FPGA Central

Posting a draft of my thought process. Welcoming comments and
corrections.



Consider an (n,k) block code. It can be decoded using 2^k matched
filters,
each matched to one code-word.

So, let us interpret this coding scheme as moving from a binary
trnasmission to
a 2^k-ary transmission.

Let E be uncoded energy per bit and Pe be uncoded error probability.
Let T_b = original bit duration.

B_b ~= 1/T_b

Required energy per bit for same error prob = ((k/n) * E) - coding
gain.

degrees of freedom is 2.B_b.T_b
Aim - to increase distance between waveforms with given energy E.

Problem : Find an M-ary signal set e_j(t) such that
norm(e_i-e_k) is as large as possible for i!=k, i<=M, k<=M

There are only 2BT orthogonal waveforms with bandwidth B and duration
T

Finding uncorrelated waveforms is easier in higher-dimensional space.
But to increase dimensions, u have to increase either B or T

There is no upper limit for time-bandwidth product - jo smith gives
the simple example of (cascaded) allpass filter.

Consider two scenarios.

1) Bandwidth is kept equal. This means that throuhput is decreased
by a factor 1- k/n.

2BT = 2 B_b . n T_b

The length of each waveform is longer by an amount (n-k)*T_b (
allowing larger
sample of noise to affect the signal ). It appears to me that
this is the factor leading to coding gain. This factor remains the
same irrespective
of the nature of code-words chosen. Would that be the reason that
"any randomly chosen
block code is good with very high probability" ?( Actually, B can also
be increased by
choosing noise-like code-words. Many code-words will lead to almost
same B.)


2) Throughput is kept same. INcrease bandwidth by factor n/k

2BT = 2 (n/k) B_b k T_b
The waveforms are noise-like and hence have lower correlation with
awgn.

Both noise and signal powers increase by same factor.

Which is better - (n,k) or (n/k, 1) code ?


Repetition coding is not good. Why ?? After all, repetition code is an
(n,1) block code and is like an ideal situation. (We use two
codewords, between which hamming distance is the maximum possible - n
).
Repetition coding reduces B by almost a factor of n. So, 2BT remains
almost same. So,the waveforms still exist in a vector space of same
dimension.

Does this mean that it is also important for the waveforms
to be as noise-like as possible ? Is the variance of correlation of
white noise with a low bandwidth waveform more than that with a high
bw w/f ? In the extreme case, if B is infinity, then this is correct.
Any two white noise instances have zero correlation.

Coding with bandwidth exapansion is as good as that without bw
expansion, because the dimension 2BT is same for both. This dimension
decides the variance of correlation of a signal with white noise.

(But with bw expansion, you dont get a larger sample of noise to
affect your signal. What is the penalty for that ? )


shankar

>
> Repetition coding is not good. Why ?? After all, repetition code is an
> (n,1) block code and is like an ideal situation. (We use two
> codewords, between which hamming distance is the maximum possible - n
> ).
> Repetition coding reduces B by almost a factor of n. So, 2BT remains
> almost same. So,the waveforms still exist in a vector space of same
> dimension.
>

Actually, it is reported that repetition coding with bandwidth expansion
( ie spreading ) is also not good in awgn channel and gives no coding gain.

Check out page 7, proposition 1 of

http://tesla.csl.uiuc.edu/~mantrava/docs/cod_spr.pdf

This answers another question - spreading does not help in awgn channel.

To conclude, no (n,1) block code gives coding gain.

shankar

jkv567 at yahoo.com

"kbc" <[email protected]> wrote in message
news:[email protected] om...
> Actually, it is reported that repetition coding with bandwidth expansion
> ( ie spreading ) is also not good in awgn channel and gives no coding
gain.

I don't understand exactly what the meaning of good in your speaking.
In other words, what point of view do you consider when you decide if it is
good?
If SNR is too low to transmit only one bit in a time, we can transmit
two bits with same information by repeatition manner. This does not
increase any hardware complexity in receiver side, so this is
one of very attractive method in such a case.
If it is not, the use of only lower rate code and higher order modulation as
possible as is to be
always best strategy. However, unfortunatley this is not true.

> This answers another question - spreading does not help in awgn channel.
>
> To conclude, no (n,1) block code gives coding gain.

Therefore, the relationship between spreading and repeation technique is not
always in-line.

"kbc" <[email protected]> wrote in message
news:[email protected] om...
> It is good if it gives coding gain. Not good otherwise.

I would agree if this changes to:
=> It is better if it gives more coding gain.

> I dont understand this argument. If you have energy to spend on two
> repeated bits, you can as well spend it on a single bit.

In Engery case, you're obviously right; however, power limited case gives
gain.
For exaample, let r, r1 and r2 as received signal as below:

] r = s + n, r1 = s1 + n1, r2 = s2 + n2,

where s, s1, s2 is transmitted signal and n, n1, n2 is received noise.
In addition, each power of s1 and s2 is half times than power of s,
respectively, where
s1 = s2 = s because of repeatition coding.

Since power of s1 added s2 is 2 times than power of s and moreover,
power of n1 added n2 is 2 times than power of n, SNR of s and n is same to
SNR of
(s1,s2) and (n1,n2) in such energy limited case.

On the other hand, the system with s1 and s2 which have same power level of
s

gives more SNR, i.e. 2 times than original case.


> Above, I am using the fact that coding gain does not depend on whether
> the implementation makes use of any bandwidth expansion or not.

I fully understand that coding gain dose not depend on the size of
bandwidth.

> Check
> http://cnx.rice.edu/content/m0094/latest/?format=pdf
> http://cnx.rice.edu/content/m0071/latest/?format=pdf
>
> shankar

I read all your proposed papers and they were greatly improved my knowledge.
However, I got a small question about your statement in your proposed paper.

> It is said, "Does any error-correcting code reduce communication errors
> when real-world constraints are taken into account? The answer now is
yes."

Do you really think that it is true in every case?

Besides, I draw the performance graph which was shown in the paper, with
add-on of some parameters, e.g. variation of code rate (R)
and performance gain (M,N|p'=M*p^N).

In short, I mentioned three points according to your proposals:
one is the definition of good in this case, and another is the gain
comparison
between energy limited and power limited case, lastly I expressed my feeling
about
your suggested papers.

Regards,
---
James ([email protected])
- Private opinions: This is not the opinion of my affiliation.

> In Engery case, you're obviously right; however, power limited case gives
> gain.
> For exaample, let r, r1 and r2 as received signal as below:
>
> ] r = s + n, r1 = s1 + n1, r2 = s2 + n2,
>
> where s, s1, s2 is transmitted signal and n, n1, n2 is received noise.
> In addition, each power of s1 and s2 is half times than power of s,
> respectively, where
> s1 = s2 = s because of repeatition coding.
>
> Since power of s1 added s2 is 2 times than power of s and moreover,
> power of n1 added n2 is 2 times than power of n, SNR of s and n is same to
> SNR of
> (s1,s2) and (n1,n2) in such energy limited case.
>
> On the other hand, the system with s1 and s2 which have same power level of
> s
>
> gives more SNR, i.e. 2 times than original case.
>

Probability of error Pe depends on the energy of the bit; not on power of
bit.
If power is increased/decreased and energy remains same, Pe remains same.

Actually , power can be decreased below noise floor without affecting Pe - this
is spread-spectrum's LPD feature.

There is only one way to decrease Pe - you have to increase Eb

There are 3 ways to increase channel capacity :
1) decrease Pe
2) increase B and throw more bits per second into the channel.
3) do both

At transmitter, energy is the constraint; not power. Notice that
life of battery is specified as Ampere-hours. Battery can support a
large instantaneous power, but for short duration.

When we speak of power, it is always average power, because it is
almost equivalent to energy.


> > Above, I am using the fact that coding gain does not depend on whether
> > the implementation makes use of any bandwidth expansion or not.
>
> I fully understand that coding gain dose not depend on the size of
> bandwidth.
>
> > Check
> > http://cnx.rice.edu/content/m0094/latest/?format=pdf
> > http://cnx.rice.edu/content/m0071/latest/?format=pdf
> >
> > shankar
>
> I read all your proposed papers and they were greatly improved my knowledge.
> However, I got a small question about your statement in your proposed paper.

It is not my paper. I dont know the author.

thanks
shankar

>
> There is only one way to decrease Pe - you have to increase Eb

Actually , 2 ways - increase Eb or do channel coding.

shankar