 some properties of the signal and is rarely used in practical OFDM systems. Adaptive channel estimation methods are typically used for a rapidly time-varying channel. In this section, we describe a generic pilot-based non-adaptive channel estimation method. In a linear time-invariant AWGN channel, the relationship between transmitted signal Xk, k= 0,1, NFFT land received signal Yk can be expressed as Yk=Hk + Zk where Zk is the frequency-domain CHAPTER 9 The IEEE 802. 16m Physical Layer (Part I) noise sampled at the kth sub-carrier frequency and Hk is the channel transfer-function sampled at the kth sub-carrier frequency. To estimate the channel, pilot symbols are used. Let's assume that every pth sub-carrier contains known pilot symbols Xpk, kp = 0, ,p, 2p, Np - 1 where Np is the total number of pilot sub-carriers across frequency. Using the known pilot symbols Xpk and the received symbols Ypk sampled at pilot sub-carriers, the channel transfer function HK sampled at pilot sub-carriers is: (9-22) where Wpk is a scaled-noise component at the pkth sub-carrier. Different methods, such as one- or two- dimensional linear interpolation can be used to estimate the channel samples at other sub-carrier frequencies. Channel estimation in OFDM systems is a two-dimensional (2-D) problem, i.e., the channel transfer function or channel impulse response is a function of time and frequency. Due to the computational complexity of 2-D estimators, one-dimensional (1-D) channel estimation methods are more practical. The idea behind 1-D estimators is to estimate the channel in one dimension (e.g., frequency) and then estimate the channel in another dimension (e.g., time), thus obtaining a 2-D channel estimate. Linear interpolation is a simple method using channel sample estimates at pilot sub- carriers. This is done by linearly interpolating the channel samples at the two nearest pilot sub-carriers. Although linear interpolation provides some limited noise reduction of the channel estimates at data locations (due to averaging functi