impulse response computing

Hello,

I'm following a course of digital image processing and a mathematical
homework is first given. I need a little help for solving this problem:

What is the impulse response of the system:
    1/ if its transfer function is G(z1,z2) =3D a + b.z1^(-1) + c.z2^(-1)
+ d.z1^(-1).z2^(-1) + e.z1.z2^(-1)
    2/ if its frequency response if G(f,g) =3D a + b.cos(2*Pi*f) +
c=2Ecos(2*Pi*g)

Are they linear ? Shift variant ? FIR or IIR ?

for the 2/, we use the inverse fourier transform, I think it's ok, but
for the 1/, I suppose we can use the inverse Z transform, but how to
find the coutours of integration ? Is there an other way to find the
impulse response ? Is it possible to find the frequency response with
the transfer function and then operate an inverse fourier transform to
find the impulse response ?

Thanks for you help !
T=F4F

ToF skrev:
> Hello,
>
> I'm following a course of digital image processing and a mathematical
> homework is first given. I need a little help for solving this problem:
>
> What is the impulse response of the system:
>     1/ if its transfer function is G(z1,z2) = a + b.z1^(-1) + c.z2^(-1)
> + d.z1^(-1).z2^(-1) + e.z1.z2^(-1)
>     2/ if its frequency response if G(f,g) = a + b.cos(2*Pi*f) +
> c.cos(2*Pi*g)
>
> Are they linear ? Shift variant ? FIR or IIR ?

I'm probably revealing my ignorance yet again when I ask in what
context such transfer functions appear. I have only encountered
z transforms with one variable in the past, here it seems to be two.

> for the 2/, we use the inverse fourier transform, I think it's ok, but
> for the 1/, I suppose we can use the inverse Z transform, but how to
> find the coutours of integration ?

The usual technique is to look in a table of transform pairs.
If you find your expression to match one of the general forms
in the table, you can derive the time-domain expression from
that.

> Is there an other way to find the
> impulse response ? Is it possible to find the frequency response with
> the transfer function and then operate an inverse fourier transform to
> find the impulse response ?

There are several ways to find the impulse response from a z transform:

- Use the tables, as indicated above
- Inverse contour integration
- Find the coefficients of the diffrence equation and impose
  a unit impulse
- Find the coefficients of the transgfer functions and use an
  inverse Fourier Transform

Rune

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:1163774914.684649.258750@h54g2000cwb.googlegroups.com...
>
> ToF skrev:
>> Hello,
>>
>> I'm following a course of digital image processing and a mathematical
>> homework is first given. I need a little help for solving this problem:
>>
>> What is the impulse response of the system:
>>     1/ if its transfer function is G(z1,z2) = a + b.z1^(-1) + c.z2^(-1)
>> + d.z1^(-1).z2^(-1) + e.z1.z2^(-1)
>>     2/ if its frequency response if G(f,g) = a + b.cos(2*Pi*f) +
>> c.cos(2*Pi*g)
>>
>> Are they linear ? Shift variant ? FIR or IIR ?
>
> I'm probably revealing my ignorance yet again when I ask in what
> context such transfer functions appear. I have only encountered
> z transforms with one variable in the past, here it seems to be two.

Rune,

Isn't it like a discrete 2-D spatial array with z^-1 the spacing as in delay 
for a time domain situation?  Also, one has to imagine that there could be 
z1^-1 in one direction and z2^-1 in the other direction (rectangular / not 
square quads in the array).

Fred 

Fred Marshall skrev:
> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> news:1163774914.684649.258750@h54g2000cwb.googlegroups.com...
> >
> > ToF skrev:
> >> Hello,
> >>
> >> I'm following a course of digital image processing and a mathematical
> >> homework is first given. I need a little help for solving this problem:
> >>
> >> What is the impulse response of the system:
> >>     1/ if its transfer function is G(z1,z2) = a + b.z1^(-1) + c.z2^(-1)
> >> + d.z1^(-1).z2^(-1) + e.z1.z2^(-1)
> >>     2/ if its frequency response if G(f,g) = a + b.cos(2*Pi*f) +
> >> c.cos(2*Pi*g)
> >>
> >> Are they linear ? Shift variant ? FIR or IIR ?
> >
> > I'm probably revealing my ignorance yet again when I ask in what
> > context such transfer functions appear. I have only encountered
> > z transforms with one variable in the past, here it seems to be two.
>
> Rune,
>
> Isn't it like a discrete 2-D spatial array with z^-1 the spacing as in delay
> for a time domain situation? Also, one has to imagine that there could be
> z1^-1 in one direction and z2^-1 in the other direction (rectangular / not
> square quads in the array).

Yeees.... but isn't it common to separate these into two 1D ZTs that
are treated separately?

Rune

Rune Allnor wrote:
> Fred Marshall skrev:

>> Isn't it like a discrete 2-D spatial array with z^-1 the
>> spacing as in delay for a time domain situation? Also, one has
>> to imagine that there could be z1^-1 in one direction and z2^-1
>> in the other direction (rectangular / not square quads in the
>> array). 
> 
> Yeees.... but isn't it common to separate these into two 1D ZTs
> that are treated separately?

It's uncommon to do that to nonseparable transfer functions (duh), 
unless you're referring to approximation by a separable filter.


Martin

-- 
Quidquid latine scriptum sit, altum viditur.

Martin Eisenberg skrev:
> Rune Allnor wrote:
> > Fred Marshall skrev:
>
> >> Isn't it like a discrete 2-D spatial array with z^-1 the
> >> spacing as in delay for a time domain situation? Also, one has
> >> to imagine that there could be z1^-1 in one direction and z2^-1
> >> in the other direction (rectangular / not square quads in the
> >> array).
> >
> > Yeees.... but isn't it common to separate these into two 1D ZTs
> > that are treated separately?
>
> It's uncommon to do that to nonseparable transfer functions (duh),

As I hinted at in my fist post, I'm on the level of ignorance where
I haven't seen too many non-separable 2D functions. As a matter
of fact I have seen none; all problems I have worked with have dealt
with separable functions. In some cases some severe simplifications
and idealizations -- "severe" in the sense that they have introduced
large errors -- have been used to obtain those separable functions.

Rune

Rune Allnor wrote:
> Martin Eisenberg skrev:
>> Rune Allnor wrote:
>> > Fred Marshall skrev:
>>
>> >> Isn't it like a discrete 2-D spatial array with z^-1 the
>> >> spacing as in delay for a time domain situation? Also, one
>> >> has to imagine that there could be z1^-1 in one direction
>> >> and z2^-1 in the other direction (rectangular / not square
>> >> quads in the array).
>> >
>> > Yeees.... but isn't it common to separate these into two 1D
>> > ZTs that are treated separately?
>>
>> It's uncommon to do that to nonseparable transfer functions
>> (duh), 
> 
> As I hinted at in my fist post, I'm on the level of ignorance
> where I haven't seen too many non-separable 2D functions. As a
> matter of fact I have seen none; all problems I have worked with
> have dealt with separable functions.

I haven't actually carried out any 2D processing myself, but I know 
that nonseparable wavelet bases are a current research topic. It's 
also sometimes useful to subject the higher-order kernels (which may 
or may not be separable) of a Volterra series to multivariate z-
transforms, though of course the z's don't correspond to spatial 
directions in that case.

> In some cases some severe simplifications and idealizations --
> "severe" in the sense that they have introduced large errors --
> have been used to obtain those separable functions. 

That's what I meant by the part of the sentence that you snipped, I 
guess. Out of interest, what systems did those functions describe? 
Did you choose separable approximations to simplify processing or for 
other reasons?


Martin

-- 
When he had finally achieved a position that
allowed him to say everything he thought, he
only thought of his position anymore.
--Gabriel Laub

Rune Allnor wrote:
> Martin Eisenberg skrev:
>> Rune Allnor wrote:
>>> Fred Marshall skrev:
>>>> Isn't it like a discrete 2-D spatial array with z^-1 the
>>>> spacing as in delay for a time domain situation? Also, one has
>>>> to imagine that there could be z1^-1 in one direction and z2^-1
>>>> in the other direction (rectangular / not square quads in the
>>>> array).
>>> Yeees.... but isn't it common to separate these into two 1D ZTs
>>> that are treated separately?
>> It's uncommon to do that to nonseparable transfer functions (duh),
> 
> As I hinted at in my fist post, I'm on the level of ignorance where
> I haven't seen too many non-separable 2D functions. As a matter
> of fact I have seen none; all problems I have worked with have dealt
> with separable functions. In some cases some severe simplifications
> and idealizations -- "severe" in the sense that they have introduced
> large errors -- have been used to obtain those separable functions.

Aside from Gaussian and mirrored exponential, what separable functions 
do you know?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Martin Eisenberg skrev:
> Rune Allnor wrote:

> > In some cases some severe simplifications and idealizations --
> > "severe" in the sense that they have introduced large errors --
> > have been used to obtain those separable functions.
>
> That's what I meant by the part of the sentence that you snipped, I
> guess. Out of interest, what systems did those functions describe?
> Did you choose separable approximations to simplify processing or for
> other reasons?

There are mainly two areas of 2D z transforms I have worked with:
Images and Partial Differential Equations.

Images are sampled on regular (n,m) points, so there separable
kernels are used by default.

Computations on PDEs are some timesdone in frequency domain
(i.e. z domain), and here one goes to great lengths to get things
to become separable. In my area of interest -- ocean acoustics --
one represent both the sea floor and sea surface as plane surfaces,
and the water column and the sea bottom as horizontally
homogeneous media. Neither are correct, but that's what it takes
to obtain separable physical domains. Of course, simplifying
the medium like that, all the interesting effects, that mess up
real-life data analyis, go away from the models. Interestingly,
there is a large an increasing gap between what happens at sea,
and what people who work at sea are concerned with, and what
the people who work exclusively with numerical models in the
labs, are interested in.

Anyway, because of those separable PDEs it was possible to do
numerical modelling of acoustic propagation literally half-way
around the world in the early '80s. With FEM or FD types programs
that would not have been possible at the time, other than possibly
at the two or three largest numerical computing facilities in the
world.

Rune

Rune Allnor wrote:
> 
> Martin Eisenberg skrev:
> > Rune Allnor wrote:
> 
> > > In some cases some severe simplifications and idealizations --
> > > "severe" in the sense that they have introduced large errors --
> > > have been used to obtain those separable functions.
> >
> > That's what I meant by the part of the sentence that you snipped, I
> > guess. Out of interest, what systems did those functions describe?
> > Did you choose separable approximations to simplify processing or for
> > other reasons?
> 
> There are mainly two areas of 2D z transforms I have worked with:
> Images and Partial Differential Equations.
> 
> Images are sampled on regular (n,m) points, so there separable
> kernels are used by default.

If you are saying that the regular sampling of an image precludes the
use of filters that are not separable into 2 1d filters - that is
incorrect. If you are saying that they are not commonly used - that is
also incorrect. The commonly used implementation of a Sharpening filter
would be one example of  a filter that can't be separated into two 1d 
filters.

-jim 



> 
> Computations on PDEs are some timesdone in frequency domain
> (i.e. z domain), and here one goes to great lengths to get things
> to become separable. In my area of interest -- ocean acoustics --
> one represent both the sea floor and sea surface as plane surfaces,
> and the water column and the sea bottom as horizontally
> homogeneous media. Neither are correct, but that's what it takes
> to obtain separable physical domains. Of course, simplifying
> the medium like that, all the interesting effects, that mess up
> real-life data analyis, go away from the models. Interestingly,
> there is a large an increasing gap between what happens at sea,
> and what people who work at sea are concerned with, and what
> the people who work exclusively with numerical models in the
> labs, are interested in.
> 
> Anyway, because of those separable PDEs it was possible to do
> numerical modelling of acoustic propagation literally half-way
> around the world in the early '80s. With FEM or FD types programs
> that would not have been possible at the time, other than possibly
> at the two or three largest numerical computing facilities in the
> world.
> 
> Rune

----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==----
http://www.newsfeeds.com The #1 Newsgroup Service in the World! 120,000+ Newsgroups
----= East and West-Coast Server Farms - Total Privacy via Encryption =----