Table Of ContentCoded OFDM by Unique Word Prefix
Mario Huemer, Senior Member, IEEE, and Christian Hofbauer Johannes B. Huber, Fellow, IEEE
Klagenfurt University University of Erlangen-Nuremberg
Institute of Networked and Embedded Systems Institute for Information Transmission
Universitaetsstr. 65-67, 9020 Klagenfurt Cauerstr. 7, D-91058 Erlangen
[email protected], [email protected] [email protected]
Abstract—In this paper we propose a novel transmit signal
structure and an adjusted and optimized receiver for OFDM
(orthogonal frequency division multiplexing). Instead of the
0 conventional cyclic prefix we us a deterministic sequence, which
1 we call unique word (UW), as guard interval. We show how
0 unique words, which are already well investigated for single
2 carrier systems with frequency domain equalization (SC/FDE),
can also be introduced in OFDM symbols. Since unique words
n
representknownsequences,theycanadvantageouslybeusedfor
a synchronization and channel estimation purposes. Furthermore, Fig.1. TransmitdatastructureusingCPs(above)orUWs(below).
J
the proposed approach introduces a complex number Reed-
8 Solomon(RS-)codestructurewithinthesequenceofsubcarriers.
ThisallowsforRS-decodingortoapplyahighlyefficientWiener • The UW is part of the DFT (discrete Fourier transfom)-
T] smoother succeeding a zero forcing stage at the receiver. We interval, whereas the CP is not.
present simulation results in an indoor multipath environment
I tohighlighttheadvantageouspropertiesoftheproposedscheme. • The CP is random, whereas the UW is a known deter-
s. ministicsequence.ThereforetheUWcanadvantageously
c beutilizedforsynchronization[4]andchannelestimation
[
I. INTRODUCTION purposes [5].
1 Both statements hold for OFDM- as well as for SC/FDE-
v In conventional OFDM signaling, subsequent symbols are
systems.However,inOFDM-differenttoSC/FDE-theintro-
8 separated by guard intervals, which are usually implemented
duction of UWs in time domain leads to another fundamental
9 as cyclic prefixes (CPs) [1]. By this, linear convolution of
2 and beneficial signal property:
the signal with the channel impulse response is transformed
1 into a cyclic convolution which allows for a low complex • A UW in time domain generates a word of a complex
.
1 equalization in frequency domain. In this paper, we propose number RS-code (cf. e. g. [6]) or of a specific coset to
0 such a code in frequency domain, i.e along the subcar-
to use known sequences, which we call unique words (UWs),
0 riers. Therefore, the UW could be exploited for error
instead of cyclic prefixes. The technique of using UWs had
1
correction or (more appropriate) for erasure correction
: alreadybeeninvestigatedin-depthforSC/FDEsystems,where
v for highly attenuated subcarriers. Another interpretation
the introduction of unique words in time domain is straight
Xi forward [2], since the data symbols are also defined in time of this fact which we prefer here, is an introduction of
correlations along the subcarriers. These correlations can
r domain. In this paper, we will show how unique words can
a advantageously be used as a-priory knowledge at the
be introduced in OFDM time domain symbols, even though
receivertosignificantlyimprovetheBER(biterrorratio)
thedataQAM(quadratureamplitudemodulation)symbolsare
behavior in frequency selective environments.
defined in frequency domain. Furthermore, we will introduce
an optimized receiver concept adjusted to the novel transmit The rest of the paper is organized as follows: In section II
signal structure. we describe our approach of how to introduce unique words
Figure 1 compares the transmit data structure of CP- and in OFDM symbols. Section III introduces an LMMSE (linear
UW-based transmission in time domain [3]. Both structures minimum mean squared error) receiver that exploits the a-
make sure that linear convolution of an OFDM symbol with prioryknowledgeintroducedatthetransmitterside.Insection
the impuls response of a dispersive (e.g. multipath) channel IV the novel UW-OFDM concept is compared to the classical
appears as a cyclic convolution at the receiver side. Neverthe- CP-OFDM by means of simulation results. For this, the IEEE
less,therearealsosomefundamentaldifferencesbetweenCP- 802.11aWLAN(wirelesslocalareanetworks)standardserves
and UW-based transmission: as reference system.
Notation
Christian Hofbauer has been funded by the European Regional Develop-
Lower-case bold face variables (a,b,...) indicate vectors,
mentFundandtheCarinthianEconomicPromotionFund(KWF)undergrant
20214/15935/23108. andupper-caseboldfacevariables(A,B,...)indicatematrices.
To distinguish between time and frequency domain variables, C(Nd+l)×(Nd+l), and form an OFDM symbol (containing
we use a tilde to express frequency domain vectors and N −N −l zero-subcarriers) in frequency domain by
d
matrices(˜a,A˜,...),respectively.WefurtheruseCtodenotethe (cid:20)d˜(cid:21)
set of complex numbers, I to denote the identity matrix, (·)T x˜=BP . (5)
˜r
todenotetransposition,(·)H todenoteconjugatetransposition,
and E[·] to denote expectation. Wewilldetailthereasonfortheintroductionofthepermution
matrix and its specific construction shortly below. Figure 2
II. GENERATIONOFUNIQUEWORDSINOFDMSYMBOLS illustrates this approach in a graphical way: The input of the
In conventional CP-OFDM, the data vector d˜ ∈ CNd×1 is IDFT block is composed of data subcarriers (d˜), zero sub-
carriers, and redundant subcarriers (˜r), which are distributed
defined in frequency domain. Typically, zero subcarriers are
over the entire non-zero part of vector x˜ as specified by the
inserted at the band edges and at the DC-subcarrier position,
permutation matrix P. The output of the IDFT block, which
which can mathematically be described by a matrix operation
corresponds to the vector x of time domain samples of an
x˜=Bd˜ (1) OFDM symbol, is composed of the random part xd, and the
zero UW 0.
with x˜ ∈ CN×1 and B ∈ CN×Nd. B consists of zero-rows
at the positions of the zero-subcarriers, and of appropriate
unit row vectors at the positions of data-subcarriers. The
vector x˜ denotes the OFDM symbol in frequency domain.
The vector of time domain samples x ∈ CN×1 is calculated
via an IDFT (inverse DFT) operation, which can conveniently
be formulated in matrix notation by x = F−1x˜. Here, F
N N
is the N-point-DFT matrix defined by F = (F ) with
N mn
F =wmn for m=0,1,...,N−1, n=0,1,...,N−1, and
mn
with w =e−j2π/N.
We now modify this conventional approach by introducing Fig. 2. Time- and frequency-domain view of an OFDM symbol in UW-
a pre-defined sequence x with x ∈ Cl×1, which we call OFDM.
u u
uniqueword,andwhichshallformthetailofthetimedomain
vector,whichwenowdenotebyx(cid:48).Hence,x(cid:48) consistsoftwo By inserting equation (5) into (4), the relation between the
partsandisgivenbyx(cid:48) =(cid:2)xT xT(cid:3)T,wherex ∈C(N−l)×1 time and the frequency domain representation of the OFDM
and x ∈ Cl×1. The vector dx reupresents the UdW of length symbol can be written as
u u
l, and thus only xd is random and affected by the data. In F−1BP(cid:20)d˜(cid:21)=(cid:20)xd(cid:21). (6)
order to simplify subsequent descriptions, but w.l.o.g. we use N ˜r 0
a two-step approach for the so-defined vector x(cid:48) :
With
• In a first step we will generate a zero UW (cid:20)M M (cid:21)
M=F−1BP= 11 12 , (7)
(cid:20)x (cid:21) N M21 M22
x= d , (2)
0 where M are appropriate sized sub-matrices, it follows that
ij
M d˜+M ˜r = 0, and hence ˜r = −M−1M d˜. With the
such that x=F−1x˜. 21 22 22 21
N matrix
• In a second step we will determine the transmit symbol T=−M−1M (8)
22 21
by
(cid:20)0(cid:21) (T∈Cl×Nd), the vector of redundant subcarriers can thus be
x(cid:48) =x+ . (3)
x determined by the linear mapping
u
We now describe the first step in detail: As in conventional ˜r=Td˜, (9)
OFDM, the QAM data symbols and the zero-subcarriers are
which (despite of the permutation) exactly corresponds to a
specifiedinfrequencydomaininvectorx˜,buthereinaddition
complex number RS-code construction along the subcarriers
thezero-wordisspecifiedintimedomainaspartofthevector
(i.e. l subsequent zeros in the transform domain). We sum-
x. As a consequence, the linear system of equations
marize, that the IDFT of a frequency domain vector as given
x=F−1x˜ (4) in equation (5), which is composed of a data part d˜, a set of
N
zero-subcarriers, and a part˜r of redundant subcarriers, which
can only be fulfilled by reducing the number N of data isdeterminedbyequation(9),resultsinatimedomainvector,
d
subcarriers, and by introducing a set of redundant subcarriers which features the zero UW 0 at its tail.
instead. We let the redundant subcarriers form the vector We notice that the construction of T, and therefore also
˜r ∈ Cl×1, further introduce a permutation matrix P ∈ the variances of the redundant subcarriers highly depend on
the positions of the redundant subcarriers within the entire We mention, that the influence of the UW could also already
frequency domain vector x˜. Hence, the permutation matrix P be eliminated earlier, by simply subtracting H˜BTx˜ from y˜.
u
hastobechosencarefully.WeselectPsuchthattrace(cid:0)TTH(cid:1) ˜s contains the data as well as the redundant subcarrier
becomesminimum[7].Thisprovidesminimumenergyonthe symbols. Since the redundant subcarrier symbols have been
redundant subcarriers on average (when averaging over all calculated out of the data symbols by equation (9), they
possible data vectors d˜). In section IV we will specify the are correlated with the data symbols and among each other.
permutation matrix P for our simulated system setup. Because of that we propose to apply an LMMSE Wiener
Notethatequation(9)introducescorrelationinthevectorx˜ smoother [8] on y˜(cid:48)(cid:48), which results in the noise reduced
of frequency domain samples of an OFDM symbol. This can estimate
advantageously be utilized in an optimized receiver structure ˜(cid:98)s=Cs˜s˜(Cs˜s˜+Cv˜v˜)−1y˜(cid:48)(cid:48), (15)
as it will be shown in the next section. In the following we
use the notation˜s with where Cs˜s˜,Cv˜v˜ ∈ C(Nd+l)×(Nd+l) denote the covariance
matrices of ˜s and v˜, respectively. Let us take a closer look
(cid:20)d˜(cid:21) (cid:20)I(cid:21)
˜s=P =P d˜ =Ud˜, (10) on these covariance matrices:
˜r T
C = E(cid:2)˜s˜sH(cid:3)
(˜s∈C(Nd+l)×1,U∈C(Nd+l)×Nd)forthenon-zeropartofx˜, s˜s˜ (cid:104) (cid:105)
such that x˜=B˜s. = E (Ud˜)(Ud˜)H
In the second step the transmit symbol x(cid:48) is generated by = E(cid:104)Ud˜d˜HUH(cid:105)
adding the unique word as described in equation (3). The
frequency domain version x˜u ∈ CN×1 of the UW is defined = UE(cid:104)d˜d˜H(cid:105)UH. (16)
by x˜ =F (cid:2)0T xT(cid:3)T. Note that x(cid:48) can also be written as
u N u
x(cid:48) =F−1(x˜ +x˜)=F−1(x˜ +B˜s). Assuming uncorrelated and zero-mean data QAM symbols
N u N u with variance σ2, we obtain a constant matrix
d
III. LMMSEUW-OFDMRECEIVER
C =σ2UUH, (17)
At the receiver side the UW may be exploited for error s˜s˜ d
and/or erasure correction as usual for RS-codes. Here, we that has to be calculated only once and can be determined in
restrict the explanations to an exploitation of correlations advance, cf. equations (8), (10). For C we have
v˜v˜
between subcarriers which proofed to be more appropriate,
C = E(cid:2)v˜v˜H(cid:3)
since the redundant subcarrier symbols - unlike the data v˜v˜
(cid:104) (cid:105)
symbols - usually do not fit to a discrete grid (e.g. odd = E (H˜−1BTF n˜)(H˜−1BTF n˜)H
N N
Gaussian integers). After the transmission over a multipath
(cid:104) (cid:105)
channel and after the common DFT operation, the non-zero = E H˜−1BTF n˜n˜HFHB(H˜−1)H
N N
part y˜ ∈ C(Nd+l)×1 of a received OFDM frequency domain = H˜−1BTF E(cid:2)n˜n˜H(cid:3)FHB(H˜−1)H
symbol can be modeled as N N
= σ2H˜−1BTF FHB(H˜−1)H
y˜ =BTF HF−1(x˜ +B˜s)+BTF n, (11) n N N
N N u N = Nσ2H˜−1BTB(H˜−1)H
n
where H denotes a cyclic convolution matrix with H ∈ = Nσ2H˜−1(H˜−1)H. (18)
CN×N, and n∈CN×1 represents a noise vector with the co- n
variancematrixσ2I.ThemultiplicationwithBT excludesthe Let H(f ) with i = 0,1,...,N +l −1 denote the channel
n i d
zerosubcarriersfromfurtheroperation.ThematrixF HF−1 frequency response at the corresponding subcarrier frequency
N N
is diagonal and contains the sampled channel frequency re- f . Now equation (18) can also be written as
i
sH˜pon∈seCo(Nnd+itls)×m(Nadi+nl)diiasgaondaol.wHn˜-siz=edBveTrFsiNonHoFf−Nt1hBe lwatittehr C =Nσ2diag(cid:40) 1 , 1 ,...(cid:41). (19)
excluding the entries corresponding to the zero-subcarriers. v˜v˜ n |H(f0)|2 |H(f1)|2
The received symbol can therefore also be written as
C depends on the noise variance σ2 and on the channel
v˜v˜ n
y˜ =H˜(BTx˜ +˜s)+BTF n. (12) frequency response. With the Wiener smoothing matrix
u N
As usual for conventional OFDM, we propose to apply a zero W˜ =C (C +C )−1 (20)
s˜s˜ s˜s˜ v˜v˜
forcing equalization by multiplying with H˜−1 from the left.
the operations performed on a received OFDM frequency
This results in
domain symbol y˜ can now be compactly written as
y˜(cid:48) =H˜−1y˜ =BTx˜ +˜s+v˜ (13)
with the noise vector v˜ = H˜−1BTuF n. After its usage for ˜(cid:98)s=W˜ H˜−1(y˜−H˜BTx˜u). (21)
N
synchronization and/or channel estimation purposes, the UW Finally, the data part d˜(cid:98) =(cid:2)I 0(cid:3)P−1˜(cid:98)s can be processed fur-
can be extracted from further operation by therasusual.Wenotice,thattheerror˜e=˜s−˜(cid:98)shaszeromean,
(cid:16) (cid:17)
y˜(cid:48)(cid:48) =y˜(cid:48) −BTx˜u =˜s+v˜. (14) and its covariance matrix is given by Ce˜e˜ = I−W˜ Cs˜s˜
Fig.3. Blockdiagramforsimulationanalysis.
[8]. C can further be used in the case when additional [10], the indices of the redundant subcarriers are chosen to be
e˜e˜
channel coding is applied. Especially, varying noise variance {2, 6, 10, 14, 17, 21, 24, 26, 38, 40, 43, 47, 50, 54, 58, 62}.
alongthesubcarrierswithinthedatavectord˜maybeexploited This choice, which can easily also be described by equation
as well known from coded transmission over time variant (5)withanappropriatelyconstructedmatrixP,minimizesthe
channels, cf. e.g. [9]. total energy of the redundant subcarriers on average (when
averaging over all possible data vectors d˜) [7].
We summarize the receiver operations per OFDM symbol:
In our approach the unique word shall take over the syn-
• Perform a DFT operation to obtain y˜.
chronizationtaskswhicharenormallyperformedwiththehelp
• Eliminate the influence of the UW by subtracting
H˜BTx˜ from y˜. of the 4 pilot subcarriers. In order to make a fair comparison,
u
the energy of the UW related to the total energy of a transmit
• Apply ZF-equalization followed by an LMMSE Wiener
symbol is set to 4/52, which exactly corresponds to the total
smoothing operation. (Of course, both operations can be
energy of the 4 pilots related to the total energy of a transmit
implemented in one combined single matrix multiplica-
symbol in the IEEE standard. Note that in conventional CP-
tion operation.)
OFDM like in the WLAN standard, the total length of an
• Extract the data part and process it further as usual.
OFDM symbol is given by T +T . However, the guard
GI DFT
Note again, that the ZF equalization and the UW elimination
interval is part of the DFT period in our approach. Therefore,
can be exchanged as described above.
both systems show comparable bandwidth efficiency.
IV. SIMULATIONRESULTS The multipath channel has been modeled as a tapped delay
line, each tap with uniformly distributed phase and Rayleigh
Figure 3 shows the block diagram of the simulated UW-
distributedmagnitude,andwithpowerdecayingexponentially.
OFDM system (equivalent complex baseband description is
Adetaileddescriptionofthemodelcanbefoundin[5].Figure
usedthroughoutthispaper).Afterchannelcoding,interleaving
4 shows one typical channel snapshot featuring an rms delay
and QAM-mapping, the redundant subcarrier symbols are
spread of 100ns. The frequency response shows two spectral
determined using equation (9). After assembling the OFDM
notches within the system’s bandwidth.
symbol, which is composed of d˜, ˜r, and a set of zero-
subcarriers, the IFFT (inverse fast Fourier transform) is per-
formed. Finally the UW is added in time domain. At the
receivertheFFT(fastFouriertransform)operationisfollowed
by a ZF equalization as in classical CP-OFDM. Next the
frequency domain version of the UW is subtracted. Then
the Wiener smoother is applied to the symbol, and finally
demapping, deinterleaving and decoding is performed. For
the soft decision Viterbi decoder the main diagonal of matrix
C is used to specify the varying noise variances along the
e˜e˜
subcarriers after equalization and Wiener filtering.
We compare our novel UW-OFDM approach with the clas-
sical CP-OFDM concept. The IEEE 802.11a WLAN standard
[10]servesasreferencesystem.Weapplythesameparameters
for UW-OFDM as in [10] wherever possible: N = 64, sam-
pling frequency f = 20MHz, DFT period T = 3.2µs,
s DFT
guard duration T = 800ns. Instead of 48 data subcarriers
GI
and 4 pilots we use Nd = 36 data subcarriers and l = 16 Fig. 4. Time- and frequency-domain representation of the used multipath
redundant subcarriers. The zero subcarriers are chosen as in channelsnapshot.
InordertoclearlydemonstratetheeffectofourUW-OFDM
approach with the derived LMMSE receiver, the following
discussions are based on results obtained for the displayed
channel snapshot. Figure 5 compares the mean squared errors
on the N +l (data + redundant) subcarriers before and after
d
the Wiener smoothing operation. We note that all subcarriers
experience a significant noise reduction by the smoother,
but the effect is impressive on the subcarriers corresponding
to spectral notches in the channel frequency response. The
subcarriers with index 15 and 46 correspond to the spectral
notches around 5MHz and -2MHz, respectively, cf. figure 4.
Fig. 6. BER comparison between the novel UW-OFDM approach and the
IEEE802.11astandardforthechannelsnapshotdisplayedabove.
V. CONCLUSION
In this work we introduced a novel OFDM signaling con-
cept, where the guard intervals are built by unique words
instead of cyclic prefixes. The proposed approach introduces
a complex number Reed-Solomon code structure within the
sequence of subcarriers. As an important conclusion we can
state, that besides the possibility to use the UW for synchro-
nization and channel estimation purposes, the novel approach
additionallyallowstoapplyahighlyefficientLMMSEWiener
Fig.5. NoisereductioneffectoftheWienersmootherinafrequencyselective smoother,whichsignificantlyreducesthenoiseonthesubcar-
environmentforEb/N0 =15dB.Above:fullscale;below:zoomedy-axis. riers, especially on highly attenuated subcarriers. Simulation
resultsillustrate,thatthenovelapproachoutperformsclassical
In figure 6 the BER-behavior of the IEEE 802.11a standard
CP-OFDM in a typical frequency selective indoor scenario.
and the novel UW-OFDM approach are compared, both in
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