From (3) and (4), it follows that (9)ϕxi(tj)=arctanx~i(tj)xi(tj)=arctan1πp.v.∫−∞+∞Ar(t′)−bi(t′)tj−t′dt′Ar(tj)−bi(tj),i=1,2;j=1,2,…,N. Combining (2) and (9), we see that phase synchrony between x1 and x2 is a function of A and defined as Rx1x2(A). From (9), we can get (10)limA→+∞ϕxi(tj)=limA→+∞arctan1πp.v.∫−∞+∞Ar(t′)tj−t′dt′−1πp.v.∫−∞+∞bi(t′)tj−t′dt′Ar(tj)−bi(tj)=limA→+∞arctan1πp.v.∫−∞+∞Ar(t′)tj−t′dt′−1πp.v.∫−∞+∞bi(t′)tj−t′dt′Ar(tj)=limA→+∞arctan1πp.v.∫−∞+∞Ar(t′)tj−t′dt′Ar(tj)=arctan1πp.v.∫−∞+∞r(t′)tj−t′dt′r(tj),j=1,2,…,N. Thus, we have limA→+∞ϕx1(tj)−limA→+∞ϕx2(tj)=arctan1πp.v.∫−∞+∞r(t′)tj−t′dt′r(tj)−arctan1πp.v.∫−∞+∞r(t′)tj−t′dt′r(tj)=0,j=1,2,…,N. As a result, from (2), we obtain limA→+∞Rx1x2(A)=limA→+∞∣1N∑j=1Nei[ϕx1(tj)−ϕx2(tj)]∣=1. Therefore, when coefficient A is large enough, the phase synchrony of two referential signals x1 and x2 will be larger than that of nonreferential signals b1 and b2.
