### Compressive sensing-based SC-FDE

The signal structure of the SC-FDE transmission system based on training sequence is shown in Fig. 1a. Where *C*_{i} = [*c*_{i, 0}, *c*_{i, 1},⋯, *c*_{i, P − 1}]^{T} is a UW with a length P. Therefore, for any *i* and *j*, *c*_{i} = *c*_{j} is established, *X*_{i} = [*x*_{i, 0}, *x*_{i, 1},⋯, *x*_{i, N − 1}]^{T} is payload data block with a length *N*. Let discrete time channel impulse response with the maximum multipath spread length *L* be *H*_{i} = [*h*_{i, 0}, *h*_{i, 1},⋯, *h*_{i, L}]^{T}, which satisfies *P* ≥ *L*. Record *M* = *P*-*L*. The received signal obtained after channel transmission, time-frequency synchronization, and the multipath interference is shown in Fig. 1b, c.

The received signal *D*_{i} = [*d*_{i, 0}, *d*_{i, 1},⋯, *d*_{i, P − 1}]^{T} corresponding to the known training sequence can be expressed as:

$$ {D}_i={\Psi}_{\mathrm{i}}{H}_i+{N}_i $$

(1)

*N*_{i} is additive white Gaussian noise and Ψ_{i} is:

$$ {\Psi}_{\mathrm{i}}={\left[\begin{array}{ccccc}{c}_{i,0}& {x}_{i,N-1}& {x}_{i,N-2}& \cdots & {x}_{i,N-L}\\ {}{c}_{i,1}& {c}_{i,0}& {x}_{i,N-1}& \cdots & {x}_{i,N-L+1}\\ {}{c}_{i,2}& {c}_{i,1}& {c}_{i,0}& \cdots & {x}_{i,N-L+2}\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}{c}_{i,L}& {c}_{i,L-1}& {c}_{i,L-2}& \cdots & {c}_{i,0}\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}{c}_{i,P-1}& {c}_{i,P-2}& {c}_{i,P-3}& \cdots & {c}_{i,P-L\hbox{-} 1}\end{array}\right]}_{P\times \left(L+1\right)} $$

(2)

It can be seen from the matrix Ψ_{i} that the last *M* samples of the received signal are not interfered by the unknown data, which constitute the inter-block-interference (IBI)-free region of the training sequence, to be recorded as *R*_{i} = [*d*_{i, L}, *d*_{i, L + 1},⋯, *d*_{i, P − 1}]^{T}. *V*_{i} is last *M* samples of *N*_{i}, then:

$$ {R}_i={\Phi}_{\mathrm{i}}{H}_i+{V}_i $$

(3)

where Φ_{i} is a Toeplitz array:

$$ {\Phi}_{\mathrm{i}}={\left[\begin{array}{ccccc}{c}_{i,L}& {c}_{i,L-1}& {c}_{i,L-2}& \cdots & {c}_{i,0}\\ {}{c}_{i,L+1}& {c}_{i,L}& {c}_{i,L-1}& \cdots & {c}_{i,1}\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}{c}_{i,P-1}& {c}_{i,P-2}& {c}_{i,P-3}& \cdots & {c}_{i,P-L\hbox{-} 1}\end{array}\right]}_{M\times \left(L+1\right)} $$

(4)

For SC-FDE of DUW, *M* = *L* + 1, the matrix Φ_{i} is column full rank, then Eq. (3) can be used to estimate the channel state using conventional algorithms such as LS and MMSE. For broadband wireless channels, the channel state *H*_{i} energy is mainly concentrated in a few elements and can be approximated as:

$$ {H}_i\left(\mathrm{n}\right)=\sum \limits_{k=0}^{K-1}{\alpha}_{i,k}\delta \left(n-{\tau}_{i,k}\right)\kern0.5em \mathrm{n}\in \left[0,\kern0.5em \mathrm{L}\right] $$

(5)

The gain of the *k*th path is *α*_{i, k}, and the delay is *τ*_{i, k}. As can be seen from the Formula (5), the element value of *H*_{i} is *α*_{i, k} at *τ*_{i, k}, the other element values are 0, and there are *K* non-zero elements. Record the path delay set as Ω_{i} = [*τ*_{i, 0}, *τ*_{i, 1},⋯, *τ*_{i, K − 1}], the path gain set as *A*_{i} = [*α*_{i, 0}, *α*_{i, 1},⋯, *α*_{i, K − 1}]. Without loss of generality, suppose 0 ≤ τ_{i,0} ≤ τ_{i,1} ≤ … ≤ τ_{i,K-1} ≤ L-1. In general, *K* < <*L*, that is, the channel has a high sparsity, so the channel estimation can be performed using CS algorithm. In CS algorithms, *R*_{i} is referred to as measurement vector, Φ_{i} is a measurement matrix, *M* is number of measurements, and *K* is sparsity of *H*_{i}. The idea of compressive sensing is to turn the problem of finding *L* unknowns *H*_{i} into a problem of estimating 2*K* unknowns Ω_{i}, *A*_{i}, the key of which lies in correct estimation of Ω_{i}. The reduced number of key unknowns enables effective improvement of performance in channel estimation. When *M* < *L* + 1, the problem of solving *H*_{i} in Formula (3) becomes a problem of solving underdetermined equation. *H*_{i} is a sparse variable, then according to compressive sensing theory, the underdetermined equation can be solved by solving the following optimization problem to obtain the estimated channel parameter *H*_{i}:

$$ {\widehat{H}}_i=\underset{H_i}{\arg \min }{\left\Vert {H}_i\right\Vert}_0\kern0.5em \mathrm{s}.\mathrm{t}.{\left\Vert {R}_i-{\Phi}_i{H}_i\right\Vert}_2\le \varepsilon $$

(6)

Where *ε* is a non-negative constant related to noise. In recent years, compressive sensing theory has been developed to provide corresponding algorithms for solution of the above problems, including basis pursuit (BP) algorithm, greedy search-based matching pursuit (MP) algorithm, and relaxation convex optimization algorithm. Among them, MP is the most widely used reconstruction algorithm with the lowest complexity and the largest number of measurements.

Take the normalized random measurement matrix proposed in literature [12] as an example. Record it as \( {G}_i={\Phi}_i^T{\Phi}_i \), *G*_{i} is a Hermitian matrix. When all elements in Φ_{i} are identically distributed random numbers and the columns are normalized (the sum of the squares of the column elements is 1), the diagonal elements of *G*_{i} are always 1, and the remaining elements are normally distributed random numbers with a mean of 0 and a variance of 1/*M*. Columns with index equal to Ω_{i} are taken from the matrix *G*_{i} to constitute matrix *B*, rows with index equal to Ω_{i} are taken from matrix *B* to constitute the sub-matrix \( I+{G}_{i{\Omega}_i} \) (*I* is a unit matrix). The remaining rows of matrix *B* make up a sub-matrix \( {G}_{i{\Omega}_i^{\hbox{'}}} \). Taking the orthogonal matching pursuit (OMP) algorithm as an example, the condition [13] for correct choice of τ_{i,k} in each iteration by OMP is that the inequation (7) is established.

$$ {\left\Vert \left(I+{G}_{i{\Omega}_i}\right){A}_i\right\Vert}_{\infty }>{\left\Vert {G}_{i{\Omega}_i^{\hbox{'}}}{A}_i\right\Vert}_{\infty } $$

(7)

All the elemental and non-diagonal elements of \( {G}_{i{\Omega}_i^{\hbox{'}}} \) are normally distributed random numbers with a mean value of 0 and a variance of 1/*M*. Therefore, for a larger *M*, the establishment probability of inequation (7) is higher. That is, the accurate rate in estimating Ω_{i} can be increased by increasing the measurements number *M*.

### Parameterized channel estimation based on priori information

Reconstruction algorithm represents the development focus of compressive sensing theory. Among the existing greedy search-based algorithms, OMP and subspace pursuit (SP) are the two most widely used ones. However, like other matching pursuit algorithms, both algorithms demand signal sparsity and channel length as priori information. These two parameters are unknown in the receiving end of wireless communication, so the practicality is limited. Literature [9] proposed a PA-CoSaMP algorithm. Based on the good autocorrelation property of the training sequence, the priori information including rough estimation of channel length and sparse level are obtained. However, since the received training sequence are contaminated by the interference caused by the preceding data block, the estimated prior information need to be compensated. This paper proposed a priori information aided (PIA) scheme, which is suitable for most matching pursuit algorithms. Several consecutive data blocks are taken as a frame according to the time correlation of wireless channel. It is considered that the path delay of the channel within a frame remains unchanged and the path gain varies slightly. In each frame, a cyclic prefix of the first training sequence is inserted as guard interval to resist IBI. The length is taken as the maximum delay spread of the channel. Then the training sequence without IBI can be used to obtain the channel priori information, which lead to a higher reconstruction of probability. The frame structure for the proposed scheme is shown in Fig. 2.

As shown in Fig. 2, *C*_{n, 0} consists of UW with a length of *P* and its cyclic prefix, let \( {C}_{n,0}={\left[{c}_{P-{L}_{\mathrm{max}}},c{}_{P-{L}_{\mathrm{max}}+1},\cdots, {c}_{P-1},{c}_0,c{}_1,\cdots, {c}_{P-1}\right]}^T \), and *L*_{max} is equal to the length of maximum delay spread that may exist in the actual channel. Set *L*_{max} = *τ*_{max} × *f*_{s}, in which *τ*_{max} is the maximum multipath delay and *f*_{s} is the sampling rate. Take *J* consecutive blocks of data as a frame, so for any *i*, *j* ∈ [1, *J*], there is *c*_{n, i} = *c*_{n, j} = [*c*_{0}, *c*_{1},⋯, *c*_{P − 1}]^{T}. The coherence time of the channel is assumed to be *T*_{c}, i.e., *T*_{c} = 1/*f*_{d}, and *f*_{d} is the Doppler frequency shift, which can be easily estimated at the receiving end [14]. When *J* = *T*_{c}/(*T*_{s}(*P* + *N*)) − 1, it is considered that the path delay of the channel remains unchanged and the path gain varies slightly within J consecutive data blocks.

The key to the PIA scheme lies in the estimation of channel length and sparsity. According to LS criterion, the last P samples of the known sequence *C*_{n, 0} without interference from unknown data can be used to obtain the channel frequency domain response \( {\widehat{H}}_n \) of the *n*th frame, and \( {\widehat{H}}_n={\left[{\widehat{H}}_{n,0},\widehat{H}{}_{n,1},\cdots, {\widehat{H}}_{n,P-1}\right]}^T \). Let *h*_{n} = [*h*_{n, 0}, *h*_{n, 1},⋯, *h*_{n, P − 1}]^{T} be the channel time domain impulse response of the nth frame, then \( {\widehat{H}}_n \) and *h*_{n} will satisfy the relationship in (8):

$$ {\widehat{H}}_n={W}_P{h}_n+{n}_p $$

(8)

Where *W*_{P} is the Fourier transform matrix in the size of *P* × *P*, *n*_{p} is the white Gaussian noise vector of *P* × 1, the variance is \( {\sigma}_{\mathrm{n}}^2 \). Owing to the unique inverse existence of *W*_{P}, *h*_{n} can be reduced to inverse Fourier transform of \( {\widehat{H}}_n \) at the point *P*, i.e., \( {\widehat{h}}_n={W_P}^{-1}{\widehat{H}}_n \). Let \( {\widehat{h}}_n={\left[{\widehat{h}}_{n,0},\widehat{h}{}_{n,1},\cdots, {\widehat{h}}_{n,P-1}\right]}^T \), then the estimate \( {\widehat{S}}_n \) of impulse response power in the nth frame can be calculated as\( {\widehat{S}}_n={\left[{\widehat{S}}_{n,0},\widehat{S}{}_{n,1},\cdots, {\widehat{S}}_{n,P-1}\right]}^T=\left[{\left|{\widehat{h}}_{n,0}\right|}^2,{\left|{\widehat{h}}_{n,1}\right|}^2,\cdots, {\left|{\widehat{h}}_{n,P-1}\right|}^2\right] \). By averaging the power of \( {\widehat{S}}_n \) within two consecutive frames, the estimated power of each path \( \widehat{S} \) can be obtained, so \( \widehat{S}=\frac{1}{2}\left({\widehat{S}}_n+{\widehat{S}}_{n+1}\right) \). \( \widehat{S} \) can be seen as a representation of the delay power spectrum which reflects statistical properties of the channel [15, 16]. In this algorithm, \( \widehat{S} \) is defined as a time observation and corresponding time parameters can be extracted from it. Let \( \widehat{S}={\left[{\widehat{S}}_0,\widehat{S}{}_1,\cdots, {\widehat{S}}_{P-1}\right]}^T \), then the elements in \( \widehat{S} \) meet the definition in (9):

$$ {\widehat{S}}_l=\frac{1}{2}\sum \limits_{i=n}^{n+1}{\left|{\widehat{h}}_{i,l}\right|}^2,l=0,1,\dots, P-1 $$

(9)

The elements in the vector \( \widehat{S} \) are arranged in descending order to obtain vector \( {\hat{S}}^{\prime }={\left[{\hat{S}}_0^{\prime },{\hat{S}}_1^{\prime },\cdots, {\hat{S}}_{P-G}^{\prime },{\hat{S}}_{P-1}^{\prime}\right]}^T \). The later *G* elements of \( {\widehat{S}}^{\prime } \) are selected to represent the noise power. In simulation experiments, *G* is selected according to different channel conditions. The latter *G* elements are averaged to obtain average noise *N*_{av}, which is subtracted from time observation \( \widehat{S} \) to obtain the vector, i.e., \( {\widehat{S}}^{{\prime\prime} }=\widehat{S}-{N}_{av} \). \( {\widehat{S}}^{{\prime\prime} } \) is searched according to (10) criterion to obtain the index:

$$ {\widehat{\Omega}}_n=\left\{l|{\widehat{S}}_l^{"}>\gamma, l=0,1,\dots, P-1\right\} $$

(10)

The elements in the index set \( {\widehat{\Omega}}_n \) obtained according to this criterion can be regarded as an estimate of the multipath delay position for the nth frame. The threshold will be determined by Formula (11) [17]:

$$ \gamma =\max \left\{{{\widehat{S}}_{\mathrm{max}}^{"}}{\ast}{10}^{-\left({\gamma}_1/10\right)},{{\widehat{S}}_{\mathrm{min}}^{"}}{\ast }{10}^{\gamma_2/10}\right\} $$

(11)

Where, \( {\widehat{S}}_{\mathrm{max}}^{"} \) and \( {\widehat{S}}_{\mathrm{min}}^{"} \) respectively represent the maximum and minimum values in \( {\widehat{S}}^{{\prime\prime} } \), \( {\gamma}_1,{\gamma}_2 \) are a representation of the energy\( {\widehat{S}}^{{\prime\prime} } \), which are determined by simulation experiments according to different systems in specific calculations. Suppose \( \widehat{L}=\max \left\{{\widehat{\Omega}}_n\right\} \) as the value of estimated channel length, and \( \widehat{K}={\left\Vert {\widehat{\Omega}}_n\right\Vert}_0 \) as the channel sparsity. According to the estimated value \( \widehat{L} \), the position of the IBI-free region in the received training sequence is determined. The input parameters of the reconstruction algorithms are initialized. The measurement matrix Φ_{i} at each moment is obtained from the channel length \( \widehat{L} \), as shown in Eq. (12).

$$ {\Phi}_{\mathrm{i}}={\left[\begin{array}{ccccc}{c}_{i,\widehat{L}}& {c}_{i,\widehat{L}-1}& {c}_{i,\widehat{L}-2}& \cdots & {c}_{i,0}\\ {}{c}_{i,\widehat{L}+1}& {c}_{i,\widehat{L}}& {c}_{i,\widehat{L}-1}& \cdots & {c}_{i,1}\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}{c}_{i,P-1}& {c}_{i,P-2}& {c}_{i,P-3}& \cdots & {c}_{i,P-\widehat{L}\hbox{-} 1}\end{array}\right]}_{M\times \left(\widehat{L}+1\right)} $$

(12)

The measurement vector is \( {R}_i={\left[{d}_{i,\widehat{L}},{d}_{i,\widehat{L}+1},\cdots, {d}_{i,P-1}\right]}^T \), and the number of measurements is \( M=P-\widehat{L} \). The subsequent steps are consistent with those of classic MP algorithms and will not be elaborated here. The OMP and SP algorithms based on prior information are referred to as PIA-OMP and PIA-SP respectively in what follows.

Regarding to the proposed PIA scheme, \( {\widehat{H}}_n \) can be efficiently calculated by *P*-point FFT to realize the LS estimation, and \( {\widehat{h}}_n \) can be calculated by *P*-point IFFT. Same as the conventional methods, the main computational burden of the PIA scheme comes from the MP algorithm.