96
3 Physical Theory of RFID System Physical Anti-Collision
If α is unknown and the transmitted signal is orthogonal to each other, the coherent
matrix of the transmitted signal R s is unit matrix. Hence, the CRB in Eq. (3.56) can
be written as
CRB(θ ) =
1
8NM
|α|
2
σ 2
(M −1)/2
k=−(M −1)/2
k 2
(π cos θ) 2 (d
2
b + d 2
a )
(3.57)
It can be shown from Eq. (3.57), when the transmitted signal is orthogonal to each
other, CRB is related to the space between two tags, antenna unit and the number
of tags. And with the increasing of the space between two tags, the CRB decreases.
Therefore, we can obtain better CRB by increasing the space between two tags.
If α is already precisely known, we only need to estimate θ and δ
2 . As estimation
of δ
2 doesn’t affect the estimation of θ, the estimation of the CRB can be expressed
as [49]
CRB(θ ) =
1
2SNR(Ma H (θ )R
T
S a(θ ) + a H (θ )R
T
S a(θ )a(θ )
2
)
(3.58)
where ||•|| is the norm of a matrix.
The CRB always increase with the estimation of more parameters, because the
steering vector considers the center of tag unit as the reference point. When the
transmitted signal is orthogonal, the item 3 of the denominator in (3.56) will be
equal to 0, the formula (3.56) will be equal to (3.58). Therefore, whether the target
range is known, it has no effect on the CRB estimation of RFID-MIMO system which
transmits orthogonal signal.
In time-division duplex (TDD) adaptive MIMO system, channel reciprocity is
an inherent characteristic of TDD system channels [38]. In TDD system, uplink
and downlink channels work in the same frequency. Therefore, on the path that
electromagnetic waves transmit, electromagnetic waves which return and go in two
directions will through the same reflection, refraction, diffraction and other physical
disturbance. Accordingly, it can be assumed that uplink and downlink channels have
the same fading characteristics, so we can put the channel state of the uplink channels
as channel state of the downlink channels, which is to say that uplink and downlink
channels have reciprocity. Let H u denote the uplink channel state matrix detected
on the uplink, and H d denote the downlink channel state matrix detected on the
downlink, so the uplink and downlink channel reciprocity of TDD
H u = H
T
d
(3.59)
where the superscript T represents the matrix transpose.
Because of the reciprocity theorem, RFID-MIMO system has a plurality of tags
and the plurality of antennas which corresponded to a plurality of output and input,
3 Physical Theory of RFID System Physical Anti-Collision
If α is unknown and the transmitted signal is orthogonal to each other, the coherent
matrix of the transmitted signal R s is unit matrix. Hence, the CRB in Eq. (3.56) can
be written as
CRB(θ ) =
1
8NM
|α|
2
σ 2
(M −1)/2
k=−(M −1)/2
k 2
(π cos θ) 2 (d
2
b + d 2
a )
(3.57)
It can be shown from Eq. (3.57), when the transmitted signal is orthogonal to each
other, CRB is related to the space between two tags, antenna unit and the number
of tags. And with the increasing of the space between two tags, the CRB decreases.
Therefore, we can obtain better CRB by increasing the space between two tags.
If α is already precisely known, we only need to estimate θ and δ
2 . As estimation
of δ
2 doesn’t affect the estimation of θ, the estimation of the CRB can be expressed
as [49]
CRB(θ ) =
1
2SNR(Ma H (θ )R
T
S a(θ ) + a H (θ )R
T
S a(θ )a(θ )
2
)
(3.58)
where ||•|| is the norm of a matrix.
The CRB always increase with the estimation of more parameters, because the
steering vector considers the center of tag unit as the reference point. When the
transmitted signal is orthogonal, the item 3 of the denominator in (3.56) will be
equal to 0, the formula (3.56) will be equal to (3.58). Therefore, whether the target
range is known, it has no effect on the CRB estimation of RFID-MIMO system which
transmits orthogonal signal.
In time-division duplex (TDD) adaptive MIMO system, channel reciprocity is
an inherent characteristic of TDD system channels [38]. In TDD system, uplink
and downlink channels work in the same frequency. Therefore, on the path that
electromagnetic waves transmit, electromagnetic waves which return and go in two
directions will through the same reflection, refraction, diffraction and other physical
disturbance. Accordingly, it can be assumed that uplink and downlink channels have
the same fading characteristics, so we can put the channel state of the uplink channels
as channel state of the downlink channels, which is to say that uplink and downlink
channels have reciprocity. Let H u denote the uplink channel state matrix detected
on the uplink, and H d denote the downlink channel state matrix detected on the
downlink, so the uplink and downlink channel reciprocity of TDD
H u = H
T
d
(3.59)
where the superscript T represents the matrix transpose.
Because of the reciprocity theorem, RFID-MIMO system has a plurality of tags
and the plurality of antennas which corresponded to a plurality of output and input,