INTRODUCTION
Ultra Wideband (UWB) transmission is a Spread Spectrum (SS) system that is
currently receiving great attention since it is a candidate for the physical
layer of several applications like sensor networks and high rate Wireless Personal
Area Networks (WPAN) (Yang and Giannakis, 2004). The
direct sequence ultrawideband (DSUWB) technology has many appealing features,
such as less complex hardware, high data transmission speed, low power, wide
bandwidth, high multipath resolution, as well as highprecision ranging capability
(Win and Scholtz, 2000) and it is one available candidate
for future shortrange indoor radio communication systems (Fontana,
2004).
For the practical application of the UWB technology, channel estimation is
a key technology that must be addressed (Zhaogan et
al., 2007). According to whether using the pilot sequence, the DSUWB
system estimation algorithm can be divided into blind channel estimation and
nonblind channel estimation. The maximum likelihood estimation (ML) algorithm
(Lee, 2010) and the Least Squares (LS) algorithm are
the most common algorithms. ML algorithm is widely used for its low complexity,
low SNR and high precision. In Dilmaghani et al.
(2004), a reverse system is employed and the flat elliptical wave is used
as the input pulse. Although this method has a relatively high accuracy, it
can not estimate the channel delay. So, the delay estimation must employ the
other devices, which will increase the system complexity. In Alizad
et al. (2005), a new UWB channel estimation method is provided, which
uses the training sequence to design a compression filter and uses it as the
input signal to get the impulse response. The performance of this method is
better, but it requires a matching filter, a sampling device and a compression
filter. All of these will increase the complexity of the algorithm and the system.
In Ertin et al. (2001), the techniques of maximumlikelihood
estimators of the channel parameters are proposed under the assumption of the
presence of a training sequence. In Zheng and Xiao (2009),
Takeda and Adachi (2005), the improved algorithms of
frequency domain equalization is provided, which have a good performance in
channel estimation. In addition, music algorithm is a good choice for the channel
estimation as is proposed in Yung et al. (2001).
In this study, we employ the ML algorithm to estimate DSUWB channel and analyze the performance of the ML algorithm. The estimation performance influenced by different assistedpilot numbers and the different Gaussian monocycle is discussed in different SNR conditions.
SYSTEM MODEL
DSUWB signal model: For the DSUWB systems, the signal is generated
usually through the following processes: first, the pseudorandom code or binary
PN code to is applied to code the sending binary code sequences; second, the
narrow pulse is modulated by amplitude. And the emission signal can be expressed
as (Yi et al., 2008):
where, d_{j}∈{1, +1} is the user’s binary information symbols, g_{n}∈{1, +1} is the user’s pseudorandom code sequence, R_{s} = 1/T_{s} is data rate and T_{s} = N_{s}T_{f} is symbol period, then each binary data symbol is expressed by N_{s} single pulses.
SV channel model: Channel model is used to represent the wireless transmission characteristics in a given environment. Signal propagation environment is one of the main factors to affect the performance of the wireless communication. So, it is important to estimate an accurate channel model for the transmission system, such as antenna diversity, equalization, coding and performance analysis issues.
SV model based on this observation: Typically, the pulse from the same multipath arrived at the receiver in the form of cluster. The arrival time of the cluster is modeled as a Poisson process with the rate of Λ:
where, T_{n} and T_{n1} are the arrival time of the n and the n1 cluster. And the first cluster’s arriving time is 0. In each clusters, the successive arrival time of multipath components also obey the Poisson distribution with the rate of Λ:
where, T_{n} and T_{n1} are the arrival times of the n and the n1 components of the kth cluster. And the first component of each cluster’s arriving time is 0.
IEEE P802.15.TG3a model: In order to make the experiment data more consistent
to the measurement data of UWB, IEEE Working Group modifies the SV model. The
impulse response of IEEE model can be expressed as (Wang
and Chang, 2007):
where, X is lognormal random variable representing the channel amplitude gain, N is the observed cluster number, K(n) is the received multipath number of the nth cluster, α_{nk }is the kth path coefficient of the nth cluster, T_{n} is the arrival time of nth cluster and τ_{nk} is the delay of k path of the nth cluster. Channel coefficients α_{nk} can be expressed as:
where, p_{nk}∈{1, +1} is the equal probability of discrete random variables and β_{nk} is channel coefficient of the kth path of the nth cluster, which is lognormal distribution. β_{nk} is given by:
where, x_{nk} is the Gaussian random variable and x_{nk} can be further decomposed into:
where, ξ_{n} and ζ_{nk} are the channel coefficient of each cluster and each component, respectively. Furthermore, with the characteristics of the cluster’s amplitude and each multipath component, we can get the μ_{nk}:
Based on the SV model, the arrival time of T_{n} and τ_{nk} obey the Poisson process and the rate are Γ and λ, respectively.
CHANNEL ESTIMATION ALGORITHM
ML estimate algorithm is one of the most common and effective estimation methods. The basic idea of the ML algorithm is that without a priori knowledge of the estimated variable, the variable is estimated by using the known parameters of the observations. Therefore, using the ML algorithm in estimation, the estimated parameters are random variables. But they are assumed to be constant and include the noise. For single user, the signal modulated by DSBPSK is expressed as Eq. 1.
Practical UWB system is likely a multiuser system. And in order to demodulate one user’s signal from the received signal, usually there are two methods. One is to use multiuser detective method. Although, this method does not require the estimated multipath channel parameters, the structure of the receiver designed according to this method is complex and the amount of the computation is large. Anther method is first to estimate the channel parameters and then the RAKE receiver is employed using the estimated channel parameters to demodulated the information.
Based on the above analysis, multiuser can be reduced to a signal user situation. The received signal is expressed as:
And the estimated signal is given by:
Then we can get the lognormal likelihood function:
For the UWB signal, when l_{1} ≠ l_{2}:
Using (10),(11) and (12), we can get:
where N_{pilot} is the number of pilot. In order to maximize ,
first
should be fixed and then changes the
to make Eq. 13 to be max and get the .
So, we should get the
partial derivative of and we get:
According to Eq. 15 and 16, we can get
and .
SIMULATION AND ANALYSIS
The affection of the number of pilot symbols on the channel estimation: This study uses the IEEE P802.15.TG3a model to simulate. First, the basic parameters of the impulse response should be defined, such as the average observation time and the cluster arrival rate, pulse average arrival rate, cluster power decay factor and so on. Second the arrival time of each cluster should be generated. From Eq. 2 we can know the arrival time of each cluster is exponentially distributed random variable. And then the multipath components should be generated. According to the Eq. 3, each cluster’s arrival time of multipath components are also exponentially distributed random variables. Finally, the impulse responses of the continuoustime are normalized.
In order to measure the performance of the channel estimation algorithm based on ML criteria, normalized mean square channel estimation error (ξ_{NMSCEE}) is employed in this study:
where, M is the number of the data, the γ(m) is the real multipath information and the (m) is the estimated multipath information. If ξ = 0, then the estimated information fixes the actual information exactly and the smaller the ξ is, the better performance of the algorithm is.
Figure 1~2 are the relationship of the
estimate channel and actual channel of CM1~CM4. CM1 is the light of sight channel
(0~4 m). CM2 and CM3 are the non light sight channels, ranging from 0~4 m and
6~10 m, respectively. And CM4 is the extreme non light sight multipath channel.
All of those are calculated in the condition of that the transmission signal
is the halfwave cosine signal, the SNR is 10dB and the pilot number is 10.
It can be seen from Fig. 1 to 2, with the
lower channel quality, the deviation between the estimation and the actual values
is increasing. Figure 3 shows the relationship curves of the
ξ_{NMSCEE} and the pilot of the four channels, with the SNR = 10
dB.
We can draw many conclusions from Fig. 3. First, the ξ_{NMSCEE}
of the four kinds of the channel decreases as the pilot number increased. Take
the CM1 for example, when the pilot number increased from 10 to 30, the ξ_{NMSCEE}
decreased rapidly from 7.3 to 2.2%. Second, when the pilot number increased
to a certain number, the decline rate of ξ_{NMSCEE} slowed significantly
and closed to unchanged. For the CM2, there is only 0.47% decline of the ξ_{NMSCEE},
when the pilot number increased from 60 to 100.

Fig. 1: 
(a) Comparison of the estimated and (b) comparison of the
estimated the actual multipath information the actual multipath information
in CM1 channel in CM2 channel 

Fig. 2: 
(a) Comparison of the estimated and (b) comparison of the
estimated the actual multipath information the actual multipath information
in CM3 channel in CM4 channel 

Fig. 3: 
The relationship of ξ_{NMSCEE} and the pilot
number for CM1 to CM4, with SNR = 10 dB 
As a result, blindly increasing the number of the pilot can not greatly improve
the performance of the algorithm, but will increase the computing complexity.
In practical application, the wise number of pilot is 50 with 1.67% of the ξ_{NMSCEE}.
Third, the estimate effect of the CM1 is the best among the four channels. And
the CM2 is the second. For the CM1 and CM4 are the limit cases, we usually only
consider CM2 and CM3 in practical application. In the following simulations,
we only discuss the CM2 and CM3.
The affection of SNR on the channel estimation: Here, the changes of
ξ_{NMSCEE} are discussed under different SNR. Figure
4 and 5 are the relationship of the ξ_{NMSCEE}
and the SNR, respectively. And the pilot number is 10, 20, 30, 50, 70 and 90.
As is shown in Fig. 4 and 5, under the
same pilot number, as the SNR increased the ξ_{NMSCEE} is gradually
reduced. And the performance of the channel estimation algorithm is improved.
When the pilot number is fewer (less than 30), the performance of the channel
estimation algorithm is greatly improved by increasing SNR. For instance, in
Fig. 5, when the SNR increased from 5 to 10 dB, the ξ_{NMSCEE}
reduced 13.25% with the pilot number = 20. In addition, when the pilot number
is greater than 50, the improvement of the performance by increasing the SNR
can be omitted.
The affection of the waveforms of the transmitted signals: For the indoor UWB transmission, the transmitted signal power must satisfy the FCC’s restrictions. Figure 6 shows the spectrums of different order Gaussian pluses which meet the FCC’s limits.
We can see that the fifth order Gaussian pulse is the minimum order pulse which satisfies the FCC’s limits. Figure 7 is the relationship of the ξ_{NMSCEE} and the different waveforms with SNR = 10 dB. And the waveforms are the halfwave cosine pulse, the fifth Gaussian pulse, the seventh Gaussian pulse and the ninth Gaussian pulse.
Figure 7 shows that the performance of the fifth order Gaussian
pulse is the best when the pilot number is less than 30.

Fig. 4: 
The relationship of ξ_{NMSCEE} and the SNR with
different pilot numbers in CM2 

Fig. 5: 
The relationship of ξ_{NMSCEE} and the SNR with
different pilot numbers in CM3 

Fig. 6: 
PSD of the highorder derivatives of the Gaussian pulse and
FCC’s restrictions 

Fig. 7: 
The relationship of the ξ_{NMSCEE} and the different
waveforms 
When the pilot number is more than 50, the ninth order Gaussian pulse has a
better performance. But there is only 0.3% improvement compared with fifth order
Gaussian pulse. Considering the compute complexity, the fifth order Gaussian
pulse is the best choice for the UWB monocycle.
All of the simulation results indicate that, compared with other methods, the
performance of the paper is better. With the same SNR, the better performance
can be got, comparing with the methods in (Dilmaghani et
al., 2004; Ertin et al., 2001). And due
to employing the ML algorithm in time domain, we can get better real time than
the frequency domain algorithm (Zheng and Xiao, 2009;
Takeda and Adachi, 2005). Moreover, even with the low
SNR, we can get the acceptable performance, which is superior to the music algorithm
(Yung et al., 2001).
CONCLUSIONS
In this study, the Maximum Likelihood algorithm is implemented in time domain DSUWB channel estimation. Compared with other estimation methods, ML algorithm can establish the channel estimation information in time domain and has less complexity in calculation. The IEEE P802.15.TG3a model is selected for the estimation simulation. To measure the estimation performance, the normalized mean square channel estimation error is employed in this paper. The simulation results indicate that the performance improves with the increasing of the pilot number and the estimation algorithm can achieve significant performance when the pilot number is about 60 in CM1, CM2 and CM3 when the SNR equals to 10 dB. In the case of pilot number less than 30, the algorithm is sensitive to SNR, that is to say the performance improves obviously with the increasing of SNR. On the other hand, in the case of pilot number more than 50, the algorithm is not sensitive to SNR and the increasing of SNR almost can not improve the estimation performance. Due to FCC’s PSD restrictions, the monocycle of the DSUWB is researched and the simulation results show that the fifth order Gaussian pulse can obtain satisfying performance.