In OFDM systems with comb-type pilot arrangement, ignoring the effects of inter-symbol interference (ISI) and inter-carrier interference (ICI), the received data at the

*k* th subcarrier (1 ≤

*k ≤ N*) of the

*n* th OFDM frame can be formulated as:

$Y\mathsf{\text{(}}n,k\mathsf{\text{)}}=X\mathsf{\text{(}}n,k\mathsf{\text{)}}\cdot H\mathsf{\text{(}}n,k\mathsf{\text{)}}+W\left(n,k\right),$

(1)

where

*X*(

*n, k*) is the transmitted OFDM symbol,

*H*(

*n, k*) is the CFR and

*W*(

*n, k*) is the AWGN noise. If

$\mathcal{P}$ denotes the set of all pilot indices, at a given pilot subcarriers

${k}_{p}\in \mathcal{P}$ and using the LS method, the CFR can be estimated as:

$\stackrel{\u0303}{H}\left(n,{k}_{p}\right)=\frac{Y\left(n,{k}_{p}\right)}{X\left(n,{k}_{p}\right)}=H\left(n,{k}_{p}\right)+\frac{W\left(n,{k}_{p}\right)}{X\left(n,{k}_{p}\right)}.$

(2)

As explained earlier, conventional methods for estimation of the CFR at non-pilot subcarriers (given the noisy measurements at pilots) are interpolation-based techniques which require relatively high sampling rates (number of pilots) to produce acceptable mean squared error (MSE). Also, the optimum structure of the pilot locations for these techniques which minimizes the MSE of the estimated channel, is the uniform distribution (equidistance) of the pilots in the spectrum.

In the sparsity-based channel estimation methods, instead of finding the CFR, the goal is to estimate the inherently sparse CIR in each OFDM frame from limited number of noisy measurements of the CFR obtained at pilot locations. The estimated CIR is then, translated into the frequency domain by means of FFT which results in an estimation of the CFR that can be used for data equalization process. In these methods, we are dealing with the following system of equations:

${\stackrel{\u0303}{\mathbf{H}}}_{p}={\mathbf{F}}_{p}\cdot \mathbf{h}+{\mathbf{n}}_{p},$

(3)

where **F**_{
p
}is the DFT submatrix with ${N}_{p}=\left|\mathcal{P}\right|$ rows associated with the pilot locations, ${\stackrel{\u0303}{\mathbf{H}}}_{p}$ is the vector of LS-estimated CFR at pilot locations, **h** is the sparse CIR vector, and **n**_{
p
}is the vector of noise values.

Generally, there are two main categories of sparsity-based methods to solve the set of equations presented in (3). One approach is to minimize the *ℓ*_{1} norm of **h** subject to (3), either directly or iteratively (such as SPGL [8]). Although the performance of such methods are considered among the bests, they are extremely slow for real-time implementation. The other approach which is considered in this article, is to use fast greedy methods such as OMP which iteratively detect and estimate the location and value of the channel taps. These methods are usually faster than *ℓ*_{1} minimization techniques by orders of magnitude while they may fall short of performance. Our simulation results confirm that their performance is acceptable for the purpose of OFDM channel estimation.

The main advantages of sparsity-based approaches can be categorized into two parts:

**(1) Decreasing MSE:** Generally, the purpose of using compressed sensing methods in solving a linear set of equations with the sparsity constraint is to achieve the Cramer Rao lower bound on MSE [9]. In extreme cases, the structured LS estimator [9] which knows the location of nonzero taps (support) through an oracle, and estimates their corresponding values using LS estimation is the best estimator. The MSE of this estimator is called CRB-S [9]. However, in general, there is no information about the location of the nonzero coefficients of **h** at the receiver and the structural LS estimator is not realizable. Simulation results indicate that we can get close to this bound by using proper sparsity-based methods.

**(2) Reducing Overhead:** Although the pilot subcarriers occupy a fraction of the spectrum, they do not convey any data. By reducing the number of pilot subcarriers, we increase the utilization efficiency of the spectrum while we may degrade the performance of the channel estimation block. As mentioned in [7], by considering the sparsity of the CIR, it is possible to capture the necessary information in the frequency domain in fewer number of pilots. The results in [5] show that *ℓ*_{1} minimization technique almost perfectly reconstructs the sparse CIR from (3) when the number of pilots is proportional to the number of channel taps. Furthermore, the reconstruction performance is independent of the location and value of the taps; i.e., unlike the interpolation-based methods, the number of required pilot subcarriers does not depend on the delay spread and degree of frequency selectivity of the channel.