One-Bit Pre-Equalization with Unstable Channel Coefficients

dk

2.3.2.2 One Bit Pre-Equalization with Unstable Channel Coefficients In the previous section, we mention that if the channel is unstable, could diverge toward infinity. Under this condition, we develop a solution to deal with the problem. Let

be designed to have the same polarity as and its magnitude equal to or exceed a specific threshold so as to combat the channel noise. Then, the following inequality holds for :

where ξ is the threshold and is a positive number. From (2.20), we have

ξ

α₀d_k^'d_k +Ψ_kd_k ≥ . (2.30)

If Ψ_kd_k ≥ξ, which means that the magnitude of Ψ_kd_k is larger than ξ. In such a

condition, the best choice is to save transmission power. Namely, the system is in the idle state without sending any signal and

' =0 due to different multipath conditions and polarities of data.

rk

Table 2.1 Summary of possible r_k

If we focus on the theme of saving transmitted power, from (2.30) there is another variable we could consider, i.e. the threshold ξ. Eq. (2.30) could be rewritten as

' ''

0 ξ

α d_kd_k +Ψ_kd_k ≥ , (2.32)

If , that means ISI caused by multipath has different polarity as the input data.

Therefore, the modified data would need addition amplitude to counteract the ISI for meeting the threshold. Under this conditon, we could lower the threshold to save power.

<0 Ψ_kd_k

''

dk

The equality holds if d_k^'' is chosen as

Table 2.1 could be renewed to summarize the six possible as shown in Table 2.2.

From Table 2.2, the first two conditions have the same property. Because ISI induced by multipath has the same polarity as the input data, it can be used directly as the transmitted signal. However, the last two conditions are contrary as the first two.

rk

Table 2.2 Summary of possible modified r_k

2.3.2.3 Two-Bits Pre-Equalization with Unstable Channel Coefficients As described in the previous section, we consider the effect of multipath imposed on the upcoming data. We utilize it to help transmission and to save transmission power. But there is another information we did not think of. At TX, we could not only obtain the multipath component, but also have the information of the upcoming data. In this section, we would take this advantage and exploit what we can do to improve system performance.

Under the condition that the channel is line-of-sight (LOS) and normalized, the maximum channel coefficient except the first one is found out first. Similar to (2.18)-(2.20), at some time instant, the anticipated received signal can be represented as

R[n]=r_k ⋅

δ

[n−k], (2.34)

k

where α_n is the maximum channel coefficient except the first one. From (2.35) and (2.36), it shows that the modified data ' would be the dominate term on determining

. From (2.36), can be written as

Therefore, from (2.37)-(2.39), it can be shown that would affect mostly, ' would seriously affect and so on. At last, ' would affect

mostly. We take 2n bits as a segment to find out the new management rule. It can be found that the former n bits could be considered as the effect terms; while the latter n bits are affected terms. Figure 2.7 illustrates the data sequences being partitioned into many segments. In each segment, ISI introduced by the effect terms would be considered if it is constructive or destructive to the affected terms in advance.

1'

+

dk d_k+n+1'

+2

dk d_k+n+2' d_k+n−1 d_{k+ n2 −1}'

L L

Fig. 2.7 Segmented data sequences

Apply the same criterion as shown in (2.30), (2.35) and (2.36) could be rewritten as

ξ

While determining , we should considered not only the interference caused by previous data, but also the interference induced by to the next n-th data. Besides the decision rule shown in Table 2.2, there is another condition that the threshold

k' destructive, the threshold in (2.40) will be lowered for saving transmission power and decreasing the ISI distortion. Therefore, Table 2.2 should be modified. Table 2.3 shows the possible under four conditions.

dk

rk

=1

dk d_k =−1

ξ

≥

Ψ_kd_k r_k =Ψ_k ≥ξ r_k =Ψ_k <−ξ

ξ

<

Ψ

≤ _kd_k

0 r_k =ξ r_k =−ξ

<0

Ψ_kd_k r_k =ξ^' <ξ r_k =−ξ^' >ξ ξ

<

Ψ

≤ _kd_k

0 and

0 ) ( <

+n n k

k d

d α

ξ ξ <

= ^'

rk r_k =−ξ^' >ξ

Table 2.3 Possible r_k under two-bit consideration

Chapter 3 Performance Analysis

3.1 System Performance

MP channel

AWGN

] [n h

detector

ni

dk d_k

'

rk

decision

Ψk

Fig. 3.1 Transmission model

Figure 3.1 illustrates the transmission model we will use in the following simulation, where is the transmitted data, is the modified data signal and is the received signal. In this model, the receiver is very simple, only the detector and the decision function are needed. We let

dk d_k^' r_k

=1

dk so that the average power of data is 1. In order to compare the performance under different channel conditions, we adopt three channel coefficients as shown in Figs. (3.2)-(3.4)

Figure 3.2 Channel impulse response A

Figure 3.3 Channel impulse response B

Figure 3.4 Channel impulse response C

In these fading channels, it can be shown that the number of multipath is channel A <

channel B < channel C. Fig. 3.5 provides the BER corresponding to three channels with different received SNR under ideal transmission condition. In ideal transmission condition, the multipath coefficients are assumed to be measured without noise, so the corresponding Ψ_k can be accurately obtained.

Figure 3.5 Performance of proposed system

In Fig. 3.5, the thick solid line is the BER of the bipolar data passing through a multipath-free channel, being interfered only by the additive Gaussian noise. The circle line corresponds to direct transmission without any ISI management, where performance is quite poor compared to the multipath-free channel. It is shown that the ISI significantly degrades system performance. Both conditions will be used as the reference for comparison. In order to compare with the case of multipath-free transmission, we define the received SNR as

) log(

10

0 BW

N

SNR_r P^r

= ⋅ , (3.1)

where P_r is the average power of r_k, BW is the bandwidth and is the noise power spectral density.

N0

In Fig. 3.5, it is shown that the BER performance of channel-A shown in Fig. 3.2 is the best. It approaches the case of multipath-free transmission. From Table 2.1, it implies that the magnitude of r_k will be equal to the specified threshold without the idle state.

While ξ =1, there are 97.52% of equal to 1 in the simulation. If the idle state occurs, the induced-ISI directly serves as , the magnitude will be larger than the threshold.

Moreover, is 1.008512 which is a little larger than 1. We record the related percentage of idle state and the received power of the simulated channel in Table 3.1.

rk

rk

Pr

channel idle state(%)

average received power

Channel A 2.48 1.008512

Channel B 16.21 1.266794

Channel C 22.49 1.677684

Table 3.1 Idle state and received power in fig. 3.5

We find that the more multiapth components, the larger percentage of the idle state as well as the average received power. Therefore, the BER for a fixed SNR is channel A <

channel B < channel C.

3.2 Analysis of System Characteristics

In the previous section, we assumed that simulation is under ideal transmission. Also, in Fig. 3.2 - 3.4, mulipath coefficients mostly appear within t=30ns. In reality, the proposed system will not be able to achieve the multipath-free transmission because the system complexity would limit the number of taps in the MP. Hence, we can only obtain the essential portion of the channel multipath coefficients instead of getting all of them.

Thus some tap length of MP would be sufficient to simplify the transmitter complexity.

We perform computer simulation for three different channels to find sufficient tap numbers. In each simulation, the threshold is equal to 1.

Figure 3.6 BER vs. SNR_r for different MP tap lengths under channel A

Fig. 3.6 depicts the BER according to different tap length of the modified MP for channel A. For L=10, the tap number is insufficient, since the performance significantly deviated from the case of multipath-free transmission. This is because there are still many multipath coefficients between L=10 and L=20. However, while L>20, as sufficient taps are included, the performance approaches to the multipath-free transmission.

Figure 3.7 BER vs. SNRr for different MP tap lengths under channel B

In the case of channel B as depicted in Fig. 3.7, while L=10, the BER performs badly. It is because the tap number is not sufficient so that ISI caused by multipath can not be eliminated efficiently. When L>20, the performance is better as L becomes large. From Fig. 3.3, it can be shown that most multipath coefficients occur before t<30ns. Hence, while L=30, the BER performance is the almost the same as L=50.

Figure 3.8 BER vs. SNR_r for different MP tap lengths under channel C

Because the multipath coefficients of channel C occur before t<30ns, Fig. 3.8 shows similarly result as Fig. 3.7. But the number of multipath components shown in channel B is less than that in channel C, the BER performance of Fig. 3.8 is thus worse than Fig.

3.7.

3.3 Adjustment of Threshold

After clarifying the system capability and characteristics, we proceed to discuss the properties of average transmitted and received power which are the average power of and , respectively. In the following simulation, we let

'

dk

rk r_k =±1 and change the

threshold ξ discussed in table 2.1. We illustrate the average power of d_k^' in Fig. 3.9.

Figure 3.9 The simulated average transmitted power under these three channels

Fig. 3.9 indicates the tendency that the average transmitted power increases as the threshold getting larger. However, if a channel has few multipath components like channel A, the change of average transmitted power is little. It is because the estimated ISI is not big enough to exceed the threshold. However, if a channel has a large number of multipath components like channel B or channel C, the change of the average transmitted power is significant. In these cases, the estimated ISI would be much bigger.

While the threshold is increased, it means that the transmitter is willing to spend more transmitted power to let equal to 1. Therefore, it is expected that BER performance would become better as the threshold getting larger. Table 3.2 records the percentage of idle state.

rk

threshold

idle state (%) (channel A)

idle state (%) (channel B)

idle state (%) (channel C)

0.2 37.15 41.97 43.83

0.4 24.89 34.61 37.98

0.6 14.05 27.75 32.39

0.8 6.57 21.52 27.15

1 2.48 16.22 22.47

2 0 3.29 8.59

3 0 0.5 3.91

4 0 0.038 2.12

5 0 0.001 1.29

6 0 0 0.81

Table 3.2 The simulated percentage of idle state

Table 3.2 demonstrates that the percentage of idle state decreases as the threshold increases. It is because if the threshold is large, the estimated ISI has less chance to satisfy d_kΨ_k≥ξ.

Figs. 3.9, 3.10 and 3.11 provide the BER performance of the cases with the threshold bigger and smaller than 1 for channel B, respectively.

In Fig 3.10, the BER becomes smaller with higher ξ, but more transmission power is needed. According to Table 3.2, if the threshold increases, the percentage of idle state decreases. From Table 2.1, if the percentage of idle state decreases, the probability of also decreases. This means more transmitted power is necessary to let the received power equal 1. Hence, the average transmitted power will increase if the

0 '= dk

threshold increases. According to (3.1), under the same SNR, if is small, will be small, too. Therefore, decision of received data would be affected less by noise as the threshold increases. So BER should be improved while

pr N₀

ξ is getting larger. Intuitively, we could adjust the threshold to transmit acceptable power to achieve a better BER.

Table 3.3 records the related average transmitted power.

Fig 3.10 BER vs. received SNR for judged threshold larger than 1

threshold

average transmitted power 1 1.9682 2 2.0977 3 2.3528 4 2.4961 5 2.5321 6 2.5336

Table 3.3 The simulated average transmitted power of judged threshold larger than 1

In Fig. 3.11, the BER becomes worse with smaller ξ , but transmitted power decreases accordingly. From Table 3.2, while the threshold decreases, the occurrence of idle states will increase. Hence, the percentage of r_k =Ψ_k and will increase, too. Since

0 '= dk

ξ is small, Ψ_k is small as well, and r_k =Ψ_k would be sensitive to the noise. Therefore, BER would be worse while the threshold is low. However, although the percentage of d_k'=0 increases, the average transmitted power is slightly decreased.

From Table 3.2, the occurrence of idle state increases while the threshold deceases, it also means that the percentage of d_k'=0 increases. This result should lower the average transmitted power. However, if ISI induced by ' before lowering the threshold is constructive to some upcoming data , the modified data

dk

n

dk₊ d_k'=0 after lowering the threshold would lead transmit more power to compensate the ISI induced by . Therefore, because of the offset between these two results, the amount of the saved transmitted power is slightly. Table 3.4 records the simulated average transmitted power for different thresholds.

n

dk₊ d_k'

Fig. 3.11BER vs. received SNR for judged threshold smaller than 1

threshold

average transmitted power

1 1.969

0.8 1.971 0.6 1.973 0.4 1.957 0.2 1.892

Table 3.4 The simulated average transmitted power of judged threshold smaller than 1

3.4 Power Saving for One-Bit System

In the previous section, transmitted power decreases while lowering the threshold.

However, BER would be much worse. If we do not adjust the threshold while ISI and the current data have different polarity, the transmitter should spend more power to let the received amplitude equal to the threshold. Therefore, if ISI is destructive to the current data, we could lower the threshold to save power. That is, we would allow the received amplitude under the threshold. In this simulation, we let ξ =1 and adjust 'ξ in Table 2.2 to see the change of BER and the average transmitted power. Fig 3.12 represents the BER according to different thresholds. Table 3.5 records the related average transmitted power.

Fig. 3.12BER vs. received SNR for modified threshold smaller than 1

threshold

average transmitted power 1 1.9679 0.9 1.6422 0.8 1.3571 0.7 1.1102 0.6 0.9029 0.5 0.7326

Table 3.5 The simulated average transmitted power of modified threshold smaller than 1

In Fig. 3.12 and Table 3.5, it can be shown that BER would be worse as the threshold getting smaller, and the related average transmitted power would decrease. If the threshold decreases, from Table 2.2, the amplitude of received data r_k =ξ' will be reduced. Therefore, the received data would be more sensitive to noise. Hence, BER

would get worse as the threshold decreases. On the other hand, the average transmitted power decreases obviously as the threshold decreases. That is because we only lower the transmitted power while ISI is destructive to the upcoming data, the average transmitted power would decrease while the threshold decreases.

3.5 Analysis of Power Saving for Two-Bit System

After clarifying the case of modifying the threshold in one-bit system, we now consider two-bit system. In Chapter 2.3.2.3, we explained that there was another information we can obtain at transmitter, which is the content of upcoming data.

Therefore, as channel coefficients are accurately measured in advance, we could find out which subsequent data would be affected mostly by the current data. And then, ISI induced by the current data would be checked if it is helpful to this mostly-affected data.

If it is not, we will reduce the threshold as the same setup of one-bit system to save power.

Fig 3.13 depicts the BER performance of different threshold and the comparison of one-bit and two-bit systems. Table 3.6 records the simulated average transmitted power and compare with one-bit system.

Fig 3.13 Comparison of one bit and two bits BER vs. received SNR

threshold

average transmitted power (one bit)

average transmitted power (two bit)

1 1.968 1.968

0.9 1.642 1.615 0.8 1.357 1.306

0.7 1.11 1.039

0.6 0.903 0.82

0.5 0.733 0.645

Table 3.6 The simulated average transmitted power under one-bit and two-bit systems

In Fig. 3.13, BER performance for two-bit system is close to that of one-bit system.

However, the BER of two-bit system is a little better than one-bit system. It is because the ISI effect of the proceeding data is reduced in advance. Also, besides modifying threshold while ISI effect is destructive to the current data, the threshold is also modified.

Therefore, average transmitted power is saved as comparing with one-bit system.

Chapter 4 Conclusions

This thesis presents a transmission scheme to solve the ISI problem while transmit speed raises to Gbps. This method utilizes measured channel coefficients to manage multipath-induced ISI and set the amplitude of noiseless received signal. The proposed system structure moves the equalization functionality to the transmitting end. The developed MP circuit at TX can first measure the channel response coefficients. Next, the modified MP can estimate the related ISI effect. According to the estimated ISI, the threshold mechanism is excuted to determine the amplitude of transmitted signal. After transmitting these signals through the multipath channel, the related output signal will be defined by a threshold. While the threshold equals to the data magnitude, the BER performance can approach the ideal multipath-free transmission. The BER could be improved by transmitting more power. The transmitted power could be reduced by lowering the threshold while ISI and the current data have different polarity. While the channel SNR is high, we can choose small threshold so as to save power but still maintain acceptable BER. At last, a two-bit system is proposed to improve BER performance and save transmitted power. Although its improvement is not obvious, it could be a useful system for future study.

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簡 歷

姓 名：陳柏任

居 住 地：台灣省台北縣

出生年月：民國七十年七月十一日 學 經 歷：

國立交通大學電信工程學系 (88 年 9 月~ 92 年 6 月)

國立交通大學電信工程學系碩士班 (93 年 9 月~ 96 年 2 月)

在文檔中 應用於短距離高速無線通訊之傳送端前置等化器 (頁 22-0)