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receivers, then a rate vector ![${\bf R} = [R_1, R_2, \ldots, R_N]$](http://windowsil.org/latexrender/pictures/bd3bc224c57f6561118b879ecc2a6301.gif "${\bf R} = [R_1, R_2, \ldots, R_N]$")
is said to be achievable on the broadcast channel if information can be transmitted reliably at rate 
from the transmitter to the receiver 
,
, simultaneously. The capacity of broadcast channel is defined as the convex hull of all achievable rate vectors. Finding the capacity of a general broadcast capacity has been a long standing open problem in information theory. In this article we will discuss the multiple input multiple output broadcast channel (MIMO-BC) whose capacity region has been found recently by Weingarten et. al.
antennas wants to transmit independent information to 
receiver is equipped with 
antennas 
. Let 
be the signal transmitted by the transmitter, then the received signal 
received by receiver ![\[{\bf y}_n = H_n{\bf x} + {\bf v}_n,\]](http://windowsil.org/latexrender/pictures/3dc6d0819d1470031d4660b271993574.gif "\[{\bf y}_n = H_n{\bf x} + {\bf v}_n,\]")
where 
is a 
channel matrix and 
is the additive white Gaussian noise (AWGN) with covariance matrix 
.
is the signal for receiver 
is transmitted by the transmitter. Dirty paper coding (DPC) is a technique developed by Costa for single input single output (SISO) AWGN channels when there is an interference signal present at the receiver together with AWGN. We assume the interference signal is known non causally at the transmitter. In this setting Costa showed that the capacity is 
which is exactly equal to the capacity of AWGN channel without any interference. Thus, DPC is shown to completely eliminate the effect of interference. Using DPC for the MIMO BC, if the signal for the first receiver is 
, then the signal 
for receiver 
is generated using 
as the interference signals and 
is transmitted from the transmitter. From now on we consider 
in this article for simplicity. With 
and 
, simultaneously achievable for receiver 
and ![\[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_1\right)}{\det\left({\bf S}_2+{\bf N}_1\right)}\]](http://windowsil.org/latexrender/pictures/d231b48745808e6900b2b7cef9bff54d.gif "\[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_1\right)}{\det\left({\bf S}_2+{\bf N}_1\right)}\]")
and ![\[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_2 + {\bf N}_2\right)} {\det\left({\bf N}_2\right)},\]](http://windowsil.org/latexrender/pictures/439320c9f3e1132578ecfaea9d24fa59.gif "\[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_2 + {\bf N}_2\right)} {\det\left({\bf N}_2\right)},\]")
where 
is the covariance matrix of ![\[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1 + {\bf N}_1\right)}{\det\left({\bf N}_1\right)}\]](http://windowsil.org/latexrender/pictures/3bbf2faa4f6d254df3cf1b10049f875f.gif "\[R_1^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1 + {\bf N}_1\right)}{\det\left({\bf N}_1\right)}\]")
and ![\[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_2\right)}{\det\left({\bf S}_1+{\bf N}_2\right)}.\]](http://windowsil.org/latexrender/pictures/916d11f440683b3dcc4808c18fb34666.gif "\[R_2^{DPC} = \frac{1}{2}\log\frac{\det\left({\bf S}_1+ {\bf S}_2 + {\bf N}_2\right)}{\det\left({\bf S}_1+{\bf N}_2\right)}.\]")
We denote the convex hull of all rate vectors achievable with DPC as 
.
, where ![\[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}_n, \ n=1,2.\]](http://windowsil.org/latexrender/pictures/8357ddee5a7849d18c5943276714697d.gif "\[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}_n, \ n=1,2.\]")
Since we are assuming 
to be a square matrix, without loss we can multiply the received signal by 
without changing the capacity. Note that ![\[{\bf y}_n = {\bf x}_n + {\bf v}_n, \ n=1,2.\]](http://windowsil.org/latexrender/pictures/41d56e0472684582a2f72d21c7a50874.gif "\[{\bf y}_n = {\bf x}_n + {\bf v}_n, \ n=1,2.\]")
Then a transformation of AMIMO-BC is considered, called degraded and aligned MIMO BC (DAMIMO-BC), where ![\[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}^{*}_n, \ n=1,2.\]](http://windowsil.org/latexrender/pictures/1b4465292912001d0491981856889816.gif "\[{\bf y}_n = {\bf H}_n{\bf x}_n + {\bf v}^{*}_n, \ n=1,2.\]")
with 
, 
denotes the covariance matrix. For any two matrices, 
,by 
we mean 
is a negative semidefinite matrix. It is easy to see that DAMIMO-BC is a degraded BC channel, i.e. one receiver receives less noisy signal than the other. For the degraded BC it is known that superposition coding is optimal, however, it is not known whether Gaussian signaling is optimal or not.
For a single input single output degraded BC (SISO-DBC), where the transmitter and each receiver has a single antenna, Bergmans showed that Gaussian signaling is optimal, however, the proof of Bergmans does not extend to the DAMIMO-BC case.
be the set of all achievable rate vectors 
with Gaussian superposition coding for DAMIO-BC and let 
and 
be the input covariance matrices achieving 
and 
. To show that any rate vector which does not belong to 
is not achievable, an enhanced DAMIMO-BC is introduced, which is defined as ![\[{\bf y}_n = {\bf x}_n + {\hat {\bf v}}_n, \ n=1,2.\]](http://windowsil.org/latexrender/pictures/c87dd9d726392c3ff8868b2f862e5a20.gif "\[{\bf y}_n = {\bf x}_n + {\hat {\bf v}}_n, \ n=1,2.\]")
where ![\[cov({\hat {\bf v}}_1) \le cov({\bf v}^{*}_1), cov({\hat {\bf v}}_1) + cov({\hat {\bf v}}_2) \le cov({\bf v}^{*}_1) + cov({\bf v}^{*}_2).\]](http://windowsil.org/latexrender/pictures/db13b5f096f8748b416db46f7bbf44e4.gif "\[cov({\hat {\bf v}}_1) \le cov({\bf v}^{*}_1), cov({\hat {\bf v}}_1) + cov({\hat {\bf v}}_2) \le cov({\bf v}^{*}_1) + cov({\bf v}^{*}_2).\]")
Clearly, any rate vector achievable on DAMIMO-BC is also achievable on the enhanced DAMIMO-BC, thus the capacity region of DAMIMO-BC is contained inside the capacity region of enhanced DAMIMO-BC.
, such that ![\[\alpha_1\left(B_1 + cov({\hat {\bf v}}_1)\right) = cov({\hat {\bf v}}_{2}).\]](http://windowsil.org/latexrender/pictures/1f40c6624992a5093cb654b9216fcbb0.gif "\[\alpha_1\left(B_1 + cov({\hat {\bf v}}_1)\right) = cov({\hat {\bf v}}_{2}).\]")
achievable on enhanced DAMIMO-BC with Gaussian signaling is equal to rate vector 
obtained with Gaussian signaling for DAMIMO-BC.
which lies outside the region 