Analysis and Comparison of Time Replica and Time Linear Interpolation for Pilot Aided Channel Estimation in OFDM Systems

Transcription

1 Int J Counications Networ and Syste Sciences doi:10436/ijcns Published Online May 010 ( Analysis and Coparison of Tie eplica and Tie inear Interpolation for Pilot Aided Channel Estiation in OFDM Systes Abstract Donglin Wang Departent of Electrical and Coputer Engineering University of Calgary Calgary Canada Eail: dowang@ucalgaryca eceived March ; revised April ; accepted May This paper analyzes and copares two tie interpolators ie tie replica and tie linear interpolator for pilot aided channel estiation in orthogonal frequency division ultiplexing (OFDM) systes The ean square error (MSE) of two interpolators is theoretically derived for the general case The equally spaced pilot arrangeent is proposed as a special platfor for these two tie interpolators Based on this proposed platfor the MSE of two tie interpolators at the virtual pilot tones is derived analytically; oreover the MSE of per channel estiator at the entire OFDM sybol based on per tie interpolator is also derived The effectiveness of the theoretical analysis is deonstrated by nuerical siulation in both the tie-invariant frequency-selective channel and the tie varying frequency-selective channel Keywords: OFDM Channel Estiation Tie eplica Tie inear Interpolation Virtual Pilots 1 Introduction Orthogonal frequency division ultiplexing (OFDM) [1-3] has been widely used in high-speed wireless counication systes such as broadband wireless local area networs (WANs) [4] wireless etropolitan area networs (WMANs) [5] and worldwide interoperability for icrowave access (WIMAX) [6] due to its advantages of transforing frequency-selective fading channels into a set of parallel flat fading sub-channels and eliinating inter-sybol interference [7] Channel estiation is one of the ost essential tass in copensating distortion fro channels and perforing coherent detection in OFDM systes Estiation is usually perfored by using pilot tones [8 9] and is based on inserting nown pilot tones in each OFDM sybol where interpolation in tie-frequency grid [10] plays an iportant role in the estiation process The usage of virtual pilot tones [11-13] and tie interpolation can reduce the redundancy and guarantee a higher transission bit rate Aong tie interpolation ethods tie replica [14 15] is widely used in tie-invariant or slow tie-varying channel which is siple to ipleent and also efficient for subcarrier usage; tie linear interpolation [16-18] is widely used in slow or fast tie-varying channel because it is siple to realize and usually can give a satisfactory perforance However soe interesting questions are raised as follows: 1) what ind of tie-varying channel is slow enough to utilize tie replica? ) Conversely what ind of tie-varying channel is so fast that we have to eploy tie linear interpolation instead of tie replica? And 3) how uch does tie linear interpolation perfor better than tie replica by for a tie-invariant channel? To answer these questions above this paper analyzes and copares the perforances of tie replica and tie linear interpolator in both the tie-invariant frequencyselective channel and the tie varying frequency-selective channel The MSE of both tie interpolators is theoretically derived for the general cases The equal spaced pilot arrangeent is eployed as a special platfor for both tie interpolators where the positions of virtual pilot tones in one OFDM sybol correspond to those of pilot tones of its last and next OFDM sybols Channel state inforation (CSI) [19] at pilot tones is estiated by least square (S) estiator CSI at virtual pilot tones in one OFDM sybol is obtained by either of tie interpolators where tie replica is to copletely replicate the CSI at pilot tones of its last OFDM sybol while tie linear interpolator is to linearly interpolate values by using the estiated CSI at the corresponding pilot tones of both its last and next OFDM sybols CSI at data Copyright 010 Scies

2 D WANG 447 tones is finally obtained by frequency linear interpolation [0] This paper is organized as follows In Section the MSEs of two interpolators ie tie replica and tie linear interpolation are theoretically derived for the general case In Section 3 the equally spaced pilot arrangeent is proposed as a special platfor for analyzing these two tie interpolators In Section 4 based on the proposed platfor the MSE of two tie interpolators at the virtual pilot tones is derived analytically; oreover the MSE of channel estiators at the entire OFDM sybol based on these two tie interpolators is also derived respectively Nuerical results are reports in Section 5 followed by conclusion in Section 6 Notation: g denotes the odulus g is the E g is the expectation operation -nor operation { } on { } El g eans the expectation on both and l Var { g } eans the variance on δ i + j( ) denotes the variation of the CSI of the th tone fro the ( i) th OFDM sybol to the ( + j) th OFDM sybol δ ( ) denotes the variation of the CSI of the th tone fro the th OFDM sybol to the ( + 1th ) OFDM sybol e ( ) and e ( ) are the channel estiation errors of the th OFDM sybol at the th tone where tie replica or tie linear interpolation are eployed for CSI estiation at the virtual pilot tones respectively MSE of Two Tie Interpolators Assue that each OFDM sybol has N subcarriers where pilots occupy P subcarriers Denote the set of pilot tones by I P By S estiation the CSI at pilot tones in th the OFDM sybol can be obtained as Y ( ) H ( ) = X ( ) (1) where X ( ) and Y ( ) are the transitted and received pilots of the th OFDM sybol respectively Assuing the pilot tones X ( ) = 1 for convenience of analysis we have H ( ) = H ( ) + W ( ) () where H ( ) represents the true value and W ( ) is a coplex-valued saple of additive white Gaussian noise (AWGN) process at the th OFDM sybol W ( )~ CN ( 0 σ ) Assuing that along the tie axis in Figure 1 the data tones in the OFDM sybol correspond to the th pilot tones in both the ( p) th and the ( + q) th OFDM sybol the CSI at the data tones in the th OFDM sybol can be obtained by tie interpolation by using the estiated CSI at the pilot tones of both the ( p) th and the ( q) th + OFDM sybol which is thus called the virtual pilot tones Denote the set of virtual tones by I In this section we will analyze and copare the MSE perforance of two tie interpolators: tie replica and tie linear interpolator 1 Tie eplica at the virtual pilot tones in the th sybol is to replicate the CSI at the pilot tones in the ( p) th sybol H ( ) = H ( ) I (3) p By () and (3) the estiation error of tie replica at the th tone can be expressed as e ( ) = H ( )- H ( ) - p = H ( )- H ( ) + W ( ) -p -p The MSE using tie replica can thus be obtained as { ( ) } { - ( )- ( ) } { δ ( ) } σ = E e = E H H + σ p = E + p Tie inear Interpolation (4) (5) However if using tie linear interpolation the estiated CSI can be obtained as follows p q H ( ) = H- p( ) + H q( ) (6) + p + q p + q for I By () and (6) the estiation error of tie th linear interpolation at the tone can be expressed as p q e( ) = ( H-p( )- H( ) ) + W-p( ) p + q p + q (7) p q = ( H+ q( )- H( ) ) + W+ q( ) p + q p + q Figure 1 The virtual pilot tones in the th OFDM sybol are tie-interpolated by using the pilot tones at both the ( p ) th and the ( + q ) th OFDM sybol Copyright 010 Scies

3 448 D WANG Based on (7) the MSE of tie linear interpolation can thus be obtained as { ( ) } = E e pδ q p + q = E + p + q p + q p + q p ( ) δ + q( ) σ ( ) (8) 3 Coparison Subtracting (8) fro (6) the difference between can be expressed as pq { ( ) } ( p + q ) = E δ + σ p E pδ p ( ) qδ + q( ) p + q p + q and (9) Fro (9) one can conclude that 1) In a tie-invariant frequency-selective channel is always lower than by 10log ( p + q) db; while pq in a tie-variant frequency-selective channel the perforance coparison depends on the specific channel variation; ) Considering a real-valued channel variation in low noise environent when δ p ( ) δ + q( ) < 0 and δ + q( ) > δ p ( ) < ; 3) Considering a real-valued channel variation in noisy environent when δ p ( ) δ + q( ) 0 or δ p ( ) δ + q( ) < 0 but δ + q( ) < δ p ( ) > 3 Special Case: Pilot Arrangeent and Channel Estiators Assue that each OFDM sybol has N subcarriers where pilots occupy P subcarriers and virtual pilots superiposed with data saples also occupy P subcarriers Figure shows the proposed pilot arrangeent as a platfor which is a special case but not loss of generality where along frequency axis the pilot spacing is and the spacing between pilot and adjacent virtual pilot is Fro Figure one can see that along tie axis the pilot spacing is and the spacing between pilot and adjacent virtual pilot is 1 Also by S estiation the CSI at pilot tones can be obtained by (1) 31 Tie Interpolation at Virtual Pilot Tones Denote the set of virtual tones by I The CSI at vir- Figure The proposed pilot arrangeent as a special platfor where the pilot tones in one OFDM sybol correspond to the virtual pilot tones in its adjacent OFDM sybol tual pilot tones is obtained by tie interpolation In this special pilot arrangeent since the virtual pilot tones at the th sybol corresponds to the pilot tones at the ( 1th ) sybol tie replica at the virtual pilot tones in one sybol is to replicate the CSI at the pilot tones of its last sybol H ( ) = H ( ) I (10) -1 On the other hand if using tie linear interpolation we can get H-1( ) + H+ 1( ) H( ) = I (11) 3 Frequency Interpolation at Data Tones Denote the set of data tones as I D Using frequency linear interpolation [0] the CSI at the whole OFDM sybol can be expressed as l l ( ) H + H( + ) H ( ) l when + P = H ( ) when = 1+ ( P 1 ) ( ) (1) where ( + l) ID IP I 1 l 1 Note that the CSI for data tones located on the right side beyond the ( 1+ P) th pilot/virtual pilot tone is decided by the edge interpolation 4 Perforance Analysis for the Special Case This section analyzes the perforance of this special case in ters of the MSEs of tie interpolators and the MSEs of the corresponding channel estiators Copyright 010 Scies

4 D WANG MSE of Tie Interpolators For this special case the MSE of tie replica in (5) becoes { } ( ) = E δ 1 + σ (13) On the other hand the MSE of tie linear interpolation in (8) becoes δ 1( ) δ( ) = E + σ (14) So based on (13) and (14) the difference between and can be obtained as follows { 1 } 1 δ 1( ) δ( ) = E δ ( ) + σ E (15) And fro (15) one can conclude that 1) in a tieinvariant frequency-selective channel is always lower than by 3 db; in a tie-variant frequencyselective channel the perforance difference depends on the specific channel variation; ) in ost situations as the general case in (9) > ie tie linear interpolation is better than tie replica 4 MSE of Channel Estiation 41 Tie eplica By S estiation on pilot tones tie replica on virtual pilot tones and frequency interpolation on data tones the corresponding MSE of channel estiation can be expressed as P P N P = P + + F (16) where P is the MSE of S estiation and F is the MSE of frequency interpolation when using tie replica at virtual pilot tones As an average of both odd and even OFDM sybols except for the right side ( 1) tones using the edge interpolation a half of other data tones with the index ( + l) ID have IP while ( + ) I for frequency linear interpolation; for the reaining data tones I while ( + l) IP for frequency linear interpolation Hence using (1) we can get F in (17) l l where ef( + l) = H( ) + H( + ) H( + l) IP I 1 1+ ( P ) and e F (1 + (P -1) + l) = H (1 + (P -1))- H (1 + (P -1) + l) are the inherent errors by frequency interpolation F is the inherent MSE of frequency interpolation By substituting (17) into (16) as the following (18) { ( )- ( ) } = E H + l H + l = F l can be expressed l l 1 El ef( l ) W( ) 1( ) N P δ l l + 1 El ef( + l ) + W( ) + δ 1( ) N P + El{ ef(1 + (P 1) + l ) } = F N P N P σ + 1 E{ δ 1( ) } 6 N P P P N P = + + P F ( 1)( N P 1) { δ 1 } + σ E + + 6N ( ) (17) (18) 4 Tie inear Interpolation By S estiation on pilot tones tie linear interpolation on virtual pilot tones and frequency interpolation on data tones the MSE of channel estiation can be expressed as P P N P = P + + F (19) where F is the MSE of frequency interpolation when using tie linear interpolation at virtual pilot tones Using (1) F can be obtained as shown in (0) Substituting (0) into (19) can be expressed as the following (1) { ( )- ( ) } = E H + l H + l = F l 1 N P l l δ El ef( + l ) + W( ) N P 1 ( ) δ( ) l l δ 1( ) δ( ) El ef( + l ) + W( ) + + El{ ef(1 + (P 1) + l ) } = N P Copyright 010 Scies

5 450 D WANG F + 1 σ 6 N P N P δ 1( ) δ( ) 1 E P P N P = + + P F + σ ( )( N P + 1) 6N δ ( ) δ( ) + E 1 43 Coparison Subtracting (1) fro (18) the difference between and can be obtained as ( )( N P + 1) P P = σ + + N N 6N (0) (1) δ 1( ) δ( ) E{ δ 1( ) } E () Fro () one can notice that P 1) Since N >> P σ is negligible and the differential MSE using () is approxiately independent N with noise; ) In a tie-invariant frequency-selective channel is approxiately equal to ; while in a tie-variant frequency-selective channel the perforance coparison depends on the specific channel variation; 3) Considering a real-valued channel variation in low noise environent when δ( ) δ 1() < 0 and δ () > δ ( ) 1 < ; 4) < while in the even OFDM sybols the pilot is inserted at the ( 5+ 8j) th tone The six-ray ultipath ayleigh fading channel is considered The average power delay profile is selected as λ = exp( l) l 5 l= 0 λ l 0 l 5 (3) Figure 3 shows the MSE perforance of tie interpolator and channel estiation in the tie-invariant frequency-selective channel where one can see that tie linear interpolator generating less noise has a 3 db lower MSE than tie replica at the virtual pilot tones However for the corresponding channel estiation at the whole OFDM tones tie linear interpolator perfors siilarly to tie replica due to a negligible noise Figure 4 shows the MSE perforance in a tie varying channel where the paraeters are E { ( ) δ 1 } = 0001 { ( ) 6 1 } 10 Var = Var 6 { δ () } = 10 δ { δ } E ( ) 000 = and respectively For interpolation at virtual pilot tones when SN 5 db tie linear interpolator perfors better than tie replica due to better noise reduction; when SN > 5 db tie replica which guarantees a ore accurate interpolation in a low noise environent perfors better than linear interpolator While for the corresponding channel estiation when SN 5 db tie linear interpolator perfors very siilarly to tie replica due to better noise reduction; when SN > 5 db tie replica also perfors better than tie linear interpolator Figure 5 shows the MSE perforance in the tie varying channel where the paraeters are E { ( ) δ 1 } MSE of tie interpolator MSE of channel estiation 5 Nuerical esults The OFDM syste under consideration is with N = 51 subcarriers and = 8 equispaced pilot tones in each sybol The length of cyclic prefix is 3 The interpolation distances p = q = 1 The odulation is QPSK The pilot tones are all 1 For 0 j 63 in the odd OFDM 1+ 8j th tone; sybols the pilot is inserted at the ( ) SN[dB] SN[dB] Figure 3 MSE of tie interpolator and channel estiation in tie-invariant frequency-selective channel Copyright 010 Scies

6 D WANG 451 MSE of tie interpolator SN[dB] MSE of channel estiation SN[dB] Figure 4 MSE of tie interpolator and channel estiation in tie-variant frequency-selective channel where the expectation is equal to E { ( ) 1 } 0001 d - = the variance is equal to { ( ) 6 1 } 10 - Var d- = the expectation E{ d ( ) } 6 =- 000 and variance Var{ d ( ) } 10 - = respectively Conclusions and tie linear interpolation were analyzed and copared especially under our proposed pilot arrangeent The MSEs of both tie interpolators were derived analytically for both interpolations at the virtual pilot tones and their corresponding channel estiation at the entire OFDM sybol Nuerical siulation results were deonstrated to reach an agreeent with theoretical analysis Fro the given results one can see that in a tie-invariant frequency-selective channel when the interpolation distances p = q =1 tie linear interpolator has a 3 db lower MSE than replica at the virtual pilot tones while they provide a siilar perforance at the entire OFDM sybol Moreover one can also see that in a tie varying frequency-selective channel tie linear interpolator outperfors tie replica except the case in a low noise environent the CSI variation fro the last OFDM sybol to the present sybol is negative to and has a saller absolute value than that fro the present sybol to the following sybol 7 Acnowledgeents MSE of tie interpolator MSE of channel estiation The author would lie to than all the anonyous reviewers of the paper The critical coents by all the reviewers have helped us to iprove the quality of our paper SN[dB] SN[dB] Figure 5 MSE of tie interpolator and channel estiation in tie-variant frequency-selective channel where the expectation is equal to E { ( ) 1 } 0001 d - = the variance is equal to { ( ) 6 1 } 10 - Var d- = the expectation E{ d ( ) } 6 = 000 and variance Var{ d ( ) } 10 - = respectively 0 51 = 0001 { ( ) 6 1 } 10 Var = E ( ) 000 δ = and 6 Var{ ( ) } 10 δ = respectively olator always perfors better than tie replica for both interpolation at the virtual pilot tones and the corresponding channel estiation at the entire tones δ { } 8 eferences [1] M Engels Wireless OFDM Systes Kluwer Acadeic Publishers New Yor 00 [] Hanzo OFDM and MC-CDMA: a Prier John Wiley & Sons Inc Hoboen 006 [3] H Schulze Theory and Applications of OFDM and CDMA: Wideband Wireless Counications John Wiley & Sons Inc Hoboen 005 [4] B Bing Wireless ocal Area Networs: The New Wire- ess evolution Wiley-Interscience New Yor 00 [5] S Methley Essentials of Wireless Mesh Networing Cabridge University Press Cabridge 009 [6] M Ma Current Technology Developents of Wiax Systes Springer Verlag New Yor 009 [7] C Pandana Y Sun and K J iu Channel-Aware Priority Transission Schee Using Joint Channel Estiation and Data oading for OFDM Systes IEEE Transactions on Signal Processing Vol 53 No 8 August 005 pp [8] Negi and J Cioffi Pilot Tone Selection for Channel Estiation in a Mobile OFDM Syste IEEE Transactions on Consuer Electrononics Vol 44 No 3 August 1998 pp Copyright 010 Scies

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