PILOT ASSISTED CHAmEL ESTIMATION FOR OFDM IN MOBILE
`CELLULAR SYSTEMS
`Fredrik Tufvesson and Torleiv Maseng
`Department of Applied Electronics, Lund University, Box 118, S-221 00 Lund, Sweden
`E-mail: Fredrik.Tufvesson@ tde.lth.se
`Abstract - The use of pilot symbols for channel estimation
`introduces overhead and it is thus desirable to keep the number
`of pilot symbols to a minimum. The number of needed pilot
`symbols for a desired bit error rate and Doppler frequency is
`highly dependent on the pilot pattern used in orthogonal fre-
`quency division multiplexed, OFDM, systems. Five different
`pilot patterns are analysed by means of resulting bit error rate,
`which is derived from channel statistics. Rearrangement of the
`pilot pattern enables a reduction in the number of needed pilot
`symbols up to a factor 10, still retaining the same performance.
`The analysis is general and can be used for performance analy-
`sis and design of pilot patterns for any OFDM system.
`
`chosen small enough to enable reliable channel estimates but
`large enough not to increase the overhead too much. This paper
`includes among all an algorithm of how to design a suitable
`pilot estimation pattern.
`In a multicarrier system there exist a unique opportunity to
`determine various parts of the channel impulse response, as
`opposed to a single-carrier system. It is no use to make efforts
`to determine already known parts. Until now it seems like no
`one has looked into the impact of the placement of the pilot
`symbols for multicarrier systems. Cavers [5] made an exhaus-
`tive theoretical analysis for single-carrier systems. He pointed
`out that it was appropriate that 14% of the sent symbols were
`pilot symbols to be able to handle large Doppler values
`uhl",=O.OS). Some pilot estimation patterns for OFDM has
`been presented in the literature, see e.g. [6], [7]. Comparisons
`between these and the one proposed here are shown later.
`
`I. INTRODUCTION
`The mobile channel introduces multipath distortion of the
`signalling waveforms. Both the amplitude and phase are cor-
`rupted and the channel characteristics changes because of
`movements of the mobile. In order to perform (coherent detec-
`tion, reliable channel estimates are required. These can be
`obtained by occasionally transmitting known data or so called
`"pilot symbols". The receiver interpolates the channel informa-
`tion derived from the pilots to obtain the channel estimate for
`the data signal, see Fig. 1.
`
`pilot
`Pa-
`Fig. l The pilot pattern is known both by the receiver and
`the transmitter. The purpose is to minimize the number of
`transmitted pilot symbols without increasing the bit error
`rate.
`OFDM, orthogonal frequency division multiplexing, is used
`and proposed for several broadcast systems and there is a
`growing interest in using the technique for the next generation
`land mobile communication system. In OFDM systems the
`information signal can be seen as divided and transmitted by
`several narrowband sub-carriers. Qpically, for practical
`OFDM systems, the frequency spacing is less than the coher-
`ence bandwidth and the symbol time is less thain the coherence
`time. This means that a receiver and pilot estimation pattern
`that take advantage of the relatively large coherence bandwidth
`and coherence time can manage with less pilot symbols,
`thereby minimizing the overhead introduced by the pilot sym-
`bols. The problem is to decide where and how often to insert
`pilot symbols. The spacing between the pilot symbols shall be
`
`II. ESTIMATION STRATEGIES
`Five different pilot patterns are analyzed, see Fig. 2.
`1. Measure all channels at the same time, compare to a broad-
`band single-carrier system.
`2. Measure the channels in increasing order, one at a time.
`3. Measure the channels one at a time in a smart, but predeter-
`mined, way. The measurement order is derived upon chan-
`nel statistics and is optimal in the sense that the total bit
`error rate is kept at a minimum each symbol time.
`4. A pilot pattern presented by T. Mueller et al. [6], where the
`pilots are located with equidistant spacings in time and fre-
`quency.
`5. A pilot pattern by P. Hoeher [7], used in [4]. The pilot sym-
`bol locations are shifted one step in frequency each pilot
`interval.
`
`Fig. 2 Examples of pilot patterns analyzed.
`
`0-7803-3659-3/97 $10.00 0 1 997 IEEE
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`For comparison the same amount of pilots is used, one of 64
`sent symbols. This means that only 1.6% extra overhead is
`introduced by the pilots, but this is not sufficient for large
`Doppler frequencies in some of the cases.
`
`111. SYSTEM
`At different signal to noise ratios the resulting bit error rate
`from each pilot pattern is evaluated using the algorithms given
`in Section VIII. A matched filter receiver and coherent BPSK
`or QPSK are used. Additive white Gaussian noise is assumed
`for every sub-channel. The channel has delays and Rayleigh
`distributed amplitudes according to the COST 207: “Typical
`Urban profile”. The complex parts of the transfer function are
`assumed to change according to a first order auto-regressive
`process as described in Section VI. The reason for using this
`model is to get a linear system which rather easy can be hand-
`led algebraically. A Kalman filter is used to estimate the fre-
`quency response of the channel. From the filter, a time depen-
`dent covariance matrix is given as described in SectionVII.
`This is used to calculate the expected bit error rate for each
`channel. See Fig. 3 for a description of the system.
`rl
`
`‘Tg-4
`
`~~i
`
`
`
`”’Ilpical channe1,H urban’
`
`./for
`
`design
`
`noise: e
`
`pilot
`
`Fig. 3 Overview of the system used to analyze the estima-
`tion patterns.
`
`IV. RESULTS
`The resulting bit error rate curves of the pilot patterns are
`presented for different Doppler frequencies in Fig. 4.
`
`BER for QPSK when different pilot patterns are used. Eb/No=lO
`
`10’’
`
`c
`
`1 6
`1 o.=
`IO.’
`1 O.*
`I O4
`fdTs
`Fig. 4 Resulting bit error rate as a function of Doppler value
`when the different pilot patterns are used. The same number
`of pilot symbols are used for all curves.
`
`1640
`
`For a given Doppler frequency the pilot pattern used sets
`the limit for the lowest pilot density to be used, alternatively
`for a given pilot density it limits the maximum Doppler fre-
`quency allowed. The calculations are made at 1800 MHz using
`10 kHz channel separation between 64 OFDM channels carry-
`ing in total 640 ksymbols/s. No intersymbol interference, per-
`fect synchronization and known Doppler frequency, fd, is
`assumed. 1.6% of the sent symbols are pilot symbols and the
`average bit error rate of the 64 channels is presented.
`The bit error rate is degraded both by imperfect channel
`estimates and noise disturbances. The pilot pattern used deter-
`mines the conversion between noise limited and estimate lim-
`ited region, see Fig. 5.
`
`BER for QPSK when different Dilot patterns are used. fdTs=O.OOl
`
`1 0”
`
`10’
`
`n:
`w l o 2
`m
`
`1 0“ 1 ........... .. .;
`
`......... ....I ...
`
`. .. .; ........... . . . ~ . .....................
`
`1
`104A
`30
`5
`10
`15
`20
`25
`EbINo
`Fig. 5 Bit error rate at different signal to noise ratios. Note
`the difference between the error floors.
`It is interesting to study the resulting pattern when estimat-
`ing the channel that gives the lowest bit error rate (strategy 3).
`A steady state pilot pattern is often achieved where only few of
`the sub-channels are measured, see Fig. 6. Channel estimates
`of the other sub-channels are achieved by filtering.
`
`PllOt pattern. m s - 0 002
`
`Pilot pattern. fdTa-0.02
`
`. . . . . . . . . . . . . . . . . . . . . . .
`
`40
`
`+++++++*+*++++++++
`
`10 . . . . . . . . . . . . . . . . . . . . . . .
`
`BO
`40
`Time
`
`80
`
`lo0
`
`‘
`
`20
`
`O
`80
`40
`Time
`
`BO
`
`i
`lo0
`
`
`20
`
`Fig. 6 Resulting pilot pattern when sending a pilot at the
`channel which gives the lowest total bit error rate. Note that
`only few channels will be used for pilot symbols.
`To see the effect of mismatch between the pilot pattern
`design parameters and the actual parameters, the optimal pilot
`patterns (pattern 3) for three Doppler values were used when
`moving at another speed. The actual Doppler frequency was as
`before assumed to be known by the estimator, just to see the
`effect of the pilot pattern without influences of the estimation
`algorithm. The designed pilot patterns for the ”typical urban”
`environment were also used in ”hilly terrain” and ”rural area”
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`specified in [2] to see the influence of the propal, ration environ-
`ment on the bit error rate, see Fig, 7.
`
`BER when using panern design& for another Domler frea. or channel. EbMo-io
`
`The bit error rates within the sub-channels differ depending
`on where the pilots are located. When minimizing the total bit
`error rate (pattern 3) the channels of the sides get higher error
`rate, see example in Fig. 8.
`
`104
`
`10’
`
`t S T s
`
`10”
`
`Fig. 7 Changes in the bit error rate due to Doppler mis-
`match and power delay profile mismatch. The pilot patterns
`designed for fdT,=0.02, 0.002 and 0.0002 were used at dif-
`ferent Doppler values and the designed pattms for typical
`urban environment were used in hilly terrain and rural area.
`As seen in the figure, the pilot patterns are robust to mis-
`matches in the design parameters.
`
`V. DISCUSSION
`To minimize the bit error rate it is desirable to spread the
`pilot symbols both in time and in frequency, as seen in Fig. 4
`and Fig. 5. Normally a worst case design is madle for the chan-
`nel estimation system and then we suggest to tailor the pilot
`pattern to each base station site. A suitable pilot pattern can be
`calculated once the propagation environment and maximum
`expected speed in the particular cell is known. In such a sys-
`tem, the pilot pattern used in the cell is among the information
`transferred to the mobile when it logs into a new cell. When
`designing the pilot pattern one also has to take the estimation
`algorithm into account. Here the estimation algorithm is used
`only for evaluation and the complexity of the used algorithm is
`not a problem. In some cases the pilot pattern has to aid the
`estimation algorithm to enable a less complex oine. The estima-
`tion algorithm used here, the Kalman filter based on an AR-
`process, has no delay and the received signal can be detected
`immediately, i.e. no future pilot symbols is taken into account
`when making the channel estimate. If the received signal is
`stored in a memory, the pilot symbols can be used in both
`”time directions” and the time spacings between pilot symbols
`can be increased, retaining the same performance.
`The degradation due to mismatch in design parameters is
`mainly caused by the estimation algorithm and therefore the
`curves for different Doppler values do not differ much. When
`the pilot pattern is designed for higher Doppler values than the
`actual one, an increased error rate is achieved since the pilot
`symbols are not located as close to each other in frequency as
`desired, In rural area the bit error rate becomes lower due to the
`increased frequency correlation while the opposite happens in
`hilly terrain. In the first case, an even better result is achieved
`with less pilots along the frequency axes and more along the
`time axes.
`
`
`Time
`
`-
`
`2
`
`‘20
`
`30
`
`-”
`
`Channel
`Fig. 8 The bit error rate becomes higher for the side chan-
`nels when the total bit error rate is kept at a minimum
`
`VI. CHANNEL MODEL
`The time dependent impulse response, h(z, I), is assumed to
`be a sum of reflections, see (1) where 6(r) denotes the dirac-
`function.
`
`N
`
`h(T, t ) = c a,(t) * 6 ( t - T n )
`
`(1)
`
`n = l
`The tap coefficients, an(t), and the tap delays, T,, , are chosen
`according to the COST 207 ”Typical Urban” model in the
`GSM specification [2]. The transfer functions, Hgt), are
`obtained by the Fourier transform and these are the functions
`we want to estimate for the different carriers. These channel
`transfer functions are regarded as flat fading and constant dur-
`ing a symbol time.
`A first order AR-process is used to model how the different
`taps may change from one time instant to another. If we look at
`all the transfer functions at the same time, it is possible to set
`up a state-space model of the form:
`H ( k + 1) = OH(k) + v ( k )
`Y ( k ) = WwW) + e(k)
`The matrix I$ is a diagonal N*N-matrix (here treated as a sca-
`lar) with elements
`
`(2)
`
`(3)
`is the symbol time including any
`that define the AR-process.
`cyclic prefix or guard space. The white noise v(k) has covari-
`ance matrix R, = F . R,,
`. FL , where R G ~ M corresponds to the
`multipath intensity profile described in [2]. The vector y(k) is
`the measured transfer functions. C(k) is an observation vector
`with ones only at the positions (channels) measured at time KT,
`and e(k) is measurement noise with a diagonal covariance
`matrix RZ.
`The parameter kAR in the auto-regressive process for the
`channel changes is chosen to adjust the ”memory” so that it
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`corresponds to the "memory" of Jakes' model. Channels corre-
`sponding to the U-shaped spectrum given in [8] were simulated
`and then estimators based on an AR-model with different km
`were used, see Fig. 9
`
`Simulated BER for PPSK. estimates based on AR-models WRh different fdTs
`0.1s1 .
`
`I
`
`I
`
`0.14
`
`I
`
`0.091
`
`' ' .
`¶ 0'
`10-1
`100
`fdTs used in eatimator (fdT- of eotimator)/(tdTi, of Dimulated channel)
`Fig. 9 Bit error rate for a simulated channel by Jakes when
`estimates are based on an AR-process. The minimum value
`is reached for kAR=0.15.
`Simulations were performed with one sub-channel (E@o= 10,
`fdT,=0.002) with every tenth symbol as a pilot symbol. In the
`figure the bit error rates of the nine data symbols are shown.
`Fig. 9 shows that the best fit, in this case, is reached for
`km"o.15.
`The adjustment of the AR-process can also be seen as an
`adjustment of the bandwidth, If we set the 90% power band-
`width of the AR process equal to the bandwidth of the Doppler
`spread, see Fig. 10, a value of km"o.158 can be calculated.
`This is the value used for all bit error rate calculations through-
`out the paper.
`
`I
`
`D o p p l r r 0ow-r .o-otrum 0' AFI-Cror-ll .L"d Channr, by .I**-.
`
`I
`
`~
`
`(6)
`
`(7)
`
`(8)
`
`K ( k ) = $P(klk- l)C(k)*
`* [C(k)P(klk- l)C(k)* + R2I-l
`P(k + 1 Ik) = $P(klk- 1)$* + RI
`-K(k)[C(k)P(klk- l)C(k)* + R2]K(k)*
`P(klk) = P(klk- 1) - P ( k l k - l)C(k)*
`. [C(k)P(klk- l)C(k)* + R2]-'C(k)P(klk- 1)
`The reconstruction error fi(klk) = ~ ( k ) - A ( k l k ) is given by:
`k ( k l k ) = [ @ - K ( k ) C ( k ) ] H ( k - l l k - l ) + v ( k - 1 ) -
`P(klk- l)C(k)*[C(k)P(klk- l)C(k)* + R21-'
`{ e ( k ) + C(k)v(k- 1))
`The Kalman filter is optimal in the sense that the variance of
`the reconstruction error is minimized. The matrix P(klk) is the
`variance matrix and this is used together with K to make an
`estimate of the bit error rate, which in turn is used to decide the
`order in which the channels are going to be measured. For pat-
`tern 3, the channel is chosen that minimizes the total bit error
`rate of the channels after the measurement. The matrices P(klk)
`and K are independent of the measured values and therefore it
`is possible to precompute the order in which the channels are
`going to be measured.
`
`(9)
`
`VIII. BIT ERROR RATE CALCULATIONS
`The bit error rate is calculated for BPSK and QPSK. A
`matched filter receiver is assumed. The sampled output of this
`filter is given by
`
`1)
`
`2x
`j - ( n -
`M
`(10)
`X , = 2 ~ H , e
`+ em
`where E is the signal energy, H,,, is the current transfer function
`at channel m, n is the signal alternative among M sent and e, is
`white gaussian noise. The bit error rate at channel m is calcu-
`lated as [ 11
`
`-2
`
`-1.6
`
`-1
`
`0
`-0.6
`trrqurnoy (f.fC)ffd
`
`0.6
`
`1
`
`,.s
`Fig. 10 Power density spectrum of the channel by Jakes and
`a first order AR-process. The latter is adjusted to have the
`90% power bandwidth equal to the Doppler spread.
`
`2
`
`VII. CHANNEL ESTIMATOR
`For the analysis and pilot pattern design a Kalman filter is
`used to estimate the transfer functions. The Kalman filter is
`causal and uses measurements up to time k to estimate the
`transfer functions, H(k). The Kalman filter is given [3] by (4)-
`(S), where x* denotes conjugate transpose of X
`A(klk) = A ( k l k - 1 ) + P(klk- l)C(k)*
`(4)
`' (C(k)P(klk- l)C(k)* + R2)-'(y(k) - C(k)A(klk- 1))
`A ( k + Ilk) = $ k ( k l k - 1) +
`(5)
`K(k)[Y(k) - C ( k ) f W l k - 111 = $A(klk)
`
`where
`
`g,,, are the outputs of the Kalman filter. These are not known in
`advance and therefore (13) - (15) are used. If matrix notation is
`used and signal alternative 1 is used for the pilot symbol (does
`affect the analysis here, but in practise different pilot symbol
`alternatives should be used), the expectations for all the chan-
`nels can be calculated as:
`
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`(13)
`
`E[X(k)fi(k)*I
`= E[{2&H(k) + N ( k ) } { H ( k ) -k(k)}*]
`= 2&E[H(k)H(k)*] - 2&E[H(k)ii(k)*]
`E[IX(k)l21 = E[{2&H(k) + N(k)}{2&H(k) + N(k)}*I (14)
`= 4 ~ ~ E [ H ( k ) H ( k ) * l + E[N(k)N(k)*]
`E[Ifw)l21 = E [ { H ( k ) - f i ( k ) W i ( k ) -&k)I*l
`(15)
`= E[ H( k)H(k)*] - 2E[ H( k)fi(k)*] + E[fi(k)ii(klk)*]
`The expected values E [ X ( k ) A ( k ) * l , EIIX(k)lzl and E[18(k)1*1 are
`independent of the measured values but dependent upon which
`channels that are measured. Therefore they are time dependent.
`The expectations can be calculated as:
`
`E[H(k)H(k)*I
`= E [ { $ H ( k - 1) + v ( k - l)}{$H(k- 1) +v(k- 1)}*1
`= q2E[H(k - l)H(k - 1)*3 + E[v(k - l)v(k - 1)*]
`= Rl/(l -q2)
`
`E[N(k)N(k)*] = R2
`E[H(k)H(k(k)*] = E[H(k)H(k)* - H(k)k(k:Ik)*]
`= E[$H(k- 1)H(k - 1)*$* + R , - RIC(k)*a*
`-$H(k- l ) k ( k - l)*$*C(k)*U"]
`= ( $ E [ H ( k - l)H(k- 1)*]$* + R1)(Z- C(k)*a*)
`where
`
`(16)
`
`(17)
`
`(18)
`
`E [ k(kl k)H( kl k)"] = P(kl k )
`a = P(klk- l)C(k)*{C(k)P(klk- l)C(k)* i k R2}-l
`(20)
`The bit error rate calculation is compared and verified by
`simulations. Proakis [ 11 gives expressions for the bit error rate
`when estimating a constant Rayleigh channel using different
`numbers of pilot symbols, see Fig. 1 1.
`
`(19)
`
`estimating a constant channel with use of (infinitely) many
`pilot symbols corresponds to that of coherent detection. Nor-
`mally in a practical system the effect of the estimation is some-
`where between these two cases.
`IX. CONCLUSIONS
`The bit error rate for pilot assisted QPSK modulation is cal-
`culated when using different pilot patterns. The ability to esti-
`mate the channel reliably when it changes due to e.g. vehicle
`movements is highly dependent on the pilot pattern used. By
`rearranging the pilot pattern it is, in some cases, possible to
`handle 10 times higher Doppler frequency alternatively possi-
`ble to reduce the number of needed pilot symbols the same
`amount, still retaining the same bit error rate. Alternatively the
`new pilot pattern could be used to reduce the bit error rate up to
`a factor 5, even more in a low noise environment. The pilot
`symbols in the proposed pilot pattern are spread out both in
`time and frequency. For a given propagation environment, e.g.
`a base station site, it is possible to pre-calculate a suitable pilot
`pattern.
`
`REFERENCES
`[ 11 J. G .Proakis, Digital Communications, third edition,
`McGraw-Hill, USA, 1995.
`[2] European Telecommunications Standard Institute,
`05.05, version 4.6.0, July 1993.
`[3] J. M. Mendel, & z, Prentice-Hall, USA,
`
`BER for BPSK. constant Rayleioh channel. EWNo=O. 10.20 and 30
`
`1995.
`[4] M. Sandell, Design and Analysis of Estimators for Multi-
`carrier Modulation and Ultrasonic Imaging, Ph.D. thesis, Lulel
`University of Technology, Sweden, 1996.
`[5] J. K. Cavers, An Analysis of Pilot Symbol Assisted Modu-
`lation for Rayleigh Fading Channels,
`vol40, no 4, pp 686-693, nov 1991.
`[6] T. Mueller, K. Brueninghaus and H. Rohling, Performance
`of Coherent OFDM-CDMA for Broadband Mobile Communi-
`cations, Wireless Personal Communications 2, pp 295-305,
`Kluwer, Netherlands, 1996.
`[7] P. Hoeher. TCM on Frequency-Selective Land-Mobile Fad-
`ing Channels, 1
`munications, Tirrenia, Italy, Sept. 1991.
`[ 81 W. Jakes, Microwave Mobile Communications, Wiley-
`Interscience, USA 1974
`d
`
`10"
`
`16'
`
`I 0"
`
`'O.1,
`
`e
`A
`;I
`Y4
`20
`YO
`Number of pilot symbols used for the estlmatu
`Fig. 11 Bit error rate for 2PSK when estimating a constant
`Rayleigh channel with several pilot symbols at different
`signal to noise ratios.
`The bit error rate when using only one pilot symbol corre-
`sponds to estimating a rapidly changing channel or the first
`estimate of an unknown channel. Then, only tlhe latest mea-
`surement is useful. In a similar way, the bit error rate when
`
`16
`
`18
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