IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

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Time Dispersion and Delay Spread Estimation for Adaptive OFDM Systems Tevfik Yücek, Student Member, IEEE, and Hüseyin Arslan, Senior Member, IEEE

Abstract—Time dispersion is one of the most important characteristics of the wireless channel. The knowledge about the time dispersion of a channel can be used to design better systems that can adapt themselves to the changing nature of the transmission medium. In this paper, algorithms to estimate the time dispersion (frequency selectivity) of the channel, which use the frequencydomain channel estimates or the frequency-domain-received signal, are proposed. The proposed algorithms estimate the power delay profile (PDP) from which the two important dispersion parameters, namely root-mean-squared (RMS) delay spread and maximum excess delay of the channel, are estimated. It is shown that synchronization errors bias the performance of the estimators based on the channel correlation, and this bias is removed by obtaining the PDP using the magnitude of the channel estimates. Moreover, the Cramér–Rao lower bound (CRLB) for the estimation of the RMS delay spread is derived, and the performances of the proposed algorithms are compared against this theoretical limit. Index Terms—Adaptation, delay spread estimation, frequency selectivity, orthogonal frequency division multiplexing (OFDM), time dispersion.

I. I NTRODUCTION

O

RTHOGONAL frequency-division multiplexing (OFDM) has been successfully applied in various wireless communication systems in the last decade [1]. Those systems, however, should be capable of efficiently working in a wide range of operating conditions, such as a large range of mobile speeds, different carrier frequencies in licensed and license-exempt bands, various delay spreads, asymmetric traffic loads in downlink and uplink, and wide dynamic signalto-noise ratio (SNR) ranges. The aforementioned reasons motivated the use of adaptive algorithms in new-generation wireless communication systems [2]. Adaptation aims at optimizing the wireless mobile radio system performance, enhancing its capacity, and utilizing the available resources in an efficient manner. However, adaptation requires a form of accurate parameter measurement. One key parameter in the adaptation of OFDM systems is time dispersion, which provides information about the frequency selectivity of the wireless channel.

Manuscript received August 14, 2006; revised August 20, 2007 and August 21, 2007. The review of this paper was coordinated by Prof. M. Juntti. T. Yücek is with Atheros Communications, Inc., Santa Clara, CA 95054 USA (e-mail: [email protected]). H. Arslan is with the Department of Electrical Engineering, University of South Florida, Tampa, FL 33613 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2007.909247

In OFDM systems, the cyclic prefix (CP) length needs to be larger than the maximum excess delay of the channel. If this information is not available, the worst-case channel condition is used for system design, which makes CP a significant portion of the transmitted data. One way to increase the spectral efficiency is to adapt the length of the CP to the changing multipath conditions, which requires the channel excess delay knowledge [3], [4]. Adaptive filtering for channel estimation is another area where the time dispersion information of the channel is useful. A 2-D Wiener filter, which is implemented as a cascade of two 1-D filters, is used for channel estimation in [5]. The bandwidth of the second filter, which is in the frequency direction, is changed, depending on the estimated delay spread of the channel to keep the noise low and, thus, to improve the channel estimation performance. Similarly, the coefficients of the frequency-domain channel estimation filter are adaptively chosen in [6], depending on the channel difference vector, which is directly related to the delay spread of the channel. The information about the time dispersion of the channel can also be used to allocate the pilot symbols in the frequency direction, whereas the allocation in time direction requires Doppler spread estimation. For example, the number of pilots and their spacing are adjusted depending on the delay spread knowledge in [7] and [8]. Other OFDM parameters that could be adaptively changed using the knowledge of the time dispersion are OFDM symbol duration and OFDM subcarrier bandwidth. Characterization of the frequency selectivity of the radio channel is studied in [9] and [10] using the level crossing rate (LCR) of the channel in the frequency domain. However, the LCR is very sensitive to noise, which increases the number of level crossings and severely deteriorates the performance of the LCR measurements [11]. Filtering the channel frequency domain (CFR) reduces the noise effect, but finding the appropriate filter parameters could be a problem. If the filter is not properly designed, one might end up smoothing the actual variation of the frequency-domain channel response. Time dispersion of the channel can be estimated using the channel impulse response (CIR) estimates as well. In [5] and [7], the CIR is obtained by taking the inverse discrete Fourier transform (IDFT) of the frequency-domain channel estimate, which is calculated at pilot locations. In [12], the instantaneous rootmean-squared (RMS) delay spread is obtained by estimating the CIR in the time domain. The detected symbols in the frequency domain are used to regenerate the time-domain signal through IDFT, and then, this signal is correlated with the received signal to obtain the CIR. Timing synchronization errors create a problem with the time-domain estimation; if synchronization is independently performed over different frames (or symbols),

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

the estimated CIRs should be time aligned as the timing errors will be different for each CIR estimate. Techniques that exploit the CP are proposed for delay spread estimation in [13] and [14]. The change of gradient of the correlation between the CP and the last part of the OFDM symbol is used as a strategy to detect the dispersion parameters in [13]. This method requires computationally complex optimization, and the accuracy of the technique can be expected to degrade for closely spaced and weak multipath components. In [14], the magnitude of each arriving tap and the corresponding delay are estimated from the calculated correlation. In [15], the RMS delay spread of the channel is obtained using the channel frequency correlation (CFC) by using the analytical relation between the RMS delay spread and the coherence bandwidth, which is obtained from the CFC. In this paper, the time dispersion of the radio channel is estimated using frequency-domain channel estimates. In OFDM systems, the channel can be easily estimated in the frequency domain, and this is usually the preferred method as both estimation and equalization in the frequency domain are simpler than their time-domain equivalents. Since the timing errors are folded into the channel estimates in the frequency domain as a subcarrier-dependent phase term, the magnitude of the channel estimates is proposed for delay spread estimation to remove the phase dependence. As a third algorithm, the magnitude of the received frequency-domain signal is used when a constant envelope modulation is employed. The proposed algorithms estimate the channel power delay profile (PDP), which is then used to extract the time dispersion parameters, i.e., RMS delay spread and maximum excess delay of the channel. We also derive the Cramér–Rao lower bound (CRLB) for RMS delay spread estimation and use it as a benchmark for testing the developed algorithms. This paper is organized as follows. The system model will be introduced in Section II, whereas the proposed algorithms will be presented in Section III. The numerical results will be given in Section IV, and the paper will be concluded in Section V.

At the receiver, the signal is received along with noise. After time and frequency synchronization, down sampling, and removal of the CP, the simplified baseband model of the received samples can be formulated as ym (n + θm ) =

L−1


xm (n − θm − l)hm (l) + wm (n)

(2)

l=0

where L is the number of sample-spaced channel taps, wm (n) is the additive white Gaussian noise sample with zero mean 2 , and the time-domain CIR for the mth and variance of σw OFDM symbol hm (l) is given as a time-invariant linear filter. The timing error θm is caused by imperfect synchronization, and its statistics depends on the SNR and the synchronization algorithm used. In this case, after taking the discrete Fourier transform (DFT) of the received signal ym (n + θm ), the samples in the frequency domain can be written as1 [16] Ym (k) = DFT {ym (n + θm )} = Xm (k)Hm (k)e−j2πkθm /N + Wm (k) 0 ≤k ≤ N − 1

(3)

where Hm and Wm are the DFTs of hm and wm , respectively. ˆ m can be The least squares (LS) estimate of the CFR H calculated using the received signal and the knowledge of transmitted symbols as ˆ m (k) = Ym (k) H Xm (k)

(4)

Wm (k) = Hm (k)e−j2πkθm /N + .    Xm (k)    ˜ (k) H m

(5)

Zm (k)

The LS channel estimation consists of the desired channel with a frequency-dependent phase term due to timing errors and an additive estimation error term due to noise. III. P ROPOSED D ELAY S PREAD E STIMATION A LGORITHMS

II. S YSTEM M ODEL The OFDM converts a serial data stream into parallel blocks of size N and modulates these blocks using IDFT. Timedomain samples of an OFDM symbol can be obtained from frequency-domain data symbols as xm (n) = IDFT {Xm (k)} N −1 1
=√ Xm (k)ej2πnk/N , N k=0

In this section, we propose algorithms to estimate the time dispersion parameters. Three methods will be given to estimate the PDP using frequency-domain information. Calculation of the RMS delay spread and the maximum excess delay from the estimated PDPs is also discussed in this section. A. Channel-Estimation-Based Algorithm

0≤n≤N −1

(1)

where Xm (k) is the transmitted data symbol at the kth subcarrier of the mth OFDM symbol, and N is the number of subcarriers. After the addition of CP and digital-to-analog conversion, the signal is passed through the mobile radio channel. In this paper, the channel is assumed to be constant over an OFDM symbol but time varying across OFDM symbols, which is a reasonable assumption for low and medium mobility.

In OFDM systems, channel estimation is usually done in the frequency domain using known training symbols. Hence, the frequency-domain channel estimates are usually available. Using these estimates, the instantaneous CFC values can be calculated as   ˆ t (k)H ˆ ∗ (k + ∆) (6) RHˆ (∆) = Et,k H t 1 It is assumed that the timing errors are small enough so that there will not be any intersymbol interference or intercarrier interference.

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where Et,k [·] is the expectation over training symbols t and over subcarriers k (averaging within an OFDM symbol), assuming the time–frequency response of the channel is widesense stationary. Assuming that the channel and noise terms are uncorrelated, the correlation given in (6) can be written as RHˆ (∆) = RH˜ (∆) + RZ (∆)

(7)

where RH˜ (∆) is the correlation of the effective channel, and RZ (∆) is the correlation of the channel estimation error. RZ (∆) becomes a delta function when the estimation errors at different subcarriers are uncorrelated, i.e., when it is white. However, as the channel estimation is a filtering operation, this noise is usually colored, and it creates a bias on the estimates obtained using RHˆ ; hence, care should be taken. When the channel estimates are obtained using the LS method given in Section II, however, the noise becomes white as the additive noise on the received signal W (k) is assumed to be white and uncorrelated with the transmitted signal as well as the channel. In this case, the correlation (6) can be written as RHˆ (∆) = RH˜ (∆) + δ(∆)σz2

(8)

where σz2 is the variance of the channel estimation error Z(k), and δ(·) is the Kronecker delta function. Using (5) and (6), the channel correlation with timing errors can be obtained as   RH˜ (∆) = RH (∆)Et ej2πθt ∆/N .

(9)

The expectation in (9) is a function of the statistics of the estimation error θt , which depends on channel conditions and the algorithm used for synchronization. This expectation can be written in terms of timing error statistics as 

j2πθt ∆/N

Et e

θmax



ej

=

2πθ∆ N

p(θ)dθ

(10)

θmin

where θmin and θmax are the minimum and maximum synchronization errors, respectively, and p(θ) is the probability density function of the synchronization error θ. In the case of perfect synchronization, it becomes a delta function, i.e., p(θ) = δ(θ). In this paper, the timing errors are assumed to have a Gaussian distribution with zero mean and variance of σθ2 [17]. As can be seen from (8) and (9), there are two impairments that affect the correlation estimates, i.e., additive noise and synchronization errors. We first assume perfect timing synchronization and concentrate on the noise term. In a similar problem for a different context, a parabola is fitted to the lags with nonzero index for finding the value at the zeroth lag of the timedomain channel correlation in [18]. In this paper, we use the same algorithm to remove the correlation term due to noise in (8). This algorithm consists of the following steps. 1) Calculate RHˆ (∆) for ∆ = 1, . . . , M . 2) Obtain the coefficients of the parabola P(∆) using the LS method.  3) Substitute P(0) into RHˆ (0), and call it RH ˜.

Fig. 1. Magnitude of CFC with perfect synchronization and with synchronization errors. The analytical [see (10)] and simulation results are shown.

Once the effect of noise is removed, the PDP can be esti mated from the CFC estimate RH ˜ by simply applying the IDFT operation as 

Pl = IDFT RH ˜ (∆) N −1 1
 √ RH˜ (∆)ej2π∆l/N , = N ∆=0

(11) 0≤l ≤L−1

(12)

where Pl = Em [|hm (l)|2 ] is the lth tap of the channel PDP. Timing error is the other impairment that degrades the performance of the channel-estimation-based algorithm. Fig. 1 shows the correlation magnitude as a function of subcarrier separation ∆ for perfect synchronization and for synchronization errors with a zero-mean Gaussian distribution and variance of 2Ts2 , i.e., σθ2 = 2Ts2 , where Ts is the sampling time. Both the analytical results obtained using (10) and the simulation results are given. Although the algorithm given in this section is simple to calculate and straightforward, timing errors affect the correlation estimates and bias the time dispersion estimation, as shown in Fig. 1. To remove this effect, we propose to use the magnitude square of the frequency-domain channel estimates, which will be discussed in the next section. B. Channel-Magnitude-Based Algorithm The correlation of the magnitude of the channel estimates can be represented as  2 2 ˆ ˆ H H (∆) = E (k) (k + ∆) (13) R|H| t . ˆ 2 t,k t By inserting (5) into (13), the correlation can be simplified to   2 4 R|H| ˆ 2 (∆) = R|H|2 (∆) + (1 + δ(∆)) 2RH (0)σz + σz (14)

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

where R|H|2 (∆) is the correlation of the magnitude of the true channel, and it can be written as (see Appendix A for a detailed derivation) R|H|2 (∆)

  = Et,k |Ht (k)|2 |Ht (k + ∆)|2

N −1 N −1 2

π(l − u)∆ = 2 Pl Pu cos2 N N u=0

(15) (16)

l=0

=

 N −1 N −1 1

2πu∆ 2πl∆ cos P P 1 + cos l u 2 N N N u=0 l=0

2πl∆ 2πu∆ + sin sin . N N

(17)

(18)

where RH (∆) is the correlation of H(k) as defined before. Note that the left-hand side (LHS) of this equation can be estimated using the received signal. The RHS is a multiplication, and when the IDFT is applied on this term, it becomes a convolution of PDP with the flipped version of itself in the time domain. This follows from the properties of DFT [19] and from the fact that the IDFT of the CFC is equal to the PDP. The resulting equation can then be written as   R|H|2 (0) IDFT R|H|2 (∆) − 2 ∗ (∆)} = IDFT {RH (∆)RH

=

u−1


Pi PN −u+i ,

1≤u≤N

The algorithms given in the previous sections can only use the training symbols for estimation. However, when the OFDM system employs a constant envelope modulation, the power of the received signal Y (k) can be used. The received signal in the frequency domain for OFDM is given in (3). Using this equation, the correlation of the magnitude of the received frequency-domain signal can be calculated as   (21) R|Y |2 (∆) = Em,k |Ym (k)|2 |Ym (k + ∆)|2 . After expanding (21), the simplified version can be written as

The effect of noise on (14) at the first correlation lag, i.e., ∆ = 0, is two times the effect of noise on the other lags. Therefore, if the value of the first correlation lag is calculated using other lags, the overall effect of the noise can be subtracted from the correlation that removes the effect of noise. The parabola-fitting algorithm described in Section III-A is used to remove the noise contribution in this method as well. Note that the first term in the right-hand side (RHS) of (17) is equal to R|H|2 (0)/2, and the second and third terms together are equal to the magnitude square of the correlation of the channel response RH (∆). Using these facts, the following equality can be obtained: R|H|2 (0) ∗ R|H|2 (∆) − = RH (∆)RH (∆) 2

C. Received-Signal-Based Algorithm

R|Y |2 (∆) = R|H|2 (∆)R|X|2 (∆)   2 4 . (22) + σw + (1 + δ(∆)) 2RH (0)RX (0)σw For constant envelope modulations, R|X|2 (∆) = RX (∆) = 1; therefore, (22) reduces to (14) as2 R|Y |2 (∆) = R|H| ˆ 2 (∆).

Therefore, the algorithm given in Section III-B can be used to ˆ estimate the PDP using Y (k) instead of H(k). This enables us to use all of the received OFDM symbols, which result in both noise averaging and getting better correlation estimates. D. Estimation of RMS Delay Spread and Maximum Excess Delay The RMS delay spread and the excess delay spread (XdB) are multipath channel parameters that can be determined from a PDP. Once the PDP of the channel is estimated using the proposed methods, these parameters can be estimated using their definitions. The RMS delay spread τRMS is the square root of the second central moment of the PDP, and the maximum excess delay (XdB) of the PDP is defined as the time delay during which the multipath energy falls to X dB below the maximum value. The RMS delay spread is defined as [20]

τRMS

(19) (20)

i=0

where u is the index of the IDFT outputs. Having the LHS of the equality calculated using the received signal, we can estimate the PDP using (20) and by solving the nonlinear set of equations. For this purpose, LS optimization is used in this paper. The number of unknowns can be limited to the number of PDP taps L. As the length of the CP is expected to be larger than the maximum excess delay, the number of unknowns is set to the length of the CP. The system of equations can then be solved as the number of unknowns is smaller than the number of equations, which is N .

(23)

  2  L−1 L−1 2   l=0 Pl τl l=0 Pl τl = − L−1 L−1 Pl Pl l l

(24)

where τl = lTs is the delay of the lth multipath component. To decrease the effect of errors in the PDP estimation, taps with a power of 25 dB below the most powerful lag are set to 0. Moreover, we consider the maximum excess delay of 25 dB, i.e., the maximum excess delay is equal to the delay of the last nonzero tap. The statistics of the channel might be changing in time because of the environmental changes or because of the mobility of the transmitter or receiver. In this case, the correlation estimates can be updated using an alpha tracker to capture 2 Without

2 = 1. Hence, σ 2 = σ 2 . loss of generality, we assume σX w z

YÜCEK AND ARSLAN: TIME DISPERSION AND DELAY SPREAD ESTIMATION FOR ADAPTIVE OFDM SYSTEMS

Fig. 2.

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Frame structure of the system used for testing the proposed algorithms.

this variation. For the channel-magnitude-based algorithm, for example, the correlation values can be updated as m m−1 R|H| 2 (∆) = (1 − α)R|Hm |2 (∆) + αR |H|2 (∆)

(25)

where R|Hm |2 (∆) is the correlation value obtained using the mth symbol. The forgetting factor 0 ≤ α ≤ 1 is a design parameter, and it should be selected depending on how fast the channel parameters are changing. To evaluate the performance of the RMS delay spread estimation obtained from the outputs of the proposed algorithms, the CRLB is used. The derivation of CRLB is given in Appendix B. The bound for the RMS delay spread estimation performance is a function of the PDP (relative delay and power of each path), and it is inversely proportional with the number of frames used for estimation. The additive noise is not used while deriving the CRLB as the effect of noise is removed in the proposed algorithms. In fact, the proposed algorithms are asymptotically SNR independent. IV. N UMERICAL R ESULTS A system similar to the OFDM physical layer (PHY) of the IEEE 802.16d standard [21] is used for the simulations. The total number of subcarriers is 256, out of which 200 subcarriers are used for transmitting data information and pilots. The center frequency carrier is set to 0, and the outermost 55 subcarriers (27 on the left and 28 on the right of the spectrum) are not used to allow for guard bands. The system bandwidth is chosen as 10 MHz, and the length of the CP is set to 32 samples (the length of the guard band is 3.2 µs). For simulating the wireless channel, Channel A of the International Telecommunication Union Radiocommunication Sector channel model [22] for vehicular environments with high antenna is used. The mobile speed used in the simulations3 is 60 km/h, and the time-varying channel is generated according to [23]. For estimating the CFR, the LS method (see Section II) is used. The estimation is assumed to be done in uplink for a user with 20 OFDM symbols in a frame with 10-ms frame duration. The frame structure of the considered system is illustrated in Fig. 2. In uplink, each user is assumed to have a training symbol for synchronization and channel estimation purposes. The channelestimation-based algorithms use the channel estimates obtained 3 Similar results are observed when tests at different mobile speeds are performed.

Fig. 3. NMSE performance of the RMS delay spread estimators as a function of the number of frames used for estimation. Ideal channel and synchronization knowledge is assumed in this figure.

using the training symbols, whereas the received signal-powerbased algorithm uses all of the 20 transmitted symbols (one training symbol and 19 data symbols) for the estimation of PDP. The first five nonzero correlation values are used for obtaining the parameters of the parabola, which is used to remove the effect of noise on correlation estimates, i.e., M = 5. The normalized mean square error (NMSE) is used as a performance measure of the estimator as it reflects both the bias and the variance of the estimator. The NMSE of τRMS is defined ∆ 2 as NMSE(τRMS ) = E[(ˆ τRMS − τRMS )2 ]/τRMS , where τˆRMS is the estimate of τRMS . Fig. 3 shows the NMSE of the RMS delay spread as a function of the number of frames used for estimation. In this figure, a perfect channel and timing information is assumed to be available, and the CRLB for the estimation [see (38)] is also presented. The multipath channel components at different frames are assumed to be independent in this figure. In this section, the labels Algorithm CE, CM, and RS correspond to channel-estimation-based, channel-magnitudebased, and received-signal-based algorithms, respectively. The channel-estimation-based algorithm yields very close results to the bound, and the performance loss in the channel-magnitudebased algorithm can be explained by the information lost by not utilizing the phase information of the channel. Figs. 4 and 5 show the NMSE of the RMS delay spread estimation as a function of the number of frames used for estimation. Fig. 4

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Fig. 4. NMSE performance of the RMS delay spread estimators as a function of the number of frames used for estimation at 5-dB SNR with perfect timing synchronization.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

Fig. 6. NMSE performance of the RMS delay spread estimators as a function of the SNR. Estimation is performed over 80 frames.

V. C ONCLUSION

Fig. 5. NMSE performance of the RMS delay spread estimators as a function of the number of frames used for estimation at 5-dB SNR with timing synchronization errors.

shows the case of perfect timing knowledge, whereas in Fig. 5, a timing mismatch with zero-mean Gaussian distribution and variance of 2Ts2 is introduced. The SNRs of the received signal for both figures are set to 10 dB. The channel-estimation-based algorithm performs better than the other algorithms in the case of no timing errors as the phase information on the channel is also used. On the other hand, it has a large error floor when there are timing errors due to inaccurate correlation estimates. The results presented in these figures show that the channelmagnitude-based algorithm (Algorithm CM) and the received signal-magnitude-based algorithm (Algorithm RS) work quite satisfactorily, even under timing errors. The NMSE as a function of SNR for 80 frames is shown in Fig. 6 for perfect timing synchronization. The performances of all three algorithms increase as the SNR increases. However, after a certain SNR level (around 10 dB), an error floor is observed as the number of frames is not large enough, and sufficient statistics are not obtained.

Delay spread estimation algorithms for OFDM systems have been proposed in this paper. The proposed algorithms estimate the PDP of the channel, which is then used to calculate the dispersion parameters. It is found that the timing errors cause an estimation error floor if the channel frequency estimates are directly used, and this problem is overcome by using the magnitude of the channel estimates. Moreover, the channelmagnitude-based algorithm is extended to a method that uses the received signal power (in the frequency domain) when the transmitted symbols have a constant envelope. The performances of the developed algorithms are tested using computer simulations, and the channel and received signal-magnitudebased algorithms are shown to perform well under different scenarios. Moreover, the CRLB for the RMS delay spread parameter is derived and used to validate the performance of the proposed methods. A PPENDIX A The frequency-domain CFR can be obtained as 1
H(k) = √ h(l)e−j2πlk/N . N l

(26)

Using (26) and the definition of correlation, the correlation of the CFR magnitude can be formulated as R|H|2 (∆)   = Em,k |Hm (k)|2 |Hm (k + ∆)|2 =

(27)

1



 E h(l)h∗ (l )h(u)h∗ (u ) N2

u l

l

u



× e−j2π[k(l−l )+(k+∆)(u−u )] . (28)

YÜCEK AND ARSLAN: TIME DISPERSION AND DELAY SPREAD ESTIMATION FOR ADAPTIVE OFDM SYSTEMS

The summands in (28) equate to zero, except for the following cases. 1) l = l = u = u . In this case • E[h(l)h∗ (l )h(u)h∗ (u )] = E[|h(l)|4 ] = 2Pl2 ;

• E[e−j2π[k(l−l )+(k+∆)(u−u )] ] = 1.   2) l = l , u = u , l = u. In this case • E[h(l)h∗ (l )h(u)h∗ (u )] = Pl Pu ;

• E[e−j2π[k(l−l )+(k+∆)(u−u )] ] = 1.   3) l = u , u = l , l = u. In this case • E[h(l)h∗ (l )h(u)h∗ (u )] = Pl Pu ;

• E[e−j2π[k(l−l )+(k+∆)(u−u )] ] = e−j2π∆(l−u) . Using these results, (28) can be written as

taps hi as independent complex Gaussian random variables. Hence, p(h; P ) can be easily formulated. When each channel tap is assumed to be independent of the others, the Fisher matrix becomes a diagonal matrix whose diagonal entries can be calculated as [I(P )]ii =

l

+

l



l

u;u=l





Pl Pu cos 2π∆(l − u)/N 

(29)

u;u=l

1

Pl Pm (1 + cos (2π∆(l − u)/N )) N2 u l 2

Pl Pm cos2 (π∆(l − u)/N ) . = 2 N u

=

A PPENDIX B The RMS delay spread is a function of the PDP, as given in (24). Let P = [P1 P2 , . . . , PL ]. Then, τRMS can be written as a function of P as τRMS = g(P ). In this case, the CRLB for τRMS can be obtained as [24] CRLB(τRMS ) =

 d d  (g(P )) I −1 (P ) g(P )T dP dP

 P2 I −1 (P ) ii = i . Nf

(37)

1
Pi2 2 Nf i=0 4τRMS   Pl τl2 τi2 ×  + l ( l Pl )2 l Pl  2 

P τ l l τi + l Pl − Pl τl . ( l Pl )3 l l L−1

CRLB(τRMS ) = (31)

(36)

Please note that the diagonal entries of I −1 (P ) give the lower bound on the estimation error of each tap’s power, i.e., E[(Pˆi − Pi )2 ] ≥ Pi2 /Nf , where Pˆi is the estimate of the ith tap. Finally, by inserting (33) and (37) into (32), the CRLB for the RMS delay spread can be obtained as

(30)

l

Nf Pi2

where Nf is the number of available CIRs, i.e., number of frames over which the estimation is performed. Consequently, the inverse of I(P ), which is a diagonal matrix as well, can be easily obtained as

R|H|2 (∆)

 1 
2

= 2 2Pl + Pl Pm N

1721

(38) R EFERENCES

(32)

where (·)T denotes the transpose, and dg(P )/dP is defined as 

∂g(P ) ∂g(P ) d ∂g(P ) (g(P )) = ··· . (33) dP ∂P1 ∂P2 ∂PL Using the definition of the RMS delay spread [see (24)], the partials with respect to each tap can be obtained as   P τ2 1 ∂g(P ) τ2  i + l l l2 = ∂Pi 2τRMS ( l Pl ) l Pl   

l Pl τl +  Pl − Pl τl . (34) τi ( l Pl )3 l l The L × L matrix I(P ) is the Fisher information matrix, which is defined by  2

∂ ln p(h; P ) [I(P )]ij = −E (35) ∂Pi ∂Pj where p(h; P ) is the likelihood function of the CIR vector h = [h1 h2 , . . . , hL ] conditioned on P . We model the channel

[1] R. Prasad and R. Van Nee, OFDM for Wireless Multimedia Communications. Boston, MA: Artech House, 2000. [2] H. Arslan and T. Yücek, “Adaptation of wireless mobile multi-carrier systems,” in Adaptation Techniques in Wireless Multimedia Networks. Commack, NY: Nova, 2006. [3] D. T. Harvatin and R. E. Ziemer, “Orthogonal frequency division multiplexing performance in delay and Doppler spread channels,” in Proc. IEEE Veh. Technol. Conf., Phoenix, AZ, May 4–7, 1997, vol. 3, pp. 1644–1647. [4] Z.-Y. Zhang and L.-F. Lai, “A novel OFDM transmission scheme with length-adaptive cyclic prefix,” J. Zhejiang Univ. Sci., vol. 5, no. 11, pp. 1336–1342, 2004. [5] F. Sanzi and J. Speidel, “An adaptive two-dimensional channel estimator for wireless OFDM with application to mobile DVB-T,” IEEE Trans. Broadcast., vol. 46, no. 2, pp. 128–133, Jun. 2000. [6] T. Onizawa, M. Mizoguchi, T. Sakata, and M. Morikura, “A simple adaptive channel estimation scheme for OFDM systems,” in Proc. IEEE Veh. Technol. Conf., Amsterdam, The Netherlands, 1999, vol. 1, pp. 279–283. [7] A. Dowler, A. Doufexi, and A. Nix, “Performance evaluation of channel estimation techniques for a mobile fourth generation wide area OFDM system,” in Proc. IEEE Veh. Technol. Conf., Vancouver, BC, Canada, Sep. 2002, vol. 4, pp. 2036–2040. [8] O. Simeone and U. Spagnolini, “Adaptive pilot pattern for OFDM systems,” in Proc. IEEE Int. Conf. Commun., Paris, France, Jun. 2004, vol. 2, pp. 978–982. [9] K. Witrisal, Y.-H. Kim, and R. Prasad, “RMS delay spread estimation technique using non-coherent channel measurements,” Electron. Lett., vol. 34, no. 20, pp. 1918–1919, Oct. 1998. [10] K. Witrisal, Y.-H. Kim, and R. Prasad, “A new method to measure parameters of frequency-selective radio channel using power measurements,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1788–1800, Oct. 2001.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

Tevfik Yücek (S’06) received the B.S. degree in electrical and electronics engineering from the Middle East Technical University, Ankara, Turkey, in 2001 and the M.S. and Ph.D. degrees in electrical engineering from the University of South Florida, Tampa, in 2003 and 2007, respectively. He is currently with Atheros Communications, Inc., Santa Clara, CA. His research interests are in signal processing techniques for wireless multicarrier systems and cognitive radio.

Hüseyin Arslan (SM’04) received the Ph.D. degree from Southern Methodist University (SMU), Dallas, TX, in 1998. From January 1998 to August 2002, he was with Ericsson Inc., Research Triangle Park, NC, where he was involved with several projects related to 2G and 3G wireless cellular communication systems. Since August 2002, he has been with the Department of Electrical Engineering, University of South Florida, Tampa. He has also been with Anritsu Company, Morgan Hill, CA, where he was a Visiting Professor during the summers of 2005 and 2006, and has been consulting part-time since August 2005. His research interests are related to advanced signal processing techniques at the physical layer with cross-layer design for networking adaptivity and quality-of-service (QoS) control. He is interested in many forms of wireless technologies, including cellular, wireless PAN/LAN/MANs, fixed wireless access, and specialized wireless data networks like wireless sensors networks and wireless telemetry. The current research interests include UWB, OFDM-based wireless technologies with emphasis on WIMAX, and cognitive and software-defined radio. Dr. Arslan is an editorial board member for the Wireless Communication and Mobile Computing journals and was the Technical Program Co-Chair of the IEEE Wireless and Microwave Conference 2004. He has also served as technical program committee member, session, and symposium organizer for several IEEE conferences.