Improved Channel Estimation For Complexity-reduced Mimo-ofdm

Improved Channel Estimation for Complexity-Reduced MIMO-OFDM Receiver by Estimation of Channel Impulse Response Length Marco Krondorf, Ting-Jung Liang, Ralf Irmer and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, D-01062 Dresden, Germany {krondorf,liang,irmer,fettweis}@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/MNS Abstract: Using a priori knowledge of maximum channel impulse response (CIR) length, time domain OFDM channel estimation systems outperforms frequency domain channel estimation. Because CIR length is an unknown quantity at the receiver side, a suitable estimation method is necessary. In this paper1 , we propose a novel algorithm named FCLI-Frequency Domain Channel Length Indicator, which estimates CIR length in order to minimize MSE. The FCLI algorithm works in OFDM systems with null-subcarriers, like the spectrum mask defined in IEEE802.11a/g and does not require to average indicators over more than one OFDM symbol. Furthermore, FCLI is designed to work in both SISO and MIMO-OFDM. The simulation results show that channel estimation performance by the FCLI algorithm can approach or outperform the performance with perfect knowledge of channel length in low or high SNR.

1. Introduction In present packet-based OFDM WLAN, such as IEEE802.11a/g, the channel estimation functions can be accomplished by a preamble and a time domain channel estimator at the receiver. This is a reliable algorithm and can be extended to MIMO-OFDM systems by sending orthogonal training sequences from different transmit antennas [6]. The time domain channel estimator with a priori knowledge of CIR length can provide a sufficient channel estimate for MIMO-OFDM systems. If the CIR length is unknown at the receiver side, it can be approximated by a certain value, such as the guard interval (GI) length. The time domain channel estimation performance degrades, if this GI approximation is much longer than the original CIR length. For channel estimate improvements, it is necessary to estimate not the actual CIR length, but the length that minimizes the MSE with respect to the instantaneous noise variance. Reliable channel estimates are essential for high-performance receive algorithms (especially MIMO-OFDM), and to allow low power pilot designs. In the literature, there are two approaches for CIR length estimation: NCLE (Noise Variance and CIR Length Estimation) algorithm [8] and a method presented by Gong and Letaief [2] that is based on SVD (Singular Value Decomposition) of the channel auto correlation matrix. Both techniques are based on averaging the estimated CIR over a long period of time and perform well in continuous streaming OFDM systems such as DVB-T, which also use no spectrum mask. The spec1 This work was partly supported by the German ministry of research and education within the project Wireless Gigabit with Advanced Multimedia Support (WIGWAM) under grant 01 BU 370. Ralf Irmer is now with Vodafone Group Research and Development, UK.

trum mask defines a set of null-subcarriers to reduce the leakage at the spectral edges. The applications of the algorithm proposed in this paper are OFDM standards such as IEEE 802.11a/g, MIMO-OFDM systems such as IEEE 802.11n or the 1 Gbit/s WLAN system WIGWAM. All these systems use a spectrum mask, and CIR averaging over multiple OFDM symbols is not always applicable because of the bursty nature of packet switched traffic. In order to avoid the problems above, we propose the FCLI-Frequency Domain Channel Length Indicator algorithm, which estimates the mean square error of the channel estimate under different hypotheses of instantaneous channel lengths and chooses the hypothesis leading to minimum MSE of the channel coefficients. The algorithm is based on only one snapshot channel estimate in the acquisition phase and the estimated MSE is calculated by channel information only in the used subset of subcarriers according to the spectrum mask. Furthermore, if there are several guard interval options in a OFDM system, such as WIGWAM or IEEE802.11n (different guard interval length for different operating scenarios), the receiver can estimate the channel impulse response length and feed it back to help a transmitter choosing an appropriate guard interval to maximize data throughput. Regarding computational complexity issues, two different orthogonal training sequences are compared. Frequency orthogonal sequences provide the same channel estimation performance as code orthogonal sequences, but have lower complexity. After the motivation of our interest in MIMO-OFDM channel estimation, the rest of the paper is organized as follows: In section 2., the MIMO-OFDM system model is introduced. Section 3. presents the time domain channel estimation algorithm and compares the computational complexity for different pilots followed by analyzing the MSE performance of the time domain channel estimator. Section 4. derives the FCLI algorithm and investigates its performance. The paper is concluded in section 5.

2. MIMO-OFDM System Model The reference model of a MIMO-OFDM system is shown in Fig. 1. It uses NT transmit antennas and NR receive antennas with respective indices t and r. A perfect time and frequency synchronization is assumed at the receiver. The OFDM system uses NC out of total NF F T subcarriers for data transmission with spectrum mask excluding DC and subcarriers at the edge, such as the spectrum mask defined in IEEE802.11a.

H 1, N R

. . .

X

Preamble N T Data N T

Y1 N NR

H NT , N R

NT

. . .

X1

. . .

Data1

. . .

Preamble1

3. Overview of Time Domain MIMOOFDM Channel Estimation and Pilot Selection

N1

H 1,1

Y NR

Figure 1: MIMO System Model The received frequency domain OFDM training symbol at receive antenna r is given by 2 Yr =

NT X

Xt Ht,r + Nr

(1)

t=1

where Yr , Ht,r , Xt and Nr are the received signal (dimension [NC × 1]) at receive antenna r, the frequency domain channel coefficients ([NC × 1]) between antennas t and r, the diagonal pilot matrix ([NC ×NC ]) whose diagonal vector is NC pilot symbols of transmit antenna t and white Gaussian noise ([NC × 1]) with variance 2 σN , respectively. The received frequency domain training symbol in Eq. (1) takes the block matrix form Yr = X Hr + Nr

(2) T

T

with the block matrices Hr = [H1,r . . . HNT ,r ]T and X = [X1 . . . XNT ]. Furthermore, the channel coefficients Ht,r can be expressed by the DFT of the time-domain channel impulse response ht,r n using a truncated fourier matrix (dimension [NC × L]), that can be written in matrix notation as Ht,r = FNC ,L ht,r . Applying the inverse discrete fourier transform to Eq. (1), the time domain received training symbol is computed as yr =

1 NF F T

NT X

t t,r + nr FH NC ,NF F T X FNC ,L h

(3)

t=1

where yr is the time domain received signal at receive t antenna r. The matrix Ct = NF1F T FH NC ,NF F T X FNC ,L denotes the circular convolution matrix of the time domain pilot symbols of transmit antenna t (dimension [NF F T × L]). The time domain received training symbols in Eq. (3) take the block matrix form yr = C hr + nr

(4)

where the block matrices C = [C1 . . . CNT ] and hr = T T [h1,r . . . hNT ,r ]T . The channel impulse responses between all transmit and receive antenna pairs are assumed to be independent. This implies that the MIMO channel estimation can be decomposed into several MISO channel estimation tasks in parallel. 2 Notation: Upper (lower) letters will be generally used for frequency-domain (time-domain) signals; boldface letters represent matrices and column vectors; letters with both boldface and underline represent block matrices or vectors in MIMO.

The time domain channel estimator can provide good estimation results in OFDM with a priori knowledge of CIR length. It can be extended to MIMO-OFDM by sending orthogonal training sequences from different transmit antennas. According to [6], Code Orthogonal (CO) and Frequency Orthogonal (FO) designs are two options of preamble training sequences for MIMOOFDM channel estimation. However, they have different computational complexity. In this work, we consider only the least squares (LS) channel estimator using a preamble with one OFDM symbol composed of FO or CO training sequences. 3.1. Channel Estimation using CO Sequences Exploiting CO sequences, such as Walsh-Hadamard codes or Phase-Shift codes [6], all Tx-antennas simultaneously transmit pilot sequences on all available carriers. The time domain LS channel estimator is derived from Eq. (4) omitting the indices t and r in the following: ˆ = C† y h

(5)

The block matrix C† = [CH C]−1 CH (dimension NT L × NF F T ) denotes the pseudo inverse of C. If Eq. (5) is executed in frequency domain using only NC available data subcarriers, the complexity of matrix multiplication can be reduced due to smaller matrix dimension. Defining the frequency domain received signal on used subcarriers Y = FNC ,NF F T y ([NC × 1]) and the block matrix Z = FNC ,NF F T C ([NC × NT L]), the same channel estimator as in Eq. (5) with lower complexity is given by ˆ = Z† Y . (6) h The functional block diagram of Eq. (6) for processˆ is shown in ing of just one estimated CIR out of h Fig. 2. Both methods above can provide frequency domain channel coefficients by an additional NF F T point ˆ = FNC ,L h ˆ mathematiFFT, which is equivalent to H cally. YN C ×1

( Z H Z ) −1 Z H

hˆ L×1

Zero Padding

( N FFT − L )

hˆ NFFT ×1

FFT

ˆ H N C ×1

N FFT Points

Figure 2: Time domain least squares channel estimation using CO pilot design 3.2. Channel Estimation using FO Sequences Different transmit antennas send training sequences in different sets of subcarriers [6]. For one specific transmit T −1 of used subcarriers are zero subcarriers. antenna, NN T This can be utilized to reduce the complexity of matrix multiplication due to smaller matrix dimension. The algorithm proposed in Fig. 3 estimates the channel coefficients on subcarriers with pilots, transforms the channel information into time domain and transforms subsequently the zero-padded CIR back by an FFT operation.

YN C ×1

Select Pilot Carriers

Y Hˆ k = k Xk

ˆ& H N PC ×1

IDFT using reduced Pseudoinverse Fourier Matrix

hˆ L×1

ˆ H N C ×1

hˆ N FFT ×1

FFT

Zero Padding

( N FFT − L )

N FFT Points

Figure 3: Time domain least squares channel estimation using FO pilot design The channel estimates on subcarriers with pilots for ˆ k = Yk ∀k ∈ P t , where Tx-antenna t are given by H Xk C P t denotes the set of NP C = N NT pilot carriers of antenna t. The estimated channel coefficient in frequency domain takes the vector/matrix form ˆ˙ = 1 X ˙ HY ˙ H 2 σX

(7)

where the dimension reduction from NC to NP C is represented by the dotted notation. The missing NC − NP C coefficients have to be interpolated. Interpolation can be done either by linear or spline-cubic interpolation directly in frequency domain [5]. An alternative approach is time domain interpolaˆ˙ tion, which calculates the corresponding CIR out of H followed by zero padding in interval [L . . . NF F T ] and ˆ L [1]: FFT to obtain the interpolated H ˆ L = FNC ,L h ˆ with H

ˆ˙ ˆ = F˙ †N ,L H h PC

(8)

2 = 1, Substituting Eq. (7) into Eq. (8) and assuming σX ˆ the interpolated HL becomes

ˆL = H

† H ˙ FNC ,L F˙ NP C ,L X˙ Y

(9)

3.3. Computational Complexity The computational complexity Ψ for time domain LS channel estimation using either FO or CO pilot designs is characterized by the number of real operations (real addition as well as real multiplication) for estimating the channel coefficients for one antenna pair (t,r). Complex operations of the channel estimator are decomposed into real operations i.e., 2 real operations for a complex addition and 6 real operations for a complex multiplication. Similarly, multiplication of the r × c complex matrix M1 with the c × r complex matrix M2 requires Ψmat = 2(c − 1)r + 6cr ≈ 8cr real operations. The NF F T point FFT requires NF F T ld(NF F T ) complex additions and NF F T ld(NF F T )/2 complex multiplications, i.e. ΨF F T = 5NF F T ld(NF F T ) real operations are required for one FFT operation. Based on Eq. (6) and Fig. 2, the complexity of CO is ΨCO = 5NF F T ld(NF F T ) + 8LNC

(10)

Similarly, based on (9) and Fig. 3, the complexity of FO yields ΨF O = 5NF F T ld(NF F T )+8LNP C +6NP C

. (11)

Eqs. (10) and (11) denote the normalized computational complexity for estimating the channel coefficients of one antenna pair. In an NT × NR MIMO system, the total complexity is NT NR times the normalized computational complexity. With typical WIGWAM parameters [4], Eq. (10) and Eq. (11) show that FO pilot design requires only 40% computational power of CO. Therefore, FO pilot design is chosen in the following analysis. 3.4. MSE Analysis The performance of the channel estimator is characterized by the mean square error (MSE) of the channel estimates averaged over all receive antennas and used subcarriers. If we perfectly know the channel length L, the MSE of the channel estimates by a least square (LS) time domain channel estimation with spectrum mask is a specific function of the perfectly known channel length 2 L and the noise power σN [6] [1], regardless whether CO or FO pilots are applied: 2 ). M SE(L) = M SEn (L, σN

(12)

In reality, L is not perfectly known and the MSE is de|CIR| Channel & Noise

0

Noise

L

Samples

Figure 4: Estimated CIR consists of two parts, channel+noise inside of CIR length [0..L] and noise only outside of CIR length fined in this case by ˆ = E[tr{(H ˆ ˆ − H)(H ˆ ˆ − H)H }] M SE(L) L L

(13)

ˆ ˆ is the channel estimate using the CIR length where H L ˆ If L is approximated by L ˆ (≥ L), the hypothesis L. ˆ is the same M SEn function (see Eq. (12)) M SE(L) ˆ and σ 2 . Conversely, if L is apwith the parameters L N ˆ ˆ proximated by L (< L), Fig. 4 shows that the M SE(L) 2 ˆ includes not only M SEn (L, σN ) induced by noise but ˆ caused by nealso an additional error part M SEch (L) glecting CIR samples inside the time domain interval ˆ . . . L]. [L ˆ = M SE(L)  ˆ σ 2 ) + M SEch (L) ˆ ,L ˆ
(14)

Our purpose is to relate the MSE of the channel estimates to some available information at the receiver, such ˆ guard inas the frequency domain channel estimate H, ˆ The terval length, noise power and CIR hypothesis L. relation between MSE and available information at the receiver is extended to the FCLI algorithm proposal in the next section. Firstly, the frequency domain channel estimate is deˆ in Eq. (9). As in traditional rived by replacing L by L

ˆ ˆ is decomposed MSE derivation, the channel estimate H L into two parts: channel information and noise:

|

{z

(15)

}

nL ˆ

It follows =

FNC ,L hL = [FNC ,Lˆ FNC ,rest ]

=

FN ,Lˆ hLˆ + FNC ,rest hrest {z } | C{z } | HL ˆ

hLˆ hrest

 (16)

Hrest

(17)

ˆ =M SEn (L, ˆ σ 2 ) + M SEch (L) ˆ M SE(L) N 2 =σN tr{ALˆ AH ˆ }+ L

(18)

It should be noted, that Eq. (18) is only an analytical representation of MSE that cannot be calculated at receiver because of unknown CIR h. Eq. (18) shows that ˆ includes both the effect of M SEn and that the M SE(L) of M SEch , as shown in Fig. 5. The M SEn is indepenˆ ≥ L or L ˆ < L. Adversely, the energy dent, no matter L of the CIR inside of the time interval rest M SEch is different in both cases. Defining the used CIR energy ˆ inside of interval [1 . . . L] ˆ as Ech (L) H H ˆ = tr{F Ech (L) ˆ hL ˆ hL ˆ FN C ,L ˆ} N C ,L

(19)

the M SEch in Eq. (18) is re-written also for the case of ˆ ≥ L: L  ˆ L ˆ
MSE ( Lˆ , σ N2 )

Lˆ2

...

L

LˆG

Samples

MSE ( Lˆ , σ ) = MSE ch ( L ) + MSE n ( Lˆ , σ ) 2 N

Time Samplesˆ

2 N

MSE ( LˆG , σ N2 ) MSE ( Lˆ1 , σ N2 ) MSE ( Lˆ2 , σ N2 ) 0

Lˆ1

Lˆ2

...

LˆG

Samples

Figure 5: Qualitative representation of MSE performance over time domain samples

4. FCLI Channel Length Estimation ˆg) The basic idea of FCLI is that we estimate M SE(L ˆ g with g ∈ [1...G] out of a given set of G CIR for all L length hypotheses by using a priori knowledge of noise 2 ˆ g with the minimum power σN . Then the hypothesis L estimated MSE is chosen ˆg L

by substituting Eq. (15) into Eq. (16) and ALˆ = FNC ,Lˆ F†N ,Lˆ . PC Subsequently, we assume that H and N are uncorrelated and substitute Eq. (17) into Eq. (13). The ˆ becomes M SE(L)

H tr{FNC ,rest hrest hH rest FNC ,rest }

Lˆ1

0

ˆ = arg min {M SE e (L ˆ g )} L

and ˆ ˆ − H = A ˆ N˙ − F H ˆ hrest N C ,L L L

MSE n ( LˆG , σ N2 )

MSE n

MSE n ( Lˆ2 , σ N2 )

Secondly, following the observation in Eq. (14), if ˆ < L, the MSE in Eq. (13) can be further extended L ˆ . . . L] by defining the time domain interval rest = [L and by separating the CIR into two subsets: channel taps ˆ and channel taps inside of time inside of hypothesis L interval rest.   hLˆ and FNC ,L = [FNC ,Lˆ FNC ,rest ] hL = hrest



MSEch

MSE n ( Lˆ1 , σ N2 )

† 2 ˙ = h ˆ . In with the pilot power σX = 1 and F˙ NP C ,Lˆ H L addition, hLˆ denotes the original CIR vector inside the ˆ and n ˆ = F˙ † ˙ length L ˆ N denotes the noise vector N P C ,L L ˆ inside of length L.

H

MSEch ( Lˆ1 ) MSE ch ( Lˆ2 )

MSE n

...

ˆˆ =F H ˆ hL ˆ + N C ,L L

† FNC ,Lˆ F˙ NP C ,Lˆ N˙

Time Samples

MSE ch

(21)

ˆ g ) denotes that the MSE is estimated by where M SE e (L ˆ g . The a priori knowledge the respective approximated L 2 of the noise power σN can be obtained by the double sliding window algorithm given in [7]. The number of tested channel length hypotheses depends on the computational power of the receiver, and is therefore a design parameter of the algorithm. It should be noted that the estimated channel length is not the maximum length of the CIR, but the length of the CIR minimizing the MSE of the channel estimate. 4.1. Calculation of MSEe Eq. (18) shows that we can separate the estimation of ˆ for a given hypothesis L ˆ into two parts. The M SE e (L) first part in Eq. (18) is ˆ σ 2 ) = σ 2 tr{A ˆ AH M SEne (L, ˆ} N N L L

(22)

where ALˆ is known. The second part in Eq. (18) is calˆ max (> L) in Eq. (20) culated by replacing L as L e ˆ e ˆ e ˆ M SEch (L) = Ech (Lmax ) − Ech (L)

(23)

e ˆ Ech (Lmax ) is equal to the whole channel energy Ech (L), e ˆ and Ech (L) denotes the estimated used channel energy ˆ inside the interval [1 . . . L]. e ˆ To avoid the effect of spectrum mask, Ech (L) should be calculated with the help of the energy of frequency ˆ = domain channel estimates EHˆ . Substituting EHˆ (L) PNC ˆ H 2 ˆ Lˆ } into Eq. (15) and conˆ ˆH ˆ (k)| = tr{HL k=1 |HL sidering the channel h as a deterministic quantity, the energy of the channel estimate inside the data subcarriers

ˆ is with CIR hypothesis L H H ˆ =tr{F E[EHˆ (L)] ˆ hL ˆ hL ˆ FN C ,L ˆ} N C ,L 2 + σN tr{ALˆ AH ˆ} L

(24)

ˆ in Eq. (19), the enNext, using the definition of Ech (L) ergy of the channel power can be derived exactly by Eq. (24). By omitting the averaging operation in Eq. (24), e ˆ the resulting Ech (L) can be considered as the estimated energy of channel power: e ˆ 2 ˆ − σN Ech (L) = EHˆ (L) tr{ALˆ AH ˆ} L

(25)

ˆ can be calculated as follows In summary, the M SE e (L) ˆ σ2 ) ˆ + M SE e (L, ˆ = M SE e (L) M SE e (L) N n ch 2 ˆ ˆ = E ˆ (Lmax ) − E ˆ (L) + 2σN tr{A ˆ AH ˆ} H

H

2 − σN tr{ALˆ max AH ˆ max }. L

L

L

(26)

channel estimator with L = 64 serves as the benchmark. The simulation results show that the FCLI algorithm outperforms the benchmark especially in low SNR and approaches the benchmark for in high SNR. The conventional channel estimator with longer approximated channel length L = 128 has a considerable performance loss compared to the benchmark. Fig. 7 uses IEEE 802.11n channel model E, whose maximum channel length is 117 samples [3] in time domain. Therefore, the performance of conventional channel estimator with L = 128 can serve as the benchmark. The simulation results show also that the FCLI algorithm outperforms the benchmark especially in low SNR and approaches the benchmark in high SNR. In contrast to Fig. 6, the conventional channel estimator with shorter approximated channel length L = 64 outperforms the benchmark in low SNR, but it has a distinct performance floor in high SNR, due to the relative effect of lost channel energy to noise energy inside of the interval [65 . . . 117] samples.

4.2. Algorithm in Pseudo Code Notation Exploiting a set of G CIR length hypotheses ˆ1 < . . . < L ˆ G ], FCLI can be formulated in pseudo [L code notation as follows:

0

10

Proposed Channel Estimation using FCLI Conventional Channel Estimation L = 128 Conventional Channel Estimation L = 64 −1

(2) for

10

2 ˆ1 < . . . < L ˆ G ], σN , Y, [L [ALˆ 1 , . . . , ALˆ G ]

i=G...1 ˆi) ˆ i = Channel Estimation(Y,L H 2 ˆ ˆ EˆH (Li ) = sum(|Hi | ) 2 ˆ i ) = σN M SEn (L tr{ALˆ i AH ˆi } L e ˆ ˆ G ) − EˆH (L ˆi) M SE (Li ) = EˆH (L ˆ −M SEn (LG ) ˆi) +2M SEn (L

MSE

(1) init:

−2

10

−3

10

−4

10

0

5

10

ˆg L

(5)

20

25

30

ˆ g )} ˆ = arg min {M SE e (L L ˆ=L ˆg ˆ=H ˆ g where L H

4.3. Simulation Results The performance of the FCLI algorithm is investigated by computer simulations with OFDM system parameters defined in the WIGWAM home/office scenario [4]. The WIGWAM OFDM system with two transmit antennas (NT = 2) uses a 1024 point FFT with 160 MHz sampling frequency (Ts = 6.25 ns and subcarrier spacing = 156.25 kHz), but only 616 (NC ) subcarriers corresponding to 100 MHz are used for data transmission. We consider only two alternatives of guard interval lengths, which correspond to 64 samples (1/16 symbol length) and 128 samples (1/8 symbol length) in time domain. Only one OFDM symbol is transmitted as the preamble for channel estimation, which is composed of frequency orthogonal sequences sent by two transmit antennas using different subcarriers with equal pilot spacing of 2 subcarriers. Specifically for the FCLI algorithm, the following set of CIR length hypotheses was used in ˆ g ∈ [20, 30, 40, 60, 80, 128] samples. the simulations: L Fig. 6 uses the IEEE 802.11n channel model D, whose maximum channel length is 64 samples [3] in time domain. Therefore, the performance of the conventional

Figure 6: Time domain least squares channel estimation using FO pilot design in 2x2 MIMO, WIGWAM 100 MHz home/office parameters and IEEE802.11n channel model D

0

10

Proposed Channel Estimation using FCLI Conventional Channel Estimation L = 128 Conventional Channel Estimation L = 64 −1

10

MSE

(4)

15

SNR in dB

end

−2

10

−3

10

−4

10

0

5

10

15

20

25

30

SNR in dB

Figure 7: Time domain least squares channel estimation using FO pilot design in 2x2 MIMO, WIGWAM 100 MHz home/office parameters and 802.11n channel model E

We can summarize that the FCLI algorithm without a priori knowledge of channel length can outperform the performance of the conventional channel estimator in low and medium SNR.

5. Conclusions In this paper, the Frequency Domain Channel Length Indicator (FCLI) algorithm is proposed to estimate the length of the channel impulse response in SISO- or MIMO-OFDM systems with a priori information of 2 noise power σN . This can be provided for instance by the double-sliding windows algorithm. The simulation results show that the FCLI algorithm can outperform the performance of conventional channel estimator with perfect knowledge of channel length in low SNR. The FCLI algorithm can work for OFDM systems with guard band in contrast with its two forerunners: NCLE algorithm and G&L algorithm. It is also shown that the FCLI algorithm requires only a snapshot channel estimate to accomplish the whole channel length estimation. The two characteristics above exhibit that the FCLI algorithm can be successfully applied in practical MIMO-OFDM systems with spectrum mask and a non-negligible amount of bursty traffic. If an OFDM system has several options of guard intervals, the estimated channel impulse response length can be fed back to help the transmitter choosing an appropriate guard interval for data transmission to maximize the data throughput. The computational complexity of time-domain channel estimation for code and frequency orthogonal pilots was compared in this paper. The proposed algorithm can be easily extended from LS to MMSE channel estimation, which was however not shown in this paper to enhance readability.

REFERENCES [1] A. Dammann G. Auer and S. Sand. Channel Estimaion for OFDM Systems with Multiple Transmit Antennas by Exploiting the Properties of Discrete Fourier Transform. In Proc. IEEE PIMRC, Beijing, China, September 2003. [2] Y. Gong and K. B. Letaief. Low Complexity Channel Estimation for SpaceTime Coded Wideband OFDM Systems. In IEEE Transactions on Wireless Communications, volume 2, September 2003. [3] IEEE802.11-03/940r2. IEEE P802.11 Wireless LANs, TGn Channel Models. January 2004. [4] R. Irmer and G. Fettweis. WIGWAM: System Concept Development for 1 Gbit/s Air Interface. In Proc. WWRF’05, San Diego, USA, July 2005. [5] Park J. Kim J. and Hong D. Performance Analysis of Channel Estimation in OFDM Systems. In IEEE Signal Processing Letters, volume 12, January 2005. [6] T.-J. Liang and G. Fettweis. MIMO Preamble Design With a Subset of Subcarriers in OFDM-based WLAN. In Proc. VTC Spring, 30.May-01.Jun 2005.

[7] J. Terry and J. Heiskala. OFDM WIRELESS LANs: A Theoretical and Practical Guide. SAMS, 2001. [8] H.-P. Kuchenbecker V. D. Nguyen and M. P¨atzold. Estimation of the Channel Impulse Response Length and the Noise Variance for OFDM Systems. In Proc. VTC Spring, 30.May-01.Jun 2005.