Two-Threshold Values in Selection Relay

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.

( )

Therefore, the inequality would be

( ) ( )

decreasing property of the exponential term, we make an approximation and change

4 _{sd rd}

integration,

f

A

( ) γ

rd would be shown as：

We could use the Q function property,

( ) ¹ ( )

From the inequality in (4.2.20), we could obtain the upper bound of the

P

_b2

1 1 exponent order 2 and 1/2. Due to a sum is dominated by the term with the lowest

diversity exponent, the diversity of

P

_b2 is 1/2.

With the results of Eq. (4.2.14), (4.2.17), (4.2.21) and (4.2.22), the

approximated BER would be

( )

From the above approximated BER, we could find out the diversity of the proposed

system under the high SNR condition would be

1.5 ≤ G

_d

≤ 2

(4.2.26) This result is consistent with that of the diversity in [7]. Based on [7], the

diversity of the uncoded Relay under Decode-and-Forward protocol would be 1.5~2

in the cooperative system with only one Relay to retransmit the Source signal.

Chapter 5

Computer Simulations

In this chapter, we show the performance of the proposed scheme under

different Relay locations. Here, our channel gains are modeled as the mutually

independent, complex Gaussian random variables, ^CN

(

^0,

^σ

^{i j2,}

)

^{, where i j}

^,

indicate the Source (s), Relay (r) and Destination (d). There are three Relay locations

which are “Relay in Middle”, “Relay close to Source”, “Relay close to Destination”.

We adjust the

σ

_{i j}²_, value to represent the different Relay locations. Here, we set the

2 ,

σ

s r as 10 and

σ

_{s d}²_,

= σ

_{r d}²_, as 1 to realize the Relay close to Source condition.

Also we set the

σ

_{r d}²_, as 10 and

σ

_{s r}²_,

= σ

_{s d}²_, as 1 to realize the Relay close to Destination condition. For the Relay in Middle condition, we set all the channel gain

variances are equal to 1. The noise power is also modeled as the mutually

independent complex Gaussian random variable. But the variance we use is equal to

1 for all receiver nodes. The power for the Phase 1 and Phase 2 is allocated by the

parameterr, i.e. P₁

= ∗

r P and

P

2

= − ( 1 r P )

. Here, the powers we use in the Phase 1 and Phase 2 are equal. The background of cooperative system is using

threshold-selection relay scheme. We use 4 threshold values which is

0.5, 1, 3 and 4

ξ

= to see the performance variation and the simulated results would be shown into three groups. We use Eq. (4.1.39) to verify the proposed

scheme simulated results. We also provide the simulations to find out the influence

of the relay threshold value on the system BER.

5.1 Under Relay in Middle Status

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.2 Threshold =1 with relay in middle

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.3 Threshold =3 with relay in middle

0 5 10 15 20 25 30 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

macx-SNR scheme MRC receiver

Theoretical BER of proposed scheme ML receiver

Figure 5.4 Threshold =4 with relay in middle

As we can see from the Figure (5.1) ~ Figure (5.4), the performance of the

proposed system could improve the performances which is close to the ML

receiver’s result under small threshold value area and the simulated results are

matched to the theoretical BER which verifies the proposed scheme. But the

performance will be same as the Selection-Relay for the large threshold value. Also

from the simulated results, we could find out the diversity order of max-SNR

detection for selection relaying. Because of the diversity order of ML receiver and

MRC receiver are 2 and 1.5 which proved in [7]. Also the simulated results lie

order of proposed scheme matches to the conclusion we made. For the threshold

value’s influence, we set the PNR to 20 dB which means after receiving signals the

destination would have 10 dB powers from the source and relay, respectively. The

computer simulation is shown as follows

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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

Threshold value

BER

ML receiver MRC receiver max-SNR scheme

Figure 5.5 Influence of threshold value with relay in middle

Because the threshold value acts like a supervisor and it would increase the

accuracy of transmitting the original symbol. With this reason, from Figure 5.5, we

could know the performances of three receivers would merge during the large

threshold utilization. But when the threshold value turns to the large value, the

utilization of the relay-destination channel would drop. With this reason, the

performances of three receivers would degrade when the threshold value increase.

Besides, when the threshold value turns to the small one, the MRC receiver’s

performance would degrade enormously due to error propagation. Hence, with the

max-SNR scheme, we could reduce the threshold value for utilizing the

relay-destination channel frequently and improve the MRC performance.

5.2 Under Relay Close to Source Status

0 5 10 15 20 25 30

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

max-SNR scheme MRC receiver

Theory BER with threshold value =0.5 ML receiver

Figure 5.6 Threshold =0.5 with relay close to source

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.7 Threshold =1 with relay close to source

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.8 Threshold =3 with relay close to source

0 5 10 15 20 25 30 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

max-SNR receiver MRC receiver

Theoretical BER of proposed scheme ML receiver

Figure 5.9 Threshold =4 with relay close to source

From Figure (5.6) ~ Figure (5.9), we could know the performances of three

receivers have similar results under relay in middle status. But the performance of

the proposed system which is so closed to the ML combining is that the decoded

signals transmitted by the Relay are the original signals due to Relay close to Source.

For the threshold value’s influence, we use the same parameter in the relay in middle

status. The simulation result of threshold value’s influence is shown as follows.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10^-5

10^-4 10^-3

Threshold value

BER

MRC receiver max-SNR scheme ML receiver

Figure 5.10 Influence of threshold value with relay close to source

Like the result in the relay in middle status, the performances of three receivers

would merge during large threshold value utilization. But due to the condition, the

utilization of the relay-destination channel would raise and the decoded signal would

be a copy of the original symbol of the source. Hence, the destination would collect

two useful data for recovering the original symbol.

5.3 Under Relay Close to Destination Status

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.11 Threshold =0.5 with relay close to destination

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.13 Threshold =3 with relay close to destination

0 5 10 15 20 25 30

Theoretical BER of proposed scheme ML receiver

Figure 5.14 Threshold =4 with relay close to destination

From Figure (5.11) ~ Figure (5.14), the performance of the proposed system

also has improved at the small threshold value area. But due to the condition that the

Relay is close to Destination, the proposed system would need more 3 dB to meet

the performance of ML combining during small threshold value utilization. Same

parameter used in the sections 5.1 and 5.2, the simulation of the threshold value’s

influence is shown at below：

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10^-5 10^-4 10^-3 10^-2

Threshold value

BER

MRC receiver max-SNR scheme ML receiver

Figure 5.15 Influence of threshold value with relay close to destination Like the result mentioned in the sections 5.1 and 5.2, the performances of three

receivers would merge during large threshold value utilization. But due to the relay

close to destination status, the BER of the relay would raise up. The destination

device.

5.4 High SNR Approximation Simulations

We also could use equation (4.2.2) to verify our simulation results. Due to the

assumption of high SNR status in section 4.2, we use the relay close to source status

as our verifications. The results are shown as follows.

0 5 10 15 20 25 30

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

max-SNR scheme Approximated BER

Figure 5.16 Threshold =0.5 for high SNR approximation

0 5 10 15 20 25 30

Figure 5.17 Threshold =1 for high SNR approximation

0 5 10 15 20 25 30

0 5 10 15 20 25 30 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

PNR in dB

BER

max-SNR scheme Approximated BER

Figure 5.19 Threshold =4 for high SNR approximation

From Figure (5.16) ~ Figure (5.19), the approximated theory BER would

approximate the simulation results during High SNR assumption.

5.5 Under 2 Threshold Values in Selection Relay

Under this concept, we make the large threshold value to 1 and small threshold

value to 0.5. With these parameters, the simulated result would be shown as follow：

0 5 10 15 20 25 30 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Relay in Middle

PNR in dB

BER

One threshold value 2 threshold values

Figure 5.20 Threshold =1 with combo scheme

From Figure 5.20 and under threshold value equals to 1, we would know the

two threshold value in selection relay improves the traditional MRC receiver in the

selection relay cooperative system. With two stages comparison, we could make

sure the quality of signal transmitted by the relay.

Chapter 6 Conclusions

In this thesis, we propose a threshold-selection relay with maximum SNR

In this thesis, we propose a threshold-selection relay with maximum SNR

在文檔中 具中繼選擇合作式通信系統之極大訊雜比檢測機制 (頁 33-0)