Max/Min of continuous signal

On 29.11.2014 9:24, glen herrmannsfeldt wrote:

>
> Next, consider all samples -1, except for two consecutive +1.
> I believe this gets the largest positive peak.
>
> -- glen
>

Glen, no. The largest positive peak is unlimited, and is achieved for 
the sequence of alternating samples 1 and -1, with a single -1 sample 
removed from the sequence, so that you have two consecutive +1.

The value of the peak depends on the length of the sequence, and each 
sample in the sequence contributes to the peak. See the details in my 
other post.

Evgeny.

On 29.11.2014 9:24, glen herrmannsfeldt wrote:
> Next, consider all samples -1, except for two consecutive +1.
> I believe this gets the largest positive peak.
>
> -- glen
>

Let's consider the digital signal you've proposed. Let the length of the 
digital signal be 20 samples. So we have:

-1,-1,-1,-1,-1,-1,-1,-1,-1, 1, 1,-1,-1,-1,-1,-1,-1,-1,-1,-1
The positive peak for its analog reconstruction is about 1.57.

Now let's replace two of -1 with 1:
-1,-1,-1,-1,-1,-1,-1, 1,-1, 1, 1,-1, 1,-1,-1,-1,-1,-1,-1,-1
Now the positive peak is around 2.08.

Keep replacing -1 with 1:
For
-1,-1,-1,-1,-1, 1,-1, 1,-1, 1, 1,-1, 1,-1, 1,-1,-1,-1,-1,-1
the positive peak is 2.37,

for
-1,-1,-1, 1,-1, 1,-1, 1,-1, 1, 1,-1, 1,-1, 1,-1, 1,-1,-1,-1
it's 2.56

for
-1, 1,-1, 1,-1, 1,-1, 1,-1, 1, 1,-1, 1,-1, 1,-1, 1,-1, 1,-1
the positive peak is 2.71.

Here you can see the plots of the considered sampled signals along with 
their analog reconstructions:

http://tinyurl.com/nwaws63

The tendency is for the value of positive peak to rise as long as you 
increase the length of the sequence of alternating -1 and 1. It's due to 
the fact that the positive peak is the sum of shifted versions of the 
sinc function multiplied by values of the respective samples. Since the 
sinc function has alternating lobes of positive and negative signs, by 
properly choosing signs of the samples you can increase the magnitude of 
the positive peak.

Also, you can always increase the value of the positive peak if you can 
add more samples. The trick is that while the integral of sinc(x)dx from 
-inf to +inf is finite, the integral of |sinc(x)|dx from -inf to +inf is 
infinite. Which means that in theory you can drive the value of the 
positive peak above any predefined positive value as long as you can 
choose the number of samples in the sequence.

Hope this helps.

Evgeny.

I believe that the OP is asking a different question. He's not looking for the maximum value that could ever be produced by reconstructing samples that are constrained to be between -1 and +1. Rather I think he's asking the question, given a finite set of samples produced by some signal of interest, what is the maximum value that can occur after reconstruction. Perhaps the OP can clarify?

Bob

On 02.12.2014 7:27, radams2000@gmail.com wrote:
> I believe that the OP is asking a different question. He's not looking for the maximum value that could ever be produced by reconstructing samples that are constrained to be between -1 and +1. Rather I think he's asking the question, given a finite set of samples produced by some signal of interest, what is the maximum value that can occur after reconstruction. Perhaps the OP can clarify?
>
> Bob
>

Bob, I agree that information about either the waveform or occupied 
spectrum of the signal of interest is required for a meaningful answer.

Indeed, there's an important aspect which makes the example I cited 
unrealistic -- namely that the analog signal has strong spectral 
components closely approaching the Nyquist frequency. In a practical 
system you'd have an anti-aliasing filter suppressing some frequency 
range near Nyquist -- which I believe limits the maximum peak value of 
reconstructed samples.

Evgeny.

On 12/2/2014 1:47 AM, Evgeny Filatov wrote:
> On 02.12.2014 7:27, radams2000@gmail.com wrote:
>> I believe that the OP is asking a different question. He's not looking
>> for the maximum value that could ever be produced by reconstructing
>> samples that are constrained to be between -1 and +1. Rather I think
>> he's asking the question, given a finite set of samples produced by
>> some signal of interest, what is the maximum value that can occur
>> after reconstruction. Perhaps the OP can clarify?
>>
>> Bob
>>
>
> Bob, I agree that information about either the waveform or occupied
> spectrum of the signal of interest is required for a meaningful answer.
>
> Indeed, there's an important aspect which makes the example I cited
> unrealistic -- namely that the analog signal has strong spectral
> components closely approaching the Nyquist frequency. In a practical
> system you'd have an anti-aliasing filter suppressing some frequency
> range near Nyquist -- which I believe limits the maximum peak value of
> reconstructed samples.

I believe he has given enough information.  He is looking for the 
max/min of the actual analog signal before sampling.  See his second 
post on 11/27.

-- 

Rick