Introduction to OFDM Equalization
Sandro Adriano Fasolo and Carlos Augusto Rocha
Telecommunications Department - Inatel - National Institute of Telecommunications
Avenida João de Camargo, 510, Santa Rita do Sapucaı́ - MG - Brasil - CEP 37540-000
[email protected], [email protected]
Abstract— How the equalization technique on Multiple
Carrier Modulation works ? This tutorial aims to present
the basic equalization technique applied in multicarrier
systems. The multicarrier system includes a cyclic prefix
and reference sub carriers in the transmitted signal to
protect it against the multipath channels. First, the cyclic
prefix prevents the intersymbol interference, while the
duration of cyclic prefix will be greater than the delay
of multipaths present in the channel. However, yet there
are the intrasymbol interference and its consequence is the
corruption of the amplitude and phase of each sub carrier
of transmitted signal. This happens because the signal interferes itself and this combination produces modifications
in the amplitude and phase of received signal. To mitigate
these effects, some reference sub carriers are transmitted
with the role of estimating the multipath channel behavior.
The interpolation technique used here for the channel
estimation is the linear interpolation. Therefore, having
the estimated behavior of the channel, the equalization can
be performed by several equalizers composed by a single
tap at each frequency of the received signal. The focus of
the paper is to show the generation of transmitted signal,
how we can include the reference sub carriers and the
cyclic prefix, the effect of multipath channel in the received
channel and the equalization procedure.
Index Terms— multiple carrier modulation, equalization,
orthogonal frequency multiplex division, digital transmission.
1. INTRODUCTION
At this time, using the modulation criterion, the systems for broadcasting of digital television can be separate
in two mayor groups: single carrier modulation (SCM)
and multiple carrier modulation (MCM). As example
of first group we have the american system, known by
ATSC (Advanced Television System Committe). In the
last group, we have the European System, DVB (Digital
Video Broadcasting), and the Japan System, ISDB-T
(Integrated Service Digital Broadcasting). The China is
studying other five systems, employing SCM, MCM
and a combination of MCM and SS technique (SpreadSpectrum). The 802.11a standard also uses the OFDM
(Orthogonal Frequency Division Multiplex). The ATSC
standard uses the 8 VSB (Vestigial Side Band with 8
level). The European and Japan standard use the OFDM
with 2k or 8k carriers. The choose so different in the
physic layer result in a natural concurrency among the
standards and stimulate a comparison between his quality
and deficiency. These differences are results of ATSC
standard uses the modulation in the time domain and the
other ones use the modulation in the frequency domain.
However, the wireless communication have the big
problem, the multipath distortion, that limits the performance in a tragic way. The SCM lets to adaptive
equalizer in the receiver the job of mitigating the multipath effect. Thus, the ATSC employs the Decision
Feedback Equalizer (DFE), an adaptive digital filter in
time domain that eliminates the multipath of channel. It
is known that the convolution in time domain between
impulse response of channel and equalizer must be the
result in the impulse function. The impulse response
of equalizer can be found using algorithms with or
without training sequence (the last one is known by
Blind Equalization). These equalizers are complex, their
implementation is based on digital filter with larger
number of coefficients (taps) and weighed algorithms for
coefficients update. The MCM modulation protects the
signal before the transmission using the cyclic prefix in
time domain and pilots carriers in the frequency domain.
The cyclic prefix avoids the interference of one OFDM
symbols in the OFDM adjacents symbols,i.e., the inter
symbols interference. However, still it will exist the
intra symbol interference. This last interference will be
resolved using the pilots carriers. Instead of one large
equalizer, it will be necessary many equalizers with just
one tap by carrier. In this paper, we present the MCM
modulation and demodulation in section 2. In section
3 is presented the cyclic prefix and the channel model.
The inter symbol interference and intra symbol one is
discussed in section 4 and 5, respectively. The simulation
is presented in section 6, that is performance designed
a multicarrier systems and using graphics, curves and
few simply equations. All the simulation was made on
MatLab Simulator. Finally, the conclusions are listed
in section 7. The authors wait with this tutorial help
the curious about the basics about the mitigation of the
effects of multipath channels in multicarrier signals.
2. OFDM MODEM
The wireless channel is responsible for main interferences and noise of telecommunications system. The
noise is mitigated increase of signal-noise ratio, coding
or choosing the most robust modulation. However, the
multipath distortion transforms a channel plane in a
frequency selective channel, with high inter symbol
interference. In SCM modulation, the channel equalizer
is employed to eliminate the multipath, although a digital
filter at symbol rate. In the SCM modulation, the symbol
occupies all the bandwidth destined for signal, Figure 1
[1].
Fig. 1.
Frequency x time for SCM and MCM.
Since that the symbol rate is high, the symbol duration
will be little, and any multipath will provoke an inter
symbol interference. In the MCM modulation, instead
of transmitting the symbol in serial mode, the idea is
to transmit the symbol in parallel mode. The symbols
in MCM modulation are transmitted in N sub carriers, where the bandwidth of each carrier is the total
bandwidth of signal divided by N. The result is that
each symbol MCM is bigger than the SCM symbol by
N factor. Now, it is possible to add a guard time on the
transmitted signal. In the SCM it is impossible to add
an guard time because it is necessary a guard time with
duration equivalent of many data symbols, the data rate
Fig. 2.
Modem OFDM.
will be low. In the MCM, the data rate will diminish,
but to acceptable values.
The MCM modulation can be transformed into a
frequency selective channel in plane channel although
the decrease of bandwidth of each sub channel. It is
made increase the symbol time duration. Of course, the
characteristics of signal (symbol duration) and channel
(time dispersion), will say whether the channel is or
not selective in frequency. Note that each SCM symbol
occurs in short time, TSCM .occupying all bandwidth.
In MCM modulation, each symbol will be occured
in TM CM = 4TSCM ,where each carrier occupies a
fraction of total bandwidth. How much bigger it will be
the number of sub carriers, bigger will be the symbol
duration and less will be the bandwidth of each sub
carrier. The basic modulator scheme is presented in
Figure 2 ( in practice, the IFFT is used) [2] [3]. This
scheme shows N modulators in quadrature, where each
carrier is modulated using a complex number. The real
part modulates the in-phase component and the imaginary part modulates the quadrature component. Using
different constellation, we can modify the data rate.
The N outputs modulators are summed to produce the
OFDM symbol. The constellation for each carrier can be
different, e. g., in data carrier are used QPSK, 16 QAM
or 64 QAM and BPSK in pilots carrier. Before presenting
the equalization technique, it is necessary to design the
system for simulation. The system will be developed in
base band. Consider the following characteristic:
•
•
Number of carriers: 8.
Duration of SCM symbol: 1ms.
Data carrier constellation: QPSK.
Data carriers: 6.
• Data carriers: 2 (first and last ones).
• Constellation for pilots carrier: BPSK.
• Guard time: 1/4.
The useful time of OFDM symbol (or, simpleness,
symbol) is given by
•
•
TM CM = N × TSCM = 8 × 1 ms = 8 ms
(1)
In the MCM system, the carriers are orthogonal, this implication means that the value for sub carrier frequency
must be multiple integer of OFDM symbol rate, that is:
fn = n
1
TM CM
= n × 125 Hz,
Fig. 4.
h(t)
1
.......
.......
....
.
......
.......
.........
.........
...
...
..
.... .
..................................................................................................
...
...
...
...
..
...
.................................................................................................................
1
τ
n = 0, 1, ..., N − 1
(2)
Note that the amplitude and phase of each carrier are
changed in OFDM symbol at rate of 125 Hz . The useful
bit rate, excluding the pilots carrier are:
Cyclic prefix.
t
Fig. 5.
Impulse response of channel.
carrier
bits
OFDM symbol
2
125
OFDM symbol carrier
second
= 1, 5 kbps
(3)
Rdc = 6
The bit rate for pilots carrier is
carrier
bits
OFDM symbol
1
125
OFDM symbol carrier
second
= 250 bps
(4)
Rpc = 2
The total bit rate is 1, 75 kbps. This rate will be
decreased by introducing of cyclic prefix, explained in
the next section. In Figure 3 it is illustrated the signals
and waveform for one OFDM symbol. The transmission
signal is the summation of co-sines and sinus, modulated
in quadrature and phase by input bits.
3. CYCLIC PREFIX
The cyclic prefix is the repetition in the begin of
OFDM symbol of final part of same OFDM symbol. The
total duration of OFDM symbol will be the useful time
plus cyclic prefix time. In the Figure 4 this scheme is
shown, with cyclic prefix equal 1/4 of useful time. But,
how does it help to mitigate the multipath channel ? First
of all, we must define a channel model to help us in this
explanation. The most simple channel with multipath is
the one composed by two rays, the line of sight and
one multipath with delay. The impulse response of this
channel and its transfer function are given by
h(t) ­ H(ω)
δ(t) + δ(t − τ ) ­ 1 + e−jωτ
and shown in the Figure 5 and Figure 6.
(5)
Fig. 6. Transfer function of response impulse with one direct path
and one multipath with τ = 5 ms.
4. INTER SYMBOL INTERFERENCE
The OFDM communication system has two kinds of
interference: Inter and intra symbol interferences. Figure
7 shows the waveform to direct path, multipath and
received signal using the channel model of last section.
The means of symbols are:
Tu1 Useful time of symbol 1.
Tg1 Cyclic prefix of symbol 1.
0
Tu1
Multipath of useful time of symbol 1.
0
Tg1
Multipath of cyclic prefix of symbol 1.
Tu2 Useful time of symbol 2.
Tg2 Cyclic prefix of symbol 2.
0
Tu2
Multipath of useful time of symbol 2.
0
Multipath of cyclic prefix of symbol 2.
Tg2
Here, is presented the signals to one carrier, but they
can be generalized to other ones. The first symbol hasn’t
any type of interference, thus, it is necessary to use the
second symbol to learn about the protection mechanism
against multipath interference.
0
C ASE Tg2 + Tu1
: If the spread delay of multipath is
less or equal that cyclic prefix duration, στ ≤ Tg , are
Fig. 3.
The MCM symbol.
will not occur interference of anterior OFDM symbol in
the actual OFDM symbol. The anterior OFDM symbol
multipath will just provoke interference in the cyclic
prefix of actual OFDM symbol. In this way, it hasn’t
inter symbol interference. In OFDM modulation it is
possible to add a time guard because the symbol duration
is higher. The impact on symbol rate is acceptable.
In the SCM modulation, it is impossible, because the
time guard will be many times the value for symbol
duration, the impact over symbol rate is unacceptable.
The new rate using the cyclic prefix must be evaluated.
The duration of cyclic prefix, called guard time, is given
by
Tg = 1/4 × Tu = 1/4 × 8 ms = 2 ms
(6)
resulting in OFDM symbol with duration of
Ts = Tu + Tg = Tu (1 + 1/4) = 8 ms + 2 ms = 10 ms
(7)
The new OFDM symbol rate is
Rs = 1/ [Tu (1 + 1/4)] = 1/10 ms = 100 sym/s (8)
Thus, the new bit rate to data carriers is
carrier
bits
OFDM symbol
Rdc = 6
2
100
OFDM symbol carrier
second
= 1, 2 kbps
(9)
Fig. 7.
Signals in antenna receiver.
The new bit rate for pilots carrier is
carrier
bits
OFDM symbol
1
100
OFDM symbol carrier
second
= 200 bps
(10)
Rpc = 2
Generalizing, the new rates can be adjusted by factor:
α=
Tu
Tu + Tg
(11)
5. INTER SYMBOL INTERFERENCE
The repetition of final part of useful symbol has
the objective of generating a continued signal, without
discontinuing.
0
C ASE Tg2 + Tg2
: the objective of this part is also to
avoid the inter symbol interference. In this example, this
part just has intra symbol interference. But this part isn’t
used in the receiver to estimate the transmitted symbol.
0
0
C ASE Tu2 + Tg2
and Tu2 + Tu2
: In the received, the
correlation operation will be evaluated over the useful
symbol, that contains the direct path and multipath
signals. The summation of two (or more) signals with
equal frequency but distinct amplitude and phase result
in other signal with same frequency but amplitude and
phase different of last two. Then, the consequence is
that the demodulated signal will have amplitude and
phase different of original transmitted signal, but with
same frequency. The channel, to produce this effect,
multiplies the transmitted signal by a complex gain, that
modifies the amplitude and phase of each sub carrier
of transmitting signal. This interference can be resolved
using a complex multiplication, inverting the gain and
phase introduced in each sub channel. Consider the
transmission of complex symbol, sn = an + jbn , and
that the frequency response of channel,H(f ) , to carrier
with frequency fn is equal to
H(fn ) = Hn e−jθn
(12)
where Hn is the gain amplitude and θn the phase in each
frequency fn . It can proved that the received complex
symbol, s0n = an + jbn , is given by
a0n = Hn [an cos(θn ) + bn sin(θn )]
b0n = Hn [bn cos(θn ) − an sin(θn )]
(13)
where a0n is the in phase and b0n is the quadrature
values. The analysis of (13) shows that exists mutual
interference between the value of quadrature and phase
components. The effect of channel is a constellation
rotation of each sub carrier, illustrated in phasor diagram
in Figure 8.
Q
b
s
. . . ....
.
.
.
.
..
..
.....
.
.
a
.
.
b ..
θ
a
Fig. 8.
I
I
Diagram of components in phase and quadrature.
Using the trigonometry, it is easy to show that the
received symbol can be found applying a phase rotation
and adjusting in amplitude given by
an = Hn [a0n cos(θn ) − b0n sin(θn )]
bn = Hn [b0n cos(θn ) + a0n sin(θn )]
6. SIMULATION USING CARRIER PILOTS
Using the OFDM system and channel presented in the
last sections, a simulation was made to illustrate these
concepts. The simulation scheme is illustrated in Figure
9. In the Figure 10 is shown the frequency response of
channel together of estimated response frequency using
the linear interpolation. In Figure 11 is shown the transmitted, received and equalized symbol’s constellation.
In Table I is shown the result of simulation and
equalization. In first column is presented the transmitted
symbol and in second columns is presented the received
symbols. The equalization of received symbols is performance using (14) and presented in last column. The
received symbols using simulation are coherent with the
values obtained applying (13) over transmitted symbols.
7. CONCLUSION
Q
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0
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.......
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n
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..................................
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n ......... ...................................... .....
..............................................................................................................................................................................................
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2) Transmission of pilot carriers in each symbol
OFDM: in all symbols are transmitted data and
training data. The estimation is made using the
pilots carrier and interpolating the frequency response among the pilots carrier. The carrier between
two pilots carriers are correct using the interpolating the gain and phase among the pilots carriers.
This last one is the most employed method.
(14)
In the OFDM system, it is necessary to know the gain
in each carrier frequency, Hn , and the phase, θn . Before
the equalization, it is necessary to estimate the frequency
response of channel, using one of these two methods:
1) Transmission of an OFDM symbol to estimate
the frequency response of channel: all the carriers
are used to make the estimation of channel. In
this method, it is possible to know the gain and
phase of channel in each carrier, without error.
But, among the transmission of training symbols,
the frequency response can change. This method
is good to slow fading.
The analysis of result shows the importance of choosing appropriate spacing among pilots carriers. The
distance between two pilots adjacent must be less than
coherence bandwidth of channel. In the linear interpolation we can see the error provoked in amplitude of
response frequency. The phase is estimated without error,
because the linear phase of channel contributes, then
the linear interpolation can get the perfect response.
The amplitude estimate is more difficult, because the
curve is non-linear. In this case, it is necessary to use
other interpolation method, such as, low-pass filter or
polinomial. After estimation of channel, the equalization
can be performed using one complex gain by sub carrier.
This gain adjusts the amplitude and phase of each
received symbol.
ACKNOWLEDGMENT
The authors would like to thank to INATEL and
FINEP/FUNTTEL for the financial support.
REFERENCES
[1] Fasolo, S.A.,Equalização de receptores para televisão digital em
alta definição para a modulação 8 VSB(in Portuguese), State
University of Campinas-UNICAMP. February, 2001.
[2] Bahai, Ahmad R. S.; SaLltzberg, Burton R., Multi-Carrier digital
communications: theory and applications of ofdm. New Jersey,
2001.
[3] Van Nee, Richard And Prasad, Ramjee; Ofdm for wireless multimedia communications. Boston: Artech House, 2000.
Fig. 9.
Simulation scheme.
Fig. 10.
Transfer function from pilots carriers.
Fig. 11.
◦ Transmitted,× received and + equalized symbol’s constellations.
TABLE I
T RANSMITTED , RECEIVED AND EQUALIZED SYMBOLS .
f(Hz)
0
125
250
375
500
625
750
875
Transmitted
Rectangular
Polar
1.000
1.000∠0°
0.707 + 0.707i
1.000 ∠ 45°
-0.707 + 0.707i 1.000 ∠ 135°
-0.707 - 0.707i
1.000 ∠ 225°
0.707 - 0.707i
1.000 ∠ 315°
0.707 + 0.707i
1.000 ∠ 45°
0.707 - 0.707
1.000 ∠ 315°
1.000
1.000 ∠0°
Received
Rectangular
Polar
1.9999
1.9999 ∠0°
1.0898 - 1.6309i
1.9615 ∠ 303.70°
-1.7071 - 0.7071i
1.8477 ∠ 202.50°
-0.3244 + 1.6309i 1.6629 ∠ 101.25°
1.4142
1.4142 ∠ 0°
-0.2168 - 1.0898i
1.1111 ∠ 258.75°
0.7071 - 0.2929i
0.7653 ∠ 337.50°
0.0761 - 0.3827i
0.3902 ∠ 281.25°
Equalized
Rectangular
Polar
0.9998
0.9998 ∠ 0°
0.7835 - 0.7835i
1.1080 ∠ -0.7854°
-0.8483 - 0.8483i
1.1996 ∠ -2.3562°
-0.8974 + 0.8974i
1.2692 ∠ 2.3562°
0.9257 + 0.9257i
1.3091 ∠ 0.7854°
0.9241 - 0.9241i
1.3068 ∠ -0.7854°
0.8725 + 0.8725i
1.2340 ∠0.7854°
0.9998
0.9998 ∠ 0°