GRAPHIC TOOL FOR A COMPARATIVE BETWEEN FRACTIONALLY
SPACED EQUALIZERS AND SYNCHRONOUS EQUALIZERS
Alexandre Carvalho Ferreira
Estevan Marcelo Lopes
Sandro Adriano Fasolo
Instituto Nacional de Telecomunicações Instituto Nacional de Telecomunicações Instituto Nacional de Telecomunicações
Av. João de Camargo, 510
Av. João de Camargo, 510
Av. João de Camargo, 510
Brasil
Brasil
Brasil
[email protected]
[email protected]
[email protected]
ABSTRACT
The objective of this paper is to implement a graphical
interface using the Matlab® software, for comparative
study of performance between fractionally spaced
equalizers and synchronous equalizers in wireless
communications systems. Traditionally, the equalization
system works with a sampling rate equal to the
transmitted symbols rate. The fractionally spaced
equalization possesses a sampling rate superior to the
symbol rate. Therefore, it is obtained a more efficient
equalization when compared with synchronous equalizers.
However the cost of this procedure is an increase in the
amount of executed calculations, or either, increase of the
computational load. The graphical interface makes
possible to the user to choose different communication
channels and to modify the equalizer parameters values.
During the simulation we observe the behavior of the
equalized data diagram, the taps gain of the equalizers and
the MSE (Mean Square Error). The implemented
technique of equalization uses LMS (Least Mean Square)
algorithm and uses a LE (Linear Equalizer) structure.
KEY-WORDS
ISI,
adaptive
equalization.
equalization,
fractionally
This effect is resulted by the convolution of the
transmitted signal with the impulsive response of the
channel with multipath, that can be modeled across the
discrete equation below:
N
z (k ) = ∑ x(k − n)ch(n) k = 1,K, N
(1)
n =1
Where x(k-n) are samples of the entrance signal and ch(n)
are samples of the discrete model of channel. To solve
this problem, is added to the receiver a system capable to
compensate or to mitigate the effect of the ISI in the
received signal. This system is called equalizer.
spaced
1. Introduction
One of the main problems that reduce the trustworthiness
of the signals received in systems of digital wireless
communications is the ISI (Inter Symbol Interference).
This appears in the majority of the communications
channels with multipath. In these channels, parcels of the
transmitted signal arrive in the receiver with different
delays and amplitude, due to the reflections of some parts
of the waves in the obstacles of the environment. The
Figure 1 illustrates this fact. The received signals across
the communication channel with multipath are modified
versions of the originally transmitted signals. The effect
of a symbol transmitted through the channels with
multipath, it is extended for an interval of time bigger
than the symbol period used to represent it. The Figure 2
illustrates the overlapping of the adjacent symbols.
Figure1 – Example of the multipath channel
Figure 2 – Graphical representation of the ISI
Therefore, to treat the subject an introduction on the basic
concepts of fractionally spaced equalization is suggested
in the section 2. In section 3 it will be presented the used
communication channels in the simulation. The graphical
interface of the developed computational tool, and its
In the equalization system, is common in many situations
to operate with a sampling rate equal sampling to the
transmitted symbols rate. It is had then an equalizer with
unitary spaces between taps in relation to the period of
symbols. T (1/T = symbols sampling rate), also known as
synchronous equalizer. However, a bigger sampling rate
than the symbol rate of 1/T can be adopted. This
implementation is known as fractionally spaced
equalization. The reason of the use of the fractionally
spaced equalizer is that, if the symbol occupies a
bandwidth bigger than the one strictly necessary, given by
the sampling theorem of Nyquist, or either, if an excess of
band will be used in relation to 1/2T, the channel behavior
will not be characterized and won’t happen the correct
channel equalization. With a 2/T fractionally spaced
equalizer, for example, it can be works with symbols with
excess of bandwidth of 100% to the cost of the increase of
100% of the filter elements and the number of
calculations growing with an equal or larger factor than 2.
An important advantage of the fractionally spaced
equalizer is that an error in the sampling phase is, in
general, less important that in synchronous equalizers.
Another advantage in the use of this type of equalizer is
the accomplishment of the optimum linear receiver, that
consists in the combination of a matched filter and a
synchronous transversal equalizer. Due to larger sampling
rate of the fractionally spaced equalizer this combination
can be better assimilated, while the synchronous equalizer
does not operate as matched filter. The coefficients of an
equalizer with T/2 can be up to date once for symbol,
based on the computed error for each symbol in particular.
The LMS (Least Mean Square) algorithm and any of its
variations can be used to adjust automatically the taps
gain.
3. Communication Channel
In the realized simulations, two communication channels
have been considered. The first channel, presented in the
Figure 3, is a fictitious channel with short length. Already
the UK long delay Static, illustrated in the Figure 4, is a
real channel that was used in the simulation to prove the
efficiency of the fractionally spaced and synchronous
equalizers in a real situation. The two channels are
detailed in Table 1.
Value
2. Fractionally Spaced Equalization
Channel #1
1
0.5
0
-0.5
1
2
3
4
5
6
Taps
Figure 3 – Channel 1: The fictitious channel
Channel UK long delay static
1
Value
characteristics, are presented in section 4. In section 5 the
results of the simulations are shown, comparing the
fractionally spaced and synchronous equalizer. To finish
the article, a conclusion is presented.
0.5
0
-0.5
10
20
30
40
50
60
70
Taps
Figure 4 – Channel: UK long delay static
4. Graphical Interface
For the accomplishment of the graphical interface,
illustrated in Figure 5, it was used the GUIDE (Graphical
User Interfaces Development Environment) tool of the
Matlab software. Through the graphical interface the
user has the option to choose different communication
channels, to simulate the behavior of the fractionally
spaced and synchronous equalizers and the number of
symbols used in the simulation. It is also possible to
modify equalizer parameters, such as the value of the
adaptation constant “µ” and the number of taps of the
filter of the equalizer. During the simulation, it can be
followed the behavior of the equalized data diagram and
the tap gains of the filter of the equalizers. Another
important graph that can be observed during the
simulation, is the MSE graph of the two equalizers. For
each equalized symbol, the MSE numerical values of the
fractionally spaced and synchronous equalizers are
graphically shown. Through the observation of these
graphs, the performance of the two equalizers can be
compared.
5. Simulation
The simulation interface operates with the two kinds of
equalizers, fractionally spaced and synchronous, so that
comparisons between the performances of each one of the
equalizers can be realized. The internal structure of
construction of the used equalizer in the simulator is the
LE. The adaptation algorithm of the tap gains used in the
simulator is known as LMS. With this algorithm, it is
possible to calculate the values of taps of the equalizer
with the objective to minimize the MSE. The symbols
used in the simulation belong to the 8 PAM
Figure 5 – Graphical interface of the computational tool for the study of the fractionally spaced equalization
( ± 1, ± 3, ± 5, ± 7 ) modulation. The greatest difficulty
founded for realizing the simulation was the programming
of the fractionally spaced equalizer. Theoretically, more
than one sample would be removed of each symbol that
arrived at the equalizer. For the simulation, was adopted a
sampling rate twice bigger than the synchronous equalizer
rate, it means, two samples per symbol. The process used
in the simulator, use as a sample of the received signal,
the average value from the adjacent samples, it means,
from the taken samples at the rate previous and
immediately posterior to the symbol rate.
The value of the first symbol is kept in the buffer
and after the second symbol arrives, an arithmetic average
will be used as the additional sample for the filter of the
equalizer. Figure 6 illustrates the process.
Mean
example, the time necessary to analyze thirty symbols in a
synchronous equalizer is 30TS . In a fractionally spaced
equalizer, that analyzes two samples for symbols, this
time is 60
TS
= 30TS .
2
5.1 First Simulation
The first simulation was performed through with the
fictitious channel #1. In this simulation was used one FIR
filter with 30 taps, an adaptation constant with value
equal to 0,0001 and twenty thousand simulated symbols.
In the Figures 7 and 8, can be observed the diagram of
equalized data of the fractionally spaced and synchronous
equalizers, respectively. In the Figure 9 the value of the
MSE in the two equalizers is represented. This is done to
compare the performance of both.
Fractionally
Equalizer
Channel
Ts
Buffer
Ts/2
Figure 6 – Sampling scheme of the fractionally spaced equalizer
Due to the inclusion of the sample that is the
average of two adjacent symbols, the number of elements
of the filter of the fractionally spaced equalizer is
multiplied by two. However, it must be observed that the
processing time of these symbols continues the same. For
Figure 7 – Diagram of equalized data of the fractionally spaced equalizer
MSE
3
Fractionally
Symbolic
2.5
Value
2
1.5
1
0.5
0
Figure 8 – Diagram of equalized data of the synchronous equalizer
Fractionally
Symbolic
Value
6
4
2
0
0
0.5
1
Symbols
1000
2000
3000
4000 5000
Symbols
6000
7000
8000
Figure 12 – MSE of the fractionally spaced and synchronous equalizers
6. Conclusions
MSE
10
8
0
1.5
2
x 10
4
Figure 9 – MSE of the fractionally spaced and synchronous equalizers
5.2 Second Simulation
In the second simulation performed through was used the
channel UK long delay static. The simulator was
configured using eight thousand symbols, 128 taps in the
FIR filter and an adaptation constant of the algorithm with
value of 0,0001. In the Figures 10 and 11 can be observed,
the diagram of equalized data in the fractionally spaced
and the synchronous equalizer respectively. The MSE
graph of the second simulation is presented in Figure 12.
Figure 10 – Diagram of equalized data of the fractionally spaced
equalizer
Figure 11 – Diagram of equalized data of the synchronous equalizer
Due to the importance of the equalization process in
channels with multipath, it is evident the necessity of a
computational tool for the study of the fractionally spaced
equalization. Analyzing the bibliographical references,
many were the advantages in the use of the fractionally
spaced equalizers. It can be worked with symbols with
bandwidth excess, the error in the sampling phase is, in
general, less important than in synchronous equalizers and
the optimum linear receiver can also be realized. After
being realized several simulations with the two kinds of
equalizers, it’s clear the benefit of the fractionally spaced
use. The Figure 9 represents the MSE graph in a
simulation with the two kinds of equalizers. The final
MSE of the fractionally spaced equalizer was
approximately 24.7% smaller than the MSE of the
synchronous equalizer using the configuration of the first
simulation, and 39.8% smaller, using the configuration of
the second simulation (Figures 9 and 12, respectively).
The Figures 7 and 8 show the equalized data diagram. It
can be noticed in the fractionally spaced equalizer graph
the effect of the equalization with a lesser number of
simulated symbols than the synchronous equalizer.
Considering the bibliographical references and the
realized simulations, one concludes that the fractionally
spaced equalizer provides a more efficient equalization
when compared with the synchronous equalizer.
However, the cost for this will be an increase in the
amount of necessary calculations, raising the
computational load.
7. References
[1] Tranter, H. WILLIAM, Theodore, Rappaport S.,
Principles of Communications Systems Simulation with
Wireless Applications, Prentice Hall, USA, 2004.
[2] Fasolo, Sandro Adriano; Equalização em Receptores
de Televisão Digital de Alta Definição Utilizando
Modulação
8
VSB,
Thesis
of
PhD,
Decom\Feec\UNICAMP, 03/2001.
[3] R. D. Gitlin and S. B. Weinstein; Fractionally-spaced
Equalization: Am improved digital transversal equalizer,
Bell Syst. Tech. J., vol. 60, Feb. 1981.
[4] R. D. Gitlin, H. C. Meadors, Jr. and S. B. Weinstein;
The Tap-Leakage Algorithm: An Algorithm for the Stable
Operation of a Digitally Implemented, Fractionally
Spaced Adaptive Equalizer, Bell Syst. Tech. J., vol. 61,
10/1982.
[5] Guimarães, Dayani Adionel; Introdução à Filtragem
Adaptativa, FEEC, Universidade Estadual de Campinas,
SP, 09/1998.
Name
UK Long
Delay
Channel #1
Description
Delay ( µ s )
[6] S. Thomas Alexander; Adaptive Signal Processing –
Theory and Applications, Springer-Verlag, New York,
1996.
Table 1 – Channels used in the simulations
Path 1
Path 2
Path 3
0
5
14
Path 4
35
Path 5
54
Path 6
75
Taps Gain
Phase (Hz)
Delay ( µ s )
1
0°
0
0.3548
0°
1
0.07943
0°
2
0.05623
0°
3
0.04467
0°
4
0.03981
0°
5
Taps Gain
Phase (Hz)
1
0°
-0.2
0°
0.3
0°
0.5
0°
0.1
0°
0.2
0°