Detection of OFDM Signals in Fast-Varying Channels With Low
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008
859
Detection of OFDM Signals in Fast-Varying Channels With Low-Density Pilot Symbols Ming-Xian Chang, Member, IEEE, and Tsung-Da Hsieh, Student Member, IEEE
Abstract—In fast-varying channels, an orthogonal frequencydivision multiplexing system needs to insert denser pilot symbols among transmitted symbols in tracking the variation of a channel. However, using denser pilot symbols reduces transmission throughput. In this paper, we propose a pseudopilot algorithm for data detection in fast-varying channels without increasing the pilot density. Our algorithm is based on a regressional model-based least-squares-fitting approach. Within a block of received symbols, we select some data symbols and regard them as pseudopilot symbols. The receiver considers all the possible patterns of the pseudopilots and associates each of them with a data sequence and a corresponding metric. The associated data sequence, whose metric is minimum, is selected as the detected data sequence. Our algorithm is not based on a decision-directed or decision-feedback architecture because the pseudopilots do not come from any detected symbols. The proposed algorithm needs to search all the possible patterns of the pseudopilots, and the complexity may increase with the number of pseudopilots and constellation size. To reduce the number of search, we further propose two modified approaches. The simulation results show that the performance of the proposed algorithms could approach a bit-error probability lower bound that is obtained by letting the receiver know the true values of the pseudopilots. Compared with the linear interpolation method, the proposed algorithm shows obvious improvement in fast-varying channels. The proposed modified approaches could also effectively reduce the number of search while maintaining the performance. We also give the complexity analysis of the proposed algorithm and an approach to determine the degree of the regression polynomial. Index Terms—Estimation, fading channels, orthogonal frequency-division multiplexing (OFDM), pseudopilot, signal detection.
I. I NTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) transmission has been widely used in modern broadband wireless systems. A wideband spectrum in OFDM transmission is divided into several narrowband subchannels, each of which suffers from nonselective fading only. By padding a guard interval of a cyclic prefix that is longer than
Manuscript received December 26, 2006; revised April 16, 2007, May 26, 2007, and May 30, 2007. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC95-2219-E-006-004. Part of this paper was presented at the IEEE Wireless Communications and Networking Conference, Hong Kong, 2007. The review of this paper was coordinated by Prof. X.-G. Xia. The authors are with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2007.906369
the maximum channel delay, OFDM can remove the intersymbol interference (ISI). If the channel is invariant within one symbol interval, then the received sample of the mth subchannel at the nth regular symbol interval can be written as [1]–[6] Ymn = Hmn Xmn + Nmn
(1)
where Ymn and Xmn are the received and transmitted symbols, respectively, Hmn accounts for the channel response (CR), and Nmn is an independent and identically distributed complex zero-mean Gaussian random variable. When the channel varies within one symbol interval, there will be interchannel interference (ICI). The ICI can be reduced by applying some available ICI-cancellation algorithms [7]–[9]. In this paper, we will still consider the signal model in (1). The aforementioned signal model also assumes that the receiver’s frequency and timing recovery subsystem is such that, within the time span of interest, the effect of the residual error can be accounted for by the CR Hmn . To estimate the transmitted data Xmn ’s, it is essential that the receiver obtains reliable estimates of the CRs. By inserting some pilot symbols among transmitted symbols, one can estimate the CRs at pilot positions. The CRs at the other positions can be estimated by interpolation, on the assumption that the channel continuously varies. Many pilot-based methods for estimating Hmn have been proposed [1]–[6]. When the channel varies fast, we need to insert denser pilot symbols to track the fast-varying CRs. However, increasing the number of pilot symbols reduces transmission throughput. In this paper, we will propose a pseudopilot data-detection algorithm and two corresponding modified approaches that do not need to increase the density of pilot symbols for data detection in fast-varying channels. Because the CRs at neighboring time slots or subchannels are usually statistically correlated, it is preferred to jointly estimate a block of CRs. Equation (1) indicates that the symbols in OFDM transmission are 2-D, as shown in Fig. 1. We can choose a block composed of symbols across time slots on a fixed subchannel or a block composed of symbols across subchannels at fixed time slots. Among a block of transmitted symbols, the transmitter inserts some pilot symbols for the CR estimation in the receiver. In this paper, we consider the block within which there are only two pilots, one on each end of the block, as shown in Fig. 2. In this scenario, if the channel quadratically varies with time (or with a subchannel), i.e., Hmn ≈ k0 n2 or Hmn ≈ k0 m2 , during the interval of the block, then we cannot obtain accurate CR estimates by applying linear interpolation (LI) between the CRs of the two pilots. Inaccurate CR estimation
0018-9545/$25.00 © 2008 IEEE
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008
Fig. 1. Two-dimensional OFDM symbols Xmn ’s, where N = N − 1.
Fig. 2. Data and pilot symbols of a block. There are only two pilot symbols within a block.
further incurs the error floor in the bit-error probability (BEP) performance. The proposed algorithm is based on a regression polynomial model that is used to describe the CR process [6]. The coefficients of the regression polynomial is determined by a least-squares-fitting (LSF) approach [6], [10]. In the proposed algorithm, the receiver selects some data symbols within the block and regards them as “pseudopilots.” Each possible pattern of pseudopilots is used together with pilots in the model-based LSF (MB-LSF) approach. By extending the function of the MB-LSF approach, we associate each pattern of pseudopilots with a data sequence and a corresponding metric. Then, the associated data sequence with the minimum metric is chosen and serves as the detected data sequence. The proposed algorithm is not based on a decision-directed or decision-feedback architecture because the pseudopilots do not come from any detected symbols. Because the complexity of the proposed algorithm is proportional to the number of possible patterns of pseudopilots, we further consider two modified approaches based on the proposed algorithm. The first modified approach uses a threshold to determine whether we should stop the search by comparing the metric with the threshold. The threshold is updated for each block. The second modified approach further applies the LI method to determine a pseudopilot pattern that is to be searched first. The simulation results show that the modified approaches could efficiently reduce the number of search while maintaining the performance almost unchanged. We compare the proposed algorithm and the modified approaches with two other methods, including the LI method and its improved version. We also consider the system in which the receiver knows the true values of pseudopilots, and both pilots and pseudopilots are used in the MB-LSF approach.
Its performance can be used as an approximate lower bound for the proposed algorithm, though it reduces the transmission throughput. The simulation results show that the proposed algorithm has performance that is close to the lower bound if the degree of the regression polynomial is appropriate. Compared with the LI method, the proposed algorithm obtains obvious improvement in fast-varying channels, and the error floors are reduced without decreasing the transmission throughput. We also give the complexity analysis based on the average number of complex multiplication for each symbol detection. The analysis shows that the complexity of the MB-LSF approach is not more than three complex multiplication per symbol, whereas for the proposed algorithm and the two modified approaches, the complexity is about 2K multiples of the MB-LSF approach, where K is the average searched number of pseudopilot patterns. The remainder of this paper is organized as follows. Section II introduces the regressional MB-LSF approach and the corresponding signal model. The proposed algorithm is developed in Section III, and the two corresponding modified approaches are given in Section IV. Section V discusses on performance lower bound, complexity analysis, degrees of the regression polynomial, and pseudopilot positions. Some simulation examples are given in Section VI. Finally, Section VII concludes this paper. II. R EGRESSIONAL MB-LSF A PPROACH In this section, we introduce the regressional MB-LSF approach [6], [10], on which we will base the algorithm that we will develop. Because the OFDM symbols Xmn ’s, where m is the index of subchannels, and n is the index of time slots, can be treated as 2-D, as shown in Fig. 1, we will consider two types of blocks. For the first type, a block is composed of received symbols across subchannels at a fixed time slot n, Y0n , Y1n , . . . , Y(L−1)n , whereas for the second type, a block is composed of received symbols on a fixed subchannel m, Ym0 , Ym1 , . . . , Ym(L−1) , where L is the number of symbols of a block. For convenience, we reserve only one index and denote a block of received symbols as Y0 , Y1 , . . . , Y(L−1) . Assume that within a block, the transmitter inserts pilot ∆ symbols {X0 , Xr , X2r , . . . , X(Np −1)r }, and we define P = {0, r, 2r, . . . , (Np − 1)r} as the set of pilot positions, where Np is the number of pilots within a block and r is the distance between pilots. Here, a pilot position (mp , np ) means that a pilot is on the mp th subchannel and at the np th time slot, and for the aforementioned two types of blocks, only one index is reserved for conciseness. Because Yn = Hn Xn + Nn , the tentative estimates of CRs at pilot positions are ˜ n = Yn = Hn + Nn , H Xn Xn
n ∈ P.
(2)
For the MB-LSF approach, the receiver models the true CRs Hn ’s as a polynomial of time slot indexes (or subchannel indexes), e.g., ∆ ˆ n = f (c, n) = H c0 + c1 n + · · · + cd nd ,
0≤n
859
Detection of OFDM Signals in Fast-Varying Channels With Low-Density Pilot Symbols Ming-Xian Chang, Member, IEEE, and Tsung-Da Hsieh, Student Member, IEEE
Abstract—In fast-varying channels, an orthogonal frequencydivision multiplexing system needs to insert denser pilot symbols among transmitted symbols in tracking the variation of a channel. However, using denser pilot symbols reduces transmission throughput. In this paper, we propose a pseudopilot algorithm for data detection in fast-varying channels without increasing the pilot density. Our algorithm is based on a regressional model-based least-squares-fitting approach. Within a block of received symbols, we select some data symbols and regard them as pseudopilot symbols. The receiver considers all the possible patterns of the pseudopilots and associates each of them with a data sequence and a corresponding metric. The associated data sequence, whose metric is minimum, is selected as the detected data sequence. Our algorithm is not based on a decision-directed or decision-feedback architecture because the pseudopilots do not come from any detected symbols. The proposed algorithm needs to search all the possible patterns of the pseudopilots, and the complexity may increase with the number of pseudopilots and constellation size. To reduce the number of search, we further propose two modified approaches. The simulation results show that the performance of the proposed algorithms could approach a bit-error probability lower bound that is obtained by letting the receiver know the true values of the pseudopilots. Compared with the linear interpolation method, the proposed algorithm shows obvious improvement in fast-varying channels. The proposed modified approaches could also effectively reduce the number of search while maintaining the performance. We also give the complexity analysis of the proposed algorithm and an approach to determine the degree of the regression polynomial. Index Terms—Estimation, fading channels, orthogonal frequency-division multiplexing (OFDM), pseudopilot, signal detection.
I. I NTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) transmission has been widely used in modern broadband wireless systems. A wideband spectrum in OFDM transmission is divided into several narrowband subchannels, each of which suffers from nonselective fading only. By padding a guard interval of a cyclic prefix that is longer than
Manuscript received December 26, 2006; revised April 16, 2007, May 26, 2007, and May 30, 2007. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC95-2219-E-006-004. Part of this paper was presented at the IEEE Wireless Communications and Networking Conference, Hong Kong, 2007. The review of this paper was coordinated by Prof. X.-G. Xia. The authors are with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2007.906369
the maximum channel delay, OFDM can remove the intersymbol interference (ISI). If the channel is invariant within one symbol interval, then the received sample of the mth subchannel at the nth regular symbol interval can be written as [1]–[6] Ymn = Hmn Xmn + Nmn
(1)
where Ymn and Xmn are the received and transmitted symbols, respectively, Hmn accounts for the channel response (CR), and Nmn is an independent and identically distributed complex zero-mean Gaussian random variable. When the channel varies within one symbol interval, there will be interchannel interference (ICI). The ICI can be reduced by applying some available ICI-cancellation algorithms [7]–[9]. In this paper, we will still consider the signal model in (1). The aforementioned signal model also assumes that the receiver’s frequency and timing recovery subsystem is such that, within the time span of interest, the effect of the residual error can be accounted for by the CR Hmn . To estimate the transmitted data Xmn ’s, it is essential that the receiver obtains reliable estimates of the CRs. By inserting some pilot symbols among transmitted symbols, one can estimate the CRs at pilot positions. The CRs at the other positions can be estimated by interpolation, on the assumption that the channel continuously varies. Many pilot-based methods for estimating Hmn have been proposed [1]–[6]. When the channel varies fast, we need to insert denser pilot symbols to track the fast-varying CRs. However, increasing the number of pilot symbols reduces transmission throughput. In this paper, we will propose a pseudopilot data-detection algorithm and two corresponding modified approaches that do not need to increase the density of pilot symbols for data detection in fast-varying channels. Because the CRs at neighboring time slots or subchannels are usually statistically correlated, it is preferred to jointly estimate a block of CRs. Equation (1) indicates that the symbols in OFDM transmission are 2-D, as shown in Fig. 1. We can choose a block composed of symbols across time slots on a fixed subchannel or a block composed of symbols across subchannels at fixed time slots. Among a block of transmitted symbols, the transmitter inserts some pilot symbols for the CR estimation in the receiver. In this paper, we consider the block within which there are only two pilots, one on each end of the block, as shown in Fig. 2. In this scenario, if the channel quadratically varies with time (or with a subchannel), i.e., Hmn ≈ k0 n2 or Hmn ≈ k0 m2 , during the interval of the block, then we cannot obtain accurate CR estimates by applying linear interpolation (LI) between the CRs of the two pilots. Inaccurate CR estimation
0018-9545/$25.00 © 2008 IEEE
860
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008
Fig. 1. Two-dimensional OFDM symbols Xmn ’s, where N = N − 1.
Fig. 2. Data and pilot symbols of a block. There are only two pilot symbols within a block.
further incurs the error floor in the bit-error probability (BEP) performance. The proposed algorithm is based on a regression polynomial model that is used to describe the CR process [6]. The coefficients of the regression polynomial is determined by a least-squares-fitting (LSF) approach [6], [10]. In the proposed algorithm, the receiver selects some data symbols within the block and regards them as “pseudopilots.” Each possible pattern of pseudopilots is used together with pilots in the model-based LSF (MB-LSF) approach. By extending the function of the MB-LSF approach, we associate each pattern of pseudopilots with a data sequence and a corresponding metric. Then, the associated data sequence with the minimum metric is chosen and serves as the detected data sequence. The proposed algorithm is not based on a decision-directed or decision-feedback architecture because the pseudopilots do not come from any detected symbols. Because the complexity of the proposed algorithm is proportional to the number of possible patterns of pseudopilots, we further consider two modified approaches based on the proposed algorithm. The first modified approach uses a threshold to determine whether we should stop the search by comparing the metric with the threshold. The threshold is updated for each block. The second modified approach further applies the LI method to determine a pseudopilot pattern that is to be searched first. The simulation results show that the modified approaches could efficiently reduce the number of search while maintaining the performance almost unchanged. We compare the proposed algorithm and the modified approaches with two other methods, including the LI method and its improved version. We also consider the system in which the receiver knows the true values of pseudopilots, and both pilots and pseudopilots are used in the MB-LSF approach.
Its performance can be used as an approximate lower bound for the proposed algorithm, though it reduces the transmission throughput. The simulation results show that the proposed algorithm has performance that is close to the lower bound if the degree of the regression polynomial is appropriate. Compared with the LI method, the proposed algorithm obtains obvious improvement in fast-varying channels, and the error floors are reduced without decreasing the transmission throughput. We also give the complexity analysis based on the average number of complex multiplication for each symbol detection. The analysis shows that the complexity of the MB-LSF approach is not more than three complex multiplication per symbol, whereas for the proposed algorithm and the two modified approaches, the complexity is about 2K multiples of the MB-LSF approach, where K is the average searched number of pseudopilot patterns. The remainder of this paper is organized as follows. Section II introduces the regressional MB-LSF approach and the corresponding signal model. The proposed algorithm is developed in Section III, and the two corresponding modified approaches are given in Section IV. Section V discusses on performance lower bound, complexity analysis, degrees of the regression polynomial, and pseudopilot positions. Some simulation examples are given in Section VI. Finally, Section VII concludes this paper. II. R EGRESSIONAL MB-LSF A PPROACH In this section, we introduce the regressional MB-LSF approach [6], [10], on which we will base the algorithm that we will develop. Because the OFDM symbols Xmn ’s, where m is the index of subchannels, and n is the index of time slots, can be treated as 2-D, as shown in Fig. 1, we will consider two types of blocks. For the first type, a block is composed of received symbols across subchannels at a fixed time slot n, Y0n , Y1n , . . . , Y(L−1)n , whereas for the second type, a block is composed of received symbols on a fixed subchannel m, Ym0 , Ym1 , . . . , Ym(L−1) , where L is the number of symbols of a block. For convenience, we reserve only one index and denote a block of received symbols as Y0 , Y1 , . . . , Y(L−1) . Assume that within a block, the transmitter inserts pilot ∆ symbols {X0 , Xr , X2r , . . . , X(Np −1)r }, and we define P = {0, r, 2r, . . . , (Np − 1)r} as the set of pilot positions, where Np is the number of pilots within a block and r is the distance between pilots. Here, a pilot position (mp , np ) means that a pilot is on the mp th subchannel and at the np th time slot, and for the aforementioned two types of blocks, only one index is reserved for conciseness. Because Yn = Hn Xn + Nn , the tentative estimates of CRs at pilot positions are ˜ n = Yn = Hn + Nn , H Xn Xn
n ∈ P.
(2)
For the MB-LSF approach, the receiver models the true CRs Hn ’s as a polynomial of time slot indexes (or subchannel indexes), e.g., ∆ ˆ n = f (c, n) = H c0 + c1 n + · · · + cd nd ,
0≤n
