) still requires the knowledge of p and R . If we discard the actual statistics and use instantaneous estimates of p and R by

( ) ( ) ( )

recursive algorithm known as least mean square (LMS) algorithm

( ) ( ) ( ) ( )

^*

ˆ

n

+ =1 ˆ

n

+

µ n e n

f f y

(3.24)

Figure 3.5 shows the signal-flow graph representation of the LMS algorithm.

( )

∑

Figure 3.5 Signal-flow graph representation of the LMS algorithm

Chapter 3 Turbo Equalization

The stability analysis [3] shows that a necessary condition for the LMS algorithm to converge to Wiener-Hopf solution is

max

0

µ

2

< <

λ

(3.25)

where

λ

_max is the largest eigenvalue of the correlation matrix

R . The convergence

of LMS algorithm is in the sense of convergence in the mean square, i.e., there is always a misadjustment exists. The convergent rate and the value of misadjustment are highly related to the step size. In the rage of (3.25), a larger step size results in a faster convergent rate but a large misadjustment. On the other hand, a smaller step size results in a slower convergent rate but a smaller misadjustment.

3.3.2 Adaptive Decision Feedback Equalization

The adaptation of an equalizer with feedforward and feedback filters can be obtained similarly. The equalizer output can be expressed in matrix form

ˆ

Thus the LMS algorithm for such system is

( )

3.4 Equalization and Decoding

For digital communication, the information bits are protected by channel codes, and the coded bits are distorted by channel and noise. The receiver’s ultimate goal is to recover the information bits. Due to the multi-layer structure, there are different structures for the receiver to achieve the ultimate goal. Figure 3.6 shows the transmitter for such system. The interleaver is placed to boost the coding gain.

Chapter 3 Turbo Equalization

Chapter 3 Turbo Equalization

在文檔中 利用適應性濾波器為設計用於多輸入多輸出之時空柵狀編碼系統之渦輪等化器 (頁 32-0)