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 than23.2σn2, 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 missdetection. 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
10−2.
-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
pComputer 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