It is assume that the number of subcarriers is *N* and the subcarriers *N*_{P} are chosen as transmitting pilot in OFDM systems. The allocation position of the pilot pattern is \( K=\left({k}_1,\cdots, {k}_{N_p}\right)\kern0.5em \left(1\le {k}_1\le \cdots \le {k}_{N_p}\le N\right) \). The relationship between the received pilot and transmitted pilot is expressed as

$$ {Y}_{N_p}={X}_{N_p}{F}_{N_p}h+{W}_{N_p}, $$

(1)

where a diagonal matrix \( {X}_{N_P}=\mathit{\operatorname{diag}}\left[X\left({k}_1\right),X\left({k}_2\right),\cdots, X\left({k}_{N_P}\right)\right] \) is composed of the transmitted pilot, the received pilot is \( {Y}_{N_P}={\left[Y\left({k}_1\right),Y\left({k}_2\right),\cdots, Y\left({k}_{N_P}\right)\right]}^T \), and the channel impulse response *h* = [*h*(1), *h*(2), ⋯, *h*(*N*)]^{T} is of the length of *N*, among which the first *L* elements contain multipath energy. \( {W}_{N_P}={\left[W\left({k}_1\right),W\left({k}_2\right),\cdots, W\left({k}_{N_P}\right)\right]}^T \) is the Gaussian white noise in the frequency domain. \( {F}_{N_P} \) is a *N*_{P} × *N* partial Fourier matrix, whose *N*_{P} row is extracted from the *N* × *N* standard Fourier matrix by the pilot pattern *K*, and defined as

$$ {F}_{N_p}=\frac{1}{\sqrt{N}}\left[\begin{array}{ccc}{f}^{k_11}& \dots & {f}^{k_1N}\\ {}\vdots & \ddots & \vdots \\ {}{f}^{k_{N_p}1}& \dots & {f}^{k_{N_p}N}\end{array}\right], $$

(2)

where *f* = *e*^{−j2π/N}. We further denote

$$ A\cong {X}_{N_p}{F}_{N_p}=\left[\begin{array}{ccc}X\left({k}_1\right){f}^{k_11}& \cdots & X\left({k}_1\right){f}^{k_1N}\\ {}\vdots & \ddots & \vdots \\ {}X\left({k}_{N_p}\right){f}^{k_{N_p}1}& \cdots & X\left({k}_{N_p}\right){f}^{k_{N_p}N}\end{array}\right]. $$

(3)

Then, (1) can be rewritten as

$$ {Y}_{N_p}= Ah+{W}_{N_p}. $$

(4)

Since the sampled interval is usually much smaller than the channel delay propagation, the channel coefficients are either zero or nearly zero, which implies that *h* is a sparse vector. According to the CS theory, the matrix *A* can be regarded as the sensing matrix, and it is essentially the weight of the transmitted pilot signal to the partial Fourier matrix. If the placement of the pilot pattern is randomly selected, the matrix *A* is a structured random matrix weighted by the transmitted pilot. Correspondingly, if the placement of the pilot pattern is deterministic, it is a deterministic sensing matrix. Therefore, the process of channel estimation based on CS is explained that the transmitted pilot is compressed to measure the impulse response *h*, and then *h* is reconstructed by the reconstruction algorithm with the received pilots. Moreover, it can clearly be seen from (3) that the pilot pattern decides the extraction of those rows of the standard Fourier transform matrix, and then determines the structure of the sensing matrix, and ultimately affects the performance of OFDM channel estimation. If the allocation of the pilot pattern is random, the corresponding sensing matrix is the structured random matrix. However, the pilot subcarrier with random distribution is not easy to be realized in the actual communication system. Therefore, it is necessary to study the deterministic sensing matrix with optimizing pilot pattern allocation so as to ensure the channel estimation performance.

When the sensing matrix satisfies the RIP, the channel impulse response can be recovered from the received pilot with a high probability. However, there is no known method to test whether a given matrix satisfies RIP in polynomial time. Alternatively, we can compute the MC of the sensing matrix *A* to verify RIP and it can be formulated as

$$ \mu (A)\kern0.5em =\underset{\underset{m\ne n}{1\le m<n\le N}}{\max}\frac{\left|\left\langle {a}_m,{a}_n\right\rangle \right|}{{\left\Vert {a}_m\right\Vert}_2{\left\Vert {a}_n\right\Vert}_2}, $$

(5)

where |〈*a*_{m}, *a*_{n}〉| denotes the inner product between the *m*th column and the *n*th column of the sensing matrix *A*. Substituting (3) to (5) can be obtained

$$ {\displaystyle \begin{array}{l}\mu (A)=\underset{\underset{m\ne n}{1\le m<n\le N}}{\max}\frac{\frac{1}{N}\left|\sum \limits_{i=1}^{N_p}\overline{X\left({k}_i\right)}{e}^{j2\pi {k}_im/N}X\left({k}_i\right){e}^{-j2\pi {k}_in/N}\right|}{\frac{1}{\sqrt{N}}{\left(\sum \limits_{i=1}^{N_p}{\left|X\left({k}_i\right){e}^{-j2\pi {k}_im/N}\right|}^2\right)}^{\frac{1}{2}}\frac{1}{\sqrt{N}}{\left(\sum \limits_{i=1}^{N_p}{\left|X\left({k}_i\right){e}^{-j2\pi {k}_in/N}\right|}^2\right)}^{\frac{1}{2}}}\\ {}\kern3.5em \\ {}\kern3.5em =\underset{\underset{m\ne n}{1\le m<n\le N}}{\max}\frac{\left|\sum \limits_{i=1}^{N_p}{\left|X\left({k}_i\right)\right|}^2{e}^{-j2\pi {k}_i\left(n-m\right)/N}\right|}{\sum \limits_{i=1}^{N_p}{\left|X\left({k}_i\right)\right|}^2}.\end{array}} $$

(6)

Since the pilots used in the OFDM system satisfy the constant amplitude, we define |*X*(*k*_{i})| = 1 (*i* = 1, 2⋯*N*_{p}). Let*d* = *n* − *m*, and Ω = {1, 2, ⋯*N* − 1}. Then (6) can be rewritten as

$$ \mu (A)=\underset{d\in \Omega}{\max}\frac{1}{N_p}\left|\sum \limits_{i=1}^{N_p}{e}^{-j2\pi {k}_id/N}\right|. $$

(7)

It can be seen from (7) that the MC of matrix *A* is determined by the pilot pattern *K*, when the number of subcarrier and pilot has been determined.

As mentioned in Section 1, the pilot pattern is optimized by the rule of minimizing the MC of the sensing matrix, and the sparse channel impulse response *h* is reconstructed based on CS algorithm. Replacing the *A* in (7) with *K* and taking it as a variable, the optimized position problem of the pilot pattern allocation can be expressed as

$$ Q=\underset{K\in \kern0.5em P}{\min}\mu (K). $$

(8)

where *P* is the set of all possible pilot patterns. In the OFDM communication system, it is unrealistic to search all pilot patterns to determine the optimal pilot pattern because the number of pilot pattern is huge and the computational complexity is very high. Therefore, it is necessary to propose an optimization method to solve Eq. (8) for reducing computational complexity and obtaining optimal pilot pattern.