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#### RESTORING BINARY SIGNAL FROM LOW FREQUENCIES

```Can someone points me to the methods to solve the following problem:

Function s(x) is binary {0,1}.  We set high frequencies above some
cuttoff limit to 0.
Resulting function is L(x). How can we restore s(x) from L(x)?

Thank you,

Yuri
```
 0
y_granik (4)
10/31/2008 4:07:30 AM
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```>Function s(x) is binary {0,1}.  We set high frequencies above some
>cuttoff limit to 0.
>Resulting function is L(x). How can we restore s(x) from L(x)?

What exactly is the binary signal?  What is the application?

Emre
```
 0
eguven (123)
10/31/2008 4:34:40 AM
```Ostap wrote:

> Can someone points me to the methods to solve the following problem:

> Function s(x) is binary {0,1}.  We set high frequencies above some
> cuttoff limit to 0.
> Resulting function is L(x). How can we restore s(x) from L(x)?

It depends.  Look in the paper:

Nyquist, Harry. "Certain factors affecting telegraph speed". Bell System
Technical Journal, 3, 324�346, 1924

It describes your problem very well.

-- glen

```
 0
gah (12851)
10/31/2008 4:36:38 AM
```On Oct 30, 9:07=A0pm, Ostap <y_gra...@yahoo.com> wrote:
> Can someone points me to the methods to solve the following problem:
>
> Function s(x) is binary {0,1}. =A0We set high frequencies above some
> cuttoff limit to 0.
> Resulting function is L(x). How can we restore s(x) from L(x)?
>
> Thank you,
>
> Yuri

Binary function is a piece-wise constant function that can have
only two values 0 or 1, say given the interval [0, 10] we define

s(x)=3D0, 0<=3Dx<2
s(x)=3D1, 2<=3Dx<7
s(x)=3D0, 7<=3Dx<=3D7.2
s(x)=3D1, 7.2<=3D10

The application is in the field of optics.

Hope this helps,

Yuri

```
 0
y_granik (4)
10/31/2008 4:42:19 AM
```On Oct 31, 12:36=A0am, Glen Herrmannsfeldt <g...@ugcs.caltech.edu>
wrote:
> Ostap wrote:
> > Can someone points me to the methods to solve the following problem:
> > Function s(x) is binary {0,1}. =A0We set high frequencies above some
> > cuttoff limit to 0.
> > Resulting function is L(x). How can we restore s(x) from L(x)?
>
> It depends. =A0Look in the paper:
>
> Nyquist, Harry. "Certain factors affecting telegraph speed". Bell System
> Technical Journal, 3, 324=96346, 1924
>
> It describes your problem very well.
>
> -- glen

What's a good source for BSTJ ???

Thanks,

John
```
 0
sampson164 (501)
10/31/2008 6:15:01 AM
```John wrote:

> On Oct 31, 12:36 am, Glen Herrmannsfeldt <g...@ugcs.caltech.edu>
> wrote:
>
>>Ostap wrote:
>>
>>>Can someone points me to the methods to solve the following problem:
>>>Function s(x) is binary {0,1}.  We set high frequencies above some
>>>cuttoff limit to 0.
>>>Resulting function is L(x). How can we restore s(x) from L(x)?

>>It depends.  Look in the paper:

>>Nyquist, Harry. "Certain factors affecting telegraph speed". Bell System
>>Technical Journal, 3, 324�346, 1924

>>It describes your problem very well.

> What's a good source for BSTJ ???

The library of a good engineering college.  Especially one around
since 1924.

http://www.worldcat.org/oclc/1519469&referer=brief_results

put in your zip code, state, province, or country and it will
find one near you.

-- glen

```
 0
gah (12851)
10/31/2008 6:47:26 AM
```
Ostap wrote:

> Can someone points me to the methods to solve the following problem:
>
> Function s(x) is binary {0,1}.  We set high frequencies above some
> cuttoff limit to 0.
> Resulting function is L(x). How can we restore s(x) from L(x)?

Is the function s(x) cyclostationary or not?

If it is cyclostationary, then the task is the typical problem of the
digital communication and there are many ways for solution depending on
L(x).

If s(x) is not cyclostationary, then the problem is non-trivial and the
solution may not be unique.

DSP and Mixed Signal Design Consultant
http://www.abvolt.com

```
 0
10/31/2008 2:18:21 PM
```>
>
>Ostap wrote:
>
>> Can someone points me to the methods to solve the following problem:
>>
>> Function s(x) is binary {0,1}.  We set high frequencies above some
>> cuttoff limit to 0.
>> Resulting function is L(x). How can we restore s(x) from L(x)?
>
>Is the function s(x) cyclostationary or not?
>
>If it is cyclostationary, then the task is the typical problem of the
>digital communication and there are many ways for solution depending on
>L(x).
>
>If s(x) is not cyclostationary, then the problem is non-trivial and the
>solution may not be unique.
>
>
>DSP and Mixed Signal Design Consultant
>http://www.abvolt.com
>

Is the pure "unmodulated" binary signals are considered cyclostationary? I
tend to believe that they are stationary process as I remember from my
digital communication readings! My understanding is that cyclostationary is
a feature in modulated signals due to the built in periodicity in the
carrier! Agree?
>
>
>
```
 0
ytachwali (18)
10/31/2008 3:48:37 PM
```>Binary function is a piece-wise constant function that can have
>only two values 0 or 1, say given the interval [0, 10] we define
>
>s(x)=3D0, 0<=3Dx<2
>s(x)=3D1, 2<=3Dx<7
>s(x)=3D0, 7<=3Dx<=3D7.2
>s(x)=3D1, 7.2<=3D10

Here is my take:  if you throw away high frequencies greater than some
cutoff, you lose the sharp transitions and get a smoother signal.  So the
result should be around 0 and 1 most of the time, assuming you keep enough
of the frequency content.

By optics, are you referring to binary optical communication?  Do you have
control over the cutoff frequency, or is there some other constraint such
as channel bandwidth?

Emre
```
 0
eguven (123)
11/1/2008 1:21:46 AM
```>Is the function s(x) cyclostationary or not?
>
>If it is cyclostationary, then the task is the typical problem of the
>digital communication and there are many ways for solution depending on
>L(x).
>
>If s(x) is not cyclostationary, then the problem is non-trivial and the
>solution may not be unique.

I don't see how cyclostationarity is relevant.  The OP did not say it
comes from a random process.  In fact his example is rather deterministic.

To the OP: there may be an extremely neat solution to your problem.  If
you have enough Fourier domain measurements, you can reconstruct your
signal *exactly* with an overwhelming probability, assuming the number of
jumps and/or the nonzero values in your signal is  small (i.e. sparse) with
respect to the total number of the samples.  See the below reference [1].

Hope this helps,

Emre

[1] E. J. Candès, J. Romberg and T. Tao. Robust uncertainty principles:
exact signal reconstruction from highly incomplete frequency information.
IEEE Trans. Inform. Theory, 52 489-509.

```
 0
eguven (123)
11/1/2008 1:44:42 AM
```>>Is the function s(x) cyclostationary or not?
>>
>>If it is cyclostationary, then the task is the typical problem of the
>>digital communication and there are many ways for solution depending on

>>L(x).
>>
>>If s(x) is not cyclostationary, then the problem is non-trivial and the

>>solution may not be unique.
>
>I don't see how cyclostationarity is relevant.  The OP did not say it
>comes from a random process.  In fact his example is rather
deterministic.

I understand a deterministic signal can also be cyclostationary; so never
mind that part.

Still, I don't see why exactly one needs this condition.  Does anyone have
a good explanation?

Emre
```
 0
eguven (123)
11/1/2008 2:34:36 AM
```On Oct 31, 6:44=A0pm, "emre" <egu...@ece.neu.edu> wrote:
> To the OP: there may be an extremely neat solution to your problem. =A0If
> you have enough Fourier domain measurements, you can reconstruct your
> signal *exactly* with an overwhelming probability, assuming the number of
> jumps and/or the nonzero values in your signal is =A0small (i.e. sparse) =
with
> respect to the total number of the samples. =A0See the below reference [1=
].
>
> Hope this helps,
>
> Emre
>
> [1] E. J. Cand=E8s, J. Romberg and T. Tao. Robust uncertainty principles:
> exact signal reconstruction from highly incomplete frequency information.
> IEEE Trans. Inform. Theory, 52 489-509.

Emre,

Perfect reference! Thank you very much. We can consider this thread to
be over.

Yuri
```
 0
y_granik (4)
11/1/2008 5:17:24 AM

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