Table 1 Variables and notations for the new sensing matrix

M

M = p ^r for prime p and a positive integer r

N ^′

N ^′ = p ^2r - 1 = M ² - 1

L

positive integer, 2 ≤ L ≤ M - 1

N

N = (M + 1)L

${\mathbb{F}}_{p^m}$

${\mathbb{F}}_{p^m}=\{0,1,\alpha ,\dots ,{\alpha }^{p^m-2}\}$ for the primitive element α

${\mathbb{Z}}_v$

${\mathbb{Z}}_v=\{0,1,2,\dots ,v-1\}$ for a positive integer v

C _s

cyclotomic coset with the coset leader s

Γ _p (v)

set of coset leaders of cyclotomic cosets modulo v over p

D

row index set, D = {d ₀,d ₁,…,d _M-1} ≡ {M,M-1,…,1} (mod M + 1)

$\tilde{\mathbf{A}}$

M × N ^′ partial Fourier matrix

A’

M × N ^′ matrix after row/column rearrangement of$\tilde{\mathbf{A}}$

σ ^(l)

M × (M + 1) submatrix of A’, 0 ≤ l ≤ L-1

${\gamma }_k^{(l)}$

${\gamma }_k^{(l)}=exp\left(j\frac{\pi d_{k}l}{M-1}\right)\times exp\left(-j\frac{\pi d_{k}l}{M+1}\right),0\le k\le M-1$

Γ ^(l)

M × M diagonal matrix with diagonal entries of${\gamma }_k^{(l)}$

${\mathbf{\text{F}}}_{M+1}^{\prime }$

M × (M + 1) DFT matrix without the first row

A

M × N new sensing matrix, A = [σ ⁽⁰⁾∣⋯∣σ ^(L-1)]