On Improved DCT Based Channel Estimation with Very Low ...

On improved DCT based Channel Estimation with very low complexity for MIMO-OFDM systems Moussa Diallo, Rodrigue Rabineau, Laurent Cariou and Maryline H´elard France Telecom R&D Division, 4 rue du Clos Courtel, 35512 Cesson-S´evign´e Cedex, France Email: {moussa.diallo,rodrigue.rabineau,laurent.cariou}@orange-ftgroup.com [email protected]

Abstract—This paper deals with a novel transform domain channel estimation (TD-CE) based on discrete cosine transform (DCT) for the MIMO-OFDM system. After the description of the conventional DCT based method, a channel estimation scheme improved in terms of both performance and complexity using small DCT blocks is proposed. Finally the efficiency of this estimator is evaluated within IEEE802.11n and 3GPP/LTE system contexts. Keywords— MIMO-OFDM, DFT, DCT, channel estimation, ”border effect”, 802.11n, 3GPP/LTE.

high frequency components in the transform domain [8] at the price of a weaker noise reduction [9]. On the top of that, the complexity and latency of the DCT calculation may cause problems. This paper proposes an improved DCT based channel estimation with very low complexity. The whole DCT window will be divided into small overlapping blocks where the length of each block is a power of 2. The complexity is reduced while the performance is improved.

I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) is an efficient candidate for high data rate wireless communications owing to its many advantages, notably its high spectral efficiency, mitigation of intersymbol interference (ISI), robustness to frequency selective fading environment, as well as the feasibility of low cost transceivers [1]. On the other hand the multi-input multi-output (MIMO) system has the potential to obtain a diversity gain and to improve the system capacity [2], [3], [4]. Hence the combination of MIMO and OFDM technologies (MIMO-OFDM) is now largely considered in the new generation of standards for wireless transmission [5]. In MIMO-OFDM systems, the channel state information (CSI) is usually required at the receiver side and thus subchannels between each antenna link have to be individually estimated. Dynamic channel estimation becomes necessary since radio channel are frequency selective and time-dependent. That is the reason why construction of training sequences has to be adapted depending on the MIMO configuration and the channel characteristics [6]. For OFDM channel estimation based on pilot insertion, different techniques can be applied: preamble method and comb-type pilot method. An efficient way to use preamble symbols is the TD-CE method. The DFT based method is considered as one of the promising methods because it can provide very good results by significantly reducing the noise on the estimated channel coefficients [7]. However, ”border effect” may occur when the number of IFFT points is different from the number of modulated subcarriers [7]. This is a real problem because the vast majority of modern multicarrier systems contain null carriers at the spectrum extremities. DCT is suggested instead of DFT for mitigating the impact of the ”border effect” owing to its capacity to reduce the

II. MIMO-OFDM S YSTEM M ODEL A simplified block diagram of the MIMO-OFDM studied system with Nt transmit antennas and Nr receive antennas is shown in Fig.1. 1

1

OFDM modulation

MIMO

OFDM demodulation Equalization &

& Nt

Frame Mapping

OFDM modulation

Nr OFDM demodulation

Detection

Channel Estimation

Fig. 1.

MIMO-OFDM block diagram.

The OFDM transmitted signal from the i-th antenna after performing IFFT (OFDM modulation) to the frequency domain signal Xi ∈ C N ×1 can be given by:  N −1 2πkn 1  Xi (k)ej N , 0 ≤ n ≤ N − 1 (1) xi (n) = N k=0

where N is the number of FFT points. The time domain channel response between the transmit antenna i and the receive antenna j under the multipath fading environments can be expressed by the following equation: hij (n) =

L−1 

hij,l δ(n − τij,l )

(2)

l=0

where L the number of paths, hij,l and τij,l the complex time varying channel coefficient and delay of the l-th path. The use of the cyclic prefix allows both the preservation of the orthogonality between the tones and the elimination of ISI (inter symbol interference) between consecutive OFDM

978-1-4244-2517-4/09/$20.00 ©2009 IEEE

symbols. Thus by using (1) and (2), the receive signal in frequency domain is given by: R(k) =

N t −1 

Xi (k)Hi (k) + Ξ(k)

DCT

Smoothing

(3)

i=0 IDCT

where Ξk the zero-mean complex Gaussian noise after the FFT process and Hi (k) the discrete response of the channel on subcarrier k between the i-th transmit antenna and the j-th receive antenna, the estimation of which is required. III. L EAST S QUARE C HANNEL E STIMATION (LS) In SISO-OFDM, without using any knowledge of the statistics on the channels, the LS estimates of the CSI can be obtained at the j-th antenna by dividing the demodulated pilots signal Rj (k) by the known pilots symbol X(k) in the frequency domain [8]. In MIMO-OFDM, from (3) an orthogonality between pilots is mandatory to get the CSI without interference from the other antennas. Assuming this orthogonality, the LS estimates can be expressed as follow: Hij,LS = Hij + (diag(X))−1 Ξ.

(4)

The orthogonality between the pilots can be obtained from different ways depending on the system: Time 000 111 000 111 000 111 000 111 111 000 000 111 000 111 000 111

11 00 000 111 00 11 000 111 00 11 000 111 00 11 000 111

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 111 000 000 111 000 111

00 11 11 00 00 11

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

00 11 11 00 00 11 11 00 11 00 00 00 11 00 11 11 11 00

00 11 11 00 11 000 111 11 00 000 111 00 00 11 00 11 000 111

From (4), it can be observed that the accuracy of LS estimated channel response is degraded by the noise component. To improve the accuracy of the channel estimation, the DCTbased method has been proposed for these two following reasons: • on the one hand, for reducing the noise component and the high frequency components in the fransform domain [8] [12]. Fig.3 illustrates the smoothing process using DCT. After removing the unused subcarriers, the estimated channel response given by (4) is first converted into the transform domain by the DCT algorithm and a smoothing filter is applied. After the smoothing, the IDCT is applied to return in the frequency domain. DCT

Null symbol Data symbol M

111 000 11 00 000 00 111 00 11 11 00 11 000 111 00 11 11 000 111 00 00 11 00 11 Tx1

Frequency

Tx2

(a) Orthogonality between pilots in 3GPP

Frequency

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111

Time 000 111 000 111 000 111 000 111 000 111 000 111 111 000 000 111 000 111 000 111 000 111 000 111

Smoothing using DCT.

IV. T IME DOMAIN CHANNEL ESTIMATION USING CLASSICAL DCT

Means negative of

Null carriers

Tx1

Frequency

Frequency

11 00 000 111 00 11 000 111 00 11 000 111 00 11 000 111

000 111 111 000 000 111 000 111

000 111 111 000 000 111 000 111

000 111 000 111 111 000 000 111 000 111 000 111

Time 00 11 11 00 11 11 00 11 00 11 00 00 11 00 11 00 00 00 11 00 11 11 11 00

111 000 11 00 000 00 111 00 11 11 00 11 000 111 00 11 11 000 111 00 00 11 00 11

Fig. 3.

Pilot symbol

00 11 11 00 00 11

11 00 11 00 11 00 00 11 000 111 11 00 000 111 00 11 000 111

Null carriers

Time 00 11 000 111 00 11 000 111 11 00 000 111 00 11 000 111

W

DFT

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 Tx2

(b) Orthogonality between pilots in 802.11n M

Fig. 2. •

•

Pilot insertion structure in 3GPP and 802.11n with Nt = 2.

In wireless LAN systems as in IEEE802.11n, the channel (TGn channel models [10]) can be assumed to be unchanged over the duration of Nt OFDM symbols due to quite slow variations. Fig. 2(b) represents the pilot insertion structure as proposed in 802.11n for a two transmit antenna system. The orthogonality between training sequences on antennas 1 and 2 is obtained by using an orthogonal matrix. The estimated channel Hij,LS can be calculated for all the M (modulated subcarriers) subcarriers due to the use of full pilots OFDM symbol. Unlike to WLAN sytems, in 3GPP system the channel (SCME typical to urban macro channel model [11]) vary too much in time due to the higher mobility. The orthogonality between training sequences in 3GPP/LTE standard is based on the transmission on each subcarrier of pilot symbols on one antenna and null symbols on the other antennas simultaneously as depicted in Fig.2(a). M Therefore LS estimates can be only calculated for N t subcarriers and interpolation has to be performed to obtain an estimation of all the subcarriers.

Fig. 4.

Comparison between DCT and DFT principle

The channel impulse response in the transform domain is then given by: M hDCT ij,LS,n = Vn

M −1  k=0

Hij,LS,k .cos

π(2k + 1)n 2M

(5)

where VnM is the coefficient of DCT depending on n.   1/M n = 0 (6) VnM = 2/M n = 0 From the DCT calculation and the multi-paths channel characteristics, the impulse response given by (5) is concentrated in the 2L first samples of hDCT n,LS . DCT method can in fact be considered as the conventional DFT method with twice more points as shown in Fig.4. In fact for a given sequence of M points, DCT conceptually extends the original M points sequence to 2M point sequence by a mirror extension of the M point sequence [13]. As a result, compared to DFT, the waveform will be smoother and more continuous in the boundary between

consecutive periods (see Fig.4). It is important to note that the level of impulse response at the order higher than 2L is not null, but it can be considered as negligible; this constitutes the great interest of the DCT. The channel response in the transform domain can be expressed by:  DCT hij,LS,n 0 ≤ n ≤ 2L − 1 DCT (7) hij,n = 0 2L ≤ n ≤ M − 1 The frequency channel response is then given by: DCT Hij,k

=

M −1 

VnM .hDCT ij,n .cos

n=0

(8)

(12)

V. N EW DCT BASED CHANNEL ESTIMATION The problem of the smoothing using DCT is that a DCT/IDCT of size M is used. Therefore the complexity and latency of this operation may cause problems. In order to reduce complexity, the whole DCT window will be divided into R blocks by using the following lemma.

1

1

1

(N−M)/3 M−N/2

N/4

(N−M)/3

N−M

N/2

(N−M)/3

M

M

M

2L

Null carriers

M

M/N t

M

Fig. 5.

DCT interpolation.

The LS estimates obtained on pilots subcarriers given by (4) is first converted into the transform domain by a DCT of length M/Nt , M Nt

= Vn

M Nt

−1

 p=0

where 0 ≤ n ≤

π(2k + 1)n 2M

New DCT approach whit R = 2 and R = 3.

A. Lemma

M/N t

hDCT ij,ip,LS,n

VnM .hDCT ij,n .cos

n=0

Fig. 6. 2L

=

M −1 

N

on the other hand, DCT can be used as an accurate interpolation method in the frequency domain when the orthogonality between training sequences is based on the transmission of pilot symbols on one antenna and null symbols on the other antennas [8] as seen in 3GPP/LTE. In that case, we have a scattered pilot symbol where the number (M/Nt ) of pilot tone subcarriers starting from i-th (antenna index) subcarrier is deployed at every Nt subcarriers.

DCT Hij,k

Null carriers

•

π(2n + 1)k 2M

obtained by the following equation after performing M points IDCT to (10).

Hij,ip,LS,k .cos

π(2p + 1)n M 2N t

If DCT windows are applied on N/R adjacent subcarriers among M in the frequency domain, useful energy of the corresponding channel impulse response will be concentrated on the first 2CP/R samples in the transform domain. In the rest of this subsection, we will prove the previous lemma. In the receiver side, the number of M points LS estimated channel response can be divided into two parts: Hij,LS = Hij + Zij

where Hij and Zij are the transfer function of the true channel and the noise component between the transmit antenna i and the received antenna j respectively. From (2) and (13), we can express Hij,LS as,

(9)

M Nt

As the impulse response representing the channel are concentrated on the 2L first elements, it is possible to apply zeropadding from 2L to M − 1 in (9) which can  DCT hij,ip,LS,n 0 ≤ n ≤ 2L − 1 = (10) hDCT ij,n 0 2L ≤ n ≤ M − 1 It is obvious from (9) that the number of available intervals between two scattered pilot subcarriers must satisfy the following condition: M (11) Nt ≤ 2L Fig.5 illustrates the interpolation process using DCT. The frequency channel response over the whole bandwidth can be

(13)

Hij,LS,k =

L−1 

hij,l e−j

2πτij,l k N

M

e−jπτij,l (1− N ) + Zij,k (14)

l=0

where 0 ≤ k ≤ M . The IDFT of the N/R first sample of Hij,LS can be calculated as, hij,IDF T,n = zij,n +  L−1 N 2πk R −jπτij,l (1− M R −1 −j N (τij,l −Rn) N ) l=0 hij,l e k=0 e N (15) . where 0 ≤ n ≤ N R N 2πk R −1 −j N (τij,l −Rn) The last term of (15) , k=0 e can be expressed as:  N τij,l = nR R −j 2π (τij,l −nR) (16) R 1−e τij,l = nR −n) −j 2π (τ 1−e

N

ij,l

L From (16), R channel taps are obviously within the first CP R samples of the channel impulse response on the transform domain but inter taps interference (ITI) is present in the others samples. By using a DCT instead of an IDFT, we will retrieve 2L R channel taps within the first 2CP R samples and less ITI in the others, due to the DCT concept illustrated in Fig. 4 and its capacity to concentrate the useful energy in the first samples. This last remark can be generalized for every N R adjacent subcarrier and every R ∈ N (see Fig.6).

B. Principle of the proposed scheme The principle of this new approach is to divide the whole DCT window into R small blocks. However the concatenation of the R overlapping blocks cannot exceed N as depicted in Fig.6. A DCT of size N/R is applied to each block. However, the greater R, the more important the ITI and thus the larger the ”border effect” on the edge of each block. For this reason, to recover the channel coefficients, we average the values in the overlapping windows between the different blocks except some subcarriers to avoid the ”border effect” (see Fig.7). 1

Average

M− N /2

1 0 0 1 0 1 0 1 0 1 0 1 0 1

N /2

M

Fig. 7.

recovery of the channel coefficients.

scheme is evaluated for the IEEE802.11n and 3GPP/LTE systems. A. System Parameters The detailed parameters of the simulation are shown in Table II. TABLE II S IMULATION PARAMETERS system Channel Model N &M CP Nt & Nr Bandwidth & CF Modulation FEC Coding Rate MIMO scheme

802.11n TGn [10] 64 ; 52 16 2;2 20MHz ; 2.4GHz QPSK conv code (7,133,171) 1/2 SDM

3GPP/LTE SCME [11] 1024 ; 600 72 4;2 15.36MHz ; 2GHz 16QAM turbo code (UMTS) 1/3 double-Alamouti

B. Simulations Results Fig.8 shows mean square error (MSE) on different subcarriers for the proposed DCT based channel estimation with R = 2; 4 and the conventional one in the 3GGP/LTE system. Firstly the ”border effect” phenomenon is still present because as described in section (IV) the level of impulse response in the transform domain at the order higher than 2L is not null. However DCT based channel estimation reduces significantly this ”border effect” in comparison to the DFT one [8] [12] [9]. On the top of that, the novel DCT method allows to obtain better MSE on different subcarriers even at the edges of the spectrum compared to the conventional one. The gain (≈ 2.3dB) between the two DCT methods calculated in subsection V-C can be observed in Fig. 9 which shows MSE versus Eb /N0 for 3GPP/LTE system. 0.05

C. Performance and complexity study

•

on the one hand, the noise power is averaged on N samples instead of M in this novel approach. Thereby N )) in comparison to the it presents a gain (10log10 ( M classical DCT based channel estimation. For instance, the gain is 0.9dB in 802.11n (N = 64 M = 52) and 2.31dB in 3GPP/LTE (N = 1024 M = 600). on the other hand, if we choose R as a power of 2, the size of the new DCT blocks will be a power of 2. The complexity will be significantly reduced by using some fast DCT algorithms [14] (see Table I).

0.04 0.035 0.03 MSE

•

DCT/IDCT DCT/IDCT R=2 DCT/IDCT R= 4

0.045

0.025 0.02 0.015 0.01 0.005 0

212

300

400

512 600 Subcarrier index

700

813

TABLE I N UMBER OF OPERATIONS Algorithm DCT New DCT

Multiplications M2 N ( 2R )log2 ( N )×R R

Additions M (M − 1) (( 3N )log2 ( N )− N + 1) × R 2R R R

VI. S IMULATION RESULTS As described early, by using the novel DCT approach, performance is improved and complexity reduced in comparison whit the conventional one. The performance of this proposed

Fig. 8.

MSE per subcarriers for 3GPP-LTE: Eb /N0 = 10dB .

Moreover, Fig.10 and Fig.11 represent the performance Eb for perfect, least square, results in terms of BER versus N 0 classical DCT and proposed DCT channel estimation with R = 2, 4, 8, 16 in IEEE802.11n and 3GPP/LTE systems respectively. For all R, the performance of the proposed scheme is improved (1dB for 3GPP/LTE and 0.4dB for IEEE802.11n) compared to classical DCT and the complexity is dramatically reduced (see Table III and IV).

In 3GPP/LTE we can note that the classical DCT based channel estimation can’t significantly improve the performance compared to the LS one. This is due to the scattering pilots symbol, the noise power is averaged in M/Nt (600/4=150) whereas the useful energy is concentrated in the first 2CP = 72 ∗ 2 = 144 samples. In other words the noise component isn’t significantly reduced. −4 DCT/IDCT DCT/IDCT R=2 DCT/IDCT R= 4

−6

−8

MSE

−10

2.3 dB

Algorithm DCT New DCT R=2 New DCT R=4 New DCT R=8 New DCT R=16

−14

−16

−18

−20 0

2

4

6

8

10

Fig. 9.

for 3GPP-LTE.

Algorithm DCT New DCT R=2 New DCT R=4 New DCT R=8

0

10

LS DCT/IDCT DCT/IDCT R=2 DCT/IDCT R=4 DCT/IDCT R=8 PERFECT

−1

BER

10

−2

−3

10

−4

4

6

8

10

12

14

16

SNR

Fig. 10.

BER versus

Eb N0

for IEEE802.11n.

0

10

LS DCT/IDCT DCT/IDCT R=2 DCT/IDCT R=4 DCT/IDCT R=8 DCT/IDCT R=16 PERFECT

−1

BER

10

−2

10

−3

10

−4

2

3

4

5

6

7

8

9

10

SNR

Fig. 11.

BER versus

Eb N0

for 3GPP/LTE.

Multiplications 2704 160 128 96

Additions 2652 414 326 224

R EFERENCES

10

2

Additions 359400 12802 11268 9736 8208

TABLE IV N UMBER OF OPERATIONS FOR 802.11 N

Eb N0

MSE versus

Multiplications 360000 4608 4096 3584 3072

12

SNR

10

In this paper, an improved DCT based channel estimation with very low complexity is proposed and evaluated in two MIMO-OFDM systems (IEEE802.11n and 3GPP/LTE). The whole DCT window is divided into R small overlapping blocks where the length of these blocks is a power of 2. The performance is improved (0.4dB on IEEE802.11n and 1dB on 3GPP/LTE) because the noise component is averaged on a larger number of subcarriers. As the complexity of the proposed scheme is largely reduced, the application of this new channel estimation scheme to any MIMO-OFDM system appears attractive. TABLE III N UMBER OF OPERATIONS FOR 3GPP/LTE

−12

10

VII. C ONCLUSION

11

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