A Null-Subcarrier-Aided Reference Symbol Mapping ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012

A Null-Subcarrier-Aided Reference Symbol Mapping Scheme for 3GPP LTE Downlink in High-Mobility Scenarios Siva D. Muruganathan, Member, IEEE, Witold A. Krzymień, Senior Member, IEEE, and Abu B. Sesay, Senior Member, IEEE

Abstract—In this paper, a new reference symbol (RS) mapping scheme for the Third-Generation Partnership Project (3GPP) long-term evolution (LTE) and LTE-Advanced (LTE-A) downlink is proposed to improve channel estimation performance in highmobility communication scenarios. The proposed scheme employs null subcarriers to guard RSs, which helps mitigate the effect of intercarrier interference (ICI) on subcarriers carrying RSs. Additionally, the proposed scheme allows the ICI gain parameters to be estimated via a simple frequency-domain estimator. Modified Cramer–Rao bound (MCRB) expressions are derived for the proposed scheme, as well as for the conventional RS mapping scheme defined in the 3GPP LTE and LTE-A standards to compare their performance at high mobile user speeds. These bounds, together with mean square errors obtained from simulations, reveal superior performance achieved by the proposed scheme in high-mobility scenarios. Additionally, at high mobile user speeds, the proposed scheme offers significant bit-error-rate (BER) performance improvement over the standard RS mapping. Index Terms—Channel frequency response estimation, intercarrier interference (ICI), modified Cramer–Rao bound (MCRB), orthogonal frequency-division multiplexing.

I. I NTRODUCTION

S

UPPORTING high mobile user speeds is one of the key requirements of the Third-Generation Partnership Project’s (3GPP) long-term evolution (LTE) and LTE-Advanced (LTE-A) standards [1]. However, the time-varying nature of the radio channel in such high-mobility communication scenarios poses a significant challenge in achieving this goal. In the 3GPP LTE/LTE-A downlink, the mobile radio channel variations within the transmit duration of one orthogonal frequencydivision multiplexed (OFDM) symbol lead to the loss of Manuscript received January 25, 2011; revised June 30, 2011 and October 20, 2011; accepted October 30, 2011. Date of publication December 7, 2011; date of current version February 21, 2012. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, by TRLabs, and by the Rohit Sharma Professorship. This paper was presented in part at the IEEE VTC-Spring, Taipei, Taiwan, May 2010. The review of this paper was coordinated by Dr. G. Bauch. S. D. Muruganathan was with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada. He is now with Research in Motion Limited, Ottawa, ON K2K 3K2, Canada. W. A. Krzymień is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada, and also with TRLabs, Edmonton, AB T5K 2M5, Canada (e-mail: [email protected]). A. B. Sesay is with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: sesay@ ucalgary.ca). Digital Object Identifier 10.1109/TVT.2011.2178621

orthogonality between different subcarriers. This will cause intercarrier interference (ICI) at the mobile receiver, which needs to be mitigated to avoid severe performance degradation [2], [3]. Furthermore, due to the presence of ICI, channel estimation at the mobile receiver becomes a formidably challenging task. Recently, various practical schemes have been studied to estimate the time-varying channel in OFDM systems. In [4], a time-domain raised-cosine interpolator and a frequencydomain raised-cosine interpolator with adaptive rolloff factor are proposed for channel estimation in a mobile digital video broadcasting handheld (DVB-H) receiver. A reducedcomplexity channel estimator for DVB-H, which exploits the banded and sparse structures of the channel matrix in the frequency and time domains, respectively, is proposed in [3] and [5]. In [6], a channel estimation scheme combining minimum mean-square-error (MMSE) interpolation and time-domain windowing is proposed to estimate the time-varying channel in DVB-H systems. More recently, in [7], the medium access control layer performance of various channel estimation algorithms has been studied in the context of the 3GPP LTE downlink. In this paper, we propose a new reference symbol (RS) mapping scheme to improve downlink channel estimation performance in high-mobility scenarios over the standard RS mapping scheme defined in the 3GPP LTE standard [8]. The proposed scheme employs null subcarriers to guard RSs, which helps mitigate the effect of ICI at subcarriers carrying RSs. In addition, the proposed scheme allows the ICI gain parameters to be estimated via a simple frequency-domain estimator. A major contribution of this paper is the derivation of modified Cramer–Rao bounds (MCRBs) to study the efficiency of the standard and the proposed RS mapping schemes in estimating the channel frequency response (CFR) gains. Generally, the MCRB is a looser bound than the standard Cramer–Rao bound (CRB) [9]–[11]. However, in the presence of nuisance or unwanted parameters in the observed signal, the MCRB is much easier to evaluate than the standard CRB [9]–[11]. In this paper, we treat the discrete transmitted symbols as the nuisance or unwanted parameters and derive the MCRBs corresponding to the standard and proposed RS mapping schemes under different user mobility scenarios. Noting that, for discrete nuisance parameters, the MCRB asymptotically approaches the standard CRB at high signal-to-noise ratios [11], we use the derived MCRBs to analytically demonstrate the performance gain achieved by the proposed RS mapping scheme over the

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MURUGANATHAN et al.: NULL-SUBCARRIER-AIDED RS MAPPING SCHEME FOR 3GPP LTE DOWNLINK

standard scheme under high-mobility conditions. Additionally, we compare the simulated mean-square-error (MSE) and biterror-rate (BER) performance of the two schemes under different mobile user speeds. These comparisons also show excellent performance improvement achieved by the proposed scheme over the standard scheme at high mobile user speeds. This paper is a significantly extended version of our earlier conference paper [17]. The additional work presented here includes the derivation of simplified expressions for CFR and ICI gains using piecewise linear approximation for channel time variations, derivation of equivalent real-valued received signal models, detailed derivation of MCRBs corresponding to the CFR gain estimates for both the standard and proposed RS mapping schemes, and verification of the tightness of the MCRBs via numerical simulations. We note that the MCRB analysis presented in this paper significantly differs from the CRB analyses presented in [18]–[21] for channel estimation/prediction in mobile OFDM systems. In [18], CRBs are derived for channel prediction under a doubly selective ray-based physical channel model. A major assumption made in the derivation of the CRBs in [18] is that the channel parameters remain constant within the estimation/ prediction time window and slowly vary beyond this time window. As a result, the derivations in [18] do not take into account the effect of ICI caused by channel variations within the transmit duration of one OFDM symbol. Similarly, the derivations in [19] assume a quasi-stationary communication environment, where the channel impulse response coefficients do not significantly change within a single OFDM symbol duration. Hence, the analyses carried out in [19] also do not take ICI into account. (It should be noted that single-tap equalizers are utilized on individual subcarriers in [19].) In this paper, we assume that the channel significantly varies within the transmit duration of one OFDM symbol and thus take into account the effect of ICI in our MCRB analysis. Unlike in [19], to mitigate the ICI caused by symbols from adjacent subcarriers, we employ MMSE equalizers for data symbol detection on individual subcarriers. Our work differs from [20] and [21] in the way time variations within an OFDM symbol duration are modeled. In [20], the Bayesian CRB is analyzed by approximating time variations of Rayleigh channel gains within an OFDM symbol by a polynomial model, and in [21], the time-varying channel is approximated as a superposition of a number of complex exponential basis functions. In our work, we utilize a piecewise-linear model to approximate the equivalent discrete channel-tap variations within an OFDM symbol duration. The rest of this paper is organized as follows: Section II presents an overview of the system and defines the channel and signal models assumed. Next, in Section III, the channel estimation method employed in the standard RS mapping scheme is briefly discussed. Details of the channel estimation improvement achieved with the proposed RS mapping scheme are then provided in Section IV. This is followed in Section V by the derivation of the MCRBs for both the standard and proposed RS mapping schemes. Numerical results are then presented in Section VI. Finally, this paper is concluded in Section VII.

625

TABLE I D ETAILS OF K EY S YSTEM PARAMETERS

Notation: Throughout this paper, the transpose operation is denoted by (•)T . Given a complex element x, we denote its real and imaginary components by {x} and {x}, respectively. Moreover, the (q, s)th element of a given matrix A is represented by [A]q,s . The q th element of a given vector p ˜ ϑ) is denoted by [p]q . Lastly, the notation f (χ| ϑ∈{κp ,∀k } X used throughout this paper is defined as follows: Let index k ˜ = {A˜1 , A˜2 , . . . , AÃ¯ }, take on values from an arbitrary set A where A¯ denotes the cardinality of the set. Additionally, ˜ ˜ ,X ˜ ˜ ,...,X ˜ ˜ } be a given set of discrete ˜κ ,X let {X p A1 A2 AA¯ transmitted symbol vectors. (Note that we alternatively rep˜ ϑ, resent this set of discrete transmitted as X symbol vectors ˜ ϑ ) denotes where ϑ ∈ {κp , ∀k }.) Then, f (χ| ϑ∈{κp ,∀k } X the conditional probability density function (pdf) of χ conditioned on all of the discrete transmitted symbols vectors ˜ ˜ ,X ˜ ˜ ,...,X ˜˜ . ˜κ ,X X p A1 A2 AA¯ II. S YSTEM OVERVIEW AND C HANNEL /S IGNAL M ODELS In this section, we provide an overview of the system parameters, the downlink reference signal type, the channel model, and the signal model assumed throughout this paper. A. System Parameters We consider a single-input–single-output downlink scenario where both the base station and the mobile receiver employ single antennas. The transmission bandwidth is chosen to be 10 MHz, which corresponds to a nominal resource block size of 50 [12]. Following the definitions in [8], frame structure type 1 consisting of 20 0.5-ms downlink slots is considered. Furthermore, the number of OFDM symbols per downlink slot is assumed to be 7. Details of other key system parameters assumed throughout this paper are provided in Table I. B. Reference Signals To facilitate channel estimation at the mobile receiver, three types of downlink reference signals are defined in the 3GPP LTE standard [8]. In this paper, we will consider user-equipment (UE)-specific RS mapping. Fig. 1 shows the

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varies over the transmit duration of one OFDM symbol (i.e., a fast fading scenario). The time variations of tap gain h (t) are governed by the autocorrelation function 2 J0 (2πfD Δt) E [h∗ (t)h (t + Δt)] = σh,

(2)

where fD denotes the maximum Doppler frequency, and J0 (•) represents the zero-order Bessel function of the first kind. It should be noted that the autocorrelation function given by (2) is derived, assuming the Clarke’s 2-D isotropic scattering model [22], and it corresponds to the classical (“bathtub”) shape of the Doppler spectrum [22]. D. Signal Model Let Xkm denote the complex symbol transmitted on subcarrier k and OFDM symbol m of an arbitrary downlink slot. (According to Section II-B and Fig. 1, Xkm can represent an RS, a data symbol, or an overhead symbol.) Following OFDM demodulation at the mobile receiver, the received signal sample Rkm corresponding to the k th subcarrier of OFDM symbol m can be expressed as [13]

Fig. 1. Mapping of UE-specific RSs to a time-frequency grid, as defined in the 3GPP LTE standard [8].

mapping of UE-specific RSs to a time-frequency grid spanning one resource block and a subframe. (Two downlink slots constitute a subframe.) In the first downlink slot of the subframe, RSs are located at predetermined subcarriers of OFDM symbols 4 and 7. RSs corresponding to the second downlink slot are transmitted through OFDM symbols 3 and 6. The transmission of RSs in different subcarriers and OFDM symbols of the timefrequency grid enables low-complexity channel estimation at the receiver based on interpolation.

m m m Hk,k X k + Wk

(3)

k =0,k =k

where K and Wkm denote the number of occupied subcarriers and a zero-mean complex additive white Gaussian noise sample 2 m , respectively. Furthermore, Hk,k in (3) reprewith variance σw sents the CFR gain at subcarrier k, which can be expressed as [3], [4] m = Hk,k

NFFT −1 L−1

1 NFFT

n=0

hm ,n exp

=0

−j2πτ k NFFT TS

(4)

wherein NFFT and TS denote the FFT and the sampling period, respectively. In addition, given an OFDM symbol transmit duration of T , hm ,n in (4) is defined as

C. Channel Model The time-varying multipath fading channel can be characterized by the tapped delay-line model with the following impulse response [13]: h(t, τ ) =

K−1

m Xkm + Rkm = Hk,k

L−1

h (t)δ(τ − τ )

(1)

hm ,n = h ([m − 1]T + nTS ) .

m Using the notation established thus far, the ICI gain Hk,k from subcarrier k (k = 0, . . . , k − 1, k + 1, . . . , k − 1) to subcarrier k in (3) can be expressed as [3], [4]

=0 m Hk,k = th

where τ and δ(•) denote the delay corresponding to the tap and the Dirac delta function, respectively. The tap gains h (t)( = 0, 1, . . . , L − 1) are modeled as independent circularly symmetric complex Gaussian random variables with zero 2 . Throughout this paper, we consider means and variances σh, the extended vehicular A (EVA) power-delay profile defined by 3GPP [12]. The root-mean-square delay spread corresponding to the EVA model is 357 ns. For this model, the number of channel taps is set to L = 9. With regard to the time variations of the channel, we consider a communication scenario where the channel significantly

(5)

1

NFFT −1 L−1

NFFT

n=0

× exp

hm ,n

=0

−j2πτ k NFFT TS

exp

−j2πn [k − k ] NFFT

.

(6)

III. C HANNEL E STIMATION W ITH S TANDARD R EFERENCE S YMBOL M APPING S CHEME Now, let κ1 , κ2 , . . . , κP represent the subcarriers in which P RSs are transmitted. (We will henceforth refer to κ1 , κ2 , . . . , κP as the RS subcarriers.) Then, from (3), the received signal


627

sample at RS subcarrier κp of the mth OFDM symbol can be expressed as Rκmp = Hκmp ,κp Xκmp + Hκmp ,k Xkm + Wκmp . (7) k =κp

Generally, the CFR gains at RS subcarriers are first estimated at the mobile receiver by ignoring the ICI [i.e., the second term on the right-hand side of (7)] [7]. Hence, the CFR gain estimate corresponding to the pth RS Xκmp is given by ˆm H κp ,κp =

Rκmp Xκmp

,

p = 1, 2, . . . , P.

(8)

Using the estimates in (8), the CFR gains at the remaining subcarriers of OFDM symbol m are obtained via frequencydomain interpolation. To estimate the CFR gains corresponding to subcarriers of OFDM symbols not carrying RSs, various interpolators in the time domain can be employed [13]. In this paper, two 1-D Wiener filters are separately employed for frequency and time interpolations [7], [13]. For the sake of simplicity, the statistical information (i.e., frequency and time correlations of the channel and the noise variance) required by the Wiener filters is assumed to be available at the mobile receiver. In practice, the required statistical information may be obtained at the mobile receiver using methods described in [23]. It should be noted that the RS subcarrier CFR estimate in (8) is reasonable for low-mobility environments, where the ICI gains Hκmp ,k (k = κp ) in (7) are relatively insignificant. However, under propagation conditions with high mobility, the ICI term in (7) [i.e., the second term on the right-hand side of (7)] becomes progressively significant with increasing maximum Doppler frequency fD . As a result, the application of (8) results in inaccurate RS subcarrier CFR estimates, which, in turn, lead to additional channel estimation errors during interpolation.

Fig. 2. Time-frequency grid illustration of the proposed RS mapping scheme for M = 1.

IV. I MPROVED C HANNEL E STIMATION W ITH P ROPOSED R EFERENCE S YMBOLS M APPING S CHEME To alleviate the problems caused by ICI under communication scenarios with high mobility, we propose a new RS mapping scheme different from the standard scheme shown in Fig. 1. The proposed scheme is based on the concept of guarding RSs with null subcarriers. The proposed RS mapping scheme is motivated by the observation that most of the ICI term energy in (7) is caused by symbols (i.e., data or other overhead symbols) located at subcarriers adjacent to the RS subcarrier [3]. Hence, to mitigate the ICI at the RS subcarriers, M adjacent subcarriers around the RS subcarrier are designated as null subcarriers in the proposed RS mapping scheme. This is equivalent to setting (9)

Fig. 3. Time-frequency grid illustration of the proposed RS mapping scheme for M = 2.

for k values satisfying |k − κp | < M . (Recall that κp is the RS subcarrier corresponding to the pth RS Xκmp .) The timefrequency grids corresponding to the proposed RS mapping scheme for M = 1 and M = 2 are shown in Figs. 2 and 3,

respectively. Comparing Figs. 2 and 3 to Fig. 1, it is noted that the figures are similar in the sense that the RSs are mapped to identical RS subcarriers. However, compared with Figs. 1–3 differ in that the RSs are guarded by null

Xkm = 0

628


subcarriers. Now, substituting (9) into (7), the received signal sample at RS subcarrier κp of the mth OFDM symbol is given by

Rκmp = Hκmp ,κp Xκmp +

Hκmp ,k Xkm + Wκmp .

(10)

|k −κp |>M

Comparing (10) with (7), it is evident that the proposed RS mapping scheme removes the deleterious contributions from the 2M significant ICI gain terms Hκmp ,k (k = κp − M, . . . , κp − 1, κp + 1, . . . , κp + M ). Hence, using (10) in (8), more accurate CFR gain estimates can be attained at RS subcarriers. Likewise, due to the placement of nulls on subcarriers adjacent to the RS subcarriers, the ICI gain terms can also be estimated in the frequency domain. For instance, let us consider the first adjacent subcarrier (κp + 1) of the mth OFDM symbol. From (3) and (9), the received signal sample corresponding to this subcarrier is given by Rκmp +1 = Hκmp +1,κp Xκmp +

Hκmp +1,k Xkm + Wκmp +1 .

|k −κp |>M

(11) In (11), the contribution from the dominant CFR gain term Hκmp +1,κp +1 is eliminated due to the null placed at subcarrier (κp + 1) [i.e., Xκmp +1 = 0]. Now, using the pth RS Xκmp and (11), the ICI gain term Hκmp +1,κp can be estimated as ˆm H κp +1,κp =

Rκmp +1 Xκmp

.

(12)

The ICI gain terms Hκmp −M,κp , . . . , Hκmp −1,κp , Hκmp +2,κp , . . . , Hκmp +M,κp can be similarly estimated. The ICI gain terms corresponding to the remaining subcarriers can be obtained via frequency and time interpolations. It should be noted that, since Xκmp +1 = 0 in the case of the standard RS mapping scheme, the contribution of the dominant CFR gain term Hκmp +1,κp +1 is not eliminated. As a result, when used with the simple ICI gain estimator of (12), the standard RS mapping scheme suffers severe performance degradation due to the interference caused by the dominant CFR gain term Hκmp +1,κp +1 . The key advantage of the proposed RS mapping scheme is that it enables higher mobile user speeds through improved downlink channel estimation. The proposed scheme allows the ICI gain parameters to be estimated via the simple estimator of (12). However, this advantage is traded off for a slight reduction in the number of complex data symbols that can be transmitted during a downlink slot. (This is due to (9), where the data symbols at subcarriers adjacent to the RS subcarrier are replaced by nulls.) To assess the effect of this tradeoff, we draw MSE and BER performance comparisons between the proposed RS mapping scheme and the standard scheme of Fig. 1 in Section VI. In the next section, we derive the MCRBs corresponding to the RS subcarrier CFR gain estimates ˆ m (p = 1, 2, . . . , P ) for the cases of the proposed RS H κp ,κp mapping scheme and the standard scheme.

V. M ODIFIED C RAMER –R AO B OUNDS FOR R EFERENCE S YMBOL S UBCARRIER CFR G AIN E STIMATES This section is organized as follows: First, in Section V-A, we simplify the CFR gain and the ICI gain expressions of (4) and (6) using the piecewise linear approximation for channel time variations within the transmit duration of one OFDM symbol. Using the simplified expressions of Section V-A, equivalent real-valued signal models of (7) and (10) are derived in Section V-B to facilitate the derivation of the MCRBs. Next, in Section V-C, we find the conditional Fisher information matrices corresponding to the standard and proposed RS mapping schemes. This is followed in Section V-D by the derivation of the MCRBs corresponding to the standard and proposed RS mapping schemes for the high-mobility scenario.

A. Simplified CFR Gain and ICI Gain Expressions m m and the ICI gain Hk,k The CFR gain Hk,k expressed in (4) and (6) depend on the NFFT L random channel taps hm ,n ( = 0, 1, . . . , L − 1; n = 0, 1, . . . , NFFT − 1). We now employ the piecewise linear approximation for channel time variations within one OFDM symbol to simplify the expressions in (4) and m m and Hk,k (6). This results in simplified expressions for Hk,k , m which only depend on the 2L random channel taps h,n (∀; n = 0, NFFT − 1). It should be noted that, for normalized Doppler values of up to 20% (fD T ≤ 0. 20), the piecewise linear model is a good approximation for channel time variations within one OFDM symbol [2], [3]. Using the piecewise linear approximation, hm ,n (∀; n = 1, 2, . . . , NFFT − 2) can be approximated as a function of hm ,0 and hm ,NFFT −1 as [2]

hm ,n

m hm ,NFFT −1 − h,0 +n NFFT n n = 1− hm . hm ,0 + NFFT NFFT ,NFFT −1

= hm ,0

(13)

In the preceding equation, n represents the time-variation index within one OFDM symbol corresponding to a given channel tap. A substitution of (13) into (4) yields

m = Hk,k

NFFT −1 L−1

1

1−

NFFT

n=0

=0

+ × exp =

×

1 NFFT L−1 =0

−j2πτ k NFFT TS NFFT −1

1−

n=0

hm ,0

exp

n NFFT

hm ,0

n NFFT

hm ,NFFT −1

n NFFT

−j2πτ k NFFT TS


+

×

1

NFFT −1

n

NFFT

n=0

NFFT

L−1

hm ,NFFT −1

exp

=0

NFFT −j2πτ k NFFT TS

Now, using the facts NFFT −1 1 n 1 1− = NFFT n=0 NFFT 2 1

NFFT −1

n

NFFT

n=0

NFFT

=

NFFT −1

1 .

(14)

1 (15) 2NFFT

+

1 1 − 2 2NFFT

(16)

m in (14), the expression for Hk,k can be further simplified as L−1 1 −j2πτ k 1 m m + Hk,k = h,0 exp 2 2NFFT NFFT TS

=

n=0

−1 NFFT

629

n NFFT

exp

−j2πn[k − k ] NFFT

−1 −j2π [k − k ] . 1 − exp NFFT

(20)

Substituting (19) and (20) in (18), the simplified expression m for Hk,k is obtained as −1 −j2π [k − k ] 1 m Hk,k = 1 − exp NFFT NFFT L−1 −j2πτ k × hm exp ,0 NFFT TS =0 L−1 −j2πτ k hm − . (21) ,NFFT −1 exp NFFT TS =0

=0

+

1 1 − 2 2NFFT

L−1

hm ,NFFT −1 exp

=0

−j2πτ k NFFT TS

m , Hk,k m Hk,k

Having derived a simplified expression for proceed toward simplifying the expression for (6). First, we substitute (13) in (6) to obtain NFFT −1 L−1 1 n m 1− hm Hk,k = ,0 NFFT n=0 NFFT =0

+

n

. (17) we next given in

hm ,NFFT −1

NFFT −j2πn[k − k ] −j2πτ k exp × exp NFFT TS NFFT NFFT −1 1 n = 1− NFFT n=0 NFFT −j2πn[k − k ] × exp NFFT L−1 −j2πτ k m h,0 exp × NFFT TS =0 NFFT −1 n −j2πn[k − k ] 1 exp + NFFT n=0 NFFT NFFT L−1 −j2πτ k m h,NFFT −1 exp . (18) × NFFT TS

=0

Now, as shown in Appendix A, the partial sums involving index n in the second equality of (18) can be expressed as NFFT −1 −j2πn[k − k ] 1 n 1− exp NFFT n=0 NFFT NFFT =

1 NFFT

1 − exp

−j2π[k − k ] NFFT

−1

(19)

B. Equivalent Real-Valued Signal Models In this section, we derive equivalent real-valued signal models of (7) and (10) using the simplified expressions found in Section V-A. The need for the equivalent real-valued models arises since it is more convenient to evaluate MCRBs involving real-valued quantities than those involving complex-valued quantities [11]. First, using (17) and (21) with k = κp , we can alternatively express (7) as L−1 NFFT + 1 m m h,0 Rκp = 2NFFT =0 −j2πτ κp NFFT − 1 × exp + NFFT TS 2NFFT L−1 −j2πτ κp m h,NFFT −1 exp × Xκmp NFFT TS =0 L−1 −j2πτ κp 1 + hm exp ,0 NFFT NFFT TS =0 L−1 −j2πτ κp m h,NFFT −1 exp − NFFT TS =0 ⎡ ⎤ −1 −j2π[κ − k ] p ×⎣ Xkm ⎦ + Wκmp . 1 − exp NFFT k =κp

(22) Let us next denote the matrix representations of hm ,n (n = m ˜ ˜ ,κ , 0, NFFT − 1) and exp(j2πτ κp /NFFT TS ) by h,n and Φ p m ˜ and Φ ˜ ,κ can be defined as respectively. Then, h p ,n ⎤ ⎡ m m h − h ,n Δ ˜m = ⎣ ,n ⎦ h (23) ,n m m h,n h,n

630


⎤ j2πτ κ j2πτ κ exp NFFTTSp − exp NFFT TSp ⎦ =⎣ j2πτ κ j2πτ κ exp NFFTTSp exp NFFTTSp ⎤ ⎡ 2πτ κ 2πτ κ cos NFFT TpS − sin NFFT TpS ⎦. =⎣ (24) 2πτ κ 2πτ κ sin NFFT TpS cos NFFT TpS ⎡

˜ ,κ Φ p

Likewise, noting that

−1 j2π[κp − k ] 1 − exp NFFT −1 2π[κp − k ] 1 2π[κp − k ] 1 = + j 1 − cos sin 2 2 NFFT NFFT

valued signal model of (10) can be obtained as follows: L−1 NFFT + 1 ˜ T ˜ m m ˜ Φ,κp h,0 Rκp = 2NFFT =0 L−1 NFFT − 1 ˜ T ˜ m ˜m Φ,κp h,NFFT −1 X + κp 2NFFT =0 L−1 L−1 T T 1 m m ˜ ˜ ˜ ˜ + Φ,κp h,0 − Φ,κp h,NFFT −1 NFFT =0 =0 ⎡ ⎤ T ˜m ˜ ˜ m⎦ ×⎣ (31) Ω κp ,k Xk + Wκp . |k −κp |>M

(25) C. Conditional Fisher Information Matrices ˜ κ ,k of [1 − we can define the matrix representation Ω p −1 exp(2π[κp − k ]/NFFT )] as (26), shown at the bottom of the page. Now, using the definitions (23)–(26) in (22), the equivalent real-valued signal model of (7) can be derived as L−1 NFFT + 1 ˜ T ˜ m m ˜ Rκp = Φ,κp h,0 2NFFT =0 L−1 NFFT − 1 ˜ T ˜ m ˜m + Φ,κp h,NFFT −1 X κp 2NFFT =0 L−1 L−1 T T 1 m m ˜ ˜ ˜ ˜ + Φ,κp h,0 − Φ,κp h,NFFT −1 NFFT =0 =0 ⎡ ⎤ T ˜m ˜ ˜ m⎦ ×⎣ (27) Ω κp ,k Xk + Wκp k =κp

where T ˜ m = Rm Rκmp R κp κp

(28)

˜ m = [ {X m } {X m } ]T , X ϑ ∈ {κp , ∀k } (29) ϑ ϑ ϑ T m ˜ m = Wm W W . (30) κp κp κp Lastly, following a procedure similar to (22)–(27) and using the notations established in (23)–(30), the equivalent real-

Furthermore, from the assumptions made in Section II-C, the (q, s)th element of the correlation matrix Cθ = E[θθ T ] of θ is defined in (33), shown at the bottom of the next page, wherein 2 is as defined in Section II-C, and the channel-tap variance σh, α = J0 (2πfD NFFT TS ). ˜ m , θ| Now, given the conditional joint pdf f (R κp ϑ∈{κp ,∀k } m 2 th ˜ element of the conditional Fisher X ) , the (q, s) ϑ

1 Note that the Fisher information matrices are conditioned on the transmitted ˜ m (ϑ ∈ {κp , ∀k }) since these symbol vectors represent the symbol vectors X ϑ nuisance or unwanted parameters in the equivalent real-valued signal models of (27) and (31). 2 Here, f (R ˜ m ) denotes the conditional joint pdf ˜ m , θ| X κp ϑ ϑ∈{κp ,∀k } m ˜ m for ϑ ∈ ˜ and θ, given the discrete transmitted symbol vectors X of R κp

ϑ

{κp , ∀k }.

−1 ⎤ j2π[κp −k ] − 1 − exp NFFT ⎢ 1 − exp ⎥ =⎢ −1 ⎥ −1 ⎣ ⎦ j2π[κp −k ] j2π[κp −k ] 1 − exp 1 − exp NFFT NFFT ⎡

˜ κ ,k Ω p

In this section, we derive the conditional Fisher information matrices, corresponding to the standard and proposed RS mapping schemes, given the discrete transmitted symbol vectors ˜ m (ϑ ∈ {κp , ∀k }). (Recall from (29) that X ˜ m is a real vector X ϑ ϑ composed of the real and imaginary parts of transmitted symbol Xϑm .)1 For notational convenience, we first define the following 4L × 1 real vector, which consists of the real and imaginary components of random channel taps hm ,n ( = 0, 1, . . . , L − 1; n = 0, NFFT − 1): m m m θ = hm 0,0 , h0,0 , . . . , hL−1,0 , hL−1,0 , m hm 0,NFFT − 1 , h0,NFFT −1 , . . . , m T hm . (32) L−1,NFFT − 1 , hL−1,NFFT − 1

⎡ =⎣ 1 2

1 − cos

j2π[κp −k ] NFFT

−1

1 2 2π[κp −k ] NFFT

−1

sin

2π[κp −k ] NFFT

−1 2

−1 ⎤ 2π[κp −k ] 2π[κp −k ] 1 − cos sin NFFT NFFT ⎦ 1 2

(26)


information matrix JR,θ| ˜ X ˜ can generally be expressed as [10], [11], [14]–[16] [JR,θ| ˜ X ˜ ]q,s

⎤ ˜m ˜ m , θ ∂ 2 ln f R κp ϑ∈{κp ,∀k } Xϑ ⎦ = −ER,θ| ˜ X ˜ ⎣ ∂[θ]q ∂[θ]s ⎡

ϑ∈{κp ,∀k }

= f

⎛

˜ m θ, ⎝R κp

scheme. Under the assumptions made in Section II, it can be shown (see Appendix B) that, for the standard RS mapping scheme ˜ m θ, ˜m X ∂ 2 ln f R κp ϑ ϑ∈{κp ,∀k } ∂[θ]q ∂[θ]s

(34)

where ER,θ| ˜ X ˜ [•] represents expectation with respect to the ˜ m , θ| ˜m conditional joint pdf f (R κp ϑ∈{κp ,∀k } Xϑ ). Since the channel-tap vector θ is statistically independent of the transmit˜ m (ϑ ∈ {κp , ∀k }), the conditional joint ted symbol vectors X ϑ ˜ m , θ| ˜m pdf f (R κp ϑ∈{κp ,∀k } Xϑ ) can be factorized as ⎛ ⎞ ˜ m , θ ˜ m⎠ f ⎝R X κp ϑ

μm κp ,Standard

+

(36)

where ER|θ, ˜ X ˜ [•] and Eθ [•] denote expectations with respect to ˜ ˜m the pdfs f (Rm κp |θ, ϑ∈{κp ,∀k } Xϑ ) and f (θ), respectively. Let us now consider the derivation of the conditional Fisher information matrix for the standard RS mapping

[Cθ ]q,s

∂μm κp ,Standard ∂[θ]γ

NFFT + 1 2NFFT

(35)

⎧1 2 ⎪ ⎪ 2 σh, q/2−1 , ⎪ ⎪ 2 ⎪ , ⎨ 12 σh, q/2−L−1 = 1 ασ 2 h, q/2−1 , 2 ⎪ ⎪ 1 ⎪ 2 ⎪ ασ , ⎪ ⎩ 2 h, q/2−L−1 0,

∂μm κp ,Standard

(37)

∂[θ]s

L−1

1 NFFT

L−1

˜m ˜T h Φ ,κp ,0

=0

NFFT − 1 2NFFT

L−1

˜m ˜T h Φ ,κp ,NFFT −1

˜m X κp

=0

˜m ˜T h Φ ,κp ,0

=0

−

L−1


=0

⎡

⎤ ˜m ˜ m θ, ∂ 2 ln f R κp ϑ∈{κp ,∀k } Xϑ ⎣ ⎦ = −E R ˜ ˜ |θ,X ∂[θ]q ∂[θ]s − Eθ

=

+

⎡

∂ 2 ln {f (θ)} ∂[θ]q ∂[θ]s

T

∂[θ]q

Substituting (35) in (34), we next rewrite the (q, s)th element of JR,θ| ˜ X ˜ as a sum of the following two parts: J R,θ ˜ |X ˜


where

ϑ∈{κp ,∀k }

q,s

2 = − 2 σw

⎞ ˜ m ⎠ f (θ). X ϑ

631

×⎣

⎤ ˜ T X ˜ m ⎦ . Ω k κp ,k

(38)

k =κp

Furthermore, in Appendix C, we show (39), shown at the bottom of the page, where am κp ,Standard Δ

=

NFFT + 1 2NFFT

1 ≤ q ≤ 2L, (2L + 1) ≤ q 1 ≤ q ≤ 2L, (2L + 1) ≤ q otherwise

⎧ T m ˜ , ⎪ ⎪ Φ γ/2−1,κp aκp ,Standard ⎪ ⎪ ⎪ T 0 −1 ⎪˜ ⎪ am ⎨ Φ γ/2−1,κp κp ,Standard , 1 0 = m ⎪Φ ˜T , ⎪

γ/2−L−1,κp b κp ,Standard ⎪ ⎪ ⎪ ⎪ 0 −1 m ⎪ ˜T ⎩Φ bκp ,Standard ,

γ/2−L−1,κp 1 0

⎡

˜m + X κp

1 NFFT

⎣

⎤ ˜ T X ˜ m ⎦ Ω k κp ,k

(40)

k =κp

s=q ≤ 4L, s = q s = q + 2L ≤ 4L, s = q − 2L

(33)

γ = 1, 3, . . . , (2L − 1) γ = 2, 4, . . . , (2L) γ = (2L + 1), (2L + 3), . . . , (4L − 1) γ = (2L + 2), (2L + 4), . . . , (4L)

(39)

632


bm κp ,Standard Δ

=

standard RS mapping scheme is derived as

NFFT − 1 2NFFT

⎡

˜m − X κp

1 NFFT

⎣

⎤ ˜ T X ˜ m ⎦ (41) Ω k κp ,k

k =κp

∂μm κp ,Standard ∂[θ]q

T

JStandard ˜ |X ˜ R,θ

∂μm κp ,Standard ∂[θ]s

(42)

where ∂μm κp ,Standard /∂[θ]γ (γ ∈ {q, s}) is as defined in (39). To evaluate the second term on the right-hand side of (36), we first note that the channel-tap vector θ is zero-mean Gaussian distributed (recall the assumptions from Section II-C) with correlation matrix Cθ , as defined in (33). Then, we have ln {f (θ)} = −2L × ln(2π) − ln det 1/2 [Cθ ] 1 − θ T (Cθ )−1 θ. 2

∂ 2 ln {f (θ)} = −eT4L,q (Cθ )−1 e4L,s ∂[θ]q ∂[θ]s

∂ 2 ln {f (θ)} = eT4L,q (Cθ )−1 e4L,s . ∂[θ]q ∂[θ]s th

∂[θ]γ

=


T


∂[θ]q

∂[θ]s

m ∂μκp ,Proposed T ∂μm 2 κp ,Proposed = JProposed ˜ |X ˜ 2 R,θ σ ∂[θ] ∂[θ]s q w q,s + eT4L,q (Cθ )−1 e4L,s

|k −κp |>M

NFFT − 1 ˜ m Xκp 2NFFT ⎡ ⎤ 1 ⎣ ˜ T X ˜ m ⎦ . (50) − Ω k κp ,k NFFT |k −κp |>M

(44)

(45)

(47)

where (48), shown at the bottom of the page, holds. m In (48), vectors am κp ,Proposed and bκp ,Proposed are defined as NFFT + 1 ˜ m m aκp ,Proposed = Xκp 2NFFT ⎡ ⎤ 1 ⎣ T ˜ ˜ m ⎦ (49) + Ω κp ,k Xk NFFT bm κp ,Proposed =

Hence, from (42), (45), and (36), the (q, s) element of for the the conditional Fisher information matrix JStandard ˜ X ˜ R,θ|

∂μm κp ,Proposed

q,s

2 2 σw

(43)

where e4L,γ (γ ∈ {q, s}) denotes a 4L × 1 vector of all zeros, except the γ th entry, which equals 1. Since the righthand side of (44) is independent of the channel-tap vector θ, the second term on the right-hand side of (36) can be expressed as −Eθ

=

for The conditional Fisher information matrix JProposed ˜ X ˜ R,θ| the proposed RS mapping scheme can be derived, following a procedure similar to that outlined in (37)–(46). Following such a procedure, it can be shown that the (q, s)th element of the conditional Fisher information matrix JProposed is ˜ X ˜ R,θ| given by

Now, taking the second derivative of (43) with respect to the elements of θ yields

+ eT4L,q (Cθ )−1 e4L,s . (46)

holds. Hence, using (39) in (37) and noting that the resulting ˜ m , we have the first term expression in (37) is independent of R κp on the right-hand side of (36) as ⎤ ⎡ ˜ m θ, ˜m ∂ 2 ln f R κp ϑ∈{κp ,∀k } Xϑ ⎣ ⎦ −E R ˜ |θ,X ˜ ∂[θ]q ∂[θ]s 2 = 2 σw

It should be noted that, due to the dependence of am κp ,Proposed and bm κp ,Proposed on the number M of null subcarriers guarding each RS, the conditional Fisher information matrix for the proposed RS mapping scheme also depends JProposed ˜ X ˜ R,θ| on M . D. MCRB Expressions for the High-Mobility Scenario First, we note that the CFR gain Hκmp ,κp is the desired parameter to be estimated in both (7) and (10). Hence, to find the MCRB expressions for the standard and proposed RS

⎧ T m ˜ ⎪ Φ ⎪

γ/2−1,κp aκp ,Proposed , ⎪ ⎪ ⎪ ⎪ ⎪ 0 −1 m ⎪ ˜T ⎪ Φ ⎪ ⎨ γ/2−1,κp 1 0 aκp ,Proposed ,

γ = 1, 3, . . . , (2L − 1) γ = 2, 4, . . . , (2L)

⎪ m ˜T ⎪ γ = (2L + 1), (2L + 3), . . . , (4L − 1) Φ ⎪

γ/2−L−1,κp bκp ,Proposed , ⎪ ⎪ ⎪ ⎪ ⎪ 0 −1 ⎪ ⎪ ˜T ⎩Φ bm

γ/2−L−1,κp κp ,Proposed , γ = (2L + 2), (2L + 4), . . . , (4L) 1 0

(48)


mapping schemes, we next define the following 2 × 1 vector composed of the real and imaginary parts of Hκmp ,κp : ⎤ ⎡ Hκmp ,κp ˜m ⎣ ⎦ H κp ,κp = Hκmp ,κp ⎤ ⎡ L−1 m h NFFT + 1 ˜ T ⎣ ,0 ⎦ = Φ ,κp 2NFFT hm =0 ,0 ⎤ ⎡ L−1 m h NFFT − 1 ˜ T ⎣ ,NFFT −1 ⎦ + Φ ,κp 2NFFT hm ,NFFT −1

=0

(51) ˜ ,κ is as defined in (24). Then, taking the derivative where Φ p m ˜ of Hκp ,κp with respect to the γ th element of θ, it can be easily shown that (52), shown at the bottom of the page, holds. We now define the following 2 × 2 conditional matrix corresponding to the estimation of Hκmp ,κp : 'T & −1 ∂ H ∂H ˜m ˜m κ κ ,κ ,κ p p p p m ˜ = J R,θ Γ Hκp ,κp |X ˜ |X ˜ ∂θ ∂θ (53) where the 2 × 4L Jacobin matrix is ˜m ∂H κp ,κp ∂θ =

˜m ∂H κp ,κp

˜m ∂H κp ,κp

∂[θ]1

∂[θ]2

···

˜m ∂H κp ,κp

˜m ∂H κp ,κp

∂[θ](4L−1)

∂[θ](4L)

.

(54) From (53), the 2 × 2 conditional matrices corresponding to the standard and proposed RS mapping schemes can be Standard obtained by replacing JR,θ| and JProposed , ˜ X ˜ with JR,θ| ˜ X ˜ ˜ X ˜ R,θ| and JProposed were derespectively. [Recall that JStandard ˜ X ˜ ˜ X ˜ R,θ| R,θ| rived earlier in (46) and (47), respectively.] Next, we average ˜ over the joint pdf of the discrete transmitted Γ(Hκmp ,κp |X) ˜ m (ϑ ∈ {κp , ∀k }) as symbol vectors X ϑ m ˜ . Γ Hκmp ,κp = EX (55) ˜ Γ Hκp ,κp |X A closed-form expression for Γ(Hκmp ,κp ) appears to be dif˜ requires the inversion ficult to obtain since Γ(Hκmp ,κp | X)

˜m ∂H κp ,κp ∂[θ]γ

633

of JR,θ| ˜ X ˜ , which, in turn, depends on the discrete transmitted ˜ m (ϑ ∈ {κp , ∀k }). Hence, in this paper, the symbol vectors X ϑ averaging in (55) is performed via the Monte Carlo method. Lastly, from (55) and using (51), the MCRB for the estimation of Hκmp ,κp is given by 1 MCRB Hκmp ,κp = Γ Hκmp ,κp 1,1 2 1 Γ Hκmp ,κp + . (56) 2,2 2 As a final remark, we note that only the MCRB analysis corresponding to the RS subcarrier CFR gain estimates is presented in this paper. MCRB analysis corresponding to nonRS subcarriers and theoretical BER analysis are not considered since these require taking into account the errors introduced during interpolation (which is beyond the scope of this paper). VI. N UMERICAL R ESULTS In this section, we compare the averages (over all RS subcarriers) of the MCRB expressions derived in Section V to MSE values obtained from simulations for cases involving the standard and proposed RS mapping schemes. To obtain the MCRB values, the averaging in (55) is performed via the Monte Carlo method with 5000 downlink slots. The MCRB and simulated MSE values are also used to demonstrate the CFR gain estimate improvements achieved by the proposed RS mapping scheme over the standard scheme at RS subcarriers. Additionally, we also compare the uncoded BERs corresponding to the standard and proposed RS mapping schemes. To take into account the slight rate loss suffered by the proposed scheme due to the insertion of null subcarriers, the MCRB, MSE, and BER performances are compared over the per-bit signal-to-noise ratio 2 . Here, Eb denotes the average energy expended (SNR) Eb /σw to transmit one data bit. Throughout this section, an MMSE equalizer [6] is utilized for data symbol detection. We first compare the simulated MSE performance and the MCRB obtained from (56) corresponding to two different high-mobility scenarios. The results for mobile user speeds v = 150 km/h and v = 300 km/h are shown in Figs. 4–7, respectively. These two mobile user(speeds are used, so that the errors caused by the ICI term k =κp Hκmp ,k Xkm in (7) on RS subcarrier CFR gain estimates are significant. Note that the number of null subcarriers guarding each RS for the proposed scheme is set to M = 1 (in Figs. 4 and 6) and M = 2

⎧ 1 NFFT +1 ˜ T ⎪ ⎪ , Φ γ/2−1,κp ⎪ 2NFFT ⎪ ⎪ 0 ⎪ ⎪ ⎪ 0 ⎪ NFFT +1 ˜ T ⎪ , Φ γ/2−1,κp ⎨ 2N FFT 1 = T ⎪ 1 NFFT −1 ˜ ⎪ ⎪ , Φ γ/2−L−1,κp ⎪ 2NFFT ⎪ 0 ⎪ ⎪ ⎪ ⎪ NFFT −1 ˜ T 0 ⎪ ⎩ 2N , Φ γ/2−L−1,κp FFT 1

γ = 1, 3, . . . , (2L − 1) γ = 2, 4, . . . , (2L) (52) γ = (2L + 1), (2L + 3), . . . , (4L − 1) γ = (2L + 2), (2L + 4), . . . , (4L)

634


Fig. 4. Simulated MSE and MCRB comparisons between the standard and proposed (M = 1) RS mapping schemes for the case with mobile user speed v = 150 km/h and normalized Doppler value fD T ≈ 0. 0185. Also included are the simulated MSE results for the case with no mobility (i.e., v = 0 km/h).




(in Figs. 5 and 7). The corresponding simulated MSE results for the case with no mobility (i.e., mobile user speed v = 0 km/h) are also shown in Figs. 4–7. From the figures, we note that the proposed RS mapping scheme performs similar to the standard scheme when ( v = 0 km/h. This is because, for v = 0 km/h, the ICI term k =κp Hκmp ,k Xkm in (7) vanishes, and guarding RSs by null subcarriers does not yield any performance improvement over the standard scheme. However, as shown in Figs. 4–7, the proposed RS mapping scheme offers notable performance improvements over the standard scheme in the high-mobility scenarios. It should be ( noted that, with increasing mobile user speeds, the ICI term k =κp Hκmp ,k Xkm in (7) becomes proportionally more significant. Due to the presence of the null subcarriers, the proposed RS mapping scheme removes the ICI caused by 2M adjacent subcarriers at each RS subcarrier [see (10)]. As a result, the proposed scheme offers better CFR gain estimates at the RS subcarriers over the standard scheme. For v = 150 km/h, the proposed scheme attains MSE improvements of 4.9 dB (when M = 1) and 8.0 dB (when M = 2) over the standard scheme at a per-bit SNR 2 = 32 dB. The corresponding improvements for the of Eb /σw case of v = 300 km/h are 5.7 dB (when M = 1) and 11.3 dB (when M = 2).

Furthermore, it is shown in Figs. 4–7 that the simulated MSE values for both the proposed and standard RS mapping schemes 2 ≥ approach the MCRB at high per-bit SNRs (i.e., Eb /σw 0 dB). It should be emphasized here that, when the nuisance parameters are discrete (which is the case in this paper since the ˜ m (ϑ ∈ {κp , ∀k }) are discrete), transmitted symbol vectors X ϑ the MCRB asymptotically approaches the standard CRB at high per-bit SNRs [11]. Hence, we can infer from Figs. 4–7 that the least-squares-type CFR estimator of (8) approaches the standard CRB at high per-bit SNRs, even in the presence of ICI. However, it should be emphasized that the MCRB values corresponding to the proposed scheme and those associated with the standard scheme are significantly different in the high-per-bit-SNR region. This is because the MCRB analysis presented in Section V-D takes into account the effect of the ICI terms in (7) and (10). Once again, the performance gain achieved by the proposed scheme over the standard RS mapping scheme in the high-per-bit-SNR region is confirmed by the MCRB curves in Figs. 4–7. Next, we note from Figs. 4–7 that the MCRB curves deviate away from the simulated MSE curves 2 ≤ 0 dB). This is because at low per-bit SNRs (i.e., for Eb /σw the MCRB is generally much looser than the standard CRB at low per-bit SNRs [11].


635

by guarding RSs with null subcarriers. Lastly, we note that, for the system parameters defined in Section II, the proposed RS mapping scheme suffers raw bit rate losses of approximately 20.1% (for the case M = 1) and 30.1% (for the case M = 2) when compared with the standard RS mapping scheme. However, given that decreasing bit rates with increasing mobility are commonly accepted in practice3 , these losses are acceptable in high-mobility scenarios and are traded off for the improved performance yielded by the proposed scheme over the standard RS mapping method. VII. C ONCLUSION

Fig. 8. BER performance comparisons between the proposed and standard RS mapping schemes for the cases with v = 0 km/h (fD T = 0) and v = 150 km/h (fD T ≈ 0. 0185).

In this paper, we have proposed a new RS mapping scheme for 3GPP LTE/LTE-A downlink to improve channel estimation performance in high-mobility scenarios. The proposed scheme employs null subcarriers to guard RSs, which helps mitigate the effect of ICI at RS subcarriers. In addition, the proposed scheme allows the ICI gain parameters to be estimated via a simple frequency-domain estimator. We derive MCRB expressions to analytically demonstrate the performance gain attained by the proposed scheme over the standard one at high mobile user speeds. Furthermore, comparisons of simulated MSE performance at RS subcarriers have been presented to reveal reduced MSEs achieved by the proposed scheme in high-mobility communication scenarios. Additionally, at high mobile user speeds, the proposed scheme offers significant BER performance improvements over the standard RS mapping scheme. A PPENDIX A D ERIVATIONS OF (19) AND (20)

Fig. 9. BER performance comparisons between the proposed and standard RS mapping schemes for the cases with v = 0 km/h (fD T = 0) and v = 300 km/h (fD T ≈ 0. 0370).

Finally, we compare the BER performance of the proposed scheme to that of the standard scheme. Figs. 8 and 9 show the BER results corresponding to the cases of v = 150 km/h and v = 300 km/h, respectively. Also shown in Figs. 8 and 9 is the BER performance achieved when v = 0 km/h. First, we note that, when v = 0 km/h, the proposed RS mapping scheme incurs per-bit SNR penalties of approximately 0.3 dB (for M = 1) and 0.5 dB (for M = 2) with respect to the standard scheme. These penalties are due to the relative rate loss suffered by the proposed scheme from guarding RSs with null subcarriers. However, despite the rate loss, the proposed scheme offers significant BER performance improvements over the standard RS mapping scheme at mobile user speeds of v = 150 km/h and v = 300 km/h. As shown in Fig. 8, when v = 150 km/h, the proposed RS mapping scheme reduces the error floor of the standard scheme by factors of 4.8 (for M = 1) and 5.6 (for M = 2). From Fig. 9, the corresponding error floor reduction factors for the case of v = 300 km/h are observed to be 2.4 (when M = 1) and 3.1 (when M = 2). These BER performance improvements achieved by the proposed RS mapping scheme are due to better CFR and ICI gain estimates obtained

First, we rewrite the left-hand side of (19) as NFFT −1 1 −j2πn[k − k ] n 1− exp NFFT n=0 NFFT NFFT =

NFFT −1

1 NFFT −

exp

n=0

1

NFFT −1

2 NFFT

n=0


n exp

n


n . (57)

Next, from the geometric series, we know that NFFT −1

γn =

n=0 NFFT −1 n=0

nγ n =

1 − γ NFFT (1 − γ)

(58)

γ − γ NFFT +1 NFFT γ NFFT − 2 (1 − γ) (1 − γ)

(59)

for γ = 1. Recalling that k = k from the initial definition of m Hk,k in (6) [note that k = k ensures the condition γ = 1 is 3 Note that the target downlink peak data rates defined by IMT-Advanced for 4G are 1 Gb/s for low-mobility scenarios and 100 Mb/s for high-mobility scenarios [1].

636


met], (58) and (59) can be utilized to evaluate the partial sums in the right-hand side of (57) as n NFFT −1 −j2π[k − k ] exp NFFT n=0 1 − exp (−j2π[k − k ]) (60) ] 1 − exp −j2π[k−k NFFT n NFFT −1 −j2π[k − k ] n exp NFFT n=0 ] ][NFFT +1] exp −j2π[k−k − exp −j2π[k−k NFFT NFFT = 2 ] 1 − exp −j2π[k−k NFFT =

−

NFFT exp (−j2π[k − k ]) . ] 1 − exp −j2π[k−k NFFT

(61)

This completes the derivation of (20). A PPENDIX B D ERIVATION OF (37) Given channel-tap vector θ and discrete transmitted sym˜ m (ϑ ∈ {κp , ∀k }), vector R ˜ m is conditionally bol vectors X κp ϑ m Gaussian with mean μκp ,Standard , as defined in (38), and correlation matrix 1 2 0 2 σw . 1 2 0 2 σw Hence, we have ⎧ ⎛ ⎞⎫ ⎨ ⎬ ˜ m θ, ˜ m⎠ ln f ⎝R X κp ϑ ⎩ ⎭ ϑ∈{κp ,∀k } T , 21 ˜m m = − ln πσw − 2 R − μ κp κp ,Standard σw ˜ m − μm × R κp κp ,Standard . (69)

Now, noting that exp

(62) exp (−j2π[k − k ]) = 1 −j2π[k − k ][NFFT + 1] −j2π[k − k ] = exp NFFT NFFT (63)

the expressions in (60) and (61) can be further simplified as n NFFT −1 −j2π[k − k ] =0 (64) exp NFFT n=0 n NFFT −1 −j2π[k − k ] −NFFT . n exp = ] NFFT 1 − exp −j2π[k−k n=0 NFFT (65) Thereupon, substituting (64) and (65) in (57), we have NFFT −1 1 −j2πn[k − k ] n 1− exp NFFT n=0 NFFT NFFT −1 −j2π[k − k ] 1 = . (66) 1 − exp NFFT NFFT This completes the derivation of (19). To derive (20), we first rewrite the left-hand side of (20) as NFFT −1 n −j2πn[k − k ] 1 exp NFFT n=0 NFFT NFFT n NFFT −1 −j2π[k − k ] 1 = n exp . (67) 2 NFFT NFFT n=0 Then, substituting (65) in (67) yields NFFT −1 n −j2πn[k − k ] 1 exp NFFT n=0 NFFT NFFT

−1 −j2π[k − k ] −1 = . 1 − exp NFFT NFFT

(68)

Then, taking the second derivative of (69) with respect to the elements of θ and noting that ∂ 2 μm κp ,Standard ∂[θ]q ∂[θ]s

= 0,

(∀q, ∀s)

yield

˜m ˜ m θ, ∂ 2 ln f R X κp ϑ ϑ∈{κp ,∀k } ∂[θ]q ∂[θ]s m ∂μκp ,Standard T ∂μm 2 κp ,Standard = − 2 . (70) σw ∂[θ]q ∂[θ]s This completes the derivation of (37). A PPENDIX C D ERIVATION OF (39)

To prove (39), we first rewrite (38) in the following form: L−1 T m m ˜ ˜ μ = am h Φ κp ,Standard

,κp

,0

κp ,Standard

=0

+

L−1


bm κp ,Standard

(71)

=0 m where am κp ,Standard and bκp ,Standard are as defined in (40) and (41). Taking the derivative of (71), with respect to [θ]γ , then yields L−1 ˜m T ∂h ∂μm κp ,Standard ,0 ˜ = am Φ κp ,Standard ,κp ∂[θ]γ ∂[θ]γ =0 L−1 ˜m T ∂h ,NFFT −1 ˜ bm + Φ,κp κp ,Standard . (72) ∂[θ]γ =0


⎧ 1 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ ˜m ∂h 0 ,0 = 1 ⎪ ∂[θ]γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 0 ⎧ 0 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ ˜m ∂h 1 ,NFFT −1 = ⎪ ∂[θ]γ ⎪ 0 ⎪ ⎪ ⎪ 0 ⎪ ⎩ 1

∂μm κp ,Standard ∂[θ]γ

0 δ , γ 1 γ/2−1, −1 δ γ/2−1, , γ 0 0 , γ 0 0 , 0 0 δ , 1 γ/2−L−1, −1 δ γ/2−L−1, , 0

= 1, 3, . . . , (2L − 1) = 2, 4, . . . , (2L)

R EFERENCES [1] E. Dahlman, S. Parkvall, and J. Sköld, 4G: LTE/LTE-A for Mobile Broadband. Kidlington, U.K.: Elsevier, 2011. [2] Y. Mostofi and D. C. Cox, “ICI mitigation for pilot-aided OFDM mobile systems,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 765–774, Mar. 2005. [3] S. Lu and N. Al-Dhahir, “Coherent and differential ICI cancellation for mobile OFDM with application to DVB-H,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4110–4116, Nov. 2008. [4] I.-W. Lai and T.-D. Chiueh, “One-dimensional interpolation based channel estimation for mobile DVB-H reception,” in Proc. IEEE ISCAS, 2006, pp. 5207–5210. [5] S. Lu and N. Al-Dhahir, “Reduced-complexity hybrid time/frequency channel estimation for DVB-H,” in Proc. IEEE Sarnoff Symp., Apr./May 2007, pp. 1–5. [6] R. Kalbasi, S. Lu, and N. Al-Dhahir, “Receiver design for mobile OFDM with application to DVB-H,” in Proc. IEEE VTC—Fall, Montréal, QC, Canada, Sep. 2006, pp. 1–5. [7] D. Martin-Sacristán, J. Cabrejas, D. Calabuig, and J. F. Monserrat, “MAC layer performance of different channel estimation techniques in UTRAN LTE downlink,” in Proc. IEEE VTC—Spring, Apr. 2009, pp. 1–5. [8] Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation, 3GPP TS 36.211 V8.7.0, 2009. [9] F. Gini, R. Reggiannini, and U. Mengali, “The modified Cramer–Rao bound in vector parameter estimation,” IEEE Trans. Commun., vol. 46, no. 1, pp. 52–59, Jan. 1998.

(73)

= (2L + 1), (2L + 2), . . . , (4L) γ = 1, 2, . . . , (2L) γ = (2L + 1), (2L + 3), . . . , (4L − 1)

(74)

γ = (2L + 2), (2L + 4), . . . , (4L)

⎧ m ⎪ ˜T Φ ⎪

γ/2−1,κp aκp ,Standard , ⎪ ⎪ ⎪ ⎪ T 0 −1 m ⎪ ˜ ⎪ aκp ,Standard , ⎨ Φ γ/2−1,κp 1 0 = m ⎪ ˜T Φ ⎪

γ/2−L−1,κp bκp ,Standard , ⎪ ⎪ ⎪ ⎪ 0 −1 ⎪ ˜T ⎪ bm ⎩Φ

γ/2−L−1,κp κp ,Standard , 1 0

Next, recalling the definitions in (23) and (32), the derivatives ˜m ˜ m /∂[θ]γ and ∂ h ∂h ,0 ,NFFT −1 /∂[θ]γ can be computed as (73) and (74), shown at the top of the page, where δ, ( ∈ { γ/2 − 1, γ/2 − L − 1}) denote the Kronecker delta function. Finally, substituting (73) and (74) in (72) yields (75), shown at the top of the page. This completes the derivation of (39).

637

γ = 1, 3, . . . , (2L − 1) γ = 2, 4, . . . , (2L) γ = (2L + 1), (2L + 3), . . . , (4L − 1)

(75)

γ = (2L + 2), (2L + 4), . . . , (4L)

[10] L.-K. Chiu and S.-H. Wu, “The modified Bayesian Cramer– Rao bound for MIMO channel tracking,” in Proc. IEEE ICC, Jun. 2009, pp. 1–5. [11] M. Moeneclaey, “On the true and the modified Cramer–Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters,” IEEE Trans. Commun., vol. 46, no. 11, pp. 1536–1544, Nov. 1998. [12] Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) Radio Transmission and Reception, 3GPP TS 36.101 V9.0.0, 2009. [13] T.-D. Chiueh and P.-Y. Tsai, OFDM Baseband Receiver Design for Wireless Communications. Hoboken, NJ: Wiley, 2007. [14] P. Tichavsky, C.H. Muravchik, and A. Nehorai, “Posterior Cramer–Rao bounds for discrete-time nonlinear filtering,” IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1386–1395, May 1998. [15] R. W. Miller and C. B. Chang, “A modified Cramer–Rao bound and its applications,” IEEE Trans. Inf. Theory, vol. IT-24, no. 3, pp. 398–400, May 1978. [16] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [17] S. D. Muruganathan, W. A. Krzymień, and A. B. Sesay, “Null subcarrier aided reference symbol mapping for improved channel estimation in 3GPP LTE downlink,” in Proc. IEEE VTC—Spring, Taipei, Taiwan, May 2010, pp. 1–5. [18] I. C. Wong and B. L. Evans, “Sinusoidal modeling and adaptive channel prediction in mobile OFDM systems,” IEEE Trans. Signal Process., vol. 56, no. 4, pp. 1601–1615, Apr. 2008. [19] J.-C. Lin, “Least-squares channel estimation for mobile OFDM communication on time-varying frequency-selective fading channels,” IEEE Trans. Veh. Technol., vol. 57, no. 6, pp. 3538–3550, Nov. 2008. [20] H. Hijazi and L. Ros, “Analytical analysis of Bayesian Cramer–Rao bound for dynamical Rayleigh channel complex gains estimation in OFDM systems,” IEEE Trans. Signal Process., vol. 57, no. 5, pp. 1889–1900, May 2009. [21] M. F. Rabbi, S.-W. Hou, and C. C. Ko, “High mobility orthogonal frequency division multiple access channel estimation using basis expansion model,” IET Commun., vol. 4, no. 3, pp. 353–367, Feb. 2010. [22] G. L. Stüber, Principles of Mobile Communication, 2nd ed. Norwell, MA: Kluwer, 2001. [23] G. Matz, “Recursive MMSE estimation of wireless channels based on training data and structured correlation learning,” in Proc. 13th Stat. Signal Process. Workshop, Jul. 2005, pp. 1342–1347.

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Siva D. Muruganathan (S’02–M’04) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2003, 2005, and 2008, respectively. From 2009 to 2010, he was a Postdoctoral Fellow with the University of Alberta, Edmonton. He is now with Research in Motion Limited, Ottawa, ON, Canada.

Witold A. Krzymien´ (M’79–SM’93) received the M.Sc. (Eng.) and Ph.D. degrees in electrical engineering from Poznań University of Technology, Poznań, Poland, in 1970 and 1978, respectively. Since April 1986, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, where he currently holds the endowed Rohit Sharma Professorship in Communications and Signal Processing. In 1986, he was one of the key research program architects of the newly launched TRLabs, Edmonton, which is Canada’s largest industry-university-government precompetitive research consortium in the Information and Communication Technology area. His research activity has been closely tied to the consortium ever since. Over the years, he has also done collaborative research work with Nortel Networks; Ericsson Wireless Communications; German Aerospace Centre (DLR), Oberpfaffenhofen, Germany; TELUS Communications; Huawei Technologies; and the University of Padova, Padova, Italy. He held visiting research appointments with the Twente University of Technology, Enschede, The Netherlands, from 1980 to 1982; Bell-Northern Research, Montréal, QC, Canada, from 1993 to 1994; Ericsson Wireless Communications, San Diego, CA, in 2000; Nortel Networks Harlow Laboratories, Harlow, U.K., in 2001; and the Department of Information Engineering, University of Padova, in 2005. His research interests include multiuser multiple-input–multiple-output (MIMO) and MIMO-OFDM systems, as well as multihop relaying and network coordination for broadband cellular applications. Dr. Krzymień is a Fellow of the Engineering Institute of Canada and a licensed Professional Engineer in the Provinces of Alberta and Ontario. He is an Associate Editor for the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY and a member of the Editorial Board of Wireless Personal Communications (Springer). From 1999 to 2005, he was the Chairman of Commission C (Radio Communication Systems and Signal Processing) of the Canadian National Committee of Union Radio Scientifique Internationale (URSI), and from 2000 to 2003, he was the Editor for Spread Spectrum and Multi-Carrier Systems of the IEEE T RANSACTIONS ON C OMMUNICATIONS. He was the recipient of a Polish national award of excellence for his Ph.D. thesis, the 1991/1992 A.H. Reeves Premium Award from the Institution of Electrical Engineers (U.K.) for a paper published in the IEE Proceedings—Part I, and the Best Paper Award at the IEEE Wireless Communications and Networking Conference in April 2008.

Abu B. Sesay (S’84–M’89–SM’01) received the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 1988 From 1986 to 1989, he was a Research Associate with McMaster University. From 1979 to 1984, he worked on various International Telecommunications Union projects. In 1989, he joined the University of Calgary, Calgary, AB, where he is currently a Full Professor with the Department of Electrical and Computer Engineering and was the Department Head from 2005 to 2011. Since 1989, he has been involved with TRLabs, Edmonton, AB, where he is currently a TRLabs Adjunct Scientist. His current research interests include space-time coding, multicarrier and code-division multiple access, multiuser detection, equalization, error correction coding, multiple-input–multiple-output systems, optical fiber/wireless communications, and adaptive signal processing. Dr. Sesay was the recipient of the IEEE 1996 Neal Shepherd Memorial Best Propagation Paper Award, the Departmental Research Excellence Award for 2002, and the 2005 Schulich School of Engineering Graduate Education Award His students have also received three IEEE conference Best Paper Awards.