Zero-forcing (or null-steering) precoding is a method of spatial signal processing by which a multiple antenna transmitter can null the multiuser interference in a multi-user MIMO wireless communication system.[1] When the channel state information is perfectly known at the transmitter, the zero-forcing precoder is given by the pseudo-inverse of the channel matrix. Zero-forcing has been used in LTE mobile networks.[2]
In a multiple antenna downlink system which comprises
Nt
K
K\leqNt
k
yk=
| T | |
| h | |
| k |
x+nk, k=1,2,\ldots,K
where
x=
| K | |
| \sum | |
| i=1 |
\sqrt{Pi}siwi
Nt x 1
nk
hk
Nt x 1
wi
Nt x 1
( ⋅ )T
\sqrt{Pi}
si
| 2) | |
| E(|s | |
| i| |
=1
The above signal model can be more compactly re-written as
y=HTWDs+n.
where
y
K x 1
H=[h1,\ldots,hK]
Nt x K
W=[w1,\ldots,wK]
Nt x K
D=diag(\sqrt{P1},\ldots,\sqrt{PK})
K x K
s=[s1,\ldots,
| T | |
| s | |
| K] |
K x 1
A zero-forcing precoder is defined as a precoder where
wi
i
hj
j
j ≠ i
wi\perphj if i ≠ j.
Thus the interference caused by the signal meant for one user is effectively nullified for rest of the users via zero-forcing precoder.
From the fact that each beam generated by zero-forcing precoder is orthogonal to all the other user channel vectors, one can rewrite the received signal as
yk=
| T | |
| h | |
| k |
| K | |
| \sum | |
| i=1 |
\sqrt{Pi}siwi+nk=
| T | |
| h | |
| k |
wk\sqrt{Pk}sk+nk, k=1,2,\ldots,K
The orthogonality condition can be expressed in matrix form as
HTW=Q
where
Q
K x K
Q
W
HT
W=\left(HT\right)+=H(HTH)-1
Given this zero-forcing precoder design, the received signal at each user is decoupled from each other as
yk=\sqrt{Pk}sk+nk, k=1,2,\ldots,K.
Quantify the amount of the feedback resource required to maintain at least a given throughput performance gap between zero-forcing with perfect feedback and with limited feedback, i.e.,
\DeltaR=RZF-RFB\leqlog2g
Jindal showed that the required feedback bits of a spatially uncorrelated channel should be scaled according to SNR of the downlink channel, which is given by:
B=(M-1)log2\rhob,m-(M-1)log2(g-1)
where M is the number of transmit antennas and
\rhob,m
To feed back B bits though the uplink channel, the throughput performance of the uplink channel should be larger than or equal to 'B'
bFBlog2(1+\rhoFB)\geqB
where
b=\OmegaFBTFB
\rhoFB
\DeltaR\leqlog2g
bFB\geq
| B | |
| log2(1+\rhoFB) |
=
| (M-1)log2\rhob,m-(M-1)log2(g-1) | |
| log2(1+\rhoFB) |
\rhob,m/\rhoFB)=Cup,dn
| * | |
| b | |
| FB,min |
=
| \lim | |
| \rhoFB\toinfty |
| (M-1)log2\rhob,m-(M-1)log2(g-1) | |
| log2(1+\rhoFB) |
=M-1
It follows from the above equation that the feedback resource (
bFB
If the transmitter knows the downlink channel state information (CSI) perfectly, ZF-precoding can achieve almost the system capacity when the number of users is large. On the other hand, with limited channel state information at the transmitter (CSIT) the performance of ZF-precoding decreases depending on the accuracy of CSIT. ZF-precoding requires the significant feedback overhead with respect to signal-to-noise-ratio (SNR) so as to achieve the full multiplexing gain.[3] Inaccurate CSIT results in the significant throughput loss because of residual multiuser interferences. Multiuser interferences remain since they can not be nulled with beams generated by imperfect CSIT.