3.3 DF DFT- Based Method and Newton’s Method
3.4.2 NSE Performance
As observed in Figure 3.5, the equivalence of the two methods is demon-strated in terms of the performance of normalized square error (NSE) be-tween true and estimated CSI.
††Thanks to the orthogonal property of STBC, extending the results in SISO-OFDM systems [36–38] to the STBC-OFDM systems should be straightforward.
Table 3.1: Simulation parameters.
Parameter Value
Carrier frequency 2.3GHz
Bandwidth 5MHz
FFT size (K) 256
Length of CP (G) 64
Number of data subcarriers (M) 200 Number of data subcarriers used (NS) 200
Modulation QPSK
Number of transmit antennas (NT) 2 Number of receive antennas (NR) 1
ITU Veh-A [34] and Channel power profiles
Jakes model [33]
0 ∼ 63 Channel delay profiles
(Uniform distribution) Normalized maximum Doppler frequency (fd) 0.01, 0.05
0 2 4 6 8 10 12 14 16 10−4
10−3 10−2 10−1 100
Eb/No (dB)
BER
DF DFT−based Method (fd=0.01, V=10) Newton’s Method (fd=0.01, V=10) DF DFT−based Method (fd=0.05, V=3) Newton’s Method (fd=0.05, V=3) DF DFT−based Method (fd=0.05, V=10) Newton’s Method (fd=0.05, V=10) Perfect CSI
Figure 3.4: BER performance of the two methods.
0 1 2 3 4 5 6 7 8 9 10
−24
−22
−20
−18
−16
−14
−12
−10
−8
V
NSE (dB)
DF DFT−based Method (Eb/No=8dB) Newton’s Method (Eb/No=8dB) DF DFT−based Method (Eb/No=16dB) Newton’s Method (Eb/No=16dB)
Figure 3.5: NSE performance of the two methods (fd= 0.05).
3.5 Summary
In this chapter, we present a derivation on the equivalence between Newton’s method and the DF DFT-based method for channel estimation in STBC-OFDM systems. The results could provide useful insights for the develop-ment of new algorithms. For example, extending the DF DFT-based method to the Levenberg-Marquardt method is quite simple through this equivalence, which is particularly helpful when the inverse for the weighting matrix does not exist [41]. As another example, a few pilot tones can be applied to form a gradient vector at the first iteration by using (3.21) and to help the DF DFT-based method jump out of local minimum, thus improving the BER performance in fast fading channels [42]. Finally, it is worth mentioning that the derivation and the relationships explored in this chapter are also valid for conventional OFDM systems since they are only simplified cases of the systems discussed in this chapter.
Chapter 4
A Refined Channel Estimation Method for STBC-OFDM
Systems in Low-Mobility Wireless Channels
4.1 Literature Survey and Motivation
OFDM has been widely applied in wireless communication systems in re-cent years due to its capability of high-rate transmission and low-complexity implementation over frequency-selective fading channels. STC is another promising technique to provide diversity gain through the use of multiple transmit antennas, especially when receive diversity is too expensive to de-ploy. In particular, STBC has received a lot of attention because a simple linear decoder can be used at the receiver side [12, 13, 40]. These advan-tages make OFDM combined with STBC, known as STBC-OFDM, an ideal
choice for several applications such as wireless metropolitan area networks (WMANs) 802.16e [43], etc. However, a high-rate STBC-OFDM system em-ploying multi-level modulation with non-constant envelope (e.g. 16QAM) generally requires accurate CSI to perform coherent detection. This in turn implies that dynamic channel estimation is a crucial factor in realizing a successful STBC-OFDM system over doubly selective channels.
Blind channel estimation, which merely relies on the received signals, is very attractive due to its bandwidth-saving advantage. Nevertheless, it requires a long data record, involves high computational complexity, and only applies to slowly time-varying channels. On the contrary, pilot-aided channel estimation, using pilot tones known to the receiver, shows great promise for applications in mobile wireless communication, even though the use of pilot tones ends up with lower data rate. DF channel estimation offers an alternative way to track channel variations; nevertheless, it is vulnerable to decision error propagation in fast time-varying channels [35–37, 44–46].
As a high quality channel estimator with low training overhead is needed for successful implementation of STBC-OFDM systems, we restrict our attention to the category of pilot-aided plus DF channel estimation methods in this chapter.
Among a wide variety of pilot-aided plus DF channel estimation meth-ods, the DFT-based channel estimation method, derived from either MMSE criterion or ML criterion, has been intensively studied for OFDM systems with preambles [36–39]. It is shown in [35–38] that the DFT-based channel estimation method using the ML criterion is simpler to implement because it
requires neither channel statistics nor operating signal-to-noise ratio (SNR).
Furthermore, as presented in [38], the performance of the ML estimator is comparable to that of the MMSE estimator at intermediate or high SNR values when the number of pilot tones is sufficiently larger than the maximal channel length (in samples). Thus, we will focus on the ML estimator in this chapter. In order to save bandwidth and improve system performance, DF data symbols are also used as pilots to track channel variations in subsequent OFDM data symbols, and this method is called the DF DFT-based channel estimation method. Recently, the mathematical equivalence between the DF DFT-based method and Newton’s method has been studied in [47], and it is concluded that even though a global solution for CSI is given as the initial value in the preamble, the DF DFT-based method is only applicable to very slowly time-varying channels because of the local optimization capability of Newton’s method.
Most mobile wireless channels are characterized by channel impulse re-sponse (CIR) consisting of a few dominant paths. The multipath delays are usually slowly time-varying. The amplitude and the phase of each path, how-ever, can vary relatively fast. In this chapter, we propose a two-stage channel estimation method by utilizing these channel characteristics. In the initial-ization stage, we employ an MPIC-based decorrelation method to identify significant paths. In the following tracking stage, we develop a refined DF DFT-based channel estimation method, in which we use a few pilot tones inserted in OFDM data symbols to form an optimal gradient vector at the first iteration. This optimal gradient vector helps the classical method jump
out of the local optimum, thus reducing the error propagation effect. The classical DF DFT-based channel estimation method is then used at the fol-lowing iterations. In addition, an approximate weighting matrix is adopted to reduce the computational complexity associated with the matrix inversion operation of the weighting matrix in the DF DFT-based channel estimation method.
The rest of this chapter is organized as follows. In Section 4.2, we de-scribe the STBC-OFDM system. In Section 4.3, we present the MPIC-based decorrelation method in the initialization stage. Next, the equivalence be-tween the DF DFT-based channel estimation method and Newton’s method is briefly reviewed, and we propose a refined DF DFT-based channel esti-mation method in the tracking stage. We then discuss the computational complexity of the proposed two-stage channel estimation method in Section 4.4. In Section 4.5, we present our computer simulation and performance evaluation results. Finally, some concluding remarks are drawn in Section 5.6.