The OFDM is a digital multi-carrier modulation scheme, which uses a large number of closely-spaced orthogonal sub-carriers that is particularly suitable for frequency-selective channels and high data rates. This technique transforms a frequency selective wide-band channel into a group of non-selective narrow-band channels, which makes its robust against large delay spreads by preserving orthogonality in the frequency domain as in frequency selective fading channels wideband signals suffer inter-symbol interference while those narrowband signals only experience at fading. Orthogonal Frequency Division Multiplexing (OFDM) systems are especially suited for channel estimation. In order to achieve the potential advantages of OFDM systems, the channel coefficients should be estimated carefully.
Channel estimation is an important and necessary function for modern wireless
receivers which can be improved using more pilot symbols (Slock,
2004). With even a limited knowledge of the wireless channel properties,
a receiver can gain insight into the information that was sent by the transmitter.
The goal of channel estimation is to measure the effects of the channel on known
or partially known transmissions, where the channel samples will be extracted
from those known pilot signal and using a proper interpolation method for accurate
estimation of the channel.
In general OFDM systems, pilots can be sent either in block or comb-type arrangements. In block-type arrangement, pilots are sent on every sub-carrier in a time-periodic manner. In comb-type, pilots are sent continuously in a frequency-periodic manner.
After the estimation of the channel transfer function of pilot tones, the channel
transpose of data tones can be interpolated according to adjacent pilot tones.
The linear interpolation has been studied in (Rinne and Renfors,
1996) and is shown to be better than piecewise constant interpolation. The
other most general methods of interpolation used for the estimation of the channel
over OFDM system is:
In this study, wavelet de-noising filter will be used in order to reduce the
effect of the noise over the estimated channel using an improved algorithm for
the channel estimation of OFDM system (Ahmed et al.,
2009) which use a as specific pilot signal arrangement different form comb
and block type as can be seen later. As the joint of improved algorithm and
wavelet de-noising can bring to more accurate estimation. Also, Viterbi detection
will be used in order to reduce the effect of the noise over the original transmitted
The transmitter and receiver of the OFDM system over pilot insertion block, TCM and wavelet de-noising filter will be described.
Transmitter structure: As shown in Fig. 1, the vector
of data s is encoded and mapped according to 2/3 convolutional coding over 8-PSK
for TCM (Ungerboeck, 1987). Then, for pilot symbol aided
channel estimation, Nf and Nt pilot symbols are inserted
periodically with the distance Df and Dt in frequency
and time grid respectively with the insertion of a single pilot sample of the
last sample of each symbol adjacent to the uniformly distributed pilot samples
symbol, as shown in Fig. 2, as the serial vector of the result
data converted to N×K vectors by a serial-to-parallel converter where N is the
number of total sub-carriers in one OFDM symbol and K is the total number of
OFDM symbols in the frame.
|| Transmitter structure of the OFDM system
||Pilot distribution for OFDM system in frequency-time grid
At this time, each OFDM symbols vector in the frame xN= [x1(k)
x2 (k) x3(k)
xN(k)] will be modulated
using N point Inverse Fast Fourier Transform (IFFT). At the terminal end of
the transmitter, the resulted OFDM vector in the frame will be sent serially
through the time varying frequency selective channel. The channel will be described
using baseband equivalent impulse response as h(k) = [ h1(k) h2(k)
hLf (k)]T where Lf is the length of channel (Coleri
et al., 2002). The channel is modeled as an impulse response h(t)
followed by the complex Additive White Gaussian Noise (AWGN) n(t):
where, αm is a complex values and Ts is the sampling
Receiver structure: At the receiver side as shown in Fig. 3, the received signal will be demodulated using N point (FFT), where the resulted signal will be as follow:
where, nn(k) is the FFT sample of the additive white Gaussian noise and Hn(k) is the frequency domain of the channel coefficient for the nth sub-carrier and the kth OFDM symbol.
The channel sample will be extracted from the received pilot signal by using LS channel estimation in frequency domain as the receiver know the sample at that signal as:
where, xp(k) is the pilot sample in the OFDM signal, is the coefficient
of the channel at that pilot sample (Biglieri et al.,
1998). Where the channel coefficient will be filtered using a wavelet de-noising
as it can be seen in Fig. 3, then estimation of the channel
will be done which is used to be combined with received signal to reduce the
channel from that signal which will be passed through ML detection using Viterbi
algorithm (Viterbi, 1967), to get the final signal.
|| Receiver structure of OFDM system
An improved algorithm for channel estimation based on pilot sample: Using those extracted channel coefficients at that pilot to estimate the channel coefficients for the OFDM signal. In order to obtain estimated channel coefficients for all sub-carriers the mean and the variance of two adjacent channel coefficient extracted from those pilot samples and then using the equation of the mean and variance to calculate the estimated channel coefficients as follow:
||Calculating of the mean and the variance of the two channel
coefficient which has been extracted from the pilot sample in the same time
period of OFDM symbol as follow:
where, Np is the number of pilot used in the equation.
||Multiplying the calculated variance by a factor ξ where
||Calculating the value of the two entire samples between the
two pilots using the following equation:
where, he is the estimated coefficient of the channel, while c1 and c2 will be calculated as follow:
||Now for the calculation of the channel coefficient between
OFDM symbol the same cited procedure will be used but with aid of the past
calculated coefficient to calculate the mean and the variance and modified
by α and β factor respectively. Where α and β factor
will be explained with aid of schemes. In this study we will propose that
those factors will be known
α and β factors: α and β factors are the modification
factor for the mean and variance of the two adjacent pilot sample of the same
sub-carrier of the OFDM symbols. Where these two factors are limited to unity
value as those factor are fluctuated around the value of one. By testing the
Probability Density Function (PDF) of these factors it was found that these
two factors limited to one as the doppler frequency fd dencrease
as shown in Fig. 4, 5 which can show the
normal distribution of 4, 6 and 8 tab of the channel length for both of α
and β factors (mean and variance factor, respectively).
Wavelet de-noising: Wavelet analysis is capable of revealing aspects
of data that other signal analysis technique as fourier analysis which miss
aspects like trends, breakdown points, discontinuities in higher derivatives
||Probability density function of α factor for different
doppler frequency of the channel
||Probability density function of β factor for different
doppler frequency of the channel
Furthermore, because it affords a different view of data than those presented
by traditional techniques, wavelet analysis can often compress or de-noise a
signal without appreciable degradation.
Fourier analysis has a serious drawback. In transforming to the frequency domain, time information is lost. When looking at a fourier transform of a signal, it is impossible to tell when a particular event took place.
Wavelet de-noising can bring to more accurate estimation for the channel by
reducing the effect of the noise on the estimated channel. As the noisy channel
coefficients pass through the wavelet de-noising filter the estimated signal
will be more accurate as shown in Fig. 6. The general wavelet
de-nosing procedure is as follows (Wang et al., 2000):
||Apply wavelet transform to the noisy signal to produce the
noisy wavelet coefficients to the level which we can properly distinguish
the PD occurrence
||Select appropriate threshold limit at each level and threshold
method (hard or soft threshold) to best remove the noises
||Inverse wavelets transform of the thresholded wavelet coefficients
to obtain a de-noised signal
The threshold is crucial when use the wavelet de-noising. Function wden in Matlab is used to remove (I +W)Xp*. The option minimaxi is used as threshold algorithm. It is presented in:
where, n is the length of the signal. It can be seen that the thr is directly proportional to the σ (D1) . So, the noisy is removed efficiently unless the threshold thr is chosen close to σ ((I +W)Xp*), the standard deviation of the noisy (I +W)Xp*. It means to minimize the Δ:
Estimated Channel over Wavelet De-noising filter, (a) Original
fading channel , (b) Noisy estimated channel and (c) Estimated channel over
D1 is the first-level detail coefficients. We choose thr to make Δmin and use wavelet de-noising method to remove (I +W)Xp* to get more accurate.
Viterbi decoder: Soft Maximum-Likelihood (ML) decoding using the Viterbi-Algorithm
(VA) is assumed (Viterbi, 1967). The Viterbi decoder
divides the input data stream into blocks, estimating the most likely data sequence
and outputting each decoded data sequence in a burst. The Viterbi algorithm
performs the trace back operation in parallel with the path calculations. The
width of the metric registers must be enlarged when using high rate punctured
codes, when the number of bits used in soft decision is large and when the Bit
Error Rate at the input of the Viterbi decoder is high. The trace back depth
should be sufficiently long to avoid loss of accuracy, therefore 20 trace back
depth was used in the simulation in order to get a good accuracy.
Simulation parameter: In this study, the simulation has been considering different values of the doppler frequency for the Rayleigh fading channel according to the velocity of the vehicle with Additive White Gaussian Noise (AWGN). The channel is assumed to be frequency selective fading, where the doppler frequency for the channel has been calculated using:
where, fd is the doppler frequency, v is the vehicle velocity and c is the light speed. Table 1, shows the parameter specifications for OFDM system based on the trellis code modulation (TCM) and the wavelet de-noising filter.
Simulation results: Now, we will test different OFDM systems over different
parameters of Rayleigh fading channel.
||Parameter of OFDM system base TCM and Wavelet de-noising filter
Where the tested OFDM systems based on
improved algorithm of channel estimation (C.C.), are OFDM system based on TCM
(CCV), OFDM system based on wavelet de-noising (CCWden) and OFDM system based
on both TCM and wavelet de-noising (CCVWden).
Figure 7 shows testing of the four OFDM systems over fixed
parameter of the channel, 100 Hz doppler frequency has been selected for the
Rayleigh fading channel in additional to AWGN for testing those different systems.
As can be seen that OFDM/CCV can improve the performance of the system in both
cases (with and without wavelet de-noising filter) by about 7.5-8 dB for all
||Performance of different OFDM systems over 100 Hz doppler
frequency of Rayleigh fading channel
||Performance of different OFDM systems over different value
of doppler frequency of Rayleigh fading channel
OFDM system based an improved algorithm for channel estimation,
TCM and wavelet de-noising over different doppler frequency of the rayleigh
While, the usage of the wavelet de-noising filter in both cases (with and
without TCM) can improve the performance limitedly, while at 40 dB of SNR, BER
will improved as for C.C. BER equal to 3×10-2, while for C.C.Wden
is about 3×10-3 so as for OFDM/C.C.V., where BER for the OFDM/C.C.V
system without Wden at 40 dB of SNR, BER will be about 8×10-2, while
with Wden will be 8×10-2.
Figure 8 shows testing of the four OFDM systems over different values of doppler frequency in addition to AWGN for testing those different systems. Where it is clear that the for the whole OFDM systems the BER of the each system increased by increasing of doppler frequency of the channel as the performance of the OFDM/CCV in both cases (with and without Wden) improve the performance of the system, while for that base Wden (with and without TCM) will increase the performance especially at high doppler frequencies of the channel.
Finally, Fig. 9 shows OFDM system based on an improved algorithm for channel estimation, TCM and wavelet de-noising over different values of doppler frequency. Where, it is obviously that the performance of that system increasing by the decreasing of the doppler frequency of the channel.
RESULTS AND DISCUSSION
From the simulation result, it was found that the OFDM/CCVWden give the pest
performance of the OFDM system over the others as it can improve the performance
by about 7-8 dB over OFDM/C.C., 6-7.5 dB over OFDM/CCWden as the performance
of the system improved up to 10 dB by using the TCM and Viterbi decoder for
error correction (Akay and Ayanoglu, 2004) and about
1.5-2 dB over OFDM/CCV.
|| Coded and uncoded OFDM system over AWGN
So, from Fig. 9, it is obvious that the OFDM/C.C.V.Wden performance increase as the doppler frequency decrease, where the estimation of the channel will be more accurate, so the BER would be decreased as the SNR increased.
Form the previous study, it was found that OFDM system over TCM encoder that
the performance of the OFDM system will be increased by about 3.5 dB as can
be seen from Fig. 10 (Fernando et
al., 1998), where the coded and uncoded OFDM has been tested over AWGN
only, where it was found that OFDM/TCM enhanced by about 3.5 dB in compared
with uncoded system.
This study present an improved algorithm of OFDM/CC system combined either with TCM, wavelet or both together. It is clear that OFDM/CCV will increase the performance of the system, obviously. OFDM/CCWden can also enhance the performance but limitedly but maximum performance can get from OFDM/CCVWden.
The authors thank the anonymous reviewers for many useful comments which help to improve the quality and readability of this study. This research was supported by Prof. Kasmiran Jumari under UKM-OUP-NBT-29-153 Project code.