Preamble and Pilot Design for OFDM

8/13/2019 Preamble and Pilot Design for OFDM

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RESEARCH Open Access

Preamble and pilot symbol design for channelestimation in OFDM systems with null subcarriersShuichi Ohno*, Emmanuel Manasseh and Masayoshi Nakamoto

AbstractIn this article, design of preamble for channel estimation and pilot symbols for pilot-assisted channel estimation inorthogonal frequency division multiplexing system with null subcarriers is studied. Both the preambles and pilotsymbols are designed to minimize the l 2 or the l norm of the channel estimate mean-squared errors (MSE) infrequency-selective environments. We use convex optimization technique to find optimal power distribution to thepreamble by casting the MSE minimization problem into a semidefinite programming problem. Then, using thedesigned optimal preamble as an initial value, we iteratively select the placement and optimally distribute powerto the selected pilot symbols. Design examples consistent with IEEE 802.11a as well as IEEE 802.16e are provided toillustrate the superior performance of our proposed method over the equi-spaced equi-powered pilot symbols andthe partially equi-spaced pilot symbols.

Keywords: Orthogonal frequency division multiplexing (OFDM), Channel estimation, Semidefinite programming(SDP), Convex optimization, Pilot symbols, Pilot design

I. IntroductionOrthogonal frequency division multiplexing (OFDM) isan effective high-rate transmission technique that miti-gates inter-symbol interference (ISI) through the inser-

tion of cyclic prefix (CP) at the transmitter and itsremoval at the receiver. If the channel delay spread isshorter than the duration of the CP, ISI is completely removed. Moreover, if the channel remains constantwithin one OFDM symbol duration, OFDM renders aconvolution channel into parallel flat channels, whichenables simple one-tap frequency-domain equalization.

To obtain the channel state information (CSI), trainingOFDM symbols or pilot symbols embedded in eachOFDM symbol are utilized. Training OFDM symbols orequivalently OFDM preambles are transmitted at thebeginning of the transmission record, while pilot sym-

bols (complex exponentials in time) are embedded ineach OFDM symbol, and they are separated from infor-mation symbols in the frequency-domain [ 1-3]. If thechannel remains constant over several OFDM symbols,channel estimation by training OFDM symbols may besufficient for symbol detection. But in the event of chan-nel variation, training OFDM symbols should be

retransmitted frequently to obtain reliable channel esti-mates for detection. On the other hand, to track the fast varying channel, pilot symbols are inserted into every OFDM symbol to facilitate channel estimation. This is

known as pilot-assisted (or -aided) channel estimation[2,4,5]. The main drawback of the pilot-assisted channelestimation lies in the reduction of the transmission rate,especially when larger number pilot symbols are insertedin each OFDM symbol. Thus, it is desirable to minimizethe number of embedded pilot symbols to avoid exces-sive transmission rate loss.

When all subcarriers are available for transmission,training OFDM preamble and pilot symbols have beenwell designed to enhance the channel estimation accu-racy, see e.g., [6] and references therein. If all the sub-carriers can be utilized, then pilot symbol sequence can

be optimally designed in terms of (i) minimizing thechannel estimate mean-squared error [ 1,3]; (ii) minimiz-ing the bit-error rate (BER) when symbols are detectedby the estimated channel from pilot symbols [ 7]; (iii)maximizing the lower bound on channel capacity withchannel estimates [ 8,9]. It has been found that equally spaced (equi-distant) and equally powered (equi-pow-ered) pilot symbols are optimal with respect to severalperformance measures.* Correspondence: [email protected]

Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan

Ohno et al . EURASIP Journal on Wireless Communications and Networking 2011, 2011 :2http://jwcn.eurasipjournals.com/content/2011/1/2

2011 Ohno et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.
mailto:[email protected]://creativecommons.org/licenses/by/2.0http://creativecommons.org/licenses/by/2.0mailto:[email protected]


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In practice, not all the subcarriers are available fortransmission. It is often the case that null subcarriersare set on both the edges of the allocated bandwidth tomitigate interferences from/to adjacent bands [ 10,11].For example, IEEE 802.11a has 64 subcarriers amongwhich 12 subcarriers, one at the center of the band (DCcomponent) and at the edges of the band are set to benull, i.e., no information is sent [ 12]. The presence of null subcarriers complicates the design of both thetraining preamble for channel estimation and pilot sym-bols for pilot-aided channel estimation over the fre-quency-selective channels. Null subcarriers may renderequi-distant and equi-powered pilot symbols impossibleto use in practice.

In the literature, several pilot symbols design techni-ques for OFDM systems with null subcarriers have beenstudied [ 11,13-16]. In [11], a method that assigns equal

power to all pilot subcarriers and utilizes the exhaustivesearch method to obtain the optimal pilot set is pro-posed. However, the approach in [ 11 ] optimizes only pilot placements of the equally powered pilot symbols.The design of pilot symbols should take into accountthe placements as well as power loading. Moreover, theexhaustive search becomes intractable for large numberof pilot symbols and/or active subcarriers even if thesearch process is carried out during the system designphase. To address the exhaustive search problem, thepartially equi-spaced pilot (PEP) scheme, which will bereferred as PEP in this article, is discussed in [ 13]. Thealgorithm in [ 13 ] is novel as it can be employed todesign pilot symbols for both the MIMO-OFDM as wellas SISO-OFDM systems. Furthermore, the design con-siders both the placements and power distribution tothe pilot symbols. However, the method does not guar-antee better performance for some channel/subcarriersconfiguration.

In [14], equi-powered pilot symbols are studied forchannel estimation in multiple antennas OFDM systemwith null subcarriers. However, they are not always opti-mal even for point-to-point OFDM system. Also in [ 15],a proposal was made that employs cubic parameteriza-tions of the pilot subcarriers in conjunction with convex

optimization algorithm to design pilot symbols. How-ever, the accuracy of cubic function-based optimizationsin [15] depends on many parameters to be selected forevery channel/subcarriers configuration which compli-cates the design. Pilot sequences designed to reduce theMSE of the channel estimation in multiple antennaOFDM system are also reported in [ 13,16] but they arenot necessarily optimal.

In this article, we optimally allocate power to the pre-amble as well as design pilot symbols to estimate the

channel in OFDM systems with null subcarriers. Eventhough there is no closed form expression relating pilotplacement with the MSE, we propose an algorithm thattakes into account both the pilot placements and powerdistribution. Our design criteria are the l 2 norm as wellas the l norm of the MSE of channel estimation in fre-quency-domain. Contrary to [ 15], where it is stated thatl is superior over l 2 , we verify that there is no signifi-cant difference in performance between the two norms.

To find the optimal power allocation, we first show that the minimization problem can be casted into asemidefinite programming (SDP) problem [ 17]. WithSDP, the optimal power allocation to minimize our cri-terion can be numerically found. We also propose aniterative algorithm that uses the designed optimal pre-amble as an initial value to determine the significantplacement of the pilot symbols and power distribution.

Finally, we present design examples under the samesetting as IEEE 802.11a and IEEE 802.16e to show theimproved performance of our proposed design over thePEPs and the equi-spaced equi-powered pilot symbols.We also made comparisons between our proposeddesign, PEP, and the design proposed by Baxley et al. in[15] for IEEE 802.16e. We demonstrated that our pro-posed design can be used as a framework to design pilotsymbols for different channel/subcarriers configurations,and it is crucial to optimally allocate power to the pilotsymbols to improve the MSE and BER performances. Itis also verified that, the conventional preamble of IEEE802.11a is comparable to the optimally designedpreamble.

II. Preamble and pilot symbols for channelestimationWe consider point-to-point wireless OFDM transmis-sions over frequency-selective fading channels. Weassume that the discrete-time baseband equivalent chan-nel has FIR of maximum length L, and remains constantin at least one block, i.e., is quasi-static. The channelimpulse response is denoted as { h0 , h1 ,..., h L-1 }. Since webasically deal with one OFDM symbol, we omit theindex of the OFDM symbol for notational simplicity.

Let us consider the transmission of one OFDM sym-bol with N number of subcarriers. At the transmitter, aserial symbol sequence { s0 , s1 ,..., s N -1 } undergoes serial-to-parallel conversion to be stacked into one OFDMsymbol. Then, an N -points inverse discrete Fouriertransform (IDFT) follows to produce the N dimensionaldata, which is parallel-to-serial converted. A CP of length N cp is appended to mitigate the multipath effects.The discrete-time baseband equivalent transmittedsignals un can be expressed in the time-domain as


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un = 1 N

N 1

k=0

ske j2 kn

N , n [0, N 1]. (1)

Assume that N cp is greater than the channel length Lso that there is no ISI between the OFDM symbols. Atthe receiver, we assume perfect timing synchronization.After removing CP, we apply DFT to the received time-domain signal y n for n [0, N - 1] to obtain for k [0, N - 1]

Y k = 1 N

N 1

n=0

y ne j2 kn

N = H k sk + W k, (2)

where H k is the channel frequency response at fre-quency 2 k / N given by

H k =L1

l=0

hle j2 kl N , (3)

and the noise W k is assumed to be i.i.d. circular Gaus-sian with zero mean and variance 2w.

For simplicity of presentation, we utilize a circularindex with respect to N where the index n of a sequencecorresponds to n modulo N . Let K be a set of activesubcarriers (i.e., non-null subcarriers), then the cardinal-ity of a set K can be represented as |K |.Take WLAN standard (IEEE 802.11a), for example,where 64 subcarriers (or slots) are available in theOFDM symbol during data transmission mode. Out of which 48 are utilized as information symbols, 4 as pilotsymbols, while the rest except for the DC subcarrierserves as spectral nulls to mitigate the interferencesfrom/to OFDM symbols in adjacent bands. Thus, K ={1,2,..., 26, 38, 39, ..., 63} and|K | = 52.The detailed structure of the OFDM packet in a time-frequency grid is shown in Figure 1. At the beginning of the transmission, two long OFDM preambles are trans-mitted to obtain CSI (see [[ 18 ], p. 600]). In IEEE802.11a standard, the first part of the preamble consistsof 10 short pilot symbols in 12 subcarriers equally

spaced at 4 subcarriers interval, which is not shown inFigure 1. The second part of the preamble initiation,which corresponds to the first two OFDM symbols of Figure 1, requires the transmission of two columns of pilot symbols in all active subcarriers in order to makeprecise frequency offset estimation and channel estima-tion possible [ 18].

For channel estimation, we place N p( |K |) pilotsymbols {p1 ,..., p N p} at subcarriersk1 , k2 ,..., k N p K (k1 < k2 < < k N p ), which areknown at the receiver. We assume that N p L so that

the channel can be perfectly estimated if there is nonoise, and we denote the index of pilot symbols asK p = {k1 ,..., k N P }.

Let diag(a ) be a diagonal matrix with the vector a onits main diagonal. Collecting the received signals havingpilot symbols as

Y = [Y k1 , . . . , Y k N p ] T , (4)

we obtain

Y = DH p p + W , (5)where DH p is a diagonal matrix with its nth diagonal

entry being H kn such that

DH p = diag( H k1 , . . . , H k N p ), (6)

and p is a pilot vector defined as

p = [p1 , . . . , p N p ] T . (7)

From Y , we would like to estimate channel frequency responses for equalization and decoding. (In this article,we consider only channel estimation by one OFDM sym-bol but the extension to multiple OFDM symbols couldbe possible). Let us define K s as an index set specifyingthe channel frequency responses to be estimated. Inother words, H k for k K s have to be estimated from Y .In a long training OFDM preamble, all subcarriers inK can be utilized for pilot symbols so that K p = K . Onthe other hand, in pilot-assisted modulation (PSAM) [ 4],a few known pilot symbols are embedded in an OFDMsymbol to facilitate the estimation of unknown channel.Thus, for PSAM, we have K s = K \K p where\representsset difference.

. 5MHz

0

-8.125MHz

f r e

q u e n c y

time

Figure 1 The time-frequency structure of an IEEE 802.11a

packet. Shaded subcarriers contain pilot symbols .


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If we can adopt equally spaced (equi-distant) pilotsymbols with equal power for channel estimation andsymbol detection, then it can be analytically shown thatthe channel mean-squared estimation error [ 1,3] as wellas the BER [7] are minimized, while the lower bound onchannel capacity [ 8,9] is maximized. But the optimality of equi-distant and equi-powered pilot symbols does notnecessarily hold true when there are null subcarriers.

In this article, for a given K , we use convex optimiza-tion technique to optimally distribute power to thesesubcarriers. Then, we propose an algorithm to deter-mine pilot set K p with significant power to be used forPSAM.

III. Mean-squared channel estimation errorLet us define F as an N N DFT matrix, the ( m + 1, n+ 1)th entry of which is e- j 2 mn / N . We denote an N L

matrix

F L = [ f 0 , . . . , f N 1] (8)

consisting of N rows and first L columns of DFTmatrix F , where is the complex conjugate transposeoperator. We also define an N p L matrix F p having f knfor kn K p as its nth row. Then, we can express (5) as

Y = Dp F ph + W , (9)where the diagonal matrix D p and channel vector h

are respectively defined as

Dp = diag( p1 , . . . , p N p ), (10)

and

h = [h0 , . . . , hL1 ] T . (11)

Let a vector having channel responses to be estimated,i.e., H k for k s, be

H s = [H k1 , . . . , H k|K s|] T . (12)

Similar to F p, we define a |K s| L matrix F s having f knfor kn K s as its nth row, where k n


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If F H s F s = cI for a non-zero constant c, then from (19),E{||E s||2} = cE{||h h||2}, where |||| denotes the Eucli-dean norm. The equation F H s F s = cI is attained if allpilot symbols have the same power and are uniformly

distributed in an OFDM symbol. But, this is not alwayspossible if there are null subcarriers in the OFDM sym-bol. As shown later, even with null subcarriers, theminimization of E{||h h||2} becomes possible.Now, our objective is to find the optimal pilot symbolsthat minimize a criterion function. Two important cri-teria are considered. One is the l 2 norm of the mean-squared channel estimation errors {r k}kK s at data sub-carriers, which is defined as

2 =k

K s

r k

12

= (trace Re)12 , (23)

Where

r k = E{|H k H k|2}. (24)The other is the maximum of { r k } defined as

= max k

K sr k, (25)

which is the l norm of {r k}kK s. I t should beremarked that 22 = E{||H s H s||2} = cE{||h h||2} if F H s F s = cI . To differentiate them, we call the former thefrequency-domain channel MSE and the latter the time-domain channel MSE.

In the long preamble of IEEE 802.11a standard, equi-powered pilot symbols are utilized but may not be opti-mal due to the existence of null subcarriers. Equi-pow-ered pilot symbols are also investigated for channelfrequency response estimation in multiple antennaOFDM system with null subcarriers [ 14]. To reduce thesum of channel MSE for multiple antenna OFDM sys-tem, pilot symbol vector p has been designed to satisfy F Hp p F p = I p in [16]. However, such pilot sequence doesnot always exist. In addition, the necessary and sufficientcondition for its existence within the active subcarrierband has not yet been fully established.

IV. Pilot power distribution with SDPFor any prescribed energy to be utilized for channel esti-mation, we normalize the sum of pilot power such that

kK p

|pk|2 = N p

k=1

k = 1. (26)

Then, our problem is to determine the optimal

= [ 1 , . . . , N p] T , (27)

that minimizes h 2 in (23) or h in (25) under the con-

straint (26).We first consider the minimization h 2 . The optimalpower distribution can be obtained by minimizing thesquared h 2 in (23) with respect to l under the con-straints that

1, . . . , 1 = 1, 0, (28)

where a 0 (or a 0) for a vector signifies that allentries of a are equal to or greater than 0 (or strictly greater than 0). As stated in the previous section, analy-tical solutions could not be found in general. As in [ 20],we will resort to a numerical design by casting our

minimization problem into a SDP problem.The SDP covers many optimization problems [ 17,21].The objective function of SDP is a linear function of a var iable x R M subject to a linear matrix inequality (LMI) defined as

F ( x ) = A0 + M

m=1

xm Am 0, (29)

where Am R M M . The complex-valued LMIs arealso possible, since any complex-valued LMI can bewritten by the corresponding real-valued LMI. Since theconstraints defined by the LMI are convex set, the glo-

bal solution can be efficiently and numerically found by the existing routines.

By re-expressing the nth row of F p as f n , our MSEminimization problem can be stated as

min

trace R1h + 1 2w

N P

n=1

n f n f H

n

1R

subject to [1, . . . , 1] 1, 0,

(30)

where R = F H s F s. This problem possesses a similar formas the transceiver optimization problem studied in [ 22],which is transformed into an SDP problem. Similar to theproblem in [ 22], our problem can be transformed into anSDP form. Now let us introduce an auxiliary Hermitematrix variable W and consider the following problem:

minW ,

trace ( W R)

subject to [1, . . . , 1] 1, 0(31)

W R1h + 1 2w

N P

n=1

n f n f H

n

1. (32)


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and is reported in [ 15] that for some special cases, it doesnot work well.

In this article, we improve the method of [ 20 ] by introducing an iterative algorithm as follows. Let N ( i)r bea positive even integer. First, we use a designed optimalpreamble with SDP and denote its l k as 01 ,...,

(0)

|K |, then,we remove N (0)r subcarriers with minimum power sym-metrically about the center (DC) subcarrier, i.e., N (0)r /2on every side of the central DC subcarrier. Then, weoptimize the remaining pilot symbols. Similarly, for theith iteration, after removing subcarriers correspondingto N ( i)r minimum power, we optimize pilot power forthe remaining set again with SDP. When the iterativealgorithm is completed, we will remain with only K psubcarrier indexes and its corresponding optimal power.

Our design procedure is as outlined by the pseudo-code algorithm below:

1) Set i = 0.2) Obtain the optimal preamble using convex optimi-zation and initialize temporary set K (i)p = K .

3) If N p < |K( i)p |, remove from K

( ip , N

(i)r subcarriers

with minimum power symmetrically with respect tothe DC subcarrier, else go to step 5.4) Optimize the power of the remaining subcarriersusing SDP and go to step 3 after updating i i + 1.5) Exit .

The value of N ( i)r ( 2) is not fixed. The number of iterations can be reduced by increasing the value of N ( i)r .However, when the number of removed subcarriers N ( ir is large, the proposed scheme may not work well forsome channel/subcarriers configuration as in [ 20 ],where the significant N p subcarriers of the optimizedpreamble are selected at once.

To obtain a better pilot set for any channel/subcar-riers configuration, the number of removed subcarriers N ( i)r should be kept smaller. There is a tradeoff betweenthe computational complexity and the estimation perfor-

mance of the resultant set. Since we can design pilotsymbols off-line, we can set the minimum for N ( i)r suchas N (i)r = 2.

VI. Design examplesIn this section, we demonstrate the effectiveness of ourproposed preamble and pilot symbols designs throughcomputer simulations. The parameters of the trans-mitted OFDM signal studied in our design examples areas in the IEEE 802.11a and IEEE 802.16e (WiMaX) stan-dards. For IEEE 802.11a, an OFDM transmission frame

with N = 64 subcarriers is considered. Out of 64 subcar-riers, 52 subcarriers are used for pilot and data trans-mission while the remaining 12 subcarriers are nullsubcarriers [[ 18], p. 600]. For IEEE 802.16e standard, anOFDM transmission frame in [[ 24], p. 429] is consid-ered. In a data-carrying symbol 200 subcarriers of the N = 256 subcarrier window are used for data and pilotsymbols. Of the other 56 subcarriers, 28 subcarriers arenull in the lower-frequency guard band, 27 subcarriersare nulled in the upper frequency guard band, and oneis the central null (DC) subcarrier. Of the 200 used sub-carriers, 8 subcarriers are allocated as pilot symbols,while the remaining 192 subcarriers are used for datatransmission.

In the simulations, the total power of each OFDMframe is normalized to one, but power distributionamong pilot symbols is not constrained to be uniform.

The diagonal element of channel correlation matrix isset to be E{hmhn} = c(m n)e0.1n for m, n [0, L -1], where () stands for Kronecker s delta, and c isselected such that trace Rh = 1.

A. Preamble designFirst, we start with the design of preamble where allactive subcarriers are considered as pilot symbols. For agiven channel length L, to design an OFDM preamble,we optimize all active subcarriers by minimizing the l 2norm or the l norm of MSEs {r k}kK s using convexoptimization package in [ 25]. Figure 2 depicts the opti-

mal power distribution to the IEEE 802.11a preambledesigned by l 2 norm when the SNR is 10 dB and thechannel length L = 4. We omit the power distributionby l norm, since it is nearly identical to that of the l 2norm-based design. Unlike the standard preamble whereequal power is allocated to all the subcarriers, our opti-mized preamble distribute power to the subcarriers suchthat the channel estimate MSE is minimized.

We also consider a case when the channel length L =8. The results in Figure 3 show the optimized preambleat 10 dB. Again, there is no significant differencebetween the design with l 2 norm and the design with l norm. However, the computational complexity of thedesign with l 2 norm is quite lower than the computa-tional complexity of the design with l norm, thereby making the former more preferable to the latter. Eventhough the design process is usually done in off-line,such minor advantage may be an important factor whendesigning preambles and pilot symbols for an OFDMframe with a large number of subcarriers.

Figures 2 and 3 show that the designed preamblesare symmetric around 0. This is due to the symmetricnature of our objective function and its constraints.There are differences in power distribution to the


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30 20 10 0 10 20 300

0.05

0.1

0.15

0.2

0.25

subcarrier

Pilot

Preamble

Figure 2 Power of the preamble and pilot symbols designed by the l 2 norm for L = N p = 4 at 10 dB (IEEE 802.11a) .

30 20 10 0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Subcarrier

Pilot

Preamble

Figure 3 Power of the preamble and pilot symbols designed by the l 2 norm for L = N p = 8 at 10 dB (IEEE 802.11a) .


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designed preambles when L = 4 and L = 8, this sug-gests that in the preamble design, equi-powered sub-carriers may not necessarily be optimal when there arenull subcarriers. This may not be well encapsulated inthe overall channel estimate MSE. However, when con-sidering the channel estimate in each subcarrier, thereis a slight difference between the proposed designs andthe standard preamble especially at the edges of theactive band.

To verify this, we compare frequency-domain chan-nel MSE h 2 obtained by the l 2 and l norm-baseddesign with the standard IEEE 802.11a preamble. By varying the channel length L, from 1 to 16, we numeri-cally obtain the channel estimate MSE for each L. Fig-ure 4 presents the frequency-domain channel MSE h 2 ,against channel length L at 10 dB. From the plot, it isobvious that there is no significant difference between

the three designs, which suggests that the standardpreamble is almost optimal in the l 2 sense even if there are null subcarriers. This is not so surprisingsince in the absence of null subcarriers, equi-poweredpreamble is optimal. Through our design approach, wenumerically corroborate that for IEEE 802.11a, thestandard preamble is nearly optimal.

To demonstrate the versatility of our method, weminimize the LS channel estimate MSE to design

preambles for the IEEE 802.16e standard. Figure 5shows the designed preamble of IEEE 802.16e for L =16. Similar to 802.11a, the distribution of power to theactive subcarriers is not uniform. This further suggeststhat equi-powered preambles are not necessarily optimalfor the OFDM systems with null subcarriers.

B. Pilot designWe employ the algorithm developed in Section V todesign pilot symbols for PSAM. Similar to the preambledesign, total power of the pilot symbols are normalizedto one. First, we consider an OFDM symbol with 64subcarriers and 4 pilot symbols, i.e., N p = 4. This com-plies with the IEEE 802.11a standard pilot symbols,where four equi-spaced and equi-powered pilot symbolsare adopted.

In general, within an OFDM symbol, the number of

pilot symbols in frequency domain should be greaterthan the channel length (maximum excess delay), whichis related to the channel delay spread (i.e., N p L) [2].When N p > L, the MSE performance will be improved aslong as the power of pilot symbols is optimally distribu-ted, but the capacity (or data rate) will be degraded.Thus, in our simulations, we use N p = L. However, itshould be remarked that except for some special cases,it still remains unclear what value of N p is optimal.

2 4 6 8 10 12 14 16

3.5

4

4.5

5

5.5

6

6.5

L

M S E

l2l

IEEE 802.11a

Figure 4 Frequency-domain channel MSE h 2 of preamble at 10 dB (IEEE 802.11a) .


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Figure 2 shows the pilot symbols designed by l 2 normat 10 dB when N p = L = 4. The designed pilot symbols

are almost equi-spaced ( K p = {8, 24)}, and the exist-ing standard allocates the equi-powered pilot symbols at(K p = {7, 21}). For OFDM systems with null subcar-riers, equi-spaced pilot symbols having the same powerare not necessarily optimal. In our proposed design, theoptimized power allocated to the pilot symbols is notuniformly distributed, which suggests that in the pre-sence of null subcarriers, equi-powered pilot symbolsmay not necessarily be optimal. This may not be wellencapsulated in the total channel estimate MSE, but ismore clearer when considering the channel estimate ineach subcarrier.

We also illustrate the performance of our proposedalgorithm by designing pilot symbols for N p = L = 8.Figure 3 presents the power distribution to the designedpilot symbols at 10 dB. The pilot power distribution isfound to be symmetric around 0. This is due to thesymmetric nature of our objective functions and the factthat pilot positions are obtained by removing the mini-mum power subcarriers symmetrically. The eight pilotsymbols are located at the subcarriers K p = {4, 12,19, 26}. Pilot symbols are well distributed within thein-band region, which ensures nearly constant estima-tion in all subcarriers.

We make a comparison of our proposed design, thePEPs scheme and the equi-spaced equi-power design

which we will refer to it as a reference design. Figure 6shows the designed pilot set for each of the three meth-ods. Both of the proposed and PEP design allocate somepilot subcarriers close to the edges. For the referencedesign, the equi-spaced and equi-powered pilot symbolsare allocated at 3, 9, 15, and 21. There are no pilotsubcarriers close to the edges of the active band. Thelack of the pilot subcarriers at the edges of the OFDMsymbol may lead to higher channel estimation errors forthe active subcarriers close to the null subcarriers.

To demonstrate the effectiveness of the pilot symbolsin Figure 6, we plot the channel estimate MSE for eachactive subcarrier. The total power allocated to the pilotsymbols is the same for all three designs. Figure 7shows the channel estimate MSE of the three designs.From the results, it is clear that, both of our proposedand the PEP design outperform the reference design,and there is no significant difference between the pro-posed design and the PEP design. The reference (equi-spaced equi-powered) design does a poor job of estimat-ing channel close to the null subcarriers, this is due tothe lack of pilot subcarriers at the edges of the OFDMsymbol. Channel estimation via extrapolation resultsinto higher errors at the edges of the OFDM symbols if

100 50 0 50 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

subcarrier

Pilot

Preamble

Figure 5 Power of the preamble and pilot symbols designed by the l 2 norm for L = N p = 16 (IEEE 802.16e) .


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30 20 10 0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Subcarrier

ProposedReferencePEP

Figure 6 Comparison of the pilot power and placement (IEEE 802.11a) .

20 10 0 10 20

0

0.5

1

1.5

2

2.5

3

Subcarrier Number

N M S E ( d B )

ProposedReferencePEP

L=1 L=8

L=4

Figure 7 Comparison of the normalized channel estimate MSE between the proposed and PEP design (IEEE 802.11a) .


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there is no pilot subcarriers with significant power atthe edges of the OFDM symbols. This suggests thatboth the pilot power and the placements need to becarefully considered in the design.

We also compare the average channel estimate MSEof our proposed design with the PEP design for differentchannel length L. We evaluate the channel estimateMSE of our proposed designs as well as the PEPscheme, by varying the channel length L, from 1 to 16.Figure 8 presents the average channel MSE againstchannel length L at 10 dB. The proposed designs, ( l 2and l ) exhibit lesser channel MSE than the PEP sym-bols except for the trivial case when L = 1. This verifiesthe potential of our proposed designs over the PEPscheme.

Figure 8 shows that the two design criteria ( l 2 and l )exhibit nearly the same h 2 for different channel lengths

L. This is contrary to [ 15], where it is stated that the l norm-based design outperforms the l 2 norm-baseddesign. Our results suggest that any of the two designcriteria can be used to design pilot symbols. However, itshould be noted that l 2 norm-based design has lesscomputational complexity as compared with the l .Since the results obtained by the two proposed methodsare almost similar for both the preamble and the pilotsymbol design, then it is reasonable to adopt the l 2

norm-based design for the preamble as well as for thepilot symbol design.

Next, we consider pilot symbol design for IEEE802.16e by minimizing the LS channel estimate MSE.Figure 5 shows the designed pilot symbols for L = N p =16. Similarly, the pilot symbols are well distributedwithin the active subcarrier band, and the power distri-bution to the pilot symbols is not uniform. The resultemphasizes on adopting non-uniform power distributionfor an OFDM frame with null subcarriers. Furthermore,the result underlines the potential of our proposedscheme in designing pilot symbols for OFDM systemswith different number of subcarriers, channel length,and dedicated number pilot subcarriers. We have provided the resul ts based on the LS est imator to verify that, in designing pilot power and placement, minimiza-tion of both the LS and the MMSE channel estimate

MSE, perform almost equally well.Figure 9 illustrates the power distribution to the pilot

symbols for the PEP, the method in [ 15], which we willrefer to it as Baxley and the proposed designs. Similarto the results in Figure 6 for IEEE 802.11a, the place-ment of pilot symbols for the PEP and our proposeddesign are almost same. However, there is a noticeabledifference in power distribution between the PEP andthe proposed design for L = N p = 16. This suggests that

2 4 6 8 10 12 14 163.5

4

4.5

5

5.5

6

6.5

L

N M S E ( d B )

l2l

PEP

Figure 8 Comparison of the channel estimate MSE h 2 between the proposed designs and the PEP at 10 dB (IEEE 802.11a) .


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the MSE performance of the two designs will be differ-ent. For Baxley s method, both the pilot placement andthe power distribution are comparable to our proposeddesign. The design in [ 15], uses exhaustive grid searchto obtain pilot set with minimum channel estimateMSE. The placement of the pilot symbols depends onthe searching granularity over the predetermineddomain of some optimizing parameters. The main chal-lenge in [15] lies in the adjustment of several parametersto obtain reasonable pilot positions. With the properselection of the optimizing parameters of the cubicfunction, the performance of Baxley s method is com-parable to our proposed design.

The prominence of optimal power distribution to thepilot symbols is further depicted in Figure 10, where forthe designed pilot symbols in Figure 9, we plot the nor-malized channel estimate MSE for each active subcar-rier. The total power allocated to the pilot symbols isthe same for all the three designs. Figure 10 depicts thechannel estimate MSE of the three designs. From theresults, it is clear that, both of our proposed and theBaxley designs outperform the PEP design, and there isno significant difference between the proposed designand the Baxley design. Unlike the results depicted inFigure 7, where the performance of the PEP design are

comparable to our proposed method, here the PEPdesign does a poor job of estimating the channel closeto the null subcarrier zone. This is not due to the lackof pilot subcarriers at the edges of the OFDM symbol,but due to insignificant power allocated to the pilotsymbols close to the null subcarriers. The results sug-gest that, considerable MSE performance improvementscan be realized by the proper distribution of power tothe pilot symbols.

Similarly, we make a comparison of the frequency-domain channel MSE between the PEP, the Baxley method, and our pilot symbol designs for IEEE 802.16e.Figure 11 presents the channel estimate MSE of the

IEEE 802.16e for N p = 16. From the results, it is clearthat there is a significant improvement in MSE perfor-mance between our proposed and the PEP designs. Thisfurther substantiates the superior performance of ourproposed method over PEP for different channel/subcar-riers configurations. The Baxley method performsequally well as our proposed design, which suggests thatthe method in [ 15] can be used for pilot symbol designsas well. However, the cubic parameterization techniqueadopted in the Baxley method obtains pilot placementfor a given set of successive active subcarriers. Thus, forthe case where some subcarriers within the active band

100 50 0 50 1000

0.01

0.02

0.03

0.04

0.05

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0.07

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subcarrier

PEPBaxleyProposed

Figure 9 Comparison of the pilot power and placement (IEEE 802.16e) .


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are reserved for other applications, the method cannotbe easily adopted. For example, to avoid the cubic para-meterization from selecting the central DC subcarrier,additional constraint is introduced. This implies that,cubic parameterization techniques call for some modifi-cations whenever active subcarriers are not consecutive.Unlike [15], our proposed design can be easily employedto obtain significant pilot set for any given set of activesubcarriers.

A good example is in [26,27], where tone reservation(TR) technique is adopted for peak-to-average powerratio (PAPR) reduction. For a system that utilizes TRmethod to mitigate PAPR, the transmitter sends dummy

symbols in some selected subcarriers within the activeband [ 26 ,27]. These reserved subcarriers change theorientation of the data subcarriers in the active bandand thereby limit direct application of the cubic parame-terization techniques in designing pilot symbols. Unlikecubic parameterization method, PEP as well as our pro-posed method can be easily adopted even when somesubcarriers within the active band are restricted.Another example is in [ 13], where PEP scheme is usedto design pilot symbols for both the MIMO-OFDM aswell as the SISO-OFDM systems. Like PEP, our pro-posed method can be easily adopted in the design of the

disjoint pilot symbols for MIMO-OFDM systems. ForMIMO-OFDM systems that utilizes pilot symbols, toreduce interference between the pilot symbols trans-mitted from different antennas, it is necessary for thepilot symbols to be orthogonal. The orthogonality of thepilot sequences for MIMO-OFDM can be established by ensuring that the pilot symbols of one transmit antennaare disjoint from the pilot symbols of any other transmitantenna or by using phase-shift (PS) codes. Baxley method cannot be directly applied to design disjointpilot sets for MIMO-OFDM systems while our methodbe can easily applied.

In the following, we explore the BER performance

gains that could be realized if the pilot symbols in IEEE802.16e are designed to conform with the proposedmethod. We demonstrate the efficacy of our proposedmethod by comparing the BER performance of the pro-posed, PEP, the Baxley and the reference design. Thefrequency-selective channel with L = N p taps is consid-ered. Each channel tap is i.i.d. complex Gaussian withzero mean and the exponential power delay profile isgiven by the vector r = [ r 0 ... r L - 1] where l = C e1/2 ,and is a constant selected such that

L1l=1

l = 1.

100 80 60 40 20 0 20 40 60 80 1000

2

4

6

8

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12

14

16

Subcarrier Number

N M S E ( d B )

ProposedBaxleyPEP

Figure 10 Comparison of the normalized channel estimate MSE for N p = 16 (IEEE 802.16e) .


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Figures 12 and 13 depict the BER performance of theOFDM system when using PSK modulation. For N p = 8,we made a comparison between the proposed design,PEP, the Baxley, and the reference design. It is notedthat conventional standard utilizes equi-spaced equi-powered pilot symbols. Figure 12 depicts the BER per-formance of the four designs for QPSK modulation. ThePEP, the Baxley, and the proposed design outperformthe reference design. The results show that, at highSNR, there is a slight improvement in BER performanceof the proposed design over the PEP and the Baxley design.

To demonstrate the flexibility of our proposed design,we provide two comparable BER curves, Proposed 1 andProposed 2, where the former is the BER performanceof our designed pilot symbols and the latter is the BERperformance of our pilot symbols optimized when wereserve subcarriers K TR = {100, 76, 47, 14} for TR.This verifies that, even for discontinuous data subcar-riers, our proposed design can still be used to designsignificant pilot symbols, while the Baxley design cannotbe directly applicable to this configuration.

For N p = 16, we only consider PEP, the Baxley, andthe proposed design as it is impossible to have 16

equally spaced pilot symbols within 200 active subcar-riers. Figure 13 depicts the BER performance for QPSK,16-PSK, and 64-PSK. The results verify that the pro-posed design provides improved BER performance overthe PEP design. This performance gap is a result of thePEP design having insignificant power distribution tothe pilot symbols at the edges of the active subcarrierband that leads to poor estimate of the channels. Theperformance of the Baxley design is similar to the pro-posed design for QPSK-modulated data. However, for16-PSK and 64-PSK modulation, our proposed designoutperforms the Baxley design. Also, for 16-PSK and64-PSK modulation, there is a slight improvement inBER performance of the Baxley method over the PEPdesign. The gain attained by our proposed method overthe PEP and the Baxley design together with the flexibil-ity of the proposed technique in designing pilot symbolsfor different channel/subcarriers configurations pro-motes our proposed design to be a candidate for pilotsymbols design in OFDM systems with null subcarriers.

VII. ConclusionWe have addressed the design of optimal preamble aswell as suboptimal pilot symbols for channel estimation

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L

M S E ( d B )

ProposedPEPBaxley

Figure 11 Comparison of the channel estimate MSE h 2 between the proposed 2 design, Baxley and the PEP for N p = 16 (IEEE802.16e) .


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0 5 10 15 20 25 30 35 40

103

102

101

SNR(dB)

B E R

ReferencePEPBaxley

Proposed 1Proposed 2

Figure 12 Comparison of the BER performance for different designs for N p = 8 (IEEE 802.16e).

0 5 10 15 20 25 30 35 40

103

102

101

SNR(dB)

B E R

ProposedPEPBaxley

16 PSKQPSK

64 PSK

Figure 13 Comparison of the BER performance for N p = 16 (IEEE 802.16e) .


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in OFDM systems with null subcarriers. First, we haveformulated the channel estimate MSE minimization pro-blem for a determined subcarrier set as an SDP, thenwe have employed convex optimization techniques toobtain near optimal power distribution to a given sub-carrier set. Design examples consistent with IEEE802.11a standard show that in terms of channel estimateMSE, the long OFDM preamble with equi-poweredactive subcarriers is nearly optimal.

In designing the pilot symbols, we have consideredpilot placement as well as power allocation. We haveproposed an iterative algorithm to determine pilot place-ments and then distribute power to the selected pilotsymbols by using convex optimization techniques. Sev-eral examples consistent with IEEE 802.11a and IEEE802.16e have been provided which show that the pro-posed methods can be used to design preamble as well

as pilot symbols for channel estimation in OFDM sys-tems with different frame size. We have also verifiedthat our proposed method is superior over the equi-spaced equi-powered pilot symbols as well as the PEPand slightly outperforms the Baxley design.

VIII. AbbreviationsCP: cyclic prefix; CSI: channel state information; IDFT:inverse discrete Fourier transform; ISI: inter-symbolinterference; LMI: linear matrix inequality; MMSE: mini-mum mean-squared error; MSE: mean-squared errors;OFDM: orthogonal frequency division multiplexing;PAPR: peak-to-average power ratio; PEP: partially equi-spaced pilot; PSAM: pilot-assisted modulation; SDP:semidefinite programming; SNR: signal-to-noise ratio;TR: tone reservation.

IX. Competing interestsThe authors declare that they have no competinginterests.

Note The material in this article was presented in part at Asia-Pacific Conference on Communications 2008 .

Received: 15 July 2010 Accepted: 3 June 2011 Published: 3 June 2011

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doi:10.1186/1687-1499-2011-2Cite this article as: Ohno et al .: Preamble and pilot symbol design forchannel estimation in OFDM systems with null subcarriers. EURASIP Journal on Wireless Communications and Networking 2011 2011 :2.


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Preamble and Pilot Design for OFDM