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Chapter 5 Proposed Path Selection Method

5.3 Refined Path Selection Method

5.3.3 Determination of the Parameter U

The determination of the value of the parameter U will depend on the probability of

false alarm and miss detection. Moreover, the probability of miss detection is related to the

strength of a channel path. In this subsection, we first analyze the influence of the miss

this path will introduce channel estimation error ∆H k( ) in the estimated channel

frequency response, i.e., we have the estimated channel frequency response:

ˆ ( ) ( ) ( )

for k =0,...,N− . From Eq.(3.2) and Eq.(5.18), the received signal after channel matching 1

can be written as

2 2

By using the assumption of Eq.(5.22), SNER of Eq.(5.21) can be approximated as

2 2

significant influence on the SNER. That is, when N µ, we have

2 2

In this thesis, N is set as 256. Hence, we will concern the probability of miss detection of

a path with energy h n( )2 =25σn2/

(

σX2N

)

as a design reference, i.e., set µ =25.

Figure 5.2 shows the CDF of the false alarm (µ = ) and the miss detection (0 µ=25)

of the variable ( )f n . According to the solid line, we have Pr(f n( ) 23.2≥ σn2)=1/64. In

other words, among f n( ) , for n G= ,...,

(

N/ 2 1

)

, the occurrence of the event of

{

f n( ) 23.2 σn2

}

is

( (

N/ 2

)

G

)

/ 64 on average. For example, when N and G are set as 256 and 64, respectively, we can acquire one point of ( )f n whose value is larger than

23.2σn2, i.e., we have max f n{ ( )} 23.2≥ σn2 and therefore Rth ≤ − ×U 23.2σn2 (in average

sense). Accordingly, Table 5.1 lists the probability of false alarm (µ = ) and miss detection 0

(µ=25) with U as a parameter. We denote RNth and f nN( ) as normalized terms, i.e.,

( )

(

2 2

)

/ /

Nth th n X

R =R σ σ N and f nN( )= f n( ) /

(

σn2/

(

σX2N

) )

. From this table, we can determine the value of U which can achieve a desired probability of false alarm and miss

detection. In this thesis, we can choose the value of U as 0.5 such that the probability of

false alarm is less than 7.55 10× 4 and the probability of miss detection is larger than

102.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Figure 5.2 The CDF of ( )f n evaluated through the empirical samples with 1,000,000 samples.

Table 5.1 The probability of false alarm and miss detection with U as a parameter.

U RNth ≤ False Alarm(µ= ): 0

The algorithm of Eq.(5.16) can be implemented by sorting the values of ( )f n , for than the two conventional path selection methods, aforementioned in the previous chapter.

{ ( )},

Figure 5.3 The proposed algorithm can be implemented by sorting the values of ( )f n , for 0,1, , 1

n= G− , in ascending order

Chapter 6 Simulation Results

In this chapter, we simulate the BER, average SE and probability of picking wrong

paths to demonstrate the performance of our proposed path selection method for channel

estimation in OFDM systems. Besides, we also compare the performance with the two

conventional path selection methods, including the number of path setting method and the

threshold setting method.

Table 6.2 Power delay profiles of channel environments ITU-Veh. A channel 0, -1, -9, -10, -15, -20 (dB) ITU-Veh. B channel -2.5, 0, -12.8, -10, -25.2, -16 (dB)

Two-path channel 0, -1 (dB)

Thirty-path exponentially decayed channel 0, -1.3029, -2.6058, -3.9087, -5.2116, -6.5144, -7.8173, -9.1202, -10.4231,

The simulation parameters are listed in Table 6.1. Throughout the simulations, carrier

frequency synchronization and symbol timing synchronization are assumed to be perfect.

Moreover, the simulations are conducted at baseband using the complex low-pass

equivalent representation. The ratio of energy between the pilot signal and the data signal

(on a subcarrier) is set to 1.

Only the small-scale fading is considered in our simulations. Besides, we use four

typical channel power delay profiles, including International Telecommunication Union

(ITU)- Vehicular A and Vehicular B fading channels, a two-path equal power fading

channel, and a thirty-path exponentially decayed fading channel, to demonstrate the

performance. The power delay profiles defined by the recommendations of the ITU are

well-established channel models for research of mobile communication systems. They

specify channel conditions for various operating environments encountered in

third-generation wireless systems, e.g the Universal Mobile Telecommunication Systems

(UMTS) Terrestrial Radio Access System (UTRA) standardised by 3GPP[12]. Both the Veh.

A and Veh. B channels are six-path channels with power delay profiles: 0, -1, -9, -10, -15,

-20 (dB) and -2.5, 0, -12.8, -10, -25.2, -16 (dB), respectively. For the two-path equal power

fading channel, the power delay profile is 0, 0 (dB). For the thirty-path exponentially

decayed fading channel, the power delay profile (linear scale) is given by [13]:

( )

delay profile of the thirty-path channel is listed in Table 6.2.

6.1 Threshold for Refined Path Selection Method

In Section 5.3, we set the value of U as 0.5 for the threshold Rth = − ×U max f n{ ( )}

in the refined path selection method. The algorithm of the refined path selection method is

that if ( )f n is smaller than the threshold Rth = −0.5×max f n{ ( )}, we say that there is a

10dB and 40dB, respectively, with U as a parameter. Figure 6.5 and Figure 6.7 show the

BER performance for the proposed path selection method in the thirty-path channel at

0

E N =10dB and 40dB, respectively, with U as a parameter. Figure 6.6 and Figure 6.8 b

show the average SE performance for the proposed path selection method in the thirty-path

channel at E N = 10dB and 40dB, respectively, with U as a parameter. Figure 6.9 and b 0

Figure 6.11 show the BER performance for the proposed path selection method in the

two-path channel at E N =10dB and 40dB, respectively, with U as a parameter. Figure b 0

6.10 and Figure 6.12 show the average SE performance for the proposed path selection

method in the two-path channel at E N = 10dB and 40dB, respectively, with U as a b 0

parameter.

We can observe that for the BER performance shown in Figure 6.1, Figure 6.3, Figure

6.5, Figure 6.7, Figure 6.9, and Figure 6.11, the threshold R which ranges from 0 to th

6 max f n{ ( )}

− × has no significant influence on BER performance of our proposed method.

As shown in Fig. 6.2 and Fig. 6.4, we can find that for the Veh. A channel, the minimum

average SE is achieved at a threshold between 0.4− ×max f n{ ( )} and 0.6− ×max f n{ ( )}.

Moreover, as shown in Figure 6.6 and Figure 6.8, we can find that for the thirty-path

channel, the minimum average SE is achieved at a threshold between 0.2− ×max f n{ ( )}

and 0.4− ×max f n{ ( )}. In Figure 6.10 and Figure 6.12, we can also observe that a threshold

between 0.6− ×max f n{ ( )} and 0.8− ×max f n{ ( )} can attain the minimum average SE in

the two-path channel. As a result, we can conclude that Rth = −0.5×max f n{ ( )} is an

appropriate value for the setting of the threshold in the refined path selection method.

Figure 6.1 The BER performance for the proposed path selection method in the Veh. A channel at E N = 10dB with threshold as a parameter. b 0

Figure 6.2 The average SE for the proposed path selection method in the Veh. A channel at

0

E N = 10dB with threshold as a parameter. b

Figure 6.3 The BER performance for the proposed path selection method in the Veh. A channel at E N = 40dB with threshold as a parameter. b 0

Figure 6.4 The average SE for the proposed path selection method in the Veh. A channel at

0

E N = 40dB with threshold as a parameter. b

Figure 6.5 The BER performance for the proposed path selection method in the thirty-path channel at E N = 10dB with threshold as a parameter. b 0

Figure 6.6 The average SE for the proposed path selection method in the thirty-path channel at E N = 10dB with threshold as a parameter. b 0

Figure 6.7 The BER performance for the proposed path selection method in the thirty-path channel at E N = 40dB with threshold as a parameter. b 0

Figure 6.8 The average SE for the proposed path selection method in the thirty-path channel at E N = 40dB with threshold as a parameter. b 0

Figure 6.9 The BER performance for the proposed path selection method in the two-path channel at E N = 10dB with threshold as a parameter. b 0

Figure 6.10 The average SE for the proposed path selection method in the two-path channel at E N = 10dB with threshold as a parameter. b 0

Figure 6.11 The BER performance for the proposed path selection method in the two-path channel at E N = 40dB with threshold as a parameter. b 0

Figure 6.12 The average SE for the proposed path selection method in the two-path channel at E N = 40dB with threshold as a parameter. b 0

6.2 System Performance in Veh. A Channel

In this section, we compare the performance of the three path selection methods in the

ITU-Veh. A channel. For the number of path setting method, the parameter N is set as 64. p

For the threshold setting method, the parameter T could be 20 or 30. dB

Figure 6.13 shows the BER performance for the three path selection methods. As shown

in Figure 6.13, the threshold setting method with TdB =20 experiences an error floor at

BER=8 10× 4. Moreover, the threshold setting method with TdB=30 performs almost the

same as the proposed path selection method and the number of path setting method at the

low E N region, while it performs a little worse at the high b 0 E N region. This b 0

implies that the BER performance for the threshold setting method is quite sensitive to the

setting of the parameter T . dB

Figure 6.14 shows the average SE for the three path selection methods. As can be seen

in this figure, for the threshold setting method, both the parameters of TdB =20 and

dB 30

T = lead to an error floor due to the loss of channel paths. Besides, the average SE of

the proposed method is about 10dB better than that of the number of path setting method for

all E N region. b 0

Figure 6.15, Figure 6.17, and Figure 6.19 show the CDF of the false alarm for the three

path selection methods at E N =10dB, 25dB, and 40dB, respectively. Figure 6.16, Figure b 0

6.18, and Figure 6.20 show the CDF of the miss detection for the three path selection

methods at E N =10dB, 25dB, and 40dB, respectively. We can find that the number of b 0

path setting method can exactly pick the six channel paths, but it also includes additional 58

fake paths. It should be noted that fake paths will increase the computation complexity of

channel tracking. As can be seen in Figure 6.15, Figure 6.17, and Figure 6.19, we can

observe that the threshold setting method with TdB=30 has much higher false alarm

probability at low E N , as compared with the proposed method. For example, for the b 0

CDF=90% and E N =10dB, the number of paths erroneously picked is 0 in the proposed b 0

method, while the number is 24 in the threshold setting method with TdB =30. This is

because the threshold setting method picks noise as channel paths more easily at low

0

E N . Even though the threshold setting method with b TdB =20 has a little less number of paths erroneously picked than the proposed method, it suffers from severe degradation on

the average SE and the BER performance. As shown in Figure 6.16, Figure 6.18, and Figure

6.20, we can observe that for the CDF of the miss detection at E N =10dB, the threshold b 0

setting method with TdB =30 performs a little better than the proposed method, whereas

the threshold setting method with TdB=20 performs much worse than the proposed

method. Moreover, we can observe that the miss detection probability of the proposed

method is almost equal to 0 at E N =25dB and 40dB. b 0

Figure 6.13 The BER performance for the three path selection methods in the Veh. A channel.

Figure 6.14 The average SE for the three path selection methods in the Veh. A channel.

Figure 6.15 The CDF of false alarm for the three path selection methods at E N = 10dB b 0 in the Veh. A channel.

Figure 6.16 The CDF of miss detection for the three path selection methods at

0

E N =10dB in the Veh. A channel. b

Figure 6.17 The CDF of false alarm for the three path selection methods at E N = 25dB b 0 in the Veh. A channel.

Figure 6.18 The CDF of miss detection for the three path selection methods at

0

E N =25dB in the Veh. A channel. b

Figure 6.19 The CDF of false alarm for the three path selection methods at E N = 40dB b 0 in the Veh. A channel.

Figure 6.20 The CDF of miss detection for the three path selection methods at

0

E N =40dB in the Veh. A channel. b

6.3 System Performance in Veh. B Channel

In this section, we compare the performance of the three path selection methods in the

ITU-Veh. B channel. Figure 6.21 and Figure 6.22 show the BER and the average SE

performance for the three path selection methods, respectively. Figure 6.23, Figure 6.25,

and Figure 6.27 show the CDF of the false alarm for the three path selection methods at

0

E N =10dB, 25dB, and 40dB, respectively. Figure 6.24, Figure 6.26, and Figure 6.28 b

show the CDF of the miss detection for the three path selection methods at E N =10dB, b 0

25dB, and 40dB, respectively. According to these figures, we can observe that the

simulation results obtained in the Veh. B channel are very similar to that in the Veh. A

channel.

Figure 6.21 The BER performance for the three path selection methods in the Veh. B channel.

Figure 6.22 The average SE for the three path selection methods in the Veh. B channel.

Figure 6.23 The CDF of false alarm for the three path selection methods at E N = 10dB b 0 in the Veh. Bchannel.

Figure 6.24 The CDF of miss detection for the three path selection methods at

0

E N =10dB in the Veh. B channel. b

Figure 6.25 The CDF of false alarm for the three path selection methods at E N = 25dB b 0 in the Veh. B channel.

Figure 6.26 The CDF of miss detection for the three path selection methods at

0

E N =25dB in the Veh. B channel. b

Figure 6.27 The CDF of false alarm for the three path selection methods at E N = 40dB b 0 in the Veh. Bchannel.

Figure 6.28 The CDF of miss detection for the three path selection methods at

0

E N =40dB in the Veh. B channel. b

6.4 System Performance in two-path Channel

In this section, we compare the performance of the three path selection methods in the

two-path channel. Figure 6.29 and Figure 6.30 show the BER and the average SE

performance for the three path selection methods, respectively. As can be seen in Figure

6.30, at high E N region, due to the fact that the energy of the two channel paths are b 0

much larger than noise energy, the average SE of the threshold setting method with

dB 30

T = is a little larger than that of the proposed method, whereas the average SE of the

threshold setting method with TdB =20 experiences an floor and is larger than that of the

proposed method. Therefore, the threshold setting method is still sensitive to the operating

value of E N . Moreover, the average SE of the proposed method is about 15dB lower b 0

than that of the number of path setting method. Figure 6.31, Figure 6.33, and Figure 6.35

show the CDF of the false alarm for the three path selection methods at E N =10dB, b 0

25dB, and 40dB, respectively. As can be seen in Figure 6.31, at CDF=99%, the threshold

setting method with TdB =30 performs much worse than the proposed method, and the

threshold setting method with TdB =20 is comparable to the proposed method. For the

CDF of the false alarm at E N =25dB, we can observe that the proposed method is b 0

slightly better than the threshold setting method with T =30 at CDF=100%. Figure 6.32,

Figure 6.34, and Figure 6.36 show the CDF of the miss detection for the three path selection

methods at E N =10dB, 25dB, and 40dB, respectively. As observed in these three b 0

figures, the miss detection probability (i.e., the value of the CDF when number of paths

erroneously picked is zero) of the proposed method is equal to 0 and is a bit better than that

of the threshold setting method.

Figure 6.29 The BER performance for the three path selection methods in the two-path channel.

Figure 6.30 The average SE for the three path selection methods in the two-path channel.

Figure 6.31 The CDF of the false alarm for the three path selection methods at E N = b 0 10dB in the two-path channel.

Figure 6.32 The CDF of miss detection for the three path selection methods at E N = b 0 10dB in the two-path channel.

Figure 6.33 The CDF of the false alarm for the three path selection methods at E N = b 0 25dB in the two-path channel.

Figure 6.34 The CDF of miss detection for the three path selection methods at E N = b 0 25dB in the two-path channel.

Figure 6.35 The CDF of the false alarm for the three path selection methods at E N = b 0 40dB in the two-path channel.

Figure 6.36 The CDF of miss detection for the three path selection methods at E N = b 0 40dB in the two-path channel.

6.5 System Performance in thirty-path Channel

In this section, we simulate the performance of the three path selection methods in the

thirty-path exponentially decaying channel in which the decaying factor is set as β =0.3.

Figure 6.37 and Figure 6.38 show the BER and the average SE performance for the

three path selection methods, respectively. As shown in Figure 6.37, since the channel paths

with smaller energy are discarded, an error floor is clearly visible at BER=5 10× 3 and

proposed method performs slightly better than that of the number of path setting method,

and the average SE performance of the proposed method is about 2dB better than that of the

number of path setting method.

Figure 6.39, Figure 6.41, and Figure 6.43 show the CDF of the false alarm for the three

path selection methods at E N =10dB, 25dB, and 40dB, respectively. From these three b 0

figures, we can find that the proposed method and the threshold setting method with

dB 20

T = is better than the threshold setting method with. For example, at E N =10dB b 0

and CDF=100%, the number of paths erroneously picked by the proposed method and the

threshold setting method with TdB =20 is 0, and the number is less than 30 incorrect paths

for the threshold setting method with TdB =30.

Figure 6.40, Figure 6.42 and Figure 6.44 show the CDF of the miss detection for the

three path selection methods at E N =10dB, 25dB, and 40dB, respectively. We can b 0

notice that for E N =10dB and CDF=90%, the number of paths erroneously picked by b 0

the proposed method is less than 14, while the number of paths erroneously picked by the

threshold setting method with TdB =20 and TdB=30 is less than 18 and 9, respectively.

However, for E N =25dB and 40dB, we can find that the proposed method performs b 0

much better than the threshold setting method at CDF=90%.

Figure 6.37 The BER performance for the three path selection methods in the thirty-path channel.

Figure 6.38 The average SE performance for the three path selection methods in the thirty-path channel.

Figure 6.39 The CDF of false alarm for the three path selection methods at E N = 10dB b 0 in the thirty-path channel.

Figure 6.40 The CDF of miss detection for the three path selection methods at E N = b 0 10dB in the thirty-path channel.

Figure 6.41 The CDF of false alarm for the three path selection methods at E N = 25dB b 0 in the thirty-path channel.

Figure 6.42 The CDF of miss detection for the three path selection methods at E N = b 0 25dB in the thirty-path channel.

Figure 6.43 The CDF of false alarm for the three path selection methods at E N = 40dB b 0 in the thirty-path channel.

Figure 6.44 The CDF of miss detection for the three path selection methods at E N = b 0 40dB in the thirty-path channel.

6.5.1 System Performance of Number of Path Setting Method with Different N

p

Computer simulations are carried out in the thirty-path exponentially decayed channel to

examine the performance of the number of path setting method under different values of

N . Here, the values of p N could be 10, 20, 30, 40, 50, and 60. Figure 6.45 and Figure p

6.46 show the BER and the average SE performance for the number of path setting method

with N as a parameter, respectively. Figure 6.45 and Figure 6.46 show that p N with the p

value less than 30 (number of channel paths existing in channel environment) causes much

more degradation than N with the value greater than or equal to 30. This result concludes p

that the event of the missing detection can degrade the BER and the average SE

performance more severely than the event of the false alarm.

Figure 6.45 The BER performance for the number of path setting method with N as a p parameter in the thirty-path channel.

Figure 6.46 The average SE for the number of path setting method with N as a parameter p in the thirty-path channel.

Chapter 7 Conclusions

DFT-based channel estimation which is derived from either ML or MMSE criterion was

intensively investigated for PA channel estimation in OFDM systems. Several kinds of path

selection methods are used to suppress noise and to further improve the performance of

channel estimation. After the path selection, the estimated channel impulse response is

transformed back into frequency-domain to obtain the estimated channel frequency

response.

However, the conventional path selection methods require knowledge of the multi-path

However, the conventional path selection methods require knowledge of the multi-path

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