Chapter 4 Conventional Path Selection Methods
4.2 Number of Path Setting Method
To improve the ML estimate of channel impulse response, the number of path setting
method first defines a parameter Np, which represents the desired number of paths. Only
Np elements with larger amplitudes in hˆML are preserved and the other paths are discarded. In consequence, the algorithm of the number of path setting method is presented
in the following: [7]
For example, let Np be 10. Then, only 10 elements with larger amplitude are said to be
the valid elements while the remaining paths are considered as noise and set as zero.
Similar to the threshold setting method, the refined estimate, ˆh , is finally transformed p
back to frequency domain to get the estimated channel frequency response, that is,
ˆ p =FFT{ }ˆp
H h . (4.5)
Chapter 5
Proposed Path Selection Method
Although the conventional path selection methods in the previous chapter, including the
threshold setting method and the number of path setting method, are widely used for
improving channel estimation, the drawback of these two methods is that they are heuristic
approach to the problem and sensitive to channel power delay profiles as well as the
operating SNR.
In general, how to set the threshold, TdB, in the threshold setting method, is relevant to
the structure of multipath power delay profiles. If the threshold is set too large, the path
selection method might pick noise. On the other hand, if the threshold is set too small, the
path selection method might lose true channel paths. Therefore, it is difficult to set a proper
threshold for all kinds of channel environments, and improper setting of the threshold will
decrease the performance of channel estimation significantly.
As to the number of path setting method, it is also difficult to know how many paths
exist in wireless channel environments. Thus, it is potentially required to estimate channel length to choose a proper parameter Np that represents the desired number of paths, or the
system may suffer from performance degradation. Similar to the threshold setting method,
Np with a too large value makes noise included in the estimated channel impulse response, and Np with a too small value excludes the true channel paths in the estimated channel
impulse response.
As shown in Figure 6.45 and 6.46 in the next chapter, the simulation results show that
inaccurate path selection raises the average SE of channel estimation, especially for the case
of losing channel paths. Furthermore, inaccurate path selection influences not only the BER
performance of the system, but also the complexity of channel tracking since the channel
paths are usually tracked path-by-path in the tracking stage. Selecting more paths than the
true channel paths existing in practical environments increases the complexity of the
channel tracking. For example, in a two-path channel, the complexity of picking 10 paths in
path selection methods is fivefold than the complexity of picking 2 paths.
According to the aforementioned discussion, we find that the conventional path
selection methods are sensitive to the setting of parameters, channel conditions and the
operating SNR. Thus, we would like to develop a novel path selection method which can
improve the BER performance, reduce the average SE of channel estimation, and increase
the probability of picking correct paths. The cost function for the proposed path selection
method is then presented in the next section.
5.1 Cost Function
Eq.(5.1) and Eq.(5.2) represent the ML estimate for channel impulse response by using
the signals on even and odd subcarriers, respectively:
2 vectors (or matrices) which involve even and odd subcarriers, respectively.
Afterward, a variable A which is a diagonal matrix of size ( , )G G is further
introduced to indicate which elements in ˆ
heven and ˆ
hodd are desired paths. Thus, the cost function for the proposed path selection method can be formulated as follows:
2 2
denoted as ˆ( )A n . If ˆ( ) 1A n = , the nth element of ˆh is considered as a valid path;
could be deemed as the estimated channel impulse response (corresponding to channel h ),
corrupted by noise. Note that the elements of z and 1 z are denoted as 2 z n1( ) and
2( )
z n , respectively, for n=0,1, ,G− , each of which is with zero-mean and variance 1
2 2
2σ σn /( XN).
The meaning of the cost function can be analyzed by substituting Eq.(5.4) and Eq.(5.5)
into Eq.(5.3). We discuss the problem with two cases: a path does or does not exist at the
nth element. When there is indeed a path at the nth element of h , if ( ) 1A n = , the cost
function with respect to the nth element of Eq(5.3) becomes as
z n1( )−z n2( )2+ z n2( )−z n1( )2 (5.6)
and if ( ) 0A n = , the cost function with respect to the nth element of Eq(5.3) becomes as
h n( )+z n1( )2+ h n( )+z n2( )2. (5.7)
As we can see, value of Eq(5.7) tends to be larger than value of Eq(5.6) when the
amplitude of a channel path is sufficiently larger than noise, and this result sides with the
solution of ( ) 1A n = . On the other hand, when there is no path at the nth element of h ,
if ( ) 1A n = , the cost function with respect to the nth element of Eq(5.3) becomes as
z n1( )−z n2( )2+ z n2( )−z n1( )2 (5.8)
and if ( ) 0A n = , the cost function with respect to the nth element of Eq(5.3) becomes as
z n1( )2+ z n2( )2. (5.9)
In this case, the variance of Eq(5.8) is twice the variance of Eq(5.9), and this result sides
with the solution of ( ) 0A n = .
5.2 Optimum Solution
To solve the minimization problem in Eq(5.3), we expand it as follows:
( ) ( ) ( ) ( )
even odd even odd odd even odd even
A
even even even odd odd even odd odd
A
Obviously, the minimization problem of Eq.(5.10) can be viewed as an ILP problem or a 0-1
programming problem; that is, we have [10]:
{ }
min f(0) (0)A + f(1) (1)A + + f L( −1) (A L−1) ,
A (5.12)
where A n( ) and f n( ) , for n=0, , -1G , are diagonal elements of A and f ,
respectively.
Since the optimum solution of a linear programming (LP) problem can only be vertices
of the feasible set [10], the ILP problem of Eq.(5.14) can be equivalently transformed into a
simple LP problem. To do this, we use 0≤A n( ) 1≤ , for n=0, , -1G instead of the
integer constraint of ( )A n , i.e., ( ) 0 or 1A n = . The process of this constraint release is
reasonable because for this new constraint, each element of the vertices in the feasible set is
either 0 or 1. Consequently, the ILP problem of Eq(5.12) can be transformed into a simple
LP. Even though the LP problem can be solved by the simplex method, the computation still
requires considerable effort to achieve the optimal solution. In order to reduce the
computation complexity, we then develop a more simple method to find the optimum
solution. In fact, it can be easily observed that the optimum solution of Eq(5.12) is to make
( )
A n corresponding to the negative value of ( )f n be one; otherwise, ( )A n is set to zero.
As a result, we have the optimum solution of Eq(5.14):
1, if ( ) 0
suppress noise and reduce the probability of false alarm by finding a proper threshold which
is insensitive to channel power delay profiles.
5.3.1 Analysis of False Alarm
By substituting Eq.(5.4) and Eq.(5.5) into Eq.(5.11), we can obtain
ˆ ˆ ˆ ˆ ˆ ˆ
According to Eq.(5.13), the analysis of false alarm can be done point-by-point due to the
fact that the vaule of ˆ( )A n merely depends on the value of ( )f n and the elements of z 1
the position n. Instead of deriving a closed-form probability density function (PDF) for the
variable ( )f n in Eq.(5.15), we use bootstrap (or resampling) techniques in Monte Carlo
methods to simulate the PDF. The basic idea of the bootstrap is to evaluate the PDF through
the empirical samples [11], and the PDF of ( )f n in Eq.(5.15) is then simulated in Figure
5.1. As shown in Figure 5.1, even if there is no path at the position ,n there is still a
certain probability for the negative value of ( )f n , resulting in the event of false alarm.
Although the range of the values for the variable ( )f n in Eq.(5.15) varies with the
variance of ( )Z k , the shape and proportion of the PDF is invariant to the value of
2/( 2 )
n XN
σ σ . To reduce the probability of false alarm, we propose a refined path selection
method in the next subsection.
Figure 5.1 The PDF of ( )f n evaluated through the empirical samples with 1,000,000 samples. generated from the random variable ( )f n . Therefore, we propose the refined path selection
method to suppress noise and to reduce the probability of false alarm in the following:
1, if ( )
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 6.23 The CDF of false alarm for the three path selection methods at E N = 10dB b 0 in the Veh. Bchannel.