Identifying this pseudo convolution operation

Hi all,

I am having trouble identifying an input/output relationship in a
communication system.

In the course of my research, I encounter an input/output relationship of
this type at time n:

y_n = \sum_{l} \sum{l'} a_{n,l,l'} x_{n-l-l'}

(The summations are over all l,l' which are finite. 
It is the result of a transmission through a doubly selective channel)

I am not sure what it represents but it almost looks like a discrete time
convolution. I am ultimately trying to put it in vector form.
I need to know if this is some well known operation or a reference on this
type of operation. It would be highly appreciated, as I cannot find
anything on it.

On May 13, 6:38&#4294967295;pm, "L13" <loic.cv@n_o_s_p_a_m.gmail.com> wrote:
> Hi all,
>
> I am having trouble identifying an input/output relationship in a
> communication system.
>
> In the course of my research, I encounter an input/output relationship of
> this type at time n:
>
> y_n = \sum_{l} \sum{l'} a_{n,l,l'} x_{n-l-l'}
>
> (The summations are over all l,l' which are finite.
> It is the result of a transmission through a doubly selective channel)
>
> I am not sure what it represents but it almost looks like a discrete time
> convolution. I am ultimately trying to put it in vector form.
> I need to know if this is some well known operation or a reference on this
> type of operation. It would be highly appreciated, as I cannot find
> anything on it.

Looks like a 2-D cross correlation to me. But there are some
inconsistancies in the notation. Your function a_{} has 3 arguments
and x_{} has just one. Did you omit some commas?

Clay

On May 14, 2:19=A0pm, Clay <c...@claysturner.com> wrote:
> On May 13, 6:38=A0pm, "L13" <loic.cv@n_o_s_p_a_m.gmail.com> wrote:
>
>
>
>
>
> > Hi all,
>
> > I am having trouble identifying an input/output relationship in a
> > communication system.
>
> > In the course of my research, I encounter an input/output relationship =
of
> > this type at time n:
>
> > y_n =3D \sum_{l} \sum{l'} a_{n,l,l'} x_{n-l-l'}
>
> > (The summations are over all l,l' which are finite.
> > It is the result of a transmission through a doubly selective channel)
>
> > I am not sure what it represents but it almost looks like a discrete ti=
me
> > convolution. I am ultimately trying to put it in vector form.
> > I need to know if this is some well known operation or a reference on t=
his
> > type of operation. It would be highly appreciated, as I cannot find
> > anything on it.
>
> Looks like a 2-D cross correlation to me. But there are some
> inconsistancies in the notation. Your function a_{} has 3 arguments
> and x_{} has just one. Did you omit some commas?
>
> Clay- Hide quoted text -
>
> - Show quoted text -

Clay,
If that were the case (i.e.g, a_{n,I,I'} and x_{n,I,I'}), wouldn't
this be a matrix multiplication yielding a vector? Where the matrices
a and x vary with time, n? Which doesn't make sense to me.

For example, let I =3D I' , with 1<=3D I <=3D3, then the sumations, for a
particular n, would be a 3-dimensional column vector (I will drop the
{} notation)

a11*x11 + a12*x12 + a13*x13
a21*x21 + a22*x22 + a23*x23
a31*x31 + a32*x32 + a33*x33


this is the same as matrix multiplication

_                   _       _                   _
| a11  a12  a13 |     | x11  x21  x31 |
| a21  a22  a23 |     | x12  x22  x32 |
| a31  a32  a33 |     | x13  x23  x33 |
--                   --      --                  --

(notice x elements are transposed) where only those elements are used
where the row number for a is the same as the column number for x (a
row 1 time x column 1, but not a row 1 time x column 2).  In other
words,

A * X with a_{I,I'} * x_{I',I} =3D 0 for I not equal to I' (notice I
switched I and I' for the x elements of X).

So it seems that to make linear algebra operator one would

A * X';  a_{I,I'} * x_{I',I} =3D 0 for I /=3D I'   (X' means X transposed)


As I said, makes no sense to me.

Maurice Givens

>On May 13, 6:38=A0pm, "L13" <loic.cv@n_o_s_p_a_m.gmail.com> wrote:
>> Hi all,
>>
>> I am having trouble identifying an input/output relationship in a
>> communication system.
>>
>> In the course of my research, I encounter an input/output relationship
of
>> this type at time n:
>>
>> y_n =3D \sum_{l} \sum{l'} a_{n,l,l'} x_{n-l-l'}
>>
>> (The summations are over all l,l' which are finite.
>> It is the result of a transmission through a doubly selective channel)
>>
>> I am not sure what it represents but it almost looks like a discrete
time
>> convolution. I am ultimately trying to put it in vector form.
>> I need to know if this is some well known operation or a reference on
thi=
>s
>> type of operation. It would be highly appreciated, as I cannot find
>> anything on it.
>
>Looks like a 2-D cross correlation to me. But there are some
>inconsistancies in the notation. Your function a_{} has 3 arguments
>and x_{} has just one. Did you omit some commas?
>
>Clay
>
Hmm it almost seems like a 2-D cross correlation.
x is an input vector, so it only has one dimension.