A Comparison of the Performance of Linearly Equalized Single Carrier And
Coded OFDM Over Frequency Selective Fading Channels Using the
Random Coding Technique
Volker Aue and Gerhard P. Fettweis
Dresden University of Technology, Germany
Reinaldo Valenzuela
Lucent Technologies, Holmdel, NJ, USA
Abstract: Different attempts have been taken to compare
performances of coded orthogonal frequency division multiplexing (COFDM) and a corresponding single carrier system
employing a linear equalizer often by comparing average bit
error rates. In this paper, the random coding technique is
used to analyze the performances of these systems for signaling over piecewise constant channels. The probabilities that
the cutoff rates are less than a desired rate are calculated
and compared. It is shown that for low SNRs and low code
rates, the cutoff rates of both systems are approximately the
same. With an increasing SNR the cutoff rate for the linearly
equalized system, however, converges faster to its maximum
value than the cutoff rate for the COFDM system. Simulation results for outage probability are presented for the
two-ray Rayleigh fading channel with QPSK as the fundamental modulation technique. For code rates close to one or
for uncoded transmission, the linearly equalized single carrier
system substantially outperforms the COFDM system. For
low to medium code rates COFDM and a linearly equalized
single carrier system perform approximately identically.
I. Introduction
Orthogonal frequency division multiplexing has received
a lot of attention, as a method for combating intersymbol
interference (ISI) due to signaling over time-dispersive communications channels. The conventional single carrier system employs an equalizer at the receiver which attempts to
mitigate the multipath effects by reconstructing the transmitted sequence. Linear equalization, decision feedback
equalization (DFE) or maximum likelihood sequence estimation (MLSE) are most commonly used [1]. The principal idea behind OFDM is to transmit blocks of data over a
piecewise constant channel, i.e., the channel is assumed to
be stationary during the transmission of a block. A stream
of high rate data is demultiplexed into several streams of
lower rate which are then transmitted in parallel over a set
of orthogonal carriers. If a sufficient number of carriers is
used, each subchannel can be regarded flat. A guard interval
consisting of a cyclic prefix (suffix) larger than the maximum
delay span precedes (succeeds) each symbol such that all ISI
is removed. The optimum carrier assignment makes use of
“water pouring” [2], but, as the channel response needs to
be known to the transmitter, feedback from the receiver is
required. Zervos and Kalet [3] give a theoretical evaluation
of this technique assuming a uniform power distribution. It
is shown that for high signal-to-noise ratios (SNRs) the per-
formance of OFDM is approximately that of a single carrier
system using a DFE [3,4]. Willing and Wittke [5] show that
OFDM can outperform a single carrier system with DFE,
if the constraint that the bit error rate (BER) is uniformly
distributed over carriers is removed and a different optimization method is used. A general treatment on DFE including
blockwise transmission using cyclic prefixes is given in [6].
A basic evaluation of the probability of system outage for
a typical office-building indoor environment of OFDM with
different carrier assignments has been carried out by Cimini
[7].
If no knowledge about the channel response can be obtained by the transmitter, e.g., for rapidly fading channels
or in a broadcast environment, the “water pouring” concept
is infeasible. Henceforth, a way to reduce the probability of
bit error is to use extensive coding to provide the redundancy
needed to overcome losses which occur on carriers where the
frequency response is subject to deep fades. This technique
usually referred to as coded OFDM (COFDM) has been proposed for different broadcasting systems [8] and is currently
under investigation for mobile wireless systems [9,10]. Sari et
al. have investigated the performance of a COFDM system
over two different frequency selective channels and report a
performance similar to that of an equivalent single carrier
system employing a linear equalizer. It is also noted by the
same authors that channel coding is a necessity in COFDM
whereas in a single carrier system it is not.
The random coding technique [11] provides a general
means to compare performances of systems which use channel coding. For environments that vary over time or place,
as it is the case for many wireless applications, it is meaningful to consider the probability that a certain requirement is
met (or not met) rather than looking at average error rates.
Assuming the channel is constant during the transmission
period of a block, the probability that the cutoff rate is less
than a desired rate then corresponds to the probability of
detecting erroneous blocks. If locations of transmitter and
receiver are fixed, e.g. in fixed wireless, this probability directly corresponds to the probability of system outage for
the anticipated code rate.
In the following section we first derive general expressions
for the cutoff rate for both systems. For a simple frequencyselective fading model, the two-ray fading channel, further
results are derived. For different code rates the probability
of outage is then evaluated by means of simulations. Results
are shown and discussed.
channel
encoder
symbol
mapping
interleaver
IFFT
add guard
interval
channel
channel
decoder
deinterleaver
FFT
remove
guard interval
(a) COFDM system model.
channel
encoder
symbol
mapping
add guard
interval
interleaver
channel
channel
decoder
deinterleaver
IFFT
equalizer
FFT
remove
guard interval
(b) Single carrier system model.
Fig. 1. System models.
II. Analysis
In order to be as general as possible we neglect implementation issues as we assume perfect interleaving, perfect
synchronization, and that the receiver has perfect knowledge
of the channel. Coherent detection is assumed. The model
for the COFDM system is shown in Fig. 1(a). The corresponding model for the linearly equalized single carrier system for which equalization is carried out in the frequency
domain is shown in Fig. 1(b). Both systems use a block signaling approach, where each block precedes a cyclic prefix.
Furthermore it is assumed that the channel is constant over
the regarded time interval. Thus, the received signal can be
viewed as being cyclicly convolved with the channel impulse
response.
As we further generalize our approach we assume an infinite block length, i.e., an infinite number of carriers for the
COFDM system, and an infinite number of symbols for the
single carrier system. For each transmitted block the overhead introduced by the guard interval is neglected as it is
identical for both systems. Furthermore, it is assumed that
the linear equalizer has infinite length or directly operates
on the channel frequency response.
A. Analysis Using the Random Coding Technique
The random coding technique is based on bounding the
average probability of bit error over all possible block codes.
The cutoff rate can be defined as a parameter of this bound
for which the ensemble average probability of error over all
possible block codes goes to zero as the block length reaches
infinity. The advantage of comparing cutoff rate rather than
capacity is that the former additionally takes the modulation
technique into consideration.
For q-ary signaling and for the AWGN channel, the cutoff
rate is derived from an exponential bound [11]. If all symbols
are equally likely, the cutoff rate is found to be
( q−1 q−1
)
1
1 XX
2
R0 = − log2
,
exp −
|sl − sk |
q2
4N0
(1)
l=0 k=0
where |sl − sk |2 is the Euclidean distance between a symbol
pair sk , sl . N0 is the noise power spectral density. For a flat
fading AWGN channel, the derivation can be carried out in
an analogous manner [12], and it can be shown that
( q−1 q−1 )
α2
1 XX
R0 = − log2
,
E exp −
|sl − sk |2
q2
4N0
l=0 k=0
(2)
where α is a random variable according to the distribution
of the magnitude of the normalized channel. E{·} denotes
expectation.
Derivation of the Cutoff Rates for COFDM and Single Carrier
(1) and (2) form the basis of our comparison. For the
COFDM system, each subchannel can be viewed as a flat
AWGN channel. Assuming perfect interleaving, all channel
coefficients can be assumed independent and identically distributed. Thus, we can use (2) to calculate the cutoff rate for
the system shown in Fig. 1(a). The remaining task is to find
the probability density function (pdf) of the channel power
density spectrum |H(f )|2 . For some channel models, the pdf
can be described analytically and (2) can be calculated in
closed form. In Section II.B we derive an expression for the
two-ray channel.
In a single carrier system each transmitted symbol occupies the entire bandwidth and thus each symbol is subject
to the same fading. The two major criteria for adjusting the
filter coefficients are zero ISI or minimum mean square error
(MMSE). The MMSE criterion trades off noise enhancement
for residual ISI, and is known to yield a better performance
in channels which exhibit deep spectral fades. For an infinite tap linear filter output the MMSE is well known [1]. The
MMSE can then be used to calculate the SNR at the output
of the equalizer [13]. For the linear equalizer, the SNR at
the equalizer output is
γe =
n
E
1
1
1+γ|H(f )|2
o − 1,
(3)
Es
is the received SNR. Es is the average symwhere γ = N
0
bol energy. If we assume perfect interleaving and a Gaussian
distribution of the output noise (3) can be used in (1) to
compute the cutoff rate for the coded linearly equalized single carrier system. We are aware that, in general, the residual
ISI is not Gaussian distributed. By means of simulating the
uncoded single carrier system employing the linear equalizer
with MMSE criterion, however, we verified that the Gaussian distribution is a valid assumption for all channels under
consideration. As the exponential bound is well above the
Q-function which gives the exact result for the error integral,
we can assume the validity of (1).
For the zero-forcing solution, the output noise is exactly
Gaussian distributed, and the output SNR γe is
γe = γ
n
E
1
1
|H(f )|2
o.
(4)
B. Special Case: Two Ray Channel
h(t) = h1 e
δ(t) + h2 e
jϕ2
δ(t − τd ),
(5)
where h1 , h2 ≥ 0 are the magnitudes of the two rays. τd is
the delay of the second path with respect to the first path.
The frequency response of this channel is given by
H(f ) = h1 ejϕ1 + h2 ej(ϕ2 −2πf τd )
(6)
and its power spectral density is
|H(f )|2 = h21 + h22 + 2h1 h2 cos(2πτd f + ϕ1 − ϕ2 ).
1
1
q
2 ,
bπ
1 − x−a
b
a − b ≤ x ≤ a + b,
(8)
h21 + h22
and b = 2h1 h2 . We introduce the normalwhere a =
−sk |2
ized pairwise distance parameter dl,k = |sl4E
and calculate
s
the expectations in (2) and (3).
For COFDM we find
E exp −γ|H(f )|2 dl,k
Z a+b
1
1
e−xγdl,k q
=
dx
bπ a−b
x−a 2
1− b
Z 1
(9)
1
1
e−(a+bξ)γdl,k p
dξ
=
π −1
1 − ξ2
= e−aγdl,k I0 (bγdl,k )
where we have set ξ = x−a
b .
For the single carrier system
1
E
1 + γ|H(f )|2
Z a+b
1
1
1
q
=
2 dx
bπ a−b 1 + γx
1 − x−a
b
Z 1
1
1
1
p
dξ
=
(10)
π −1 1+aγ + ξ
1 − ξ2
bγ
= r
1
1+aγ
bγ
2
.
−1
Finally, we obtain the general expressions for the cutoff rates
for the two-ray channel. For COFDM
( q−1 q−1
)
1 X X −aγdl,k
e
I0 (bγ dl,k ) .
(12)
R0 = − log2
q2
and for the linearly equalized single carrier system
R0 =
(
−log2
)
q−1 q−1
i
h p
1 XX
2
2
2
.
exp −
(1 + aγ) − b γ − 1 dl,k
q2
l=0 k=0
(13)
Discussion
√
For simplicity we set xi = hi γ, i = {1, 2}, and define ã =
x21 + x22 and b̃ = 2x1 x2 , and consider the terms
(7)
If the bandwidth occupied by the signal is large with respect
to 1/τd , the distribution of |H(f )|2 is
px (x) =
(11)
l=0 k=0
For the two-ray fading channel (1) and (2) can be calculated in closed form as shown as follows. The channel impulse
response for the piecewise constant channel is
jϕ1
Substituting the result in (3) yields
p
γe = (1 + aγ)2 − b2 γ 2 − 1.
A(ã, b̃, dl,k ) = e−ãdl,k I0 (b̃ dl,k )
(14)
and
q
2
2
B(ã, b̃, dl,k ) = exp −
(1 + ã) − b̃ − 1 dl,k
(15)
The cutoff rate increases as these terms decrease. As expected, in case of a single path, either h1 or h2 are zero, and
thus b̃ = 0, for which (12) and (13) yield the same result. For
b̃ dl,k close to zero, I0 (b̃ dl,k ) ≈ 1 and (1 + ã)2 − b̃2 ≈ (1 + ã)2 ,
and hence A ≈ B. For b̃ and b̃ dl,k large, we can use the
asymptotic formula
1
I0 (b̃ dl,k ) ≈ q
eb̃dl,k ,
2π b̃dl,k
(16)
for which
1
A(ã, b̃, dl,k ) ≈ q
e−(ã−b̃)dl,k
2π b̃dl,k
It is easily verified that the negative exponent in B is larger
than (ã − b̃) dl,k . Therefore, it is expected that B < A for b̃
large1 , i.e., we can expect the cutoff rate to converge faster
to its maximum value for the linear equalizer, as the SNR
increases. In case that both rays carry the same power, i.e.,
whereas B
h1 = h2 , b̃ = ã, A converges only with √ 1
converges with e−
√
2ãdl,k
2πãdl,k
for b̃ large. Notethat the equivalent
q
2
2
term for the zero-forcing equalizer is exp − (ã − b̃ ) dl,k
which equals one for ã = b̃ and does not converge.
For dl,k = 1, A(x1 , x2 ) and B(x1 , x2 ) are analyzed numer1 ,x2 )
ically. The ratio B(x
A(x1 ,x2 ) is shown in Fig. 2. It is seen that
1
In all practical systems, the normalized Euclidean distance dl,k is considerably larger than zero. Thus, it is sufficient to assume b̃ large.
probability that cutoff rate is less than desired rate
1E+00
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
x1
LE R=0.01
OFDM R=0.01
LE R=1/2
OFDM R=1/2
LE R=3/4
OFDM R=3/4
LE R=0.999
OFDM R=0.999
1E-01
1E-02
0
3
0.5
0
1.5
1
2
2.5
B(x1 ,x2 )
.
A(x1 ,x2 )
for x1 , x2 large, B(x1 , x2 ) converges considerably faster to
zero than A(x1 , x2 ) does as the ratio goes to zero as either
x1 or x2 become large. For small values of x1 , x2 and for
x1 = x2 , B(x1 , x2 ) ≈ A(x1 , x2 ). It can be seen that for values around x1 , x2 ≈ 0.6, A(x1 , x2 ) > B(x1 , x2 ). It has been
found numerically that the maximum relative deviation occurs at xi = 0.654, for which A(x1 , x2 ) exceeds B(x1 , x2 ) by
3.4 percent. Thus, if a high likelihood exists for x1 , x2 to
be in that range, the cutoff rate for COFDM can exceed the
cutoff rate of the single carrier system. We can conclude that
for low SNRs, the cutoff rate of the COFDM system can be
expected to be similar to the cutoff rate of the equivalent
linearly equalized single carrier system using the MMSE criterion. For higher SNRs, the cutoff rate of the single carrier
system can be expected to be higher.
C. Results for QPSK
As an example, we consider the cutoff rates for COFDM
and the equivalent single carrier system with quarternary
phase shift keying (QPSK) for signaling over two-ray
Rayleigh fading channels, and calculate the probability that
the cutoff rate is below a desired rate, i.e., the outage probabilities of these ideal systems. For QPSK, the cutoff rate for
COFDM is
R0 = − log2
1 1 h − 12 γ|H(f )|2 i 1 h −γ|H(f )|2 i
+ E e
+ E e
,
4 2
4
(17)
and for the two-ray channel,
R0 = − log2
1 1 − 1 γa
+ e 2 I0
4 2
10
15
20
25
SNR per bit [dB]
30
35
40
Fig. 3. Two-ray Rayleigh fading channel.
x2
Fig. 2. Ratio
5
3
1
1
γb + e−γa I0 (γb) .
2
4
(18)
For the linearly equalized single carrier system, the cutoff
rate is given by
1 1 − 1 γe 1 −γe
+ e 2 + e
,
(19)
R0 = − log2
4 2
4
where γe is given by (3) and (11), respectively. For the tworay Rayleigh fading channel, the magnitudes h1 and h2 have
the pdf
2
ph (h) =
h
h − 2σ
2
h ,
e
σh2
h ≥ 0,
(20)
where σh2 = 1/4 for the normalized channel, i.e., for E(h21 +
h22 ) = 1.
Here, we consider code rates of Rc = 0.01, 1/2, 3/4 and
0.999. Rc = 1/2 and 3/4 are chosen as an example for code
rates which are often associated with convolutional codes.
Rc = 0.999 in practice corresponds to systems that have to
operate with little or without channel coding. Rc = 0.01
has been chosen for the limiting case where almost all transmitted bits are used for coding. The desired transmission
rate (in bits/dimension) is Rd = 2Rc , since two coded bits
are transmitted on each QPSK symbol. The symbol energy
Es = 2Rc Eb , where Eb is the bit energy.
The results obtained by Monte Carlo simulations are
shown in Fig. 3. As expected, for the very low code rate
(Rc = 0.01) the performance of COFDM and the equivalent
single carrier system is equal. For the code rate of 1/2, the
COFDM system outperforms the single carrier system by an
insignificant factor. For an SNR per bit of 15 dB, the cutoff rate of the COFDM system falls only in 1 percent of all
cases below the desired rate. For a code rate of 3/4, 1 dB
SNR more is required for the COFDM system for the same
outage probability. At the other extreme where almost no
channel coding is used, the single carrier system can operate
at 20 dB SNR per bit whereas for the COFDM system 33 dB
are needed. This confirms the well known fact that OFDM
without “water pouring” should not be used without coding
[8].
We have also computed the outage probabilities of
COFDM and single carrier with QPSK over three and five
ray Rayleigh fading channels with same average power, i.e.,
Z ∞
E
|h(t)|2 dt = 1.
is approximately the same as of the COFDM system. The
overall system complexity further depends on synchronization issues, channel estimation, decoder complexity and on
linearity requirements for the amplifiers.
For these channels, (17) and (3) have been computed numerically. The results show the same characteristics as the
results for the two-ray channel of Fig. 3 with the only difference that the required SNR per bit for a desired outage
probability is less than for the two-ray Rayleigh fading channel. This, however, is expected, as the probability, that a
signal level falls below a certain threshold decreases as the
number of rays increases.
The authors gratefully thank G. J. Foschini for his suggestions on carrying out this work.
−∞
D. Other Signal Sets
Employing signal sets other than QPSK as the fundamental modulation technique, e.g., to enable trellis coded modulation can usually yield different cutoff rates for both systems. As the basic terms (14) and (15) are independent
of the modulation technique, it is expected that the behavior of COFDM with regard to the single carrier system
does not change significantly. Moving more points ã, b̃, dl,k
for which A(ã, b̃, dl,k ) ≈ B(ã, b̃, dl,k ) into a region for which
B(ã, b̃, dl,k ) < A(ã, b̃, dl,k ) can increase the difference in cutoff rate between the COFDM system and the single carrier
system.
III. Conclusions
We have analyzed the performance of coded OFDM and
the corresponding linearly equalized single carrier system
for piecewise constant frequency-selective channels using the
random coding technique for which we calculated the probabilities that the cutoff rate is less than a desired threshold.
For linear equalizer, the MMSE criterion should be used as
the preferred criterion for adjusting the filter coefficients.
We have shown that despite the differences in signaling
and signal processing, the performance of both systems is
approximately identical for low to medium code rates. For
code rates close to one, the linearly equalized system using
the MMSE criterion substantially outperforms the COFDM
system. This, however, is expected, since in a single carrier
system, the SNR is evenly distributed over symbols whereas
in a COFDM system it is not. Hence, a high error probability exists for symbols on carriers for which the SNR is low.
Simulations showed that the outage probability is approximately the same for both systems and for code rates up to
one half. Further simulations reveal that these results can
be generalized for channels with more than two rays.
Thus, it is shown that for low SNRs preferring COFDM
over a linearly equalized single carrier system is an implementation issue. If less transmitted bits are to be used for
coding or at high SNRs, a single carrier system employing
a linear equalizer with the MMSE criterion can perform significantly better, and should be the preferred solution.
As the linear equalizer can be implemented in the frequency domain the complexity of the single carrier system
Acknowledgments
References
[1] S. U. H. Qureshi, “Adaptive equalization,” Proceedings
of the IEEE, vol. 53, pp. 1349–1387, Sept. 1985.
[2] J. A. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Communications Magazine, pp. 5–22, May 1990.
[3] N. A. Zervos and I. Kalet, “Optimized decision feedback
equalization versus optimized orthogonal frequency division multiplexing for high-speed data transmission
over the local cable network,” IEEE International Conference on Communications, vol. 2, (Boston), pp. 1080–
1085, June 1989.
[4] I. Kalet, “The multitone channel,” IEEE Transactions
on Communications, vol. 37, no. 3, pp. 119–124, Feb.
1989.
[5] T. J. Willink and P. H. Wittke, “Optimization and
performance evaluation of multicarrier transmission,”
IEEE Transactions on Information Theory, vol. 43,
no. 2, pp. 426–440, Mar. 1997.
[6] J. M. Cioffi and G. D. Forney, “Canonical packet transmission on the ISI channel with Gaussian noise,” Proc.
of IEEE GLOBECOM 1996, vol. 2, (London), pp. 1405–
1410, Nov. 1996.
[7] L. J. Cimini, Jr., “Performance studies for high-speed
indoor wireless communications,” Wireless Personal
Communications, vol. 2, pp. 67–85, Kluwer Academic
Publishers, 1995.
[8] H. Sari, G. Karam, and I. Jeanclaud, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Communications Magazine, vol. 33, no. 2,
pp. 100–109, Feb. 1995.
[9] J. P. Aldis, M. P. Althoff, and R. Van Nee, “Physical
layer architecture and performance in the WAND user
trial system,” ACTS Mobile Communications Summit
’96, pp. 196–203, Nov. 1996.
[10] B. Stantchev, J. Kuehne, M. Bronzel, and G. Fettweis,
“An integrated FSK-signaling scheme for OFDM-based
advanced cellular radio,” Proceedings of the 47th IEEE
Vehicular Technology Conference, vol. 3, pp. 1629–1633,
Apr. 1997.
[11] J. M. Wozencraft and I. M. Jacobs, Principles of communication engineering. New York: Wiley, 1965.
[12] S. H. Jamali and T. Le-Ngoc, Coded-Modulation Techniques For Fading Channels. Kluwer Academic Publishers, 1994.
[13] J. G. Proakis, Digital Communications. McGraw-Hill,
second ed., 1989.