Shannon Limit for 2-PSK Depending on Code Rate

Hello,

Assume, I have a BPSK transmission with a given Forward Error
Correction with Code Rate R.
From many papers, I know that the minimum EbN0 for the transmission at
R=1/2 is 0.19 dB, but I have problems calculating this value (and -of
course- for other rates R).
My approach:
http://users.ecs.soton.ac.uk/rm/wp-content/CapSNR.jpg gives me a clue
about the channel capacity for BPSK depending on EsN0.
If I want to verify the result for R=1/2 as a test:
EsN0 = SNR = EbN0*R
SNR_dB = EbN0_dB + R_dB => SNR_dB = 0.19 dB - 3.01 dB = -2.82 dB

Using the chart above (or a simulation), I get a capacity result of
0.4093 bit/Hz. This does not meet my expectations of having the limit
at the point where the capacity meets the code rate.
Perhaps anyone can help me correcting the result or give me a
calculation for e.g. R=4/7.

Thanks in advance!

ven arc  <va20384@googlemail.com> wrote:

>Assume, I have a BPSK transmission with a given Forward Error
>Correction with Code Rate R.
>From many papers, I know that the minimum EbN0 for the transmission at
>R=1/2 is 0.19 dB, but I have problems calculating this value (and -of
>course- for other rates R).

Good question.  There is a calculation presented in Heegard and Wicker
that the BPSK capcity for R = 1/2 is 0.0 dB, and I have never been
able to rationalize this value with the often-stated value of 0.19 dB.
Since it's only two tenths of a dB I have not lost any sleep over
the discrepancy.

What would be really nice is to unambiguously know the capacities for not 
only BPSK at any rate, but higher-order QAM, and also for each of these 
with finite block lengths (using something like Fabrizio Pollara's 
sphere-packing method).  Then you'd be able to know how close to capacity 
just about any coded modulation system is performing.  Whoever is able 
to go through this entire set of calculations definitively gets a prize.

Steve

Thank you for your comment on that topic. So as I read in a book about
that topic (they don't show up the calculations), it is only possible
to calculate the values numerically and there is a chart of EbN0 limit
over code rate.
My first approach (and I'm not a professional) was to start from the
channel capacity which is slightly lower in BPSK, but as it converges
at smaller SNRs where the Shannon limits for the different code rates
lie, so the difference should be negligible.
I don't even get the point how we reach a value for the EbN0 limit in
general, e.g. like Shannon limit in SNR region ->  converting to EbN0
by using the spectral efficiency and the code rate. But I never get
the correct values.
I know that Eric Jacobsen (active in this Newsgroup) used these limits
in his tutorial here: http://www.dsprelated.com/showarticle/136.php
Perhaps he or someone else can give a clue what is the correct way to
calculate this - I've been trying for half a day.

Thank you for your comment on that topic. As I read in a book about
that topic (they don't show up the calculations), it is only possible
to calculate the values numerically and there is a chart of EbN0
limit
over code rate.
My first approach (and I'm not a professional) was to start from the
channel capacity which is slightly lower in BPSK, but as it converges
at smaller SNRs where the Shannon limits for the different code rates
lie, so the difference should be negligible.
I don't even get the point how we reach a value for the EbN0 limit in
general, e.g. like Shannon limit in SNR region ->  converting to EbN0
by using the spectral efficiency and the code rate. But I never get
the correct values.
I know that Eric Jacobsen (active in this Newsgroup) used these
limits
in his tutorial here: http://www.dsprelated.com/showarticle/136.php
Perhaps he or someone else can give a clue what is the correct way to
calculate this - I've been trying for half a day.

Volker L&#4294967295;cken  <volker.luecken@googlemail.com> wrote:

>Thank you for your comment on that topic. So as I read in a book about
>that topic (they don't show up the calculations), it is only possible
>to calculate the values numerically and there is a chart of EbN0 limit
>over code rate.
>My first approach (and I'm not a professional) was to start from the
>channel capacity which is slightly lower in BPSK, but as it converges
>at smaller SNRs where the Shannon limits for the different code rates
>lie, so the difference should be negligible.
>I don't even get the point how we reach a value for the EbN0 limit in
>general, e.g. like Shannon limit in SNR region ->  converting to EbN0
>by using the spectral efficiency and the code rate. But I never get
>the correct values.
>I know that Eric Jacobsen (active in this Newsgroup) used these limits
>in his tutorial here: http://www.dsprelated.com/showarticle/136.php
>Perhaps he or someone else can give a clue what is the correct way to
>calculate this - I've been trying for half a day.

There's a calculation of this around page 8 of the following document.
Note that his curve needs to be compensated by subtracting off 
-10log10(code rate) to convert from Es/No to Eb/No.

http://ieee802.org/16/tgm/contrib/C80216m-07_097.pdf

I'm pretty sure the expression for mutual information capacity he is using 
is the same expression everyone else is using; what I'm not sure is whether 
it's actually exactly correct.  But it's close enough.

Steve

On 10/20/2011 07:54 PM, Steve Pope wrote:
> ven arc  <va20384@googlemail.com> wrote:
> 
>> Assume, I have a BPSK transmission with a given Forward Error
>> Correction with Code Rate R.
>>From many papers, I know that the minimum EbN0 for the transmission at
>> R=1/2 is 0.19 dB, but I have problems calculating this value (and -of
>> course- for other rates R).
> 
> Good question.  There is a calculation presented in Heegard and Wicker
> that the BPSK capcity for R = 1/2 is 0.0 dB, and I have never been
> able to rationalize this value with the often-stated value of 0.19 dB.
> Since it's only two tenths of a dB I have not lost any sleep over
> the discrepancy.
> 

0.0 dB is the rate-1/2 capacity for the unconstrained-input AWGN channel
(i.e., with Gaussian input, which maximizes C). The BPSK capacity is
indeed 0.18dB at rate 1/2. There seems to be a misunderstanding or typo
in your book.

Laurent

On Oct 20, 1:54�pm, spop...@speedymail.org (Steve Pope) averred:

> Since it's only two tenths of a dB I have not lost any sleep over
> the discrepancy.

Near the Shannon limit, fractions of a dB matter a lot.
As the atheist said to the monotheist, "You and I are
not that different: we differ by only one in the number
of gods whose existence we deny."

Dilip Sarwate
who, as a Hindu, is far from the Shannon limit
on the number of gods one should believe in

Laurent Schmalen  <loron@gmx.de> wrote:

>On 10/20/2011 07:54 PM, Steve Pope wrote:

>> ven arc  <va20384@googlemail.com> wrote:
>> 
>>> Assume, I have a BPSK transmission with a given Forward Error
>>> Correction with Code Rate R.
>>>From many papers, I know that the minimum EbN0 for the transmission at
>>> R=1/2 is 0.19 dB, but I have problems calculating this value (and -of
>>> course- for other rates R).
>> 
>> Good question.  There is a calculation presented in Heegard and Wicker
>> that the BPSK capcity for R = 1/2 is 0.0 dB, and I have never been
>> able to rationalize this value with the often-stated value of 0.19 dB.
>> Since it's only two tenths of a dB I have not lost any sleep over
>> the discrepancy.
>> 
>
>0.0 dB is the rate-1/2 capacity for the unconstrained-input AWGN channel
>(i.e., with Gaussian input, which maximizes C). 


Um, no that is not correct.

Steve

On Oct 21, 12:46�pm, spop...@speedymail.org (Steve Pope) claimed:

>
> >0.0 dB is the rate-1/2 capacity for the unconstrained-input AWGN channel
> >(i.e., with Gaussian input, which maximizes C).
>
> Um, no that is not correct.


So what *is* the correct answer
(to within 0.19 dB, of course!  :-)  )

dvsarwate  <dvsarwate@yahoo.com> wrote:

>On Oct 21, 12:46&#4294967295;pm, spop...@speedymail.org (Steve Pope) claimed:
>
>>
>> >0.0 dB is the rate-1/2 capacity for the unconstrained-input AWGN channel
>> >(i.e., with Gaussian input, which maximizes C).
>>
>> Um, no that is not correct.
>
>
>So what *is* the correct answer
>(to within 0.19 dB, of course!  :-)  )

I think -1.0 dB is within 0.19 dB 

S.