ML Combining Method

The ML (Maximum Likelihood) combining method is the optimum solution for

the Destination node from [3], [5] and [7]. First, we assume the symbol transmitted

from the Source node has equal probability. Then the optimum receiver can be

designed by the log-likelihood ratio of the received signal posterior probability. In

[9], for BPSK the log-likelihood ratio can be defined as：

( ( ) )

to represent the received signals directly from the Source and the Relay,

respectively. With these two definitions, the Destination ML decision rule can be

written as

, ,

ˆ arg max

^s _{s d} ^r _{r d}

m = l

^⎛⎜⎝

y

^⎞⎟⎠

+ l

^⎛⎜⎝

y

^⎞⎟⎠ (2.3.3) With the Eq, (2.3.3), we could use the similar concept in [5] and [9] to derive

the ML receiver. Because of the Decode-and-Forward protocol, the relay may make

decision errors. We should reconsider about

^l

^r

( ) ^y

more seriously. First, based on the relay decision errors, the log-likelihood can be shown as：

( ) ( )

And the transition probability of the Relay making decision errors is defined

as： According to these two parameters, the numerator of

^l

^r

( ) ^y

can be rewritten as：

Similarly, the denominator has a similar form as Eq. (2.3.6). Therefore, we can

extend

^l

^r

( ) ^y

, and design the receiver based on this result. Therefore, the ML receiver will contain one set of the weighting gain and follow a non-linear mapping

function block which agrees with the result in [5]. The set of the weighting gains and

the non-linear mapping can be shown as followed：

*

“

ε

_r”, mean the transition probability of the Relay making decision errors.

Based on these two results, the receiver scheme would be shown as：

X

Figure 2.3 The destination ML receiver scheme

Compared to the Figure 2.2, the ML receiver has an extra nonlinear mapping

function for the relay signal. The function f t( ) serves as a limiter which

minimizes the contribution from the Relay when it is unreliable. Therefore, how to

design a receiver at the Destination with comparable performance as the ML

receiver without using any non-linear mapping function is the major challenge in the

cooperative system.

Chapter 3

Maximum SNR Detection for Selection Relay

At the end of the Chapter 2, we say the non-linear function works like a limiter.

Therefore, it will minimize the contribution from the Relay when it is unreliable.

The Figure 3.1 shows the performance comparison between the MRC and ML

receivers and the channel gains h_{s d,} ,h_{s r,} ,h_{r d,} we use are mutually independent

zero-mean, complex Gaussian random variables with variances set to 1. The noises

, , _{s r}, , ,

s d r d

z z z are also mutually independent zero-mean, complex Gaussian random

variables with variances set to 1. Also the PNR means the total system power

( P= +P₁ P₂ ) to noise power ratio in dB. The result is shown below ：

0 5 10 15 20 25 30 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

Relay in the Middle and Equal power

MRC combining ML combining

Figure 3.1 Performance comparisons of MRC and ML combining methods Because of knowing the relay decision error probability, we know that the ML

receiver uses this information to mitigate the error propagation. Therefore, from the

Figure 3.1, the performance has greatly improved by using ML receiver than using

MRC receiver. There are several works which improve the Destination receiver

performance by different ways. We describe [5] and [6] they use in the next sections.

3.1 The Selection Relay Method

In [6], they propose a selection Relay under the Decode-and-Forward protocol

at the cooperative system. The Relay uses the threshold value to evaluate the

received signal reliability. If the received signal power is larger than the threshold

value, then the Relay will decode the received signal and forward the decoded signal

to the Destination. Otherwise, the Relay just suspends the Decode-and-Forward

protocol. The Figure 3.2 shows the algorithm.

No Yes

2 2

ysr

σ ^>ξ

Phase 1

Phase 1 Phase 2

Figure 3.2 Threshold-selection relay system

Under this algorithm, the Destination node would have two types of the

received signal and they could be shown as follows：

When the relay received signal power is smaller than the threshold value, the relay

would stop the Decode-and-Forward protocol. With this condition, the destination

received signal would be：

1

sd sd sd

y = P h x n +

(3.1.1) When the relay received signal power is larger than the threshold value, the relay

stop to function the Decode-and-Forward protocol. Therefore, the destination

1 The Destination uses the MRC combining method to estimate the symbol

transmitted by the Source. Under this algorithm, the simulation results show

improved performance than using the traditional cooperative system. Also, the

performance is getting better and better when the threshold value increases. Figure

3.3 reflects the conclusions they made.

0 5 10 15 20 25 30

Relay in Middle & Equal power

PNR in dB

BER

ML receiver

Selection Relay with threshold = 3 Selection Relay with threshold = 1 MRC receiver

Figure 3.3 Improvement of selection relaying

The Performance analysis is provided for complex system with the BPSK

signal. The reason that the small threshold value performance is worse than the large

threshold value is the small threshold value will cause the Relay tends to forward the

most of its received information which in turn increases the chance that incorrect

symbol is sent to the Destination node. Therefore, the threshold-selection Relay

improves the traditional MRC performance and is getting close to the ML

performance when the threshold value increases.

But the performance will be bounded when the threshold value and power ratio

is large enough. They discuss about this phenomenon and also choose a case that the

location of the Relay is close to the Destination. And the simulation result is shown

at the Figure 3.4 and Figure 3.5.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10^-4 10^-3 10^-2

Relay close to Destination

Threshold value

BER

Threshold-selection Relay

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10^-4

10^-3 10^-2 10^-1 10⁰

Relay close to Destinaiton and threshold value =3

Power Ratio

BER

Threshold-selection Relay

Figure 3.5 Boundary of the power ratio

As we can see, in the Figure 3.4 the performance is bounded when the

threshold value is large than 3. Also in the Figure 3.5, the performance is bounded

when the power ratio is large than 0.85. Therefore, when the Relay is close to

Destination, they claim the optimum threshold value and power ratio are 3 and 0.85.

3.2 Maximum SNR Detection

There is another way to improve the traditional MRC performance. In [5], the

Relay always executes the Decode-and-Forward protocol without any selection

function. And they focus on the receiver at the Destination node based on the rule

which maximizes the SNR of the slicer input. Therefore, the Destination receiver

structure change into a set of MRC weighting gain and a gain function block for the

received signal y_{r d,} . Figure 3.6 shows the structure.

Figure 3.6 Maximum SNR detection structure

For obtaining the Gain

α

, they first examine the Relay decision

ˆx

^{. The}

Relay decision

ˆx

can be written as：

ˆ

_s

x = + x e

(3.2.1) In Eq. (3.2.1), e is a random variable capturing the effects of decision errors.

Also, if the symbol x_s is using the equal probability BPSK signal

andx_s∈

{

x x1, ₋1

}

.

Therefore, we can yield the random variable properties by using some

calculations.

( )

And from the Eq. (3.2.2), the variance of the random variable is

( )

²

2

1 1

e

p * x x

σ = −

₋ (3.2.3) According to these properties, the new signal constellation of the Relay

decoded symbol can be written as：

1 1 1 According to this new constellation, the

ˆx

transmitted by the Relay can be

rewritten as

x ˆ = + x

_s

e

, where From Eq. (3.2.6), we can get the weighting gain for this received signal, i.e.

( )

We can extract the traditional MRC weighting gain and the gain function from

the Eq. (3.2.7). The result is shown below：

( )

According to the Eq. (3.2.8) and the traditional MRC weighting gain w_{s d,} , the

Gain structure improves the performance of the traditional MRC scheme. The

simulation result is shown in Figure 3.7.

0 5 10 15 20 25 30

Relay in Middle and Equal power

Maximum SNR detector MRC receiver

ML receiver

Figure 3.7 Improvement of maximum SNR detector

This method can improve the Destination receiver performance. But around the

high PNR area, the new method seems to be worse than the ML method by about 3

or 4 dB.

3.3 The Proposed Architecture

According to these two methods in sections 3.1 and 3.2, they have their own

advantage. The threshold-selection Relay only needs a threshold value to evaluate

the received signal good enough and the maximum SNR detection method use a

gain

α

to adjust the influence of the Relay decision errors. The disadvantage for first method is the smaller threshold value, the worse performance. Therefore, we

could combine these two methods to combat the disadvantage in the first method.

Figure 3.8 shows the combined scheme.

No Yes

2 2

ysr

σ >ξ

Phase 1

Phase 1 Phase 2

Figure 3.8 Proposed architecture

×

Figure 3.9 max-SNR MRC with correction weighting

From the Figure 3.8 and 3.9, we can use the advantage of second method to

mitigate the small threshold phenomenon in first method. Table 3.1 shows the

comparison between the proposed system and previous works.

Functionality

Threshold value at the Relay Gain factor at Destination receiver

Ref. [5]

Table 3.1 Differences between three methods

3.4 Two-Threshold Values in Selection Relay

We also offer another method to improve the selection-relay performance. The

concept of this method is using the two different threshold values to measure the

received signal’s quality. There are three working status in the relay node. When the

don’t work. When the relay received signal power is larger than large threshold value,thr_high, the relay would decode and forward the signal. Otherwise, the relay

would amplify and forward the received signal. For the relay works in the

amplify-and-forward protocol, we use the amplified factor in [5]. To satisfy the

output power constraint, the relay amplifier can operate at a maximum gain

satisfying

The corresponding destination weighting can be implemented as

*

With these equations, we could easily use the MRC combiner as the destination

receiver. For the relay works in the decode-and-forward protocol, we use the same

method in section 3.3.

Chapter 4

Theoretical BER of The Max-SNR Selection-Relay Architecture

In this chapter, we derive the theoretical BER of the max-SNR selection-relay

scheme in detail. Also, we find the approximation for the theoretical BER to obtain

the diversity order.

4.1 Theoretical BER Analysis

Before we derive the theoretical BER, we should make some assumptions first.

The signal model of the cooperative system is just the same as the Chapter 2. We

assume the channel gains h_{s d,} ,h_{r d,} ,h_{s r,} are mutually independent complex

the noises are also mutually independent complex Gaussian with zero-mean and

variances equal to

σ

². The signal we use is the BPSK symbol withx_s∈

{

x x1, ₋1

}

. With these assumptions, we can derive the theoretical BER.

4.1.1 Classification of The Proposed System

Scenarios

From the Figure3.9, the working status of the Relay depends on the Relay

received signal power which is transmitted from the Source. Therefore, there are two

scenarios of the received signals at the Destination in the proposed system. The

scenario 1 means the Relay received signal power lower than the threshold value.

The scenario 2 is the opposite side, i.e. the Relay received signal power is larger

than the threshold value. In the scenario 1 the Destination only has one signal which

is transmitted from the Source. In the scenario 2, the Destination will have two

signals and these two signals come from different ways. One is from the Source. The

other one is from the Relay.

Hence, in the Scenario 1 the Destination receiver can be regarded as Figure 4.1.

*

Figure 4.1 Scenario 1 destination receiver

In Scenario 2, the Destination receiver can be regarded as Figure 3.9. It is very

important to know the Destination receiver type while we derive the theoretical BER.

With these two receiver type, we will know the proposed system theory BER which

depends on the received signal power at the Relay. And the theoretical BER would

be shown as follows：

( )

₁ power is small than threshold value and

|1

Pe_φ denotes the system BER when the relay received signal power is small.

( )

1 and |1

P

φ

Pe_φ mean the occurrence and the system BER of the opposite scenario.

4.1.2 Occurrence probability of two Scenarios

We use the similar derivation in [6] and [8] to obtain the theoretical BER. First,

know the Relay received signal model. Because

h

_{s r,} and

n

_{s r,} are the complex

Gaussian random variables with zero-mean and different variances. The received

signal

y

_{s r,} is also the complex Gaussian random variable with zero-mean and variance equals to P₁

σ

_{s r}²_, +

σ

². Therefore, the received signal power y_{s r,} ²is an exponential distribution random variable with ₁

2 2

1 ,

1 P

s r

λ ⁼ σ + σ

. With this property,

we can easily obtain the occurrence probability of the low-SNR scenario 1

( )

The occurrence probability of the high-SNR scenario 2 is

( ) ^{( )}

4.1.3 Error probability of two Scenarios

First we can use the Figure 4.2 to analyze the error probability of the scenario

1.

*

Figure 4.2 Scenario 1 destination receiver

From Eq. (2.1.2), we could know the destination received signal model. If the

symbol

x = 1

and the channel gain h_{s d,} is known, the properties of

y

₁ can be obtained easily. The detailed derivation is shown as follows：

1 , , and variance equal to

2 2

2 According to these two statistical properties, we can know the probability

density function of the

y

₁.

Figure 4.3 Illustration ofy ’s PDF ₁

From Figure 4.3, we can realize the error probability of y is the painted area if

the symbol

x

equals to 1. Hence, the conditional error probability is shown below.

1 the channel effect：

1

Before we start to derive the error probability of the scenario 2, we should analyze the component of

|1

Figure 4.4 Scenario 2 destination receiver scheme The received signals are rewritten from Eq. (2.1.2) and (2.1.3) here：

, 1 , ,

( )

where

p

denotes the relay BER. Based on the Eq. (4.1.11), we realize that we

need two forms to analyze the

y

₂. That is,

y

₂ is rewritten as： a correct decision, and the Eq. (4.1.13) will accompany with a probability

p

which

the Relay make a wrong decision. Besides, the gain factor

α

in [5] is defined as With these three equations in (4.1.11), (4.1.12) and (4.1.13) we know the error

probability of the scenario 2 will depend on the error probability of the Relay. Hence,

the error probability of the scenario 2 could be shown as follows：

1

( )

1 2

|

1

_{b b}

P

e_φ

= − p P + pP

(4.1.15)

From Eq. (4.1.15), we should know the error probability of the Relay first

before we derive the error probability of the scenario 2.

To derive the error probability

p

of the Relay, we can use Figure 4.5 to

analyze.

Figure 4.5 Relay receiver scheme

First, we could start to analyze

y

_{s r,} from Eq. (2.1.1). Given the channel gain

with zero mean and variance equals to

2 2

,

2 hs r

σ

. Therefore, we can know the

2 Based on these properties, the probability density function of

y

₃ is

λ=0 ²

The conditional error probability can be derived by calculating the painted area.

That is,

Then we average the channel gain effect to obtain the error probability of the

Relay .

When we average the channel gain effect, we should also notice that the received

signal power should be larger than the threshold value. Therefore, the lower limit of

the integration in the Eq. (4.1.20) should meet this criterion. From the statistical

viewpoint, the received power could be calculated as 2 2 With Eq. (4.1.22), the lower limit of the integral in Eq. (4.1.20) would be

limit Therefore, the error probability of the Relay is shown as follows：

when

ξ

≤1,

After we obtain the error probability of the Relay, we derive the error

Gaussian random variables with zero mean and variances equal to

2 2

individually.

Therefore,

y

₂’s statistical parameters are

( )

Based on these properties, we can know the probability density function and

obtain the error probability

P

_b1.

λ=0 μ^2,correct

We can obtain the conditional error probability by calculating the painted area.

, , , 0

From Eq. (4.1.30), we use the MatLab to calculate the double integration of the

expectation.

We can use the same method to derive the error probability

P

_b2. From Eq.

(4.1.18), it is obviously known that the difference between Eq. (4.1.12) & (4.1.13) is

the amplitude of the symbol. Hence, from Eq. (4.1.13) the mean of

y

₂’s statistical

2, 2,

From Eq, (4.1.31) and (4.1.32), we would know the variance is the same as

2 2,correct

σ

but the mean changes. With these two results, the probability density function can be shown as follows

λ=0 μ^2,wrong

Figure 4.8 Illustration ofy_2,wrong’s PDF

We can derive the error probability by these properties and Eq. (4.1.29). We can

obtain

Same as Eq. (4.1.30), we use the MatLab to calculate the expectation in Eq, (4.1.33).

With equations (4.1.2), (4.1.3), (4.1.8), (4.1.24), (4.1.25), (4.1.30), (4.1.33),

equation (4.1.1) would be shown as

( ) ( ) ( )

From the above theoretical error probability, we could know when

α

equals to 1 the theoretical error probability would meet the traditional MRC result in [6].

4.2 High SNR Approximation Analysis

In this section, we try to approximate the theoretical BER. Therefore, we could

get the diversity order from the simplified theoretical BER.

4.2.1 The Approximated Theoretical BER

We use the high SNR condition to obtain the diversity order of proposed

scheme. Before we start to derive, we should know the meaning of the diversity

order. In [8], the definition of the diversity order is the negative exponent of the

( )

^d

e s

P ∝ γ

⁻G

where γ

_s

→∞

(4.2.1) With this definition, we parameterize on the three SNRs first, i.e.

1 _s

,

2 _s

,

_sr 3 _s

sd

c

rd

c c

γ = γ γ = γ γ = γ

where

γ

_s denotes the average SNR. We also assume the Relay is close to the Source to achieve the high SNR status. With this assumption, we would know the relay

BER

p

is almost zero and the correction weighting

α

in Eq. (4.1.14) is close to 1. With the above assumption, we could approximate equation (4.1.1) namely the

theoretical BER of the proposed system. That is,

( )

₁

Therefore, the occurrence probability of the scenario 1 could be simplified by

using Taylor Expansion. That is,

( )

With this property, Eq. (4.1.3) could be shown as follows：

3

( ) ( )

¹

¹

^{3 s}

P c

φ ξ

≈ γ

+

. (4.2.4) With Eq. (4.2.4), we could know the occurrence probability of the scenario 2. That

is,

According to the Eq. (4.1.12), the error probability of the scenario 1 can be

simplified by using the similar method. That is,

1

( )

Now we can approximate the error probability of the scenario 2 from the Eq.

(4.2.1), i.e. _{1 2}

| b b

P

e_φ

≈ P + pP

. With the assumption of the negligible relay BER

BER

P

b1^and

P

_b2. The reason why we make this assumption is we hope the conditional error probability

P

_b^h₁^{s d, ,hr d, and}

P

_b^h₂^{s d, ,hr d,} which have a simplified form like

Q ( x )

^{, where}

x

is an integer. Based on these assumptions, we can start to simplify

1

( )

¹³²

(

^{1 2}

)

²³²

We can simplfy 2

The error probability of the Relay could be simplified as follows. From the

results of Eq. (4.1.24) and (4.1.25), we know the relay BER

p

has two types.

Therefore, we could use the similar method to simplified the Eq. (4.1.24) and it can

be shown：

From Eq. (4.1.25), we could simplify the equation by using Taylor expansion

we could obtain the simplified form of the Eq. (4.1.25)：

when

ξ

>1and

γ

_s 1,

With these approximated equations (4.2.4), (4.2.5), (4.2.6), (4.2.7), (4.2.8),

(4.2.9), (4.2.10), we can substitute into equation (4.2.2) to obtain the final form of

the approximated theoretical BER.

( )

( ) ( ) ^{( )}

From Eq. (4.2.11) and Eq. (4.2.1), we could know the diversity order of first

term is 2 but we can’t know the exactly diversity order in the second term. Therefore

we need make an approximation again to obtain the diversity order of the second

term.

4.2.2 Derivation of Proposed System Diversity Order

For the purpose of knowing the diversity order of the proposed system, we need

to approximate the theoretical BER in Eq. (4.2.2) much further. We use the similar

method in [8] to approximate the second term of the simplified theoretical BER.

For

P

_b1 , we could use the Chernoff bound to obtain another form of the

where

f

X

( ) x

denotes the Gaussian probability density function,

^{u x} ( )

^{means the} There’s a minimum value in Eq. (4.2.12) by differentiating with respect to t and

occurs when t equals the destination received SNR. With this result, Eq. (4.2.12)

would be simplified into

1 After we average the channel effect, the Eq. (4.2.13) would be

( ) ( )

For the error probability of the Relay, we use the Q-function property to obtain

the approximation. The property is

( ) ¹

²²

¹ ₂

²²

With this inequality (4.2.15), the conditional approximated error probability is

2 1

Hence, we average the channel effect of Eq. (4.2.16), we could obtain

3 1 (4.2.17), we know the diversity order of the Relay error probability is 1.

For

P

_b2, we also use the Q-function inequality in (4.2.15) to obtain the upper

bound. Therefore,

P

_b^h₂ would be

From the inequality (4.2.18), we could obtain the

P

_b2 by averaging the channel

( )

From the inequality in (4.2.19), we could use the long division method to

obtain another form in the exponential part, i.e.

obtain another form in the exponential part, i.e.

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