Randy is keeping me thinking here and I wanted to comment on this
again.

My comments below were mostly with spatial multiplexing in mind, and
channel correlation isn't a big deal for beamforming cases.   The
upshot is that MIMO channel capacity computation, well, depends.

So I poked around a bit more and turned up some interesting things.

1. Signal Processing magazine had an interesting article in the May 05
issue (p83) by Petre Stoica et al.  (For some reason a lot of
Scandinavian names turn up regarding MIMO capacity analysis.)   This
particular treatment leans heavily on the covariance matrices of the
input and output vectors.

2. I thought this analysis was especially nice:
http://www.cwc.oulu.fi/nrs04/slides/mimo_capacity.pdf
Again, a Scandihoovian.  This is nice because it points out that MIMO
capacity analysis changes significantly depending on the assumptions
about the channel.  A particularly astute observation is that "The
evaluation of the fading MIMO channel capacity is complicated".  ;)

One thing I am noticing, though, is that analyses tend to simplify the
fading channels by assigning a single fading vector to each element of
the channel matrix, in other words, using the average SNR of each
channel rather than using per-channel water-filling or something like
that.   Regardless, the analyses do use the H matrix or some variant
in the capacity computations so capacity is definitely dependent on
the channel instance.   I think it would be more so if a channel
response vector were used for each entry in the H matrix rather than a
single coefficient.

So, yeah, it's complicated.   And, yeah, it's channel dependent.

I'd be really, really happy if knowledgable folks could shed some more
light on this stuff.

On Thu, 26 Jan 2006 11:31:50 -0700, Eric Jacobsen
<eric.jacobsen@ieee.org> wrote:

>On Sun, 15 Jan 2006 19:10:41 GMT, Randy Yates <yates@ieee.org> wrote:
>
>>Eric Jacobsen <eric.jacobsen@ieee.org> writes:
>>> [...]
>>> I think you're missing the gist of the MIMO issues completely because
>>> in order for MIMO to work the channels _cannot_ be flat.  
>>
>>Do you mean that the noise cannot be flat (white)? If so, then I don't
>>understand why not because I thought that the decorrelation afforded
>>by the (de)spreading processing makes adjacent interference look like
>>uncorrelated noise.
>
>Sorry to be so long in responding, I was off the continent for a
>while.
>
>The noise is still white, but the channel response is frequency
>selective.   I guess one could argue that after equalization the noise
>isn't white any more, but since that happens in the receiver
>processing it isn't really a channel effect.
>
>And spreading and despreading isn't necessary for MIMO...most of the
>MIMO systems of which I'm aware don't use any spreading.
>
>>> It's also kind of moot, in a way, because capacity needs to be
>>> computed for each, specific channel instance.  It's not as
>>> generalized as Shannon's expression.
>>
>>I'm amazed you would say this, Eric. There's probably a
>>misunderstanding somewhere (most likely my own).
>
>No problemo.   We can sort it out of if you're really interested.
>
>>If I were a SISO comm system designer who wanted to get the most
>>capacity possible out of a channel, and MIMO design afforded a higher
>>aggregate rate, then I would partition the single input into multiple
>>inputs on the transmit side and concatenate the multiple outputs
>>together to a single output on the receive side in order to take
>>advantage of the higher rate. Then, voila - a different channel
>>capacity.
>>
>>This is why I assert that MIMO is semantics.  What am I missing?
>
>MIMO is definitely not semantics.   The amount of capacity available
>from a MIMO system depends completely on the channel responses in the
>spatial streams.   The more independent the fading in the streams, the
>higher the capacity.   The more correlated the channel responses, the
>less the capacity.   If all of the spatial streams are completely
>correlated (i.e., identical), then the system is essentially a SISO
>system and SISO capacity will apply.
>
>This is why I said that capacity has to be computed on a
>channel-by-channel basis.   It isn't a generalized approach like we
>get with Shannon's formula, you get what the channel response affords
>you, and it changes every time the channels change.
>
>In a practical MIMO comm system it is then essential that the transmit
>and receive antennas be separated in space far enough that there is a
>reasonable expectation that the channel responses will be
>decorrelated.   For very small devices, say, a handset, this can be
>difficult to achieve because the device just may not be physically
>large enough to get adequate separation in the antennas.
>
>The partitioning scheme you describe is a common model for a MIMO
>system.   There's one input and one output, and the system splits the
>data up across the spatial streams and then recombines it in the
>receiver.   I just got back from the IEEE 802.11 meeting where,
>finally, TGn confirmed acceptance of the single, merged MIMO-based
>proposal.   In the proposed system each spatial stream can be
>modulated independently (sort of a spatial water-filling scheme) in
>order to try to get best use out of the spatial channels.
>
>MIMO is an interesting beast.  It seems to work pretty well, but I
>don't think it's going to be as ubiquitous as some might claim.
>
>Eric Jacobsen
>Minister of Algorithms, Intel Corp.
>My opinions may not be Intel's opinions.
>http://www.ericjacobsen.org
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

On Sun, 15 Jan 2006 19:10:41 GMT, Randy Yates <yates@ieee.org> wrote:

>Eric Jacobsen <eric.jacobsen@ieee.org> writes:
>> [...]
>> I think you're missing the gist of the MIMO issues completely because
>> in order for MIMO to work the channels _cannot_ be flat.  
>
>Do you mean that the noise cannot be flat (white)? If so, then I don't
>understand why not because I thought that the decorrelation afforded
>by the (de)spreading processing makes adjacent interference look like
>uncorrelated noise.

Sorry to be so long in responding, I was off the continent for a
while.

The noise is still white, but the channel response is frequency
selective.   I guess one could argue that after equalization the noise
isn't white any more, but since that happens in the receiver
processing it isn't really a channel effect.

And spreading and despreading isn't necessary for MIMO...most of the
MIMO systems of which I'm aware don't use any spreading.

>> It's also kind of moot, in a way, because capacity needs to be
>> computed for each, specific channel instance.  It's not as
>> generalized as Shannon's expression.
>
>I'm amazed you would say this, Eric. There's probably a
>misunderstanding somewhere (most likely my own).

No problemo.   We can sort it out of if you're really interested.

>If I were a SISO comm system designer who wanted to get the most
>capacity possible out of a channel, and MIMO design afforded a higher
>aggregate rate, then I would partition the single input into multiple
>inputs on the transmit side and concatenate the multiple outputs
>together to a single output on the receive side in order to take
>advantage of the higher rate. Then, voila - a different channel
>capacity.
>
>This is why I assert that MIMO is semantics.  What am I missing?

MIMO is definitely not semantics.   The amount of capacity available
from a MIMO system depends completely on the channel responses in the
spatial streams.   The more independent the fading in the streams, the
higher the capacity.   The more correlated the channel responses, the
less the capacity.   If all of the spatial streams are completely
correlated (i.e., identical), then the system is essentially a SISO
system and SISO capacity will apply.

This is why I said that capacity has to be computed on a
channel-by-channel basis.   It isn't a generalized approach like we
get with Shannon's formula, you get what the channel response affords
you, and it changes every time the channels change.

In a practical MIMO comm system it is then essential that the transmit
and receive antennas be separated in space far enough that there is a
reasonable expectation that the channel responses will be
decorrelated.   For very small devices, say, a handset, this can be
difficult to achieve because the device just may not be physically
large enough to get adequate separation in the antennas.

The partitioning scheme you describe is a common model for a MIMO
system.   There's one input and one output, and the system splits the
data up across the spatial streams and then recombines it in the
receiver.   I just got back from the IEEE 802.11 meeting where,
finally, TGn confirmed acceptance of the single, merged MIMO-based
proposal.   In the proposed system each spatial stream can be
modulated independently (sort of a spatial water-filling scheme) in
order to try to get best use out of the spatial channels.

MIMO is an interesting beast.  It seems to work pretty well, but I
don't think it's going to be as ubiquitous as some might claim.

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

Eric Jacobsen <eric.jacobsen@ieee.org> writes:
> [...]
> I think you're missing the gist of the MIMO issues completely because
> in order for MIMO to work the channels _cannot_ be flat.  

Do you mean that the noise cannot be flat (white)? If so, then I don't
understand why not because I thought that the decorrelation afforded
by the (de)spreading processing makes adjacent interference look like
uncorrelated noise.

> It's also kind of moot, in a way, because capacity needs to be
> computed for each, specific channel instance.  It's not as
> generalized as Shannon's expression.

I'm amazed you would say this, Eric. There's probably a
misunderstanding somewhere (most likely my own).

If I were a SISO comm system designer who wanted to get the most
capacity possible out of a channel, and MIMO design afforded a higher
aggregate rate, then I would partition the single input into multiple
inputs on the transmit side and concatenate the multiple outputs
together to a single output on the receive side in order to take
advantage of the higher rate. Then, voila - a different channel
capacity.

This is why I assert that MIMO is semantics.  What am I missing?
-- 
%  Randy Yates                  % "How's life on earth? 
%% Fuquay-Varina, NC            %  ... What is it worth?" 
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates wrote:

> I did a little back-of-the-napkin analysis. For a 2-user system
> we can model the signal power as S and the noise as N = n + S, 
> where n is the "natural channel noise."

You assumed that one user appears to another user as a decorrelated 
noise source. This is actually the worst case. Usually, the users are 
more or less orthogonal (or can be orthogonalized) to each other.

> Then the capacity
> for a single user is 
> 
>     C1 = W * log(1 + S / n)
> 
> and the capacity for each of two users is
> 
>     C2 = W * log(1 + S / (n + S)),
> 
> and thus the total capacity for two users is
> 
>     CT = 2 * C2.
> 
> For S = n = 1 I get
> 
>     C1 = W
>     CT = 1.17 * W,
> 
> i.e., we gain a little total capacity out of the channel.

The total power in the channel is twice the power of each user, that's 
why there is a gain.

> However, if S = 127 and n = 1, then
> 
>     C1 = 7 * W
>     CT ~= 2 * W,
> 
> i.e., we lose total capacity.  
> 
> Is this correct? Is this what they call the "near/far" problem?

Your math is correct.
However the "near/far" is typically referred to more particular problems 
with the dynamic range.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

On Sat, 14 Jan 2006 16:40:10 GMT, Randy Yates <yates@ieee.org> wrote:

>I did a little back-of-the-napkin analysis. For a 2-user system
>we can model the signal power as S and the noise as N = n + S, 
>where n is the "natural channel noise." Then the capacity
>for a single user is 
>
>    C1 = W * log(1 + S / n)
>
>and the capacity for each of two users is
>
>    C2 = W * log(1 + S / (n + S)),
>
>and thus the total capacity for two users is
>
>    CT = 2 * C2.
>
>For S = n = 1 I get
>
>    C1 = W
>    CT = 1.17 * W,
>
>i.e., we gain a little total capacity out of the channel.
>However, if S = 127 and n = 1, then
>
>    C1 = 7 * W
>    CT ~= 2 * W,
>
>i.e., we lose total capacity.  
>
>Is this correct? Is this what they call the "near/far" problem?

Okay, so I decided to look more closely.  But only because I like
you...  ;)

I think you're missing the gist of the MIMO issues completely because
in order for MIMO to work the channels _cannot_ be flat.  So starting
with the flat channel, AWGN capacity equation will be misleading from
the beginning.

Also, how much of the co-channel signal comes back in to contribute as
interference energy will also be a bit channel dependent, since for
each sub-slice the fading will be different, and a different amount of
interference energy will be present.

Computing capacity in MIMO channels isn't trivial.  There wasn't even
agreement on how to do it for a while, although I think that's more or
less settled down now.   It's also kind of moot, in a way, because
capacity needs to be computed for each, specific channel instance.
It's not as generalized as Shannon's expression.
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

On Sat, 14 Jan 2006 16:40:10 GMT, Randy Yates <yates@ieee.org> wrote:

Not sure enough to comment about you capacity analysis (actually, I'm
too mathematically tired to give it a close look), but:

>Is this what they call the "near/far" problem?

Generally not.  Near/far has to do with being able to simultaneously
(or thereabouts) deal with a strong, high SNR signal from a nearby
user as well as a very low power, low SNR signal from a far away user.
Normally you'd set the AGC different for each, but in a multi-user
system you don't get that luxury, so you just need boatloads of
dynamic range and the ability to process both.
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

"John E. Hadstate" <jh113355@hotmail.com> writes:

> "Randy Yates" <yates@ieee.org> wrote in message 
> news:zmlzchvo.fsf@ieee.org...
>>> Another shortcoming of the usual capacity equation is 
>>> that it is
>>> really for a "single use" of the channel.  If you want to 
>>> use MIMO,
>>> you can squeeze more out of the channel, since each 
>>> spatial channel
>>> response will be different.
>>
>> Single use/MIMO could be semantics. If the same "user" is 
>> simply
>> using, e.g., multiple codes in a CDMA system. In that 
>> case, Shannon
>> lied?!? Really, this doesn't make sense.
>
> Shannon didn't lie.  His equation applies to each stream 
> individually.  

That's good to know. ...

> However, you have to account for the fact 
> that each stream (depending on how well you can isolate 
> them) is noise to the other streams.  Consequently, the 
> combined channel capacity will not equal the sum of the 
> capacities of each stream by itself.

I did a little back-of-the-napkin analysis. For a 2-user system
we can model the signal power as S and the noise as N = n + S, 
where n is the "natural channel noise." Then the capacity
for a single user is 

    C1 = W * log(1 + S / n)

and the capacity for each of two users is

    C2 = W * log(1 + S / (n + S)),

and thus the total capacity for two users is

    CT = 2 * C2.

For S = n = 1 I get

    C1 = W
    CT = 1.17 * W,

i.e., we gain a little total capacity out of the channel.
However, if S = 127 and n = 1, then

    C1 = 7 * W
    CT ~= 2 * W,

i.e., we lose total capacity.  

Is this correct? Is this what they call the "near/far" problem?
-- 
%  Randy Yates                  % "I met someone who looks alot like you,
%% Fuquay-Varina, NC            %             she does the things you do, 
%%% 919-577-9882                %                     but she is an IBM."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

"Randy Yates" <yates@ieee.org> wrote in message 
news:zmlzchvo.fsf@ieee.org...
>> Another shortcoming of the usual capacity equation is 
>> that it is
>> really for a "single use" of the channel.  If you want to 
>> use MIMO,
>> you can squeeze more out of the channel, since each 
>> spatial channel
>> response will be different.
>
> Single use/MIMO could be semantics. If the same "user" is 
> simply
> using, e.g., multiple codes in a CDMA system. In that 
> case, Shannon
> lied?!? Really, this doesn't make sense.

Shannon didn't lie.  His equation applies to each stream 
individually.  However, you have to account for the fact 
that each stream (depending on how well you can isolate 
them) is noise to the other streams.  Consequently, the 
combined channel capacity will not equal the sum of the 
capacities of each stream by itself.

Eric Jacobsen <eric.jacobsen@ieee.org> writes:

> On Fri, 13 Jan 2006 00:40:23 GMT, Randy Yates <yates@ieee.org> wrote:
>
>>We all know and love Shannon's original channel capacity theorem
>>
>>  C = W * log_2 (1 + P/N).
>>
>>However, this assumes a white noise spectral density. Is there a
>>formulation for channel capacity which takes into account a colored
>>noise spectrum?
>
> As other's have pointed out, you can do a piece-wise computation
> across the band.   Doing this and taking advantage of it is usually
> called "water filling" the channel, and the "water filling" capacity
> is the best you can do in this case.
>
> Some multi-carriers systems (e.g., ADSL, HomePlug) use adaptive bit
> loading across the subcarriers to take advantage of this.

Ahh, yes, I'd heard of that but hadn't made the connection all the
way back to Shannon. Cool.

> Another shortcoming of the usual capacity equation is that it is
> really for a "single use" of the channel.  If you want to use MIMO,
> you can squeeze more out of the channel, since each spatial channel
> response will be different.

Single use/MIMO could be semantics. If the same "user" is simply
using, e.g., multiple codes in a CDMA system. In that case, Shannon
lied?!? Really, this doesn't make sense. 
-- 
%  Randy Yates                  % "Watching all the days go by...    
%% Fuquay-Varina, NC            %  Who are you and who am I?"
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://home.earthlink.net/~yatescr