(i) Shift transformation matrix: (12).
Z is the lower shift matrix of order N, (10).
For the phase-shift-based selection, we compute the phase shift matrix through (19) given each.
Thus one solution to the phase shift matrix can be expressed as [10, Theorem 2]: (19).
where S is the q×q shift matrix and the operations are in (mathbb {F}_{2}^{q}).
where matrices WTX, Hdelay, ADoppler, and ARX are the transmission connection matrix, the impulse response matrix, the Doppler shift matrix, and the reception channel matrix, respectively.
The Doppler shift matrix ADoppler is the L × L diagonal matrix for the addition of the Doppler frequency shift to each probe antenna branch signal.
An up-conversion of the kth subcarrier to its respective subcarrier frequency is performed by the shift matrix mathbf{P}^{ k)}=mathbf{Psi}left(mathbf{p}^{ k)}right)otimesmathbf{I}_{M}, (11).
After substituting (12) into (11), the desired signal correlation matrix R s with 4 N × 4 N dimension can be rewritten as R s = C ˜ 1 1 2 σ 1 2 H 1 H 1 H + H 2 H 2 H C ˜ 1 H (13). Since H1 and H2 are CFO-independent, R s can only be affected by the frequency shift matrix C ˜ 1.
To transform the interval ([-1,1]) to [0, T], we apply the ((n+1 times (n+1)) shift matrix R, which is defined by begin{aligned} R_{ij}=left{ begin{array}{ll} left( begin{array}{c} i j end{array}right) gamma _{1}^{i-j}gamma _{2}^{j} ;&quad j=0,1,ldots,i,quad i=0,1,ldots,n, 0 ;& i<j, end{array}right.
