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"James K." <txdiversity@hotmail.com> wrote in message =
news:br0o9q$lto$1@news1.kornet.net...
For further understanding, I show the examples of the pdf functions in =
(2), which is often used in the evaluation of a SU-MIMO capacity. Note =
that SU is abbreviated from Single User.
- For ZF receiver: p(r) =3D Exp[-r]
- Optimal processing: p(r) =3D Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}]=20
> I thought that you wanted to see just some applications of MIMO on the =
> frequency selective channel. Regarding to the capacity of MIMO on the=20
> frequency selective channel, it looks like really new area. As matter =
of=20
> fact, the capacity of even single path MIMO channel seems to have more =
> things to be researched, yet.
>=20
> In practice, as far as I know, they (G. J. Foschini and M. J., Gans)=20
> researched starting from the ergodic capacity so I'm not sure yet =
about=20
> its exact formulae for instatant case. The ergodic capacity is =20
>=20
> E[C] =3D E[Sum[Log[2,1+(P/N)*r[i]]],{i,1,min[M,N]}], -- (1)
>=20
> where E[.] is the expectation function, Sum[.,{i,1,min[M,N]}] is the=20
> sumation operator ranging i=3D1 to minimum value of [M,N], Log[2,.] is =
> the logarithm function with base 2, P is SNR (Signal to Noise power=20
> Ratio) when the noise distribution is normalized to ~N(0,1), r[i] is=20
> i-th eigen value of the channel matrix multiple H'H, and M, N are the=20
> number of Rx and Tx antenna elements, respectively. This equation (1) =
is=20
> simply solved to the equation (2) when we assume M=3DN,
>=20
> E[C] =3D Integrate[N*Log[2,1+(P/N)*r]*p(r), {r,0,Infinity}], -- (2)
> =20
> where surprisingly p(r) is the generalized pdf of SNR for the single =
path=20
> Rayleigh channel. For the instant channel, we need to know values of =
not=20
> only 1st moment but also other higher order moment. Thus, this case is =
> not clear yet to me.
In practice, we evaulate (2) by often using the p(r) as:
- For ZF receiver: p(r) =3D Exp[-r],
- Optimal processing: p(r) =3D =
Sum[Exp[-r]*LaguerreL[k,r]^2,{k,0,N-1}],=20
where LaguerreL[k,r] is the Laguerre polynomial of oder k=20
=20
LaguerreL[k,r] =3D (Exp[r]/k!)*D[Exp[-r]*x^k,{x,k}]. -- (3) =20
> In brief, I have seen a study of the capacity formulae in SISO =
frequncy=20
> selective channel, e.g. OFDM analysis. Its extention to MIMO supposed =
to=20
> be still *Big Deal* for us to study, as it was noted by Johnathn.
>=20
> BR,
> ------
> James K. (txdiversity@hotmail.com)
> - Any remarks, proposal and/or indicator would be greatly respected.
> - Private opinions: These are not the opinions from my affiliation.
> [Home] http://home.naver.com/txdiversity
> --
BR,
------
James K. (txdiversity@hotmail.com)
- Any remarks, proposal and/or indicator would be greatly respected.
- Private opinions: These are not the opinions from my affiliation.
[Home] http://home.naver.com/txdiversity
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<BODY bgColor=3D#ffffff background=3D""><FONT face=3D굴림 =
size=3D2>"James K."=20
<txdiversity@hotmail.com> wrote in message news:</FONT><A=20
href=3D"mailto:br0o9q$lto$1@news1.kornet.net"><FONT =
face=3D굴림=20
size=3D2>br0o9q$lto$1@news1.kornet.net</FONT></A><FONT =
face=3D굴림=20
size=3D2>...<BR><BR>For further understanding, I show the examples of =
the pdf=20
functions in (2), which is often used in the evaluation of a SU-MIMO =
capacity.=20
Note that SU is abbreviated from Single User.<BR> - For ZF =
receiver:=20
p(r) =3D Exp[-r]<BR> - Optimal processing: p(r) =3D=20
Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}] <BR><BR>> I thought that you =
wanted to=20
see just some applications of MIMO on the <BR>> frequency selective =
channel.=20
Regarding to the capacity of MIMO on the <BR>> frequency selective =
channel,=20
it looks like really new area. As matter of <BR>> fact, the capacity =
of even=20
single path MIMO channel seems to have more <BR>> things to be =
researched,=20
yet.<BR>> <BR>> In practice, as far as I know, they (G. J. =
Foschini and M.=20
J., Gans) <BR>> researched starting from the ergodic capacity so I'm =
not sure=20
yet about <BR>> its exact formulae for instatant case. The ergodic =
capacity=20
is <BR>> <BR>> E[C] =3D=20
E[Sum[Log[2,1+(P/N)*r[i]]],{i,1,min[M,N]}], -- (1)<BR>> =
<BR>> where=20
E[.] is the expectation function, Sum[.,{i,1,min[M,N]}] is the <BR>> =
sumation=20
operator ranging i=3D1 to minimum value of [M,N], Log[2,.] is <BR>> =
the=20
logarithm function with base 2, P is SNR (Signal to Noise power <BR>> =
Ratio)=20
when the noise distribution is normalized to ~N(0,1), r[i] is <BR>> =
i-th=20
eigen value of the channel matrix multiple H'H, and M, N are the =
<BR>> number=20
of Rx and Tx antenna elements, respectively. This equation (1) is =
<BR>>=20
simply solved to the equation (2) when we assume M=3DN,<BR>>=20
<BR>> E[C] =3D Integrate[N*Log[2,1+(P/N)*r]*p(r), =
{r,0,Infinity}],=20
--=20
(2)<BR>> &n=
bsp; &nb=
sp; =20
<BR>> where surprisingly p(r) is the generalized pdf of SNR for the =
single=20
path <BR>> Rayleigh channel. For the instant channel, we need to know =
values=20
of not <BR>> only 1st moment but also other higher order moment. =
Thus, this=20
case is <BR>> not clear yet to me.<BR><BR>In practice, we evaulate =
(2) by=20
often using the p(r) as:<BR><BR> - For ZF receiver: p(r) =3D=20
Exp[-r],<BR> - Optimal processing: p(r) =3D=20
Sum[Exp[-r]*LaguerreL[k,r]^2,{k,0,N-1}], <BR><BR>where LaguerreL[k,r] is =
the=20
Laguerre polynomial of oder k <BR> =20
<BR> LaguerreL[k,r] =3D (Exp[r]/k!)*D[Exp[-r]*x^k,{x,k}]. -- =
(3) <BR><BR>> In brief, I have seen a study of the =
capacity=20
formulae in SISO frequncy <BR>> selective channel, e.g. OFDM =
analysis. Its=20
extention to MIMO supposed to <BR>> be still *Big Deal* for us to =
study, as=20
it was noted by Johnathn.<BR>> <BR>> BR,<BR>> ------<BR>> =
James K.=20
(txdiversity@hotmail.com)<BR>> - Any remarks, proposal and/or =
indicator would=20
be greatly respected.<BR>> - Private opinions: These are not the =
opinions=20
from my affiliation.<BR>> <STRONG><FONT =
color=3D#ff0000>[Home]</FONT></STRONG>=20
http://home.naver.com/txdiversity<BR>> =
--<BR><BR>BR,<BR>------<BR>James K.=20
(txdiversity@hotmail.com)<BR>- Any remarks, proposal and/or indicator =
would be=20
greatly respected.<BR>- Private opinions: These are not the opinions =
from my=20
affiliation.<BR><STRONG><FONT color=3D#ff0000>[Home]</FONT></STRONG>=20
http://home.naver.com/txdiversity</FONT></BODY></HTML>
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capacity of MIMO frequency selective channel
Started by ●December 6, 2003
Reply by ●December 9, 20032003-12-09
Reply by ●December 9, 20032003-12-09
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For further understanding, I show the examples of the pdf functions in =
(2), which is often used in the evaluation of a SU-MIMO capacity. Note =
that SU is abbreviated from Single User.
- For ZF receiver: p(r) =3D Exp[-r]
- Optimal processing: p(r) =3D Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}]=20
---
"James K." <txdiversity@hotmail.com> wrote in message =
news:br0o9q$lto$1@news1.kornet.net...
I thought that you wanted to see just some applications of MIMO on the =
frequency selective channel. Regarding to the capacity of MIMO on the =
frequency selective channel, it looks like really new area. As matter of =
fact, the capacity of even single path MIMO channel seems to have more =
things to be researched, yet.
In practice, as far as I know, they (G. J. Foschini and M. J., Gans) =
researched starting from the ergodic capacity so I'm not sure yet about =
its exact formulae for instatant case. The ergodic capacity is =20
E[C] =3D E[Sum[Log[2,1+(P/N)*r[i]]],{i,1,min[M,N]}], -- (1)
where E[.] is the expectation function, Sum[.,{i,1,min[M,N]}] is the =
sumation operator ranging i=3D1 to minimum value of [M,N], Log[2,.] is =
the logarithm function with base 2, P is SNR (Signal to Noise power =
Ratio) when the noise distribution is normalized to ~N(0,1), r[i] is =
i-th eigen value of the channel matrix multiple H'H, and M, N are the =
number of Rx and Tx antenna elements, respectively. This equation (1) is =
simply solved to the equation (2) when we assume M=3DN,
E[C] =3D Integrate[N*Log[2,1+(P/N)*r]*p(r), {r,0,Infinity}], -- (2)
=20
where surprisingly p(r) is the general pdf of SNR for the single path =
Rayleigh channel. For the instant channel, we need to know values of not =
only 1st moment but also other higher order moment. Thus, this case is =
not clear yet to me.
+ For further understanding, I show the examples of the pdf functions in =
(2), which is often used in the evaluation of a + SU-MIMO capacity. Note =
that SU is abbreviated from Single User.
+ - For ZF receiver: p(r) =3D Exp[-r]
+ - Optimal processing: p(r) =3D =
Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}]=20
In brief, I have seen a study of the capacity formulae in SISO frequncy =
selective channel, e.g. OFDM analysis. Its extention to MIMO supposed to =
be still *Big Deal* for us to study, as it was noted by Johnathn.
BR,
------
James K. (txdiversity@hotmail.com)
- Any remarks, proposal and/or indicator would be greatly respected.
- Private opinions: These are not the opinions from my affiliation.
[Home] http://home.naver.com/txdiversity
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<STYLE></STYLE>
</HEAD>
<BODY bgColor=3D#ffffff background=3D"">For further understanding, I =
show the=20
examples of the pdf functions in (2), which is often used in the =
evaluation of a=20
SU-MIMO capacity. Note that SU is abbreviated from Single =
User.<BR> - For=20
ZF receiver: p(r) =3D Exp[-r]<BR> - Optimal processing: p(r) =3D=20
Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}] <BR><BR>---<BR>"James K."=20
<txdiversity@hotmail.com> wrote in message=20
news:br0o9q$lto$1@news1.kornet.net...<BR><BR>I thought that you wanted =
to see=20
just some applications of MIMO on the frequency selective channel. =
Regarding to=20
the capacity of MIMO on the frequency selective channel, it looks like =
really=20
new area. As matter of fact, the capacity of even single path MIMO =
channel seems=20
to have more things to be researched, yet.<BR><BR>In practice, as far as =
I know,=20
they (G. J. Foschini and M. J., Gans) researched starting from the =
ergodic=20
capacity so I'm not sure yet about its exact formulae for instatant =
case. The=20
ergodic capacity is <BR><BR> E[C] =3D=20
E[Sum[Log[2,1+(P/N)*r[i]]],{i,1,min[M,N]}], -- (1)<BR><BR>where =
E[.] is=20
the expectation function, Sum[.,{i,1,min[M,N]}] is the sumation operator =
ranging=20
i=3D1 to minimum value of [M,N], Log[2,.] is the logarithm function with =
base 2, P=20
is SNR (Signal to Noise power Ratio) when the noise distribution is =
normalized=20
to ~N(0,1), r[i] is i-th eigen value of the channel matrix multiple H'H, =
and M,=20
N are the number of Rx and Tx antenna elements, respectively. This =
equation (1)=20
is simply solved to the equation (2) when we assume M=3DN,<BR><BR> =
E[C] =3D=20
Integrate[N*Log[2,1+(P/N)*r]*p(r), {r,0,Infinity}], --=20
(2)<BR> =
&=
nbsp; =20
<BR>where surprisingly p(r) is the general pdf of SNR for the single =
path=20
Rayleigh channel. For the instant channel, we need to know values of not =
only=20
1st moment but also other higher order moment. Thus, this case is not =
clear yet=20
to me.<BR><BR>+ For further understanding, I show the examples of the =
pdf=20
functions in (2), which is often used in the evaluation of a + SU-MIMO =
capacity.=20
Note that SU is abbreviated from Single User.<BR>+ - For ZF=20
receiver: p(r) =3D Exp[-r]<BR>+ - Optimal processing: p(r) =
=3D=20
Sum[Exp[-r]*LaguereL[k,r]^2,{k,0,N-1}] <BR><BR>In brief, I have seen a =
study of=20
the capacity formulae in SISO frequncy selective channel, e.g. OFDM =
analysis.=20
Its extention to MIMO supposed to be still *Big Deal* for us to study, =
as it was=20
noted by Johnathn.<BR><BR>BR,<BR>------<BR>James K.=20
(txdiversity@hotmail.com)<BR>- Any remarks, proposal and/or indicator =
would be=20
greatly respected.<BR>- Private opinions: These are not the opinions =
from my=20
affiliation.<BR><STRONG><FONT color=3D#ff0000>[Home]</FONT></STRONG>=20
http://home.naver.com/txdiversity</BODY></HTML>
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