How do you interpret ADC result?

Hello,

I search for any professional solution/opinion on the following task:

Initial Conditions:
Consider, an ideal positive (input from 0 to Uref) 8-bit (output form 0 to
255, inclusive) A/D converter produced a sample with a value of [x]
(0<=x<=255). The reference applied to the ADC is of [+Uref].

Question:
That intepretation method (or formulae) do you use (or recommend) to
reconstruct (in fact, guess) an original signal level (U) which the
produced sample value (x) is corresponding to?

I'm asking about this, because i saw no such an information in DSP related
books.

Beforehand thanks,
Alex

On 11/7/2010 6:11 AM, sanaysan wrote:
> Hello,
>
> I search for any professional solution/opinion on the following task:
>
> Initial Conditions:
> Consider, an ideal positive (input from 0 to Uref) 8-bit (output form 0 to
> 255, inclusive) A/D converter produced a sample with a value of [x]
> (0<=x<=255). The reference applied to the ADC is of [+Uref].
>
> Question:
> That intepretation method (or formulae) do you use (or recommend) to
> reconstruct (in fact, guess) an original signal level (U) which the
> produced sample value (x) is corresponding to?
>
> I'm asking about this, because i saw no such an information in DSP related
> books.
>
> Beforehand thanks,
> Alex
>
>
>

One way or another, you need to calibrate the A/D conversion process.
That is, what voltage input yields what numeric code output?

Maybe you can do that by reviewing the A/D converter spec sheet.
And, certainly, you can do that by inputting various stable voltages 
(i.e. dc) and observing the output codes.

Once calibrated, you will be able to relate an actual output code to an 
input voltage.  Neglecting noise and other anomalous effects of course.

It seems a bit strange that you use the term "Uref".  And you have 
already stated:
"(input from 0 to Uref) 8-bit (output form 0 to 255"

So, it rather sounds like an input of voltage valued "Uref" will yield a 
255 output.  Assuming no compression, i.e. linear input/output 
relationship, then a particular code output would relate to a voltage 
input like this:

voltage in = Uref * code out / 255
and depending on the A/D specs, somehting like +/- 0.5/255 representing 
the quantization relative to the analog input.



So, if that's correct, then code out = 255 means an input of Uref.

If that's not correct, then there is some other input voltage Vmax that 
*just* yields 255 output (anything larger than 255.5*Vmax/255 more or 
less may still yield 255 output and would be "saturated").

Then:

voltage in = Vmax * code out / 255.

This is just a more general formulation that that using Uref and if 
Vmax=Uref then they both work the same.

Let's say that Vmax = 1.000 volts.
Then each bit represents 1.000/255 = 3.9216mv
And, any output xxx decimal would represent an input voltage of

xxx / 255 * 3.9216 * 10^-3  +/- 3.9216/2.

I've not run into any A/D's that threshold so that it would be
xxx/255 * Vmax/255 -0, +Vmax/255 or
xxx/255 * Vmax/255 -Vmax/255, +0
But such devices may exist.....

Fred

On Nov 7, 9:11&#4294967295;am, "sanaysan" <sanaysan@n_o_s_p_a_m.gmail.com> wrote:
> Hello,
>
> I search for any professional solution/opinion on the following task:
>
> Initial Conditions:
> Consider, an ideal positive (input from 0 to Uref) 8-bit (output form 0 to
> 255, inclusive) A/D converter produced a sample with a value of [x]
> (0<=x<=255). The reference applied to the ADC is of [+Uref].
>
> Question:
> That intepretation method (or formulae) do you use (or recommend) to
> reconstruct (in fact, guess) an original signal level (U) which the
> produced sample value (x) is corresponding to?
>
> I'm asking about this, because i saw no such an information in DSP related
> books.
>
> Beforehand thanks,
> Alex

Are you making this too hard?  Is this a homework question or a take
home test?

This is a simple linear relationship between the input voltage and the
numeric representation 'x'.  The data sheet for your ADC will give you
the details, but I expect you don't have a data sheet for an ADC on a
test.  The instructor gave you all the info you need.

Rick

>On Nov 7, 9:11=A0am, "sanaysan" <sanaysan@n_o_s_p_a_m.gmail.com> wrote:
>> Hello,
>>
>> I search for any professional solution/opinion on the following task:
>>
>> Initial Conditions:
>> Consider, an ideal positive (input from 0 to Uref) 8-bit (output form 0
t=
>o
>> 255, inclusive) A/D converter produced a sample with a value of [x]
>> (0<=3Dx<=3D255). The reference applied to the ADC is of [+Uref].
>>
>> Question:
>> That intepretation method (or formulae) do you use (or recommend) to
>> reconstruct (in fact, guess) an original signal level (U) which the
>> produced sample value (x) is corresponding to?
>>
>> I'm asking about this, because i saw no such an information in DSP
relate=
>d
>> books.
>>
>> Beforehand thanks,
>> Alex
>
>Are you making this too hard?  Is this a homework question or a take
>home test?
>
>This is a simple linear relationship between the input voltage and the
>numeric representation 'x'.  The data sheet for your ADC will give you
>the details, but I expect you don't have a data sheet for an ADC on a
>test.  The instructor gave you all the info you need.
>
>Rick
>

Could it be the linear relationship     

(Vref+ - Vref-)/2^n = (5-0)/256 [Volts/Step]   Maybe that's enough of a
hint to get you in the right direction.

Hi, Fred,

The method you wrote:

>voltage in = Uref * code out / 255

or, in common form,

  voltage in = Uref * [X] / [Xmax]  where  0<=[X]<=Xmax  Xmax=2^n-1 
  (n is ADC bit number) 

is erroneous.  It can be proven very simple:

Consider the ADC with Uref=1, n=2, and normal (not shifted) transfer
function.  A code of 0 ([X]=0) will correspond with an interval (of input
voltages) from 0 to 0.25, and you interpret it as 0/3=0.0, etc... The all 4
cases can be represented as the following table:

Input voltage     Intepretation  Comment
interval          result
[0.00; 0.25)      0/3=0.00       in interval, error -0.00/+0.25
[0.25; 0.50)      1/3=0.33       in interval, error -0.08/+0.17
[0.50; 0.75)      2/3=0.67       in interval, error -0.17/+0.08
[0.75; 1.00]      3/3=1.00       in interval, error -0.25/+0.00
            
LSB=0.25, no shift                    maximum error +0.25/-0.25
                                                     -LSB/+LSB

But any computational algotith and the DSP theory awaits from you accuracy
of LSB/2 !!!  Is not it?

Then, your expression for the error with n=8
> +/- 0.5/255
is not rigth.

Now, look at the shifted transfer function case:

Input voltage     Intepretation  Comment
interval          result
[0.000; 0.125)    0/3=0.000      in interval
[0.125; 0.375)    1/3=0.333      in interval
[0.375; 0.625)    2/3=0.667      (!!!) OUT OF INTERVAL (!!!)
[0.625; 1.000]    3/3=1.000      in interval
            
LSB=0.25, shift=-LSB/2=-0.125      

One point is out of interval.  (It is for any n >=2.)
Did you think about this earlier? 

Also, you can state that the interpetation result is the same in both
cases: it is totally alienated from the ADC matter (i.e. transfer
function). 

I know that such a solution is very popular today, but it is comletely
naked.

About calibration.  I described that the ADC is ideal.  Therefore, it is
not necessary to calibrate it.  (Calibration is another process, but during
it, you also must take into account that an ADC result, or code,
corresponds with an interval of input volatges inssted of a single point.)
 
Thanks a lot for your help and explanation.

Alex  

Well, Rick,

look at any ADC datasheet, but make it carefully.  That do you see?

Do you see the ADC transfer function?  If yes, you can state that each code
corresponds with an interval (definable by two points: lower and higher
limits) of input levels.

For an ideal ADC, there is one and only one code that corresponds to the
input level being converted.  But, in backward direction, there are
unlimited number of input levels that can corespond to a single code.

It is simple to solve the problem in forward direction: the transfer
function provides only one solution.  But, how to solve the problem
effectively and right in backward direction?  This is what i asked in this
post. 

This was not a homework, nor a test.  Algthouth, it could be a test.  How
do you think, Rick, did you pass it?

Alex

P.S.  Datasheet for an ideal ADC -- very funny :-)


>
>Are you making this too hard?  Is this a homework question or a take
>home test?
>
>This is a simple linear relationship between the input voltage and the
>numeric representation 'x'.  The data sheet for your ADC will give you
>the details, but I expect you don't have a data sheet for an ADC on a
>test.  The instructor gave you all the info you need.
>
>Rick
>

So what you're asking is: how do you figure out to infinite resolution 
the input to a finite resolution system?

Be sure to let us all know when you've figured it out.

On 11/9/2010 4:14 AM, sanaysan wrote:
> Well, Rick,
>
> look at any ADC datasheet, but make it carefully.  That do you see?
>
> Do you see the ADC transfer function?  If yes, you can state that each code
> corresponds with an interval (definable by two points: lower and higher
> limits) of input levels.
>
> For an ideal ADC, there is one and only one code that corresponds to the
> input level being converted.  But, in backward direction, there are
> unlimited number of input levels that can corespond to a single code.
>
> It is simple to solve the problem in forward direction: the transfer
> function provides only one solution.  But, how to solve the problem
> effectively and right in backward direction?  This is what i asked in this
> post.
>
> This was not a homework, nor a test.  Algthouth, it could be a test.  How
> do you think, Rick, did you pass it?
>
> Alex
>
> P.S.  Datasheet for an ideal ADC -- very funny :-)
>
>
>>
>> Are you making this too hard?  Is this a homework question or a take
>> home test?
>>
>> This is a simple linear relationship between the input voltage and the
>> numeric representation 'x'.  The data sheet for your ADC will give you
>> the details, but I expect you don't have a data sheet for an ADC on a
>> test.  The instructor gave you all the info you need.
>>
>> Rick
>>


-- 
Rob Gaddi, Highland Technology
Email address is currently out of order

On 11/9/2010 4:14 AM, sanaysan wrote:
> Well, Rick,
>
> look at any ADC datasheet, but make it carefully.  That do you see?
>
> Do you see the ADC transfer function?  If yes, you can state that each code
> corresponds with an interval (definable by two points: lower and higher
> limits) of input levels.
>
> For an ideal ADC, there is one and only one code that corresponds to the
> input level being converted.  But, in backward direction, there are
> unlimited number of input levels that can corespond to a single code.
>
> It is simple to solve the problem in forward direction: the transfer
> function provides only one solution.  But, how to solve the problem
> effectively and right in backward direction?  This is what i asked in this
> post.
>
> This was not a homework, nor a test.  Algthouth, it could be a test.  How
> do you think, Rick, did you pass it?
>
> Alex
>
> P.S.  Datasheet for an ideal ADC -- very funny :-)
>
>
>>
>> Are you making this too hard?  Is this a homework question or a take
>> home test?
>>
>> This is a simple linear relationship between the input voltage and the
>> numeric representation 'x'.  The data sheet for your ADC will give you
>> the details, but I expect you don't have a data sheet for an ADC on a
>> test.  The instructor gave you all the info you need.
>>
>> Rick
>>

So what you're asking is: how do you determine with infinite resolution 
the input to a finite resolution system?

Be sure to let us know when you've figured it out.

-- 
Rob Gaddi, Highland Technology
Email address is currently out of order

On 11/9/2010 3:37 AM, sanaysan wrote:
> Hi, Fred,
>
> The method you wrote:
>
>> voltage in = Uref * code out / 255
>
> or, in common form,
>
>    voltage in = Uref * [X] / [Xmax]  where  0<=[X]<=Xmax  Xmax=2^n-1
>    (n is ADC bit number)
>
> is erroneous.  It can be proven very simple:
>
> Consider the ADC with Uref=1, n=2, and normal (not shifted) transfer
> function.  A code of 0 ([X]=0) will correspond with an interval (of input
> voltages) from 0 to 0.25, and you interpret it as 0/3=0.0, etc... The all 4
> cases can be represented as the following table:
>
> Input voltage     Intepretation  Comment
> interval          result
> [0.00; 0.25)      0/3=0.00       in interval, error -0.00/+0.25
> [0.25; 0.50)      1/3=0.33       in interval, error -0.08/+0.17
> [0.50; 0.75)      2/3=0.67       in interval, error -0.17/+0.08
> [0.75; 1.00]      3/3=1.00       in interval, error -0.25/+0.00
>
> LSB=0.25, no shift                    maximum error +0.25/-0.25
>                                                       -LSB/+LSB
>
> But any computational algotith and the DSP theory awaits from you accuracy
> of LSB/2 !!!  Is not it?
>
> Then, your expression for the error with n=8
>> +/- 0.5/255
> is not rigth.
>
> Now, look at the shifted transfer function case:
>
> Input voltage     Intepretation  Comment
> interval          result
> [0.000; 0.125)    0/3=0.000      in interval
> [0.125; 0.375)    1/3=0.333      in interval
> [0.375; 0.625)    2/3=0.667      (!!!) OUT OF INTERVAL (!!!)
> [0.625; 1.000]    3/3=1.000      in interval
>
> LSB=0.25, shift=-LSB/2=-0.125
>
> One point is out of interval.  (It is for any n>=2.)
> Did you think about this earlier?
>
> Also, you can state that the interpetation result is the same in both
> cases: it is totally alienated from the ADC matter (i.e. transfer
> function).
>
> I know that such a solution is very popular today, but it is comletely
> naked.
>
> About calibration.  I described that the ADC is ideal.  Therefore, it is
> not necessary to calibrate it.  (Calibration is another process, but during
> it, you also must take into account that an ADC result, or code,
> corresponds with an interval of input volatges inssted of a single point.)
>
> Thanks a lot for your help and explanation.
>
> Alex

By "calibration" I meant that you had to relate the input voltages to 
the output codes.  Being "ideal" doesn't relieve that need.  But, it may 
be simple indeed!

Fred

On Nov 9, 12:37&#4294967295;pm, "sanaysan" <sanaysan@n_o_s_p_a_m.gmail.com> wrote:

> About calibration. &#4294967295;I described that the ADC is ideal. &#4294967295;Therefore, it is
> not necessary to calibrate it. &#4294967295;(Calibration is another process, but during
> it, you also must take into account that an ADC result, or code,
> corresponds with an interval of input volatges inssted of a single point.)

The ADC is only one, tiny part in the aqcuisition system.
There are any number of analog amplifiers and losses
throughout, so even if you can describe the ADC in detail,
you don't have any clue about how the voltages the ADC
discretizes relate to whatever goes on in the real world.

You need to calibrate your system in order to relate the
numbers out of the ADC to any measurable quantity of the
real world.

Rune