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Int. J. Communications, Network and System Sciences, 2010, 3, 446-452 doi:10.4236/ijcns.2010.35059 Published Online May 2010 (http://www.SciRP.org/journal/ijcns/) Copyright © 2010 SciRes. IJCNS Analysis and Comparison of Time Replica and Time Linear Interpolation for Pilot Aided Channel Estimation in OFDM Systems Donglin Wang Department of Electrical and Computer Engineering, University of Calgary, Calgary, Canada Email: dowang@ucalgary.ca Received March 10, 2010; revised April 11, 2010; accepted May 12, 2010 Abstract This paper analyzes and compares two time interpolators, i.e., time replica and time linear interpolator, for pilot aided channel estimation in orthogonal frequency division multiplexing (OFDM) systems. The mean square error (MSE) of two interpolators is theoretically derived for the general case. The equally spaced pilot arrangement is proposed as a special platform for these two time interpolators. Based on this proposed plat- form, the MSE of two time interpolators at the virtual pilot tones is derived analytically; moreover, the MSE of per channel estimator at the entire OFDM symbol based on per time interpolator is also derived. The ef- fectiveness of the theoretical analysis is demonstrated by numerical simulation in both the time-invariant frequency-selective channel and the time varying frequency-selective channel. Keywords: OFDM, Channel Estimation, Time Replica, Time Linear Interpolation, Virtual Pilots 1. Introduction Orthogonal frequency division multiplexing (OFDM) [1-3] has been widely used in high-speed wireless com- munication systems, such as broadband wireless local area networks (WLANs) [4], wireless metropolitan area networks (WMANs) [5] and worldwide interoperability for microwave access (WIMAX) [6], due to its advan- tages of transforming frequency-selective fading chan- nels into a set of parallel flat fading sub-channels and eliminating inter-symbol interference [7]. Channel estimation is one of the most essential tasks in compensating distortion from channels and perform- ing coherent detection in OFDM systems. Estimation is usually performed by using pilot tones [8, 9] and is based on inserting known pilot tones in each OFDM symbol, where interpolation in time-frequency grid [10] plays an important role in the estimation process. The usage of virtual pilot tones [11-13] and time interpolation can reduce the redundancy and guarantee a higher transmis- sion bit rate. Among time interpolation methods, time replica [14, 15] is widely used in time-invariant or slow time-varying channel, which is simple to implement and also efficient for subcarrier usage; time linear interpola- tion [16-18] is widely used in slow or fast time-varying channel, because it is simple to realize and usually can give a satisfactory performance. However, some inter- esting questions are raised as follows: 1) what kind of time-varying channel is slow enough to utilize time rep- lica? 2) Conversely, what kind of time-varying channel is so fast that we have to employ time linear interpolation instead of time replica? And 3) how much does time lin- ear interpolation perform better than time replica by for a time-invariant channel? To answer these questions above, this paper analyzes and compares the performances of time replica and time linear interpolator in both the time-invariant frequency- selective channel and the time varying frequency-selec- tive channel. The MSE of both time interpolators is theo- retically derived for the general cases. The equal spaced pilot arrangement is employed as a special platform for both time interpolators, where the positions of virtual pilot tones in one OFDM symbol correspond to those of pilot tones of its last and next OFDM symbols. Channel state information (CSI) [19] at pilot tones is estimated by least square (LS) estimator. CSI at virtual pilot tones in one OFDM symbol is obtained by either of time interpo- lators, where time replica is to completely replicate the CSI at pilot tones of its last OFDM symbol while time linear interpolator is to linearly interpolate values by using the estimated CSI at the corresponding pilot tones of both its last and next OFDM symbols. CSI at data D. L. WANG Copyright © 2010 SciRes. IJCNS 447 tones is finally obtained by frequency linear interpolation [20]. This paper is organized as follows. In Section 2, the MSEs of two interpolators, i.e. , time replica and time linear interpolation, are theoretically derived for the gen- eral case. In Section 3, the equally spaced pilot arrange- ment is proposed as a special platform for analyzing these two time interpolators. In Section 4, based on the proposed platform, the MSE of two time interpolators at the virtual pilot tones is derived analytically; moreover, the MSE of channel estimators at the entire OFDM symbol based on these two time interpolators is also de- rived, respectively. Numerical results are reports in Sec- tion 5, followed by conclusion in Section 6. Notation: 2 g denotes the modulus. 2 g is the 2-norm operation. { } k E g is the expectation operation on k. { } ,kl E g means the expectation on both k and l. { } k Var g means the variance on k. , () mimj k −+ δ denotes the variation of the CSI of the th k tone from the ( ) th mi − OFDM symbol to the ( ) th mj + OFDM symbol. () m k δ denotes the variation of the CSI of the th k tone from the th m OFDM symbol to the ( ) 1th m+ OFDM symbol. ( ) R m ek and ( ) L m ek are the channel estimation errors of the th m OFDM symbol at the th k tone where time replica or time linear interpo- lation are employed for CSI estimation at the virtual pilot tones, respectively. 2. MSE of Two Time Interpolators Assume that each OFDM symbol has N subcarriers where pilots occupy P subcarriers. Denote the set of pilot tones by P I . By LS estimation, the CSI at pilot tones in the th m OFDM symbol can be obtained as () ˆ() () m mm Yk Hk Xk = (1) where () m Xk and () m Yk are the transmitted and re- ceived pilots of the th m OFDM symbol, respectively. Assuming the pilot tones ()1 m Xk = for convenience of analysis, we have ˆ ()()() mmm HkHkWk =+ (2) where () m Hk represents the true value and () m Wk is a complex-valued sample of additive white Gaussian noise (AWGN) process at the th m OFDM symbol, ( ) 2 ()~0, m WkCN σ . Assuming that along the time axis in Figure 1, the da- ta tones in the th m OFDM symbol correspond to the pilot tones in both the ( ) th mp− and the ( ) th mq+ OFDM symbol, the CSI at the data tones in the th m OFDM symbol can be obtained by time interpolation by using the estimated CSI at the pilot tones of both the ( ) th mp−and the ( ) th mq+ OFDM symbol, which is thus called the virtual pilot tones. Denote the set of vir- tual tones by PP I . In this section, we will analyze and compare the MSE performance of two time interpolators: time replica and time linear interpolator. 2.1. Time Replica Time replica at the virtual pilot tones in the th m sym- bol is to replicate the CSI at the pilot tones in the ( ) th mp− symbol, ˆ ˆ ()(), mmpPP HkHkkI − =∈. (3) By (2) and (3), the estimation error of time replica at the th k tone can be expressed as - -- ˆ ()()-() ()-()(). R mmpm mpmmp ek Hk Hk Hk Hk Wk = =+ (4) The MSE using time replica can thus be obtained as { } { } { } 22 2 - 22 , ()()-() (). R Rkmkmpm kmpm EekEHk Hk Ek − ==+ =+ ξσ δσ (5) 2.2. Time Linear Interpolation However, if using time linear interpolation, the estimated CSI can be obtained as follows, - ˆ ˆˆ ()()() mmpmq pq HkHkHk p qp q+ =+ ++ (6) for PP kI ∈. By (2) and (6), the estimation error of time linear interpolation at the th k tone can be expressed as ( ) ( ) -- ()()-()() ()-()(). L mmpmmp mqmmq pq ekHk HkWk p qp q pq Hk HkWk p qp q ++ =+ ++ =+ ++ (7) Figure 1. The virtual pilot tones in the th m OFDM sym- bol are time-interpolated by using the pilot tones at both the ( ) th −mp and the ( ) th +mq OFDM symbol. D. L. WANG Copyright © 2010 SciRes. IJCNS 448 Based on (7), the MSE of time linear interpolation can thus be obtained as { } ( ) 2 222 ,, 2 2 () ()() . L Lkm mpmmmq k Eek pkqk pq Ep qp qp q −+ = + =−+ ++ + ξ δδ σ (8) 2.3. Comparison Subtracting (8) from (6), the difference between R ξ and L ξ can be expressed as { } ( ) 2 2 ,2 2 ,, 2 () ()() . RLkmpm mpmmmq k pq Ek p q pkqk Ep qp q − −+ −=+ + −− ++ ξξδσ δδ (9) From (9), one can conclude that 1) In a time-invariant frequency-selective channel, L ξ is always lower than R ξ by ( ) 2 10log 2 p q pq +dB; while in a time-variant frequency-selective channel, the per- formance comparison depends on the specific channel variation; 2) Considering a real-valued channel variation, in low noise environment, when ,, ()()0 mpmmmq kk −+ < δδ and ,, ()() mmqmpm kk +− >δδ, RL < ξξ ; 3) Considering a real-valued channel variation, in noisy environment, when ,, ()()0 mpmmmq kk −+ ≥ δδ or ,, ()()0 mpmmmq kk −+ < δδ but ,, ()() mmqmpm kk +− <δδ , RL > ξξ . 3. Special Case: Pilot Arrangement and Channel Estimators Assume that each OFDM symbol has N subcarriers where pilots occupy P subcarriers and virtual pilots su- perimposed with data samples also occupy P subcarriers. Figure 2 shows the proposed pilot arrangement as a platform, which is a special case but not loss of general- ity, where along frequency axis, the pilot spacing is 2L and the spacing between pilot and adjacent virtual pilot is L. From Figure 2, one can see that along time axis, the pilot spacing is 2 and the spacing between pilot and ad- jacent virtual pilot is 1. Also, by LS estimation, the CSI at pilot tones can be obtained by (1). 3.1. Time Interpolation at Virtual Pilot Tones Denote the set of virtual tones by PP I . The CSI at vir- Figure 2. The proposed pilot arrangement as a special plat- form, where the pilot tones in one OFDM symbol correspond to the virtual pilot tones in its adjacent OFDM symbol. tual pilot tones is obtained by time interpolation. In this special pilot arrangement, since the virtual pilot tones at the th m symbol corresponds to the pilot tones at the ( ) 1th m− symbol, time replica at the virtual pilot tones in one symbol is to replicate the CSI at the pilot tones of its last symbol, -1 ˆ ˆ ()(), mmPP HkHkkI =∈ . (10) On the other hand, if using time linear interpolation, we can get -11 ˆˆ ()() ˆ(), 2 mm mPP HkHk HkkI + + =∈ . (11) 3.2. Frequency Interpolation at Data Tones Denote the set of data tones as D I . Using frequency linear interpolation [20], the CSI at the whole OFDM symbol can be expressed as ( ) ( ) ˆ ˆ ()() 1122 ˆ() ˆ () 121. mm m m Lll HkHkL LL whenkLP Hkl Hk whenkLP − ++ ≤≤+− += =+− (12) where () D klI +∈ , PPP kII ∈∪ , 11 lL ≤≤− . Note that the CSI for data tones located on the right side be- yond the ( ) 1th PL+ pilot/virtual pilot tone is decided by the edge interpolation. 4. Performance Analysis for the Special Case This section analyzes the performance of this special case in terms of the MSEs of time interpolators and the MSEs of the corresponding channel estimators. D. L. WANG Copyright © 2010 SciRes. IJCNS 449 4.1. MSE of Time Interpolators For this special case, the MSE of time replica in (5) be- comes { } 2 2 1 (). Rkm Ek − =+ ξδσ (13) On the other hand, the MSE of time linear interpola- tion in (8) becomes 2 2 1()() . 2 mm Lk kk E − − =+ δδ ξσ (14) So, based on (13) and (14), the difference between R ξ and L ξ can be obtained as follows, { } 2 221 1()() 1 (). 22 mm RLkmk kk EkE − − − −=+− δδ ξξδσ (15) And, from (15), one can conclude that 1) in a time- invariant frequency-selective channel, L ξ is always lower than R ξ by 3 dB; in a time-variant frequency- selective channel, the performance difference depends on the specific channel variation; 2) in most situations, as the general case in (9), RL > ξξ , i.e., time linear inter- polation is better than time replica. 4.2. MSE of Channel Estimation 4.2.1. Time Replica By LS estimation on pilot tones, time replica on virtual pilot tones and frequency interpolation on data tones, the corresponding MSE of channel estimation can be ex- pressed as 2 , LRLPRRF PPNP NNN − =++ ξξξξ (16) where P ξ is the MSE of LS estimation and RF ξ is the MSE of frequency interpolation when using time replica at virtual pilot tones. As an average of both odd and even OFDM symbols, except for the right side ( ) 1 L − tones using the edge interpolation, a half of other data tones with the index ( ) D klI +∈ have P kI ∈ while ( ) PP kLI +∈ for fre- quency linear interpolation; for the remaining data tones, PP kI ∈ while ( ) P klI +∈ for frequency linear inter- polation. Hence, using (12), we can get RF ξ in (17), where ()()()() Fmmm Lll eklHkHkLHkl LL − +=++−+ , PPP kII ∈∪ , ( ) 1122 kLP ≤≤+− , and (1(2-1)) F e LP l ++ (1(2-1))-(1(2-1)) mm H LP H LP l =+++ , are the inher- ent errors by frequency interpolation, F ξ is the inherent MSE of frequency interpolation. By substituting (17) into (16), LRL ξ can be expressed as the following (18), { } { } 2 , 2 ,1 2 ,1 2 ˆ ()-() 11 1()()() 22 11 1()()() 22 1 (1(21)) 2 211 1 62 RFklmm klFmm klFmm lFF EHkl Hkl LLll Eekl Wk k NPLL LlLl Eekl Wk k NPLL L EeLPl NP LL LNP − − =++= −− −+++ − −− +−+++ − − ++−+= − −− +− − ξ δ δ ξ { } 2 2 1 211 1(), 62 km LL Ek LNP − −− +− − σ δ (17) ()( ) { } 2 21 2 2121 (). 6 LRLPRF km PPNP NNN LNPL Ek NL − − =++ −−−+ ++ ξξξξ σδ (18) 4.2.2. Time Linear Interpolation By LS estimation on pilot tones, time linear interpolation on virtual pilot tones and frequency interpolation on data tones, the MSE of channel estimation can be expressed as 2 , LLLPRLF PPNP NNN − =++ ξξξξ (19) where LF ξ is the MSE of frequency interpolation when using time linear interpolation at virtual pilot tones. Us- ing (12), LF ξ can be obtained as shown in (20). Substituting (20) into (19), LLL ξ can be expressed as the following (21), { } { } 2 , 2 1 , 2 1 , 2 ˆ()-() 11 1 22 ()() ()() 2 11 1 22 ()() ()() 2 1(1(21)) 2 LFklmm mm klFm mm klFm lF EHkl Hkl L NP kk Lll Eekl Wk LL L NP kk lLl Eekl Wk LL L EeLPl NP ξ δδ δδ − − =++= − − − − − +++ − +− − − − +++ − ++−+= − D. L. WANG Copyright © 2010 SciRes. IJCNS 450 2 2 1 211 1 62 ()() 211 1. 622 F mm k LL LNP kk LL E LNP ξσ δδ − −− +− − − −− +− − (20) ()( ) 2 21 2 2121 6 ()() . 2 LLLPRF mm k PPNP NNN LNPL NL kk E − − =++ −−−+ + − + ξξξξ δδ σ (21) 4.2.3. Comparison Subtracting (21) from (18), the difference between LRL ξ and LLL ξ can be obtained as ( ) ( ) { } 2 2 21 1 2121 26 ()() (). 2 LRLLLL mm kmk LNPL PP NNNL kk EkE− − −−−+ −=++ − ×− ξξσ δδ δ (22) From (22), one can notice that 1) Since N >> P, 2 2 P N σ is negligible and the dif- ferential MSE using (22) is approximately independent with noise; 2) In a time-invariant frequency-selective channel, LRL ξ is approximately equal to LLL ξ ; while in a time-variant frequency-selective channel, the perform- ance comparison depends on the specific channel varia- tion; 3) Considering a real-valued channel variation, in low noise environment, when 1 ()()0 mm kk − < δδ and () m k > δ 1 () m k − δ , LRLLLL < ξξ ; 4) LRLLLLRL −<− ξξξξ . 5. Numerical Results The OFDM system under consideration is with N = 512 subcarriers, and 2L = 8 equispaced pilot tones in each symbol. The length of cyclic prefix is 32. The interpola- tion distances p = q = 1. The modulation is QPSK. The pilot tones are all 1. For 063 j ≤≤ , in the odd OFDM symbols, the pilot is inserted at the ( ) 18th j + tone; while in the even OFDM symbols, the pilot is inserted at the ( ) 58th j + tone. The six-ray multipath Rayleigh fading channel is considered. The average power delay profile is selected as 5 0 exp() l l l l = − = ∑ λ λ , 05 l ≤≤ . (23) Figure 3 shows the MSE performance of time inter- polator and channel estimation in the time-invariant fre- quency-selective channel, where one can see that time linear interpolator generating less noise has a 3 dB lower MSE than time replica at the virtual pilot tones. However, for the corresponding channel estimation at the whole OFDM tones, time linear interpolator performs similarly to time replica due to a negligible noise. Figure 4 shows the MSE performance in a time vary- ing channel where the parameters are { } 1 () km Ek − δ 0.001 = , { } 6 1 ()10 km Vark − − =δ, { } ()0.002 km Ek=−δ, and { } 6 ()10 km Vark − =δ, respectively. For interpolation at vir- tual pilot tones, when 25 SNR ≤ dB, time linear inter- polator performs better than time replica due to better noise reduction; when SNR > 25 dB, time replica, which guarantees a more accurate interpolation in a low noise environment, performs better than linear interpolator. While for the corresponding channel estimation, when 25 SNR ≤ dB, time linear interpolator performs very similarly to time replica due to better noise reduction; when SNR > 25 dB, time replica also performs better than time linear interpolator. Figure 5 shows the MSE performance in the time va- rying channel where the parameters are { } 1 () km Ek − δ 010 2030 -95 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of time interpolator Time replica Time linear interp 010 20 30 -95 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of channel estimation Time replica Time linear interp Figure 3. MSE of time interpolator and channel estimation in time-invariant frequency-selective channel. D. L. WANG Copyright © 2010 SciRes. IJCNS 451 010 20 30 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of time interpolator Time replica Time linear interp 010 20 30 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of channel estimation Time replica Time linear interp Figure 4. MSE of time interpolator and channel estimation in time-variant frequency-selective channel, where the ex- pectation is equal to { } 1 ()0.001 km Ekδ − =, the variance is equal to { } 6 1 ()10 km Varkδ − −=, the expectation { } () km Ek δ 0.002 =− , and variance { } 6 ()10 km Varkδ − =, respectively, 0512 k≤≤ . 01020 30 -95 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of time interpolator Time replica Time linear interp 01020 30 -95 -90 -85 -80 -75 -70 -65 -60 SNR[dB] MSE[dB] MSE of channel estimation Time replica Time linear interp Figure 5. MSE of time interpolator and channel estimation in time-variant frequency-selective channel, where the ex- pectation is equal to { } 1 ()0.001 km Ekδ − =, the variance is equal to { } 6 1 ()10 km Varkδ − −= , the expectation { } () km Ek δ 0.002 =, and variance { } 6 ()10 km Varkδ − =, respectively, 0512 k≤≤ . 0.001 = , { } 6 1 ()10 km Vark − −=δ, { } ()0.002 km Ek=δ, and { } 6 ()10 km Vark − =δ, respectively. Time linear interpola- tor always performs better than time replica for both in- terpolation at the virtual pilot tones and the correspond- ing channel estimation at the entire tones. 6. Conclusions Time replica and time linear interpolation were analyzed and compared, especially under our proposed pilot ar- rangement. The MSEs of both time interpolators were derived analytically for both interpolations at the virtual pilot tones and their corresponding channel estimation at the entire OFDM symbol. Numerical simulation results were demonstrated to reach an agreement with theoreti- cal analysis. From the given results, one can see that, in a time-invariant frequency-selective channel, when the interpolation distances p = q =1, time linear interpolator has a 3 dB lower MSE than replica at the virtual pilot tones while they provide a similar performance at the entire OFDM symbol. Moreover, one can also see that, in a time varying frequency-selective channel, time lin- ear interpolator outperforms time replica except the case, in a low noise environment, the CSI variation from the last OFDM symbol to the present symbol is negative to and has a smaller absolute value than that from the pre- sent symbol to the following symbol. 7. Acknowledgements The author would like to thank all the anonymous re- viewers of the paper. The critical comments by all the reviewers have helped us to improve the quality of our paper. 8. References [1] M. 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