EURASIP Journal on Wireless Communications and Networking
Volume 2009 (2009), Article ID 807549, 10 pages
doi:10.1155/2009/807549
Research Article

Degenerated-Inverse-Matrix-Based Channel Estimation for OFDM Systems

Makoto Yoshida

Fujitsu Laboratories Limited, YRP R&D Center, 5-5, Hikari-no-Oka, Yokosuka, 239-0847, Japan

Received 21 January 2009; Accepted 14 April 2009

Academic Editor: Dmitri Moltchanov

Copyright © 2009 Makoto Yoshida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper addresses time-domain channel estimation for pilot-symbol-aided orthogonal frequency division multiplexing (OFDM) systems. By using a cyclic sinc-function matrix uniquely determined by 𝑁 𝑐 transmitted subcarriers, the performance of our proposed scheme approaches perfect channel state information (CSI), within a maximum of 0.4 dB degradation, regardless of the delay spread of the channel, Doppler frequency, and subcarrier modulation. Furthermore, reducing the matrix size by splitting the dispersive channel impulse response into clusters means that the degenerated inverse matrix estimator (DIME) is feasible for broadband, high-quality OFDM transmission systems. In addition to theoretical analysis on normalized mean squared error (NMSE) performance of DIME, computer simulations over realistic nonsample spaced channels also showed that the DIME is robust for intersymbol interference (ISI) channels and fast time-invariant channels where a minimum mean squared error (MMSE) estimator does not work well.

1. Introduction

Orthogonal frequency division multiplexing (OFDM) is well known as an anti-multipath-fading technique for broadband wireless systems and is used as a standard in digital broadcasting and wireless LAN systems. Increased demand for a better broadband wireless system—a fourth-generation (4G) mobile system—has stemmed from the use of this technique [1].

Coherent OFDM detection requires the channel state information (CSI) to be estimated accurately because signals received over a multipath fading channel have unknown amplitude and phase variations. Known pilot symbols are therefore inserted into the transmitted data stream periodically and channel estimation is performed by interpolating them.

Various pilot-symbol-aided channel estimation schemes have been investigated for OFDM [2–4] and multiinput-multioutput (MIMO) OFDM systems [5, 6]. OFDM with high-order modulation schemes, such as multilevel QAM (M-QAM), requires more accurate channel estimation than does OFDM with PSK modulation because it is more sensitive to noise. An adaptive OFDM technique [7] uses M-QAM and is essential for highly efficient communications. OFDM systems with multiple transmitting antennas, including MIMO-OFDM systems, also need accurate channel estimation. When different signals are transmitted from different transmit antennas simultaneously, the received signal can be considered as the superposition of these signals, which have higher-order signal constellations than does the original signal.

The minimum mean squared error (MMSE) estimator has been proposed as an optimal solution for a pilot-symbol-aided channel estimation scheme [8]. The MMSE estimator, however, requires huge computational resources, and the performance deteriorates significantly for fast time-invariant channels where the convergence algorithm cannot work well within the observation duration.

Our proposed scheme uses a cyclic sinc-function matrix uniquely determined by Nc transmitted subcarriers. Since this sinc-function (“time response of a subcarrier” in a broad sense) is a deterministic and known vector, the inverse matrix (IM) approach can be used for high-precision estimation without supplementary information such as knowledge of the channel statistics and operating SNR, which are required in the MMSE estimator.

Time-domain channel estimators have problems of energy leakage over nonsample spaced channels [5, 8, 9] and high computational complexity. Our proposed scheme solves not only these two problems but also that of residual noise by introducing a degenerated inverse matrix (DIM) and oversampling technique simultaneously.

This paper shows that the degenerated inverse matrix estimator (DIME) can estimate CSI in a fast-fading environment almost perfectly no matter what the subcarrier modulation scheme and delay spread of the channel are. In Section 2 we describe the system model, and in Section 3 we discuss DIME, comparing it with other estimators based on zero-forcing (ZF), ZF with averaging in the frequency domain (ZF-FAV) [1], and MMSE. In Section 4 we compare these techniques in terms of computational complexity and performance, including theoretical analysis on normalized mean squared error (NMSE) performance of DIME through computer simulations under the specifications that we assumed for 4G mobile systems. Our conclusions are given in Section 5.

2. System Description

Figure 1 shows the frame format for OFDM signals. The data symbols are time-multiplexed with the pilot symbols. This time-division-multiplexing (TDM) type of pilot symbol is used in the 4G system proposed by Atarashi et al. [1].

807549.fig.001
Figure 1: OFDM system frame format.

Thus, in this paper, pilot symbols mapped over all subcarriers are interpolated only in the time direction. We assume that 𝑇 𝑆 is the sampling interval and that a guard interval (GI) of time length 𝑇 𝐺 is used to eliminate intersymbol interference (ISI). Thus the OFDM symbol duration is 𝑇 = 𝑁 𝑇 𝑆 + 𝑇 𝐺 , where N is the size of the fast Fourier transform (FFT) used in the system.

Consider the OFDM system shown in Figure 2, where 𝑥 𝑛 are the transmitted symbols, 𝑔 ( 𝑡 ) is the channel impulse response (CIR), 𝑤 ( 𝑡 ) is the additive white Gaussian noise (AWGN), and 𝑦 𝑛 are the received symbols. In this system the transmitted symbols 𝑥 𝑛 ( 0 ≤ 𝑛 ≤ 𝑁 𝑐 − 1 ) assigned to 𝑁 𝑐 subcarriers are fed into an 𝑁 -point ( 𝑁 𝑐 < 𝑁 ) inverse FFT, where 𝑁 - 𝑁 𝑐 subcarriers (virtual carriers) are not used at the edges of the spectrum to avoid aliasing problems at the receiver. Note that in this paper, 𝐅 𝐅 𝐓 𝑁 and 𝐈 𝐅 𝐅 𝐓 𝑁 , respectively, denote an 𝑁 -point F F T and 𝑁 -point inverse F F T given by 𝐅 𝐅 𝐓 𝑁 1 ( 𝐱 ) = √ 𝑁 𝑁 − 1 𝑘 = 0 𝑥 ( 𝑘 ) 𝑒 − 𝑗 2 𝜋 𝑘 𝑛 / 𝑁 , 𝐈 𝐅 𝐅 𝐓 𝑁 1 ( 𝐱 ) = √ 𝑁 𝑁 − 1 𝑛 = 0 𝑥 ( 𝑛 ) 𝑒 𝑗 2 𝜋 𝑘 𝑛 / 𝑁 . ( 1 )

807549.fig.002
Figure 2: Block diagram of OFDM system.

The CIR is expressed by

𝑔 ( 𝑡 ) = 𝐿 − 1 𝑙 = 0 𝛼 𝑙 𝛿 𝑡 − 𝜏 𝑙 , ( 2 ) where L is the total path number and 𝛼 𝑙 and 𝜏 𝑙 are the complex amplitude and time delay of the 𝑙 th path. Thus we can say that the maximum excess delay 𝜏 m a x = 𝜏 𝐿 − 1 . We also assume that the entire CIR lies inside the guard interval, that is, 0 ≤ 𝜏 m a x ≤ 𝑇 𝐺 . Therefore, the cyclic convolution of the received sequence over the FFT window is preserved. The N-dimensional received symbol vector

𝑦 𝐲 = 0 ⋯ 𝑦 𝑁 𝑐 / 2 − 1 𝑦 𝑁 𝑐 / 2 ⋯ 𝑦 𝑁 − 𝑁 𝑐 / 2 − 1 𝑦 𝑁 − 𝑁 𝑐 / 2 ⋯ 𝑦 𝑁 − 1 𝑇 ( 3 )

is given as

𝐲 = 𝐅 𝐅 𝐓 𝑁 𝐈 𝐅 𝐅 𝐓 𝑁 𝐠 ( 𝐱 ) ⊗ √ 𝑁 + 𝐰 , ( 4 ) where ⊗ denotes cyclic convolution. The N -dimensional transmitted symbol vector with N -Nc zeros, the CIR vector after sampling of 𝑔 ( 𝑡 ) , and the AWGN vector after sampling of 𝑤 ( 𝑡 ) are, respectively, given by

𝑥 𝐱 = 0 ⋯ 𝑥 𝑁 𝑐 / 2 − 1 𝑥 0 ⋯ 0 𝑁 − 𝑁 𝑐 / 2 ⋯ 𝑥 𝑁 − 1 𝑇 , 𝑔 𝐠 = 0 𝑔 1 ⋯ 𝑔 𝑁 − 1 𝑇 , 𝑤 𝐰 = 0 𝑤 1 ⋯ 𝑤 𝑁 − 1 𝑇 ( 5 )

(the superscript [ ⋅ ] 𝑇 indicates vector transpose). Note that 𝑦 𝑛 ( 𝑁 𝑐 / 2 ≤ 𝑛 ≤ 𝑁 − 𝑁 𝑐 / 2 − 1 ) are discarded at the receiver as virtual subcarriers.

The k th element of vector 𝐠 can be expressed by

𝑔 𝑘 = 1 √ 𝑁 𝐿 − 1 𝑙 = 0 𝛼 𝑙 𝑒 − 𝑗 ( 𝜋 / 𝑁 ) 𝑘 + ( 𝑁 − 1 ) 𝜏 𝑙 / 𝑇 𝑆 ⋅ s i n 𝜋 𝜏 𝑙 / 𝑇 𝑆 ( 𝜏 s i n 𝜋 / 𝑁 ) 𝑙 / 𝑇 𝑆 − 𝑘 . ( 6 )

Equation (6) indicates that if 𝜏 𝑙 / 𝑇 𝑆 is not an integer, the energy will leak to all taps 𝑔 𝑘 . This is a serious problem for the time-domain channel estimator. Furthermore, the time response (sinc function in our system) of an OFDM signal with virtual carriers leaks to all taps, superposed on the leakage of (6).

We can rewrite the right-hand side of (4) in matrix notation [8] as follows:

𝐲 = 𝐗 𝐅 𝑁 𝐠 + 𝐰 , ( 7 )

where 𝐗 is the N-dimensional diagonal matrix

𝑥 𝐗 = d i a g 0 ⋯ 𝑥 𝑁 𝑐 / 2 − 1 0 ⋯ 0 𝑥 𝑁 − 𝑁 𝑐 / 2 ⋯ 𝑥 𝑁 − 1 , ( 8 )

𝐅 𝑁 is an N × N-dimensional FFT matrix with entries

𝐅 𝑁 𝑘 , 𝑛 = 1 √ 𝑁 𝑒 − 𝑗 2 𝜋 𝑘 𝑛 / 𝑁 0 ≤ 𝑘 , 𝑛 ≤ 𝑁 − 1 , ( 9 )

and 𝐰 = 𝐅 𝑁 𝐰 .

3. Channel Estimation

We describe several estimators based on the system model described in the previous section. The goal is to derive estimates of the channel transfer function (CTF) 𝐡 , which is the Fourier transform of the CIR. That is, 𝐡 = 𝐅 𝑁 𝐠 .

Note that in pilot-symbol-aided channel estimation, since a pilot symbol is used only as a known transmitted symbol, the receiver can easily estimate the CTF in the frequency domain.

3.1. ZF and ZF-FAV Estimators

The ZF estimator, or least square (LS) estimator, uses the pilot symbol sequence to generate the estimated CTF

̂ 𝐡 Z F = 𝐙 𝐲 = 𝐙 𝐗 𝐅 𝑁 𝐠 + 𝐰 = 𝐙 𝐗 𝐅 𝑁 𝐠 + 𝐙 𝐰 , ( 1 0 ) where 𝐙 is the N-dimensional diagonal matrix

𝐙 = d i a g 1 / 𝑥 0 ⋯ 1 / 𝑥 𝑁 𝑐 / 2 − 1 0 ⋯ 0 1 / 𝑥 𝑁 − 𝑁 𝑐 / 2 ⋯ 1 / 𝑥 𝑁 − 1 , , 𝐙 𝐗 = d i a g 1 ⋯ 1 0 ⋯ 0 1 ⋯ 1 ( 1 1 )

and the second term is the residual noise term of ZF.

The ZF-FAV estimator [1] uses the CTF averaged over the adjacent 2 𝐷 + 1 subcarriers in the frequency domain for noise suppression. This averaging process is done after getting the CTF for each subcarrier. The estimated CTF at the 𝑛 th subcarrier ( 0 ≤ 𝑛 ≤ 𝑁 𝑐 / 2 − 1 , 𝑁 − 𝑁 𝑐 / 2 ≤ 𝑛 ≤ 𝑁 − 1 ) is given by

ℎ 𝑍 𝐹 − 𝐹 𝐴 𝑉 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ( 𝑛 ) = 2 𝐷 + 1 ( 𝑛 + 𝐷 ) m o d 𝑁 𝑑 = ( 𝑛 − 𝐷 + 𝑁 ) m o d 𝑁 ℎ 𝑍 𝐹 𝑁 ( 𝑑 ) 0 ≤ 𝑛 < 𝑐 2 𝑁 − 𝐷 , 𝑁 − 𝑐 2 1 + 𝐷 ≤ 𝑛 < 𝑁 , 𝑁 𝑐 / 2 − 𝑛 − 𝐷 𝑁 𝑐 / 2 − 1 𝑑 = ( 𝑛 − 𝐷 + 𝑁 ) m o d 𝑁 ℎ 𝑍 𝐹 𝑁 ( 𝑑 ) 𝑐 2 𝑁 − 𝐷 ≤ 𝑛 < 𝑐 2 , 1 𝑛 − 𝑁 − 𝑁 𝑐 / 2 + 𝐷 + 1 ( 𝑛 + 𝐷 ) m o d 𝑁 𝑑 = 𝑁 − 𝑁 𝑐 / 2 ℎ 𝑍 𝐹 𝑁 ( 𝑑 ) 𝑁 − 𝑐 2 𝑁 ≤ 𝑛 < 𝑁 − 𝑐 2 + 𝐷 . ( 1 2 )

Since this algorithm employs the property of a coherent bandwidth, its performance degrades when the channel has a large delay spread.

3.2. MMSE Estimator

The MMSE estimator is proposed as an optimum solution for pilot-symbol-aided channel estimation. If the vector 𝐠 is uncorrelated with the vector 𝐰 , the estimated CTF is given by [8]

̂ 𝐡 M M S E = 𝐅 𝑁 𝐑 𝐠 𝐲 𝐑 − 1 𝐲 𝐲 𝐲 , ( 1 3 )

where

𝐑 𝐠 𝐲 = E 𝐠 𝐲 H = 𝐑 𝐠 𝐠 𝐅 H 𝑁 𝐗 H , 𝐑 𝐲 𝐲 = E 𝐲 𝐲 H = 𝐗 𝐅 𝑁 𝐑 𝐠 𝐠 𝐅 H 𝑁 𝐗 H + 𝜎 2 𝐈 𝑁 ( 1 4 )

( [ ⋅ ] H and 𝐈 𝑁 indicate a Hermitian matrix and an N × N-dimensional identity matrix, resp.).

Since (13) requires the autocovariance matrix of 𝐠 , 𝐑 𝐠 𝐠 = E ⟨ 𝐠 𝐠 H ⟩ , and the noise variance, 𝜎 2 = E ⟨ | 𝑤 𝑖 | 2 ⟩ , this algorithm is not suitable for a fast-fading environment where these two quantities cannot be converged.

To solve the problem of high computational complexity, a modification of the MMSE has been proposed [8]. The modified MMSE reduces the size of 𝐑 𝐠 𝐠 by considering a given area, for example, the number of taps in a guard interval.

3.3. DIM Estimator (DIME)

The proposed estimator, DIME, which is based on time-domain signal processing, solves not only the problems of energy leakage and computational complexity but also that of residual noise, by introducing a degenerated inverse matrix and oversampling technique.

The zero-insertion CTF for 𝑀 -fold oversampling is first formed by inserting MN -Nc zeros in the middle frequency indices:

̂ ̆ 𝐡 Z F = ⎧ ⎪ ⎨ ⎪ ⎩ ℎ Z F ( 𝑛 ) 0 ≤ 𝑛 < 𝑁 𝑐 / 2 , 0 𝑁 𝑐 / 2 ≤ 𝑛 < 𝑀 𝑁 − 𝑁 𝑐 ℎ / 2 , Z F ( 𝑛 − ( 𝑀 − 1 ) 𝑁 ) 𝑀 𝑁 − 𝑁 𝑐 / 2 ≤ 𝑛 < 𝑀 𝑁 , ( 1 5 ) where the superscript ̆ 𝑎 denotes oversampling.

Then the oversampled CIR is solved after calculating the IFFT of ̂ ̆ 𝐡 Z F :

̂ ̆ 𝐠 Z F = 𝐅 H 𝑀 𝑁 ̂ ̆ 𝐡 Z F ̆ ̆ = 𝐒 𝐠 + 𝐰 , ( 1 6 ) where ̆ ‌ 𝐰 = 𝐅 H 𝑀 𝑁 ̆ 𝐰 , and the time-response matrix 𝐒 is given by

𝐒 = 𝐅 H 𝑀 𝑁 ̆ 𝐙 ̆ 𝐗 𝐅 𝑀 𝑁 . ( 1 7 )

Using the inverse matrix, 𝐒 − 1 , which is a definitive and known matrix, the IM estimator generates

̂ ̆ 𝐠 𝐼 𝑀 = 𝐒 − 1 ̂ ̆ 𝐠 Z F = 𝐒 − 1 𝐒 ̆ ̆ 𝐠 + 𝐰 = ̆ 𝐠 + 𝐒 − 1 ̆ ‌ 𝐰 . ( 1 8 )

Next, we degenerate 𝐒 to reduce the size of the matrix and suppress residual noise.

A property of the time-response matrix 𝐒 formed by the cyclic sinc-function is that significant energy is concentrated in the diagonal elements. Since the correlation between ̆ 𝐠 and ̆ g Z F is high, we can form the degenerated inverse matrix 𝐒 ′ − 1 as follows. First, in (16), we discard the 𝑘 th sample ̂ ̆ g Z F ( 𝑘 ) with lower energy than a given threshold, the corresponding ̆ g ( 𝑘 ) , and the corresponding noise term ̆ ‌ w ( 𝑘 ) . Next, the matrix size of 𝐒 is reduced by generating a submatrix 𝐒 ′ in which both the row and column corresponding to the discarded samples are eliminated. If the shape of the CIR is cluster-like, multiple degenerated inverse matrices can be generated on a cluster-by-cluster basis. This significantly reduces the computational complexity (evaluated quantitatively in Section 4).

The degenerated ̂ ̆ 𝐠 Z F is given as

̂ ̆ 𝐠 Z F = 𝐒 ̆ ̆ 𝐠 ′ + 𝐰 , ( 1 9 ) and we get

̂ ̆ 𝐠 D I M = 𝐒 − 1 ̂ ̆ 𝐠 Z F = ̂ ̆ 𝐠 + ̆ 𝐗 𝐅 ′ 𝑀 𝑁 − 1 𝐅 𝑀 𝑁 ̆ 𝐰 . ( 2 0 )

Then the MN -dimensional (full size) vector ̆ 𝐠 D I M is reformed by zero-padding all the discarded samples. Zero-padding also has the function of suppressing residual noise.

By calculating the FFT of ̂ ̆ 𝐠 D I M , the DIME generates

̂ ̆ 𝐡 D I M = 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 D I M , ( 2 1 ) and then the 𝑇 𝑆 -sampled CTF is given as

̂ 𝐡 D I M = ⎧ ⎪ ⎨ ⎪ ⎩ ̆ ℎ D I M ( 𝑛 ) 0 ≤ 𝑛 < 𝑁 𝑐 ̆ ℎ / 2 , D I M ( 𝑛 + ( 𝑀 − 1 ) 𝑁 ) 𝑁 − 𝑁 𝑐 / 2 ≤ 𝑛 < 𝑁 . ( 2 2 )

4. Performance Evaluation

4.1. System Parameters

The main simulation parameters assumed for a broadband mobile communications system are listed in Table 1. We assumed that both symbol-time and sample-time were synchronized perfectly.

tab1
Table 1: Simulation parameters.

We then examined the BER performance using a general 𝐿 -ray Rayleigh fading channel with exponential decay paths (Figure 3). The average received power of the l th path decreased by ( 𝑙 × Δ 𝑃 ) dB relative to the first path, where 𝑙 = 0 , 1 , … , 𝐿 − 1 . We set Δ 𝑃 = 1 and 𝐿 = 1 2 . The path interval was given as Δ 𝜏 = 𝜏 𝑖 − 𝜏 𝑖 − 1 . In this simulation, we defined a tapped-delay-line channel model for a nonsample spaced channel with a tap interval of 𝑇 𝑆 / 4 and a first tap delay of 𝜏 0 = 𝑇 𝑆 / 4 . This channel model was used to evaluate the worst case for the DIME with two-fold oversampling (M=2).

807549.fig.003
Figure 3: Channel model with exponential decay paths.
4.2. Complexity Analysis

We first analyze the computational complexity of the channel estimators described in Section 3. We assume that the complexity is the sum of the number of complex multiplications, complex divisions, and complex additions. Table 2 shows the computational complexity for five algorithms.

tab2
Table 2: Computational complexity.

The main complexity of the ZF estimator is 1024-point FFT operation. The ZF-FAV estimator requires 3-subcarrier averaging in addition to that of the ZF estimator. Since these estimators do not require time-domain processing, they have very low complexity.

The main complexity of the MMSE-based estimator is the inverse matrix operation. In this paper, we assume that the autocovariance matrix 𝐑 𝐠 𝐠 is calculated on a pilot–by-pilot basis to follow fast time-invariant channels and the noise variance is assumed to be known. The size of the degenerated autocovariance matrix 𝐑 𝐠 𝐠 for the modified MMSE estimator is set as 2 2 0 × 2 2 0 ( ≈ 𝑇 𝐺 × 𝑇 𝐺 ). Since our parameters require a very large matrix for the MMSE, this level of complexity is unfeasible and we discarded the full-MMSE from the performance comparison.

For the inverse matrix operation, the DIME with 𝑀 =2 needs to calculate two complex F F T operations: 2 𝑁 -point I F F T and F F T . Although the F F T is required to be large, the resulting complexity is very small because the DIM technique is used. Using the threshold for DIM, the 2 0 4 8 × 2 0 4 8 full matrix was degenerated to a 4 3 × 4 3 -dimensional matrix. The complexity of the DIME is 8.6 times that of the ZF and 1/20000 that of MMSE estimators.

4.3. MSE Performance

We then examined the NMSE [10] performance as a function of 𝐸 𝑏 / 𝑁 0 in two different channel models that included paths within ( 𝜎 / 𝑇 = 0 . 0 4 3 ) and beyond GI ( 𝜎 / 𝑇 = 0 . 1 2 3 ).

NMSE performances with 𝜎 / 𝑇 = 0 . 0 4 3 ( 𝜎 = 0 . 6 7 𝜇 𝑠 ) are shown in Figure 4. The normalized rms delay spreads of 𝜎 / 𝑇 = 0 . 0 4 3 correspond to 𝜏 m a x ≈ 𝑇 𝐺 . The maximum Doppler frequency 𝑓 𝑑 = 4 8 0  Hz, corresponding to the normalized Doppler frequency of 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 , was also examined. In a within-GI case without the occurrence of ISI, we can get the analytical N M S E in a relatively easy way. The analytical N M S E s for D I M and Z F estimator are thus plotted simultaneously.

807549.fig.004
Figure 4: NMSE performance with 𝜎 / 𝑇 = 0 . 0 4 3 and 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 .

Analytical N M S E for D I M estimator, N M S E D I M , is given as

𝑁 𝑀 𝑆 𝐸 D I M = ∑ 𝑁 c / 2 − 1 𝑛 = 0 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 + ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 + 𝜆 𝜎 𝑀 𝑁 2 , ( 2 3 ) where [ ⋅ ] 𝑛 denotes the n th element of a vector, 𝜆 is the number of the samples greater than a given threshold (see Appendix A). In (23), ̂ ̆ 𝐠 and 𝜆 were previously given by going through simple simulation.

Analytical NMSE for ZF estimator, NMSEZF, is also given as

N M S E Z F = 𝜎 2 , ( 2 4 ) (see Appendix B).The noise suppression gain by the first order linear interpolation, − 1 . 9  dB, was also considered (see Appendix C).

We can see that the analytical results are in excellent agreement with the computational simulation results. The slight differences in ZF at higher 𝐸 𝑏 / 𝑁 0 and in DIME at lower 𝐸 𝑏 / 𝑁 0 are due to the channel estimation error ( = 𝜀 ), which is not considered in the assumption.

For the former case, since the NMSE performance of ZF depends only on 𝐸 𝑏 / 𝑁 0 ( = 𝜎 2 ), we can see the effect of channel estimation error at higher 𝐸 𝑏 / 𝑁 0 , where 𝜎 2 < 𝜀 . For the latter case, since 𝜆 is independent of 𝐸 𝑏 / 𝑁 0 , we can see the effect of erroneous selection of effective samples at lower 𝐸 𝑏 / 𝑁 0 .

Next, NMSE performances with 𝜎 / 𝑇 = 0 . 1 2 3 ( 𝜎 = 1 . 9 2 𝜇 𝑠 ) , 𝜏 m a x ≈ 3 𝑇 𝐺 are shown in Figure 5. The DIME performs well at the target 𝐸 𝑏 / 𝑁 0 lower than 10 dB (Figures 6 and 7), regardless of the channel model since it has a powerful noise suppression capability. The modified MMSE estimator works well only in both a high 𝐸 𝑏 / 𝑁 0 and within-GI environment that can generate 𝐑 𝐠 𝐠 accurately on a pilot-by-pilot basis. Based on computational complexity and performance, three estimators other than the modified MMSE estimator are examined in the following sections.

807549.fig.005
Figure 5: NMSE performance with 𝜎 / 𝑇 = 0 . 1 2 3 and fdT=0.0075.
4.4. Performance in Modulation Scheme

We examined the BER performance as a function of 𝐸 𝑏 / 𝑁 0 in two different subcarrier modulation schemes: 16 QAM and 64 QAM.

BER performances with 16-QAM-OFDM and 64-QAM-OFDM are shown in Figures 6 and 7, respectively. 𝜎 / 𝑇 = 0 . 0 4 3 and 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 were examined.

807549.fig.006
Figure 6: BER performance with 16-QAM-OFDM.
807549.fig.007
Figure 7: BER performance with 64-QAM-OFDM.

We can see that the DIME achieved a good performance for 16-QAM-OFDM, degradation within 0.2 dB (compared to perfect CSI), and for 64-QAM-OFDM, degradation within 0.4 dB, even in a nonsample spaced channel. The frequency-domain estimators, both ZF and ZF-FAV, were obviously not sensitive to the chosen nonsample tap position. The subcarrier averaging in the frequency domain, employed for ZF-FAV, had a certain effect on 16-QAM-OFDM but had no effect on 64-QAM-OFDM. For 16-QAM-OFDM the performance gain of the DIME relative to ZF-FAV was almost 1.5 dB at a BER of 1 0 − 3 , and for 64-QAM-OFDM it was more than 2.0 dB at a BER of 1 0 − 3 .

4.5. Performance in Delay Spread

We next examined the relation between a normalized rms delay spread 𝜎 / 𝑇 and the average received 𝐸 𝑏 / 𝑁 0 performance needed for an average BER of 1 0 − 3 .

The effects of the delay spread in different channel estimators with 16-QAM-OFDM and 64-QAM-OFDM are shown in Figures 8 and 9, respectively. A normalized Doppler frequency of 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 was examined and the delay spread was varied by changing Δ 𝜏 in the channel model depicted in Figure 3. Although ZF-FAV performed well when the delay spread was small, its performance deteriorated significantly as the delay spread increased, or the coherent bandwidth became narrower. Also in this situation, the DIME performance was maintained regardless of the delay spread of the channel, even in ISI channels with 𝜎 / 𝑇 > 0 . 0 4 3 , where a delayed path occurs beyond the GI.

807549.fig.008
Figure 8: Required 𝐸 𝑏 / 𝑁 0 performance versus normalized rms delay spread with 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 (16-QAM-OFDM).
807549.fig.009
Figure 9: Required 𝐸 𝑏 / 𝑁 0 performance versus normalized rms delay spread with 𝑓 𝑑 𝑇 = 0 . 0 0 7 5 (64-QAM-OFDM).
4.6. Performance in Doppler Frequency

We examined the relation between the normalized Doppler frequency 𝑓 𝑑 𝑇 and the average received 𝐸 𝑏 / 𝑁 0 performance needed for an average BER of 1 0 − 3 .

The effects of the normalized Doppler frequency on different channel estimators with 16-QAM-OFDM and 64-QAM-OFDM are shown in Figures 10 and 11, respectively. A normalized delay spread of 𝜎 / 𝑇 = 0 . 0 4 3 was examined and the maximum Doppler frequency, 𝑓 𝑑 , was changed from 64 Hz to 480 Hz, corresponding to a vehicle speed of 14 km/h to 104 km/h at a carrier frequency of 5 GHz. The performance of all channel estimators was maintained from low to high Doppler frequencies because they do not require any channel statistics that need a specific time coherency, such as averaging in the time direction. The DIME is also a decision-directed estimator operating on a pilot–by-pilot basis.

807549.fig.0010
Figure 10: Required 𝐸 𝑏 / 𝑁 0 performance versus normalized Doppler frequency with 𝜎 / 𝑇 = 0 . 0 4 3 (16-QAM-OFDM).
807549.fig.0011
Figure 11: Required 𝐸 𝑏 / 𝑁 0 performance versus normalized Doppler frequency with 𝜎 / 𝑇 = 0 . 0 4 3 (64-QAM-OFDM).

5. Conclusion

This paper described a novel channel estimation scheme for OFDM systems. The proposed channel estimator, DIME, uses a cyclic sinc-function matrix that is uniquely determined by Nc transmitted subcarriers and is composed of a deterministic and known vector. The computational complexity required for time-domain processing was reduced by taking a submatrix approach. Our setup reduced the matrix size by 1/20000 compared to the MMSE estimator.

For realistic nonsample spaced channels, including ISI channels, the DIME performed very well—yielding nearly the actual CSI—regardless of the delay spread, Doppler frequency, and subcarrier modulation scheme. We also showed that an oversampling size of 𝑀 = 2 for the DIME was sufficient to maintain this performance in arbitrary nonsample spaced channel models.

Appendices

A. NMSE for DIM Estimator

The normalized mean squared error (NMSE) is generally given by

E ∑ N M S E = 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | | ℎ 𝑛 − ℎ 𝑛 | | | 2 + ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | | ℎ 𝑛 − ℎ 𝑛 | | | 2 E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝑥 𝑛 | | 2 + ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | 𝑥 𝑛 | | 2 , ( A . 1 ) where ̂ 𝐡 is the N -dimensional estimated CTF vector and E { ⋅ } denotes the expectation operator.

The CTF of DIM estimator is redefined as

̂ 𝐡 D I M = ⎧ ⎪ ⎨ ⎪ ⎩ ̆ ℎ D I M ( 𝑛 ) 0 ≤ 𝑛 < 𝑁 𝑐 ̆ ℎ / 2 , D I M ( 𝑛 + ( 𝑀 − 1 ) 𝑁 ) 𝑁 − 𝑁 𝑐 / 2 ≤ 𝑛 < 𝑁 , ( A . 2 )

where

̂ ̆ 𝐡 D I M = 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 D I M , ( A . 3 ) and ̆ 𝐠 D I M is reformed by zero-padding all the discarded samples of

̂ ̆ 𝐠 D I M = 𝐒 − 1 ̂ ̆ 𝐠 Z F = ̂ ̆ 𝐠 + ̆ 𝐗 ′ 𝐅 𝑀 𝑁 − 1 𝐅 𝑀 𝑁 ̆ 𝐰 . ( A . 4 )

From (A.1) to(A.4), NMSE for DIM estimator is given as

N M S E D I M = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | | ℎ D I M 𝑛 − [ ℎ ] 𝑛 | | | 2 𝑁 𝑐 + E ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | | ℎ D I M 𝑛 − [ ℎ ] 𝑛 | | | 2 𝑁 𝑐 ( A . 5 )

The first term in (A.5) is derived as

E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | | ℎ D I M 𝑛 − [ ℎ ] 𝑛 | | | 2 𝑁 𝑐 = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 D I M 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 + 𝐅 𝑀 𝑁 ̆ 𝐗 ′ 𝐅 𝑀 𝑁 − 1 𝐅 𝑀 𝑁 ̆ 𝐰 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | | 2 𝑁 𝑐 = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 + 𝜆 𝜎 2 𝑀 𝑁 2 , ( A . 6 ) where [ ⋅ ] 𝑛 denotes the 𝑛 th element of a vector, 𝜆 is the number of the samples greater than a given threshold. Since the second term in (A.5) can also be derived similarly, NMSE for DIM estimator is defined as (A.7) In (A.7), the first term and the second term are channel estimation errors, and the third term is the residual noise.

N M S E D I M = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 + E ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | 𝐅 𝑀 𝑁 ̂ ̆ 𝐠 𝑛 − 𝐅 𝑀 𝑁 ̆ 𝐠 𝑛 | | 2 𝑁 𝑐 + 𝜆 𝜎 𝑀 𝑁 2 , ( A . 7 )

B. NMSE for ZF Estimator

The CTF of ZF is redefined as

̂ 𝐡 𝐙 𝐅 = 𝐙 y = 𝐙 𝐗 𝐅 𝑁 𝐠 + 𝐰 = 𝐙 𝐗 𝐅 𝑁 𝐠 + 𝐙 𝐰 . ( B . 1 )

From (A.1) and (B.1), NMSE for ZF estimator is given as

N M S E Z F = E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | | ℎ Z F 𝑛 − [ ℎ ] 𝑛 | | | 2 E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝑥 𝑛 | | 2 + ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | 𝑥 𝑛 | | 2 + E ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | | ℎ Z F 𝑛 − [ ℎ ] 𝑛 | | | 2 E ∑ 𝑁 𝑐 / 2 − 1 𝑛 = 0 | | 𝑥 𝑛 | | 2 + ∑ 𝑁 − 1 𝑛 = 𝑁 − 𝑁 𝑐 / 2 | | 𝑥 𝑛 | | 2 = E 𝐅 t r 𝑁 𝐠 + 𝐙 𝐰 − 𝐅 𝑁 𝐠 𝐅 𝑁 𝐠 + 𝐙 𝐰 − 𝐅 𝑁 𝐠 𝐻 𝑁 𝑐 = E t r ( 𝐙 𝐰 ) ( 𝐙 𝐰 ) 𝐻 𝑁 𝑐 = 𝜎 2 , ( B . 2 ) where t r { ⋅ } denotes the trace operator.

C. The Noise Suppression Gain by the First Order Linear Interpolation

The pilot symbol interval is a 14-OFDM-symbol in our OFDM system frame format (see Figure 1 and Table 1). The CTF of mth data symbol for a subcarrier, ℎ 𝑚 , can be derived by the first order interpolation method that is given as

ℎ 𝑚 = ℎ 𝑝 1 − ℎ 𝑝 0 ℎ 1 5 𝑚 + 𝑝 0 , ( C . 1 )

where ℎ 𝑝 0 and ℎ 𝑝 1 are the estimated CTFs of pilot symbols for a subcarrier that has sandwiched data symbols.

Since the noise is included in ℎ 𝑝 0 and ℎ 𝑝 1 and can be assumed to be independent, the noise suppression gain is calculated as

1 1 0 ∗ l o g 1 0 1 4 1 4 𝑚 = 1 𝑚 1 5 2 + 1 5 − 𝑚 1 5 2 ≈ − 1 . 9 d B . ( C . 2 )

Acknwoledgments

This work was supported, in part, by the National Institute of Information and Communications Technology (NICT) of Japan under contracted research entitled “The research and development of advanced radio signal processing technology for mobile communication systems.” The author would like to thank T. Taniguchi, T. Saito and T. Takano of Fujitsu Laboratories Limited for their encouragement and suggestions throughout this work. The author also thanks Y. Amezawa of Mobile Techno Corp. for his help in developing the software used in the computer simulation.

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