# Deconvolution problem

Started by October 8, 2008
```Hi guys,
I have a theoretical relationship that variable z = convolution(x, y ).
I now have vectors of z and x and I am trying to deconvolute them to get
vector y. I tried to to FFT to solve for y. The problem is that I also got
some oscillation in the y vector while there should be none. The shape of
the vectors z and x all look like bell shape and they look pretty smooth.
They are close to zero(but no zeros there) at the tails. Anybody has any
idea how should I fix this problem, either using filter or other method?

Neil

```
```Neil wrote:
> Hi guys,
> &#2013266080; &#2013266080;I have a theoretical relationship that variable z = convolution(x, y ).
> I now have vectors of z and x and I am trying to deconvolute them to get
> vector y. I tried to to FFT to solve for y. The problem is that I also got
> some oscillation in the y vector while there should be none. The shape of
> the vectors z and x all look like bell shape and they look pretty smooth.
> They are close to zero(but no zeros there) at the tails. Anybody has any
> idea how should I fix this problem, either using filter or other method?
>
> Neil

If the frequency domain approach via FFT doesn't work for you, try
time domain inversion via Wiener SysID Filter. Remember that (assuming
that x[n] = y[n] = 0 for n <0)

z[n] = sum_k=0^n x[k] y[n-k]

or written out:

z = x y
z = x y + x y
z = x y + x y + x y
...

and so on. This is linear in the unknown y, and you can use least-
squares (for noisy observations z) or minimax (for smooth z) inversion
techniques to find y.

Regards,
Andor
```
```>Neil wrote:
>> Hi guys,
>> =A0 =A0I have a theoretical relationship that variable z =3D
convolution(=
>x, y ).
>> I now have vectors of z and x and I am trying to deconvolute them to
get
>> vector y. I tried to to FFT to solve for y. The problem is that I also
go=
>t
>> some oscillation in the y vector while there should be none. The shape
of
>> the vectors z and x all look like bell shape and they look pretty
smooth.
>> They are close to zero(but no zeros there) at the tails. Anybody has
any
>> idea how should I fix this problem, either using filter or other
method?
>>
>> Neil
>
>If the frequency domain approach via FFT doesn't work for you, try
>time domain inversion via Wiener SysID Filter. Remember that (assuming
>that x[n] =3D y[n] =3D 0 for n <0)
>
>z[n] =3D sum_k=3D0^n x[k] y[n-k]
>
>or written out:
>
>z =3D x y
>z =3D x y + x y
>z =3D x y + x y + x y
>...
>
>and so on. This is linear in the unknown y, and you can use least-
>squares (for noisy observations z) or minimax (for smooth z) inversion
>techniques to find y.
>
>Regards,
>Andor
>

I don't have DSP background, do you know any good reference on 'Wiener
SysID Filter'? and I don't understand the '3D' in your formula.

Neil
```
```On 8 Okt., 16:16, "niuer" <iam...@gmail.com> wrote:
> >Neil wrote:
> >> Hi guys,
> >> =A0 =A0I have a theoretical relationship that variable z =3D
> convolution(=
> >x, y ).
> >> I now have vectors of z and x and I am trying to deconvolute them to
> get
> >> vector y. I tried to to FFT to solve for y. The problem is that I also
> go=
> >t
> >> some oscillation in the y vector while there should be none. The shape
> of
> >> the vectors z and x all look like bell shape and they look pretty
> smooth.
> >> They are close to zero(but no zeros there) at the tails. Anybody has
> any
> >> idea how should I fix this problem, either using filter or other
> method?
>
> >> Neil
>
> >If the frequency domain approach via FFT doesn't work for you, try
> >time domain inversion via Wiener SysID Filter. Remember that (assuming
> >that x[n] =3D y[n] =3D 0 for n <0)
>
> >z[n] =3D sum_k=3D0^n x[k] y[n-k]
>
> >or written out:
>
> >z =3D x y
> >z =3D x y + x y
> >z =3D x y + x y + x y
> >...
>
> >and so on. This is linear in the unknown y, and you can use least-
> >squares (for noisy observations z) or minimax (for smooth z) inversion
> >techniques to find y.
>
> >Regards,
> >Andor
>
> I don't have DSP background, do you know any good reference on 'Wiener
> SysID Filter'?

Any book on adaptive / optimal filters, or more generally, statistical
signal processing.
> and I don't understand the '3D' in your formula.
> Would you please explain more?

That's funny. Everytime I type "=" it seems to show up as "=3D" in
your newsreader. I've been posting many "="'s here in this group, and
you are the first one to attach a 3D, so I assume that this a problem

Essentially, I just wrote out the convolution sum and noted that it is
linear in the unknown variable and suggeste how to solve for the
variable. What's not clear?

Regards,
Andor
```
```Andor wrote:

...

> That's funny. Everytime I type "=" it seems to show up as "=3D" in
> your newsreader. I've been posting many "="'s here in this group, and
> you are the first one to attach a 3D, so I assume that this a problem