Algorithms and Applications of Channel Impulse ... - Semantic Scholar

Algorithms and Applications of Channel Impulse Response Length Estimation for MIMO-OFDM Marco Krondorf, Ting-Jung Liang, Martin Goblirsch and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, D-01062 Dresden, Germany {krondorf,liang,goblir,fettweis}@ifn.et.tu-dresden.de, http://www.vodafone-chair.de/MNS Abstract— The performance of the widely applied timedomain channel estimation for SISO- and MIMO-OFDM systems strongly depends on the preciseness of information regarding maximum channel impulse response (CIR) length. In practice the required CIR length is usually approximated by the length of the cyclic prefix which is an upper bound for most of the actual instantaneous CIR length. In this article1 , we introduce an appropriate channel length estimation method - named as FCLI (Frequency Domain Channel Length Indicator) which estimates the CIR length minimizing the estimated mean square error of an instantaneous channel estimate. The simulation results present that the FCLI outperforms the approximation of cyclic prefix length. In addition, we investigate two applications of FCLI on 1) cyclic prefix adaptation OFDM systems in which we minimize the overhead of the cyclic prefix by the estimated CIR length and 2) cyclic prefix free OFDM systems in which we directly re-construct the structure of cyclic repetition with reduced error. Compared to the traditional channel estimation with the approximation of cyclic prefix length, the overall system throughput of either cyclic prefix adaptation or cyclic prefix free OFDM transmission scheme is significantly enhanced regardless of perfect synchronization or practical distribution of time offsets.

I. I NTRODUCTION In present packet-based OFDM WLAN, such as IEEE802.11a/g, the channel estimation functions can be accomplished by a preamble [1] and a time domain channel estimation [13] 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 [12]. The time domain channel estimator with a-priori knowledge of CIR length can provide sufficiently good channel estimates for MIMO-OFDM systems, but in practice the required CIR length is usually approximated by the length of the cyclic prefix and the time domain channel estimation performance degrades significantly, if the approximated CIR length is much longer than the real CIR length. The reasons motivate us to find an appropriate method to estimate the CIR length. In the literature, there are already two important approaches for CIR length estimation: NCLE (Noise Variance and CIR Length Estimation) algorithm [14] and a method presented by Gong and Letaief [4] based on SVD (Singular Value 1 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

Decomposition) of the channel auto-correlation matrix. Both techniques average the estimated CIR over a long period of time and perform well in continuous streaming OFDM systems such as DVB-T. However, averaging CIR over multiple OFDM symbols is not always applicable in practical SISOand MIMO-OFDM systems, such as IEEE 802.11a/g, IEEE 802.11n or the 1 Gbit/s WIGWAM [6] systems because of the bursty nature of packet switched traffic. Additionally, the spectrum mask which defines a set of null-subcarriers to reduce the leakage at the spectral edges is not considered in their algorithms. In order to avoid the problems above, we introduced the FCLI-Frequency Domain Channel Length Indicator algorithm firstly in [11], which estimates the mean square error (MSE) of the channel estimate under different hypotheses of instantaneous channel lengths and the hypothesis leading to minimum MSE is chosen as an appropriate CIR length. It should be noted that the FCLI algorithm is based on only one snapshot of channel estimate in the acquisition phase and the estimated MSE is calculated by channel information inside of the used subset of subcarriers defined by a spectrum mask. In addition, the cyclic prefix (CP) of OFDM symbols contains a cyclic extension of the transmitted data and prevents from inter symbol and inter carrier interference (ISI & ICI). This technique allows a simple FFT-based frequency equalization and gives rise to low cost terminals, which are highly suitable for consumer electronics and mobile applications, but - as expected - the drawback of inserting cyclic prefix before OFDM symbols is the loss of spectral efficiency. Based on the findings of FCLI, it is possible to enhance the overall system throughput by either truncating the CP to a minimal necessary amount or by applying cyclic prefix free OFDM systems in which we directly re-construct the structure of cyclic repetition with reduced error. In this article, we investigate the system throughput of both throughput enhanced methods above using FCLI or approximation of cyclic prefix length for time-domain channel estimation in OFDM receivers taking into account residual time offsets. After the motivation of our interest in channel impulse response length estimation, the rest of this article is organized as follows: In Sec.II a discrete baseband MIMO-OFDM system model is presented. Sec.III summarizes the preamble-based MIMO-OFDM channel estimation algorithms and analyzes the mean square error of channel estimates with given hypothesis

of CIR lengths followed by the derivation of FCLI algorithm in Sec.IV. In Sec.V and Sec.VI, we depict how FCLI is applied for cyclic prefix adaptation and cyclic prefix free OFDM systems followed by throughput comparison in Sec. VII and conclusions in Sec. VIII. II. MIMO-OFDM S YSTEM M ODEL Generally, we consider MIMO-OFDM systems shown in Fig.1 which uses NT transmit and NR receive antennas (respective indices t and r) and NF F T points FFT/IFFT, where NC subcarriers are used for data transmission and the other NF F T −NC subcarriers, in addition to DC, forming a spectrum mask at the spectral edges to reduce outer band radiation.

OFDM symbol vector Yi,r of size [NC × 1] can be written as follows NT X Yi,r = Xi,t Hr,t + Ni,r (2) t=1

where Xi,t denotes the i-th [NC × NC ] diagonal matrix containing the frequency domain data symbols of transmit antenna t. The Ni,r denotes the i-th AWGN vector at receive antenna r. The Hr,t represents the [NC ×1] vector of frequency domain channel coefficients between antennas r and t and the channel coefficient at subcarrier position n is given by Hr,t (n) =

L−1 X

hr,t (k)e

−j2π N nk

FF T

(3)

k=0

OFDM

1

OFDM

2

1

OFDM -1

2

OFDM -1

H

OFDM

NR

NT

Fig. 1.

Demodulation

. . .

Antenna Mapping

Decoding

. . .

Coding Modulation

Antenna Demapping

OFDM -1

MIMO OFDM System Model

The OFDM blocks are transmitted seamlessly and continuously in time as presented in Fig. 2. One OFDM block is composed of NB OFDM data symbols and one preamble, which contains one cyclic prefix and NT identical Long Training Symbols (LTS; of size one OFDM symbols without CP) designed for channel estimation. Time

OFDM Preamble Symbol 1 Fig. 2.

...

OFDM Preamble Symbol N 4B

...

Block structure of OFDM Transmission

The i-th time domain transmitted OFDM symbol xi,t from the t-th transmit antenna is the inverse FFT of frequencydomain signal Xi,t pre-pended with length NCP cyclic prefix. The i-th time domain OFDM symbol at receive antenna r can be formulated as   NT L−1 X X  hr,t (j)xi,t (k − j) + ni,r (k) (1) yi,r (k) = t=1

j=0

with time index k and L represents the length of the corresponding channel impulse response2 hr,t . The length of cyclic prefix NCP ≥ L is chosen to prevent inter-symbol-interference and preserve subcarrier orthogonality. For further details on OFDM please see [8]. When perfect time and frequency synchronisation is assumed at receive antenna r, the i-th received frequency domain 2 Notation:

Upper (lower) letters will be generally used for frequencydomain (time-domain) signals; boldface letters represent matrices and column vectors; letters with both boldface and underline represent block matrices or vectors in MIMO.

During one OFDM block, the channel impulse responses between all transmit and receive antenna pairs are assumed to be static, therefore we omit symbol index i in channel coefficients Hr,t (n). III. OVERVIEW OF T IME D OMAIN MIMO-OFDM C HANNEL E STIMATION The time domain channel estimation provides sufficiently accurate channel estimates with a-priori knowledge of CIR length [13] for SISO-OFDM and, based on the work in [3] and [12], the same concept can be extended to MIMO-OFDM by sending orthogonal training sequences from different transmit antennas, such as Code Orthogonal (CO) and Frequency Orthogonal (FO) designs of preamble training sequences. In this work, we consider only the least squares (LS) channel estimation using a preamble composed of FO pilots. This is due to the fact, that FO pilot designs can achieve the same estimation performance [11] as CO pilot designs but require a much lower computational complexity for channel estimation. In addition, MIMO-OFDM channel estimation using FO pilots can be decomposed into NT NR SISO channel estimation problems in parallel. Therefore, we omit the antenna indices t and r in the following sections. A. Channel Estimation using FO Sequences The FO design means that different transmit antennas transmit training sequences in different sets of equally distributed NP C = NC /NT subcarriers and, for one specific transmit antenna, no pilot signals are transmitted on the other NC − NP C subcarriers. This characteristics of smaller matrix dimension for one specific transmit antenna can be utilized to reduce the complexity of matrix multiplication. Fig. 3 shows an example of time-domain channel estimation using FO pilots and the estimated frequency-domain channel coefficient in Fig. 3 takes the following vector/matrix form ˆ˙ = 1 X˙ H Y ˙ H 2 σX

(4)

where the dimension reduction from NC to NP C is represented by the dotted notation. Therefore, the missing NC − NP C channel coefficients have to be interpolated. This can be done either by linear / spline-cubic interpolation directly in

Y[ N C ×1]

Select Pilot Carriers

&ˆ H [ N PC ×1]

Y Hˆ k = k Xk

|CIR| IDFT to calculate CIR of Length L

Channel & Noise

Noise

hˆ [ L×1] ˆ H [ N C ×1]

FFT

hˆ [ N FFT ×1]

Zero Padding

( N FFT − L )

N FFT Points

0

Fig. 3. Time domain least squares channel estimation using FO pilot design

frequency domain [7] or by time domain interpolation shown ˆ˙ in Fig. 3, which calculates the corresponding CIR out of H followed by zero padding in interval [L . . . NF F T ] and FFT ˆ L [2] by to obtain the interpolated H ˆ L = FNC ,L h ˆL H

with

ˆ˙ ˆL = F˙ †N ,L H h PC

(5)

where [.]† denotes the pseudo-inverse matrix operation and F˙ NP C ,L represents the truncated Fourier-matrices of dimension [NP C × L] containing the elements   nk ˙ [FNP C ,L ]n,k = exp −j2π (6) NF F T with 0 ≤ k ≤ L − 1 and n ∈ P t . The P t denotes the index set of the NP C pilot carriers of Tx-antenna t. Similarly, matrix FNC ,L of dimension [NC × L] has entries   nk ˙ (7) [FNC ,L ]n,k = exp −j2π NF F T with 0 ≤ k ≤ L − 1 and data-subcarrier index n. Substituting 2 Eq. (4) into Eq. (5) and assuming σX = 1, the interpolated ˆ HL becomes ˆ L = FNC ,L F˙ †N ,L X ˙ HY ˙ H PC

(8)

ˆ L − H)(H ˆ L − H)H ]} M SE(L) = E{tr[(H

(9)

where tr[.] and E{.} denotes the trace and expectation operaˆ L represents the channel estimates using the CIR length tor. H L that can be obtained by Eq.(8). Furthermore, Fig. 4 shows the absolute value of the estiˆ ≥ L, mated CIR vs. time samples. If L is approximated by L ˆ MSE is only affected by AWGN inside of [0 : L−1]. Hence we ˆ = M SEn (L, ˆ σ 2 ) with parameters L ˆ and σ 2 , have M SE(L) N N

Samples

Fig. 4. Estimated CIR consists of two parts, channel+noise inside of CIR length [0...L] and noise only outside of CIR length

2 where M SEn (L, σN ) denotes the MSE part, directly induced 2 ˆ − 1]. by AWGN of variance σN inside of [0 : L ˆ < L, the M SE(L) ˆ Conversely, if L is approximated by L ˆ σ 2 ) induced by noise but also includes not only M SEn (L, N ˆ caused by neglecting an additional error part M SEch (L) ˆ . . . L]. The CIR samples inside the time domain interval [L phenomenon discussed above can be summarized as

ˆ = M SE(L)  ˆ σ 2 ) + M SEch (L) ˆ ,L ˆ