Getting the most information in a limited number of transitions

> With the current equipment that I have to work with I cannot use the entire >bandwidth available to me, because I cannot exercise the modulator to its >fullest. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com

I must not understand the question because the answer seems to be trival...

encode a 1 as 1  and encode a 0 as -1

doesn't that yield the least amount of transistions?

If you want no DC component then xor your data with a PRBS stream before encoding?

I must be missing something in your question.

Mark

On Fri, 08 Nov 2013 08:17:44 -0800, makolber wrote:

>> With the current equipment that I have to work with I cannot use the
>> entire >bandwidth available to me, because I cannot exercise the
>> modulator to its >fullest. -- Tim Wescott Wescott Design Services
>> http://www.wescottdesign.com
> 
> I must not understand the question because the answer seems to be
> trival...
> 
> encode a 1 as 1  and encode a 0 as -1
> 
> doesn't that yield the least amount of transistions?
> 
> If you want no DC component then xor your data with a PRBS stream before
> encoding?
> 
> I must be missing something in your question.
> 
> Mark

Ignoring DC balance for the moment:

If I separate a time period into 10 slots and allow two transitions, then 
I can encode 36 different possibilities, or about 5 bits with two 
transitions.  I pay heavily in terms of how well I can receive in a noisy 
environment, but I gain greatly in terms of how many bits I can encode 
per transition.

If I separate a time period into 21 slots and allow four transitions, 
then I can encode 4845 different possibilities, for 12 bits with just 
four transitions.

Trying to achieve DC balance then complicates things, and I'm obviously 
far more noise sensitive than if I just transmit binary.

There's lots of avenues to pursue, most of which are dead ends, which is 
why I was hoping to find someone who'd already been there and done that.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

Welcome to information theory.  Elegant subject, but the tools available for computing and designing signals are very limited.  You are probing around and while that's fine, you can get vastly different answers as you add more and more constraints.  

What many have suggested is to work in the differential domain.  In discrete time, let indicators denote the locations of transitions.  What is the minimum number of indicators to encode B bits?  Why, it is 1.  Define x to be the indicator vector of length 2^B.  Then for any B bits, you can encode with only one indicator, and thus mapping back from differential domain you get only one transition.  However, this is basically a pulse-position modulation scheme with HUUUUUUUUGE alphabet size.  Of course performance is poor.  

However, you should know that there are domains where such schemes are used, in particular in systems with significant nonlinearities, detector constraints, or electro-mechanical constraints.  Look up "Combinatorial PPM" as a keyword to find some interesting ideas. 

You described a transmitter that can send bilevel signal, but you haven't said anything about the propagation channel, noise, receiver constraints.  So we can all continue to conjecture, and try to fish further constraints out of you, or you can describe the entire system and help us help you. 

So, what it is that you are trying to do again? 

Julius 


On Friday, November 8, 2013 11:45:51 AM UTC-5, Tim Wescott wrote:
> 
> Ignoring DC balance for the moment:
> 
> 
> 
> If I separate a time period into 10 slots and allow two transitions, then 
> 
> I can encode 36 different possibilities, or about 5 bits with two 
> 
> transitions.  I pay heavily in terms of how well I can receive in a noisy 
> 
> environment, but I gain greatly in terms of how many bits I can encode 
> 
> per transition.
> 
> 
> 
> If I separate a time period into 21 slots and allow four transitions, 
> 
> then I can encode 4845 different possibilities, for 12 bits with just 
> 
> four transitions.
> 
> 
> 
> Trying to achieve DC balance then complicates things, and I'm obviously 
> 
> far more noise sensitive than if I just transmit binary.
> 
> 
> 
> There's lots of avenues to pursue, most of which are dead ends, which is 
> 
> why I was hoping to find someone who'd already been there and done that.
> 
> 
> 
> -- 
> 
> Tim Wescott
> 
> Control system and signal processing consulting
> 
> www.wescottdesign.com

Hi,

maybe the original problem statement needs more detail.
Otherwise, the amount of information in a single transition (or two if you
need to establish a timing reference) approximates infinity, when the
distance between the transitions ("symbol length") isn't bounded.

Assume the detectable time resolution is 1 (similar to discrete amplitude
levels in a conventional system, i.e. QAM). Then I can encode 8 bits if the
max. symbol length is 256, 16 bits for 65536 max. duration etc.



	 

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On Fri, 08 Nov 2013 13:03:34 -0600, mnentwig wrote:

> Hi,
> 
> maybe the original problem statement needs more detail.
> Otherwise, the amount of information in a single transition (or two if
> you need to establish a timing reference) approximates infinity, when
> the distance between the transitions ("symbol length") isn't bounded.
> 
> Assume the detectable time resolution is 1 (similar to discrete
> amplitude levels in a conventional system, i.e. QAM). Then I can encode
> 8 bits if the max. symbol length is 256, 16 bits for 65536 max. duration
> etc.

I'm not at liberty to detail the whole channel, so I'm trying to just 
shine a light on the relevant parts.

There is noise, and there is distortion that requires equalization.  So I 
can't just have an infinitely finely-spaced PWM.

I had been hoping that there was some developed application, and one 
person mentioned something -- so I'll pursue that and see what I can see.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

On 08/11/13 21:50, Tim Wescott wrote:
> On Fri, 08 Nov 2013 13:03:34 -0600, mnentwig wrote:
>
>> Hi,
>>
>> maybe the original problem statement needs more detail.
>> Otherwise, the amount of information in a single transition (or two if
>> you need to establish a timing reference) approximates infinity, when
>> the distance between the transitions ("symbol length") isn't bounded.
>>
>> Assume the detectable time resolution is 1 (similar to discrete
>> amplitude levels in a conventional system, i.e. QAM). Then I can encode
>> 8 bits if the max. symbol length is 256, 16 bits for 65536 max. duration
>> etc.
>
> I'm not at liberty to detail the whole channel, so I'm trying to just
> shine a light on the relevant parts.
>
> There is noise, and there is distortion that requires equalization.  So I
> can't just have an infinitely finely-spaced PWM.
>
> I had been hoping that there was some developed application, and one
> person mentioned something -- so I'll pursue that and see what I can see.

You may need to be very pragmatic about the degree to
which you pursue perfection and the degree to which
you accept non-ideal performance.

I suspect that without a background in transmissions
through non-ideal channels, you might make some
elementary mistakes. It is one of those areas that can
take decades to understand and a large engineering
resource to implement. Seriously consider picking
off-the-shelf "components" and re-purposing them.

On 11/8/2013 3:50 PM, Tim Wescott wrote:


> There is noise, and there is distortion that requires equalization.  So I
> can't just have an infinitely finely-spaced PWM.
>
> I had been hoping that there was some developed application, and one
> person mentioned something -- so I'll pursue that and see what I can see.

With DC balance requirement and minimum transitions, symbol could be a 
period of square wave with variable phase from 0 to Pi (excluding 
endpoints). Gray coded data is obvious choice.
Frequency of the square wave is another degree of freedom; however there 
should be integer relationship between possible frequencies. Otherwise 
the receiver would be very complex and inefficient.

This frequency + phase modulation is probably as efficient as it could 
be. Sync and equalization shouldn't be a problem.


Vladimir Vassilevsky
DSP and Mixed Signal Designs
www.abvolt.com