Question about the z-transform for ARMA modelling

Hello!

I have some questions about the z-transform and what to use it for in
for example ARMA-filters. I know it is used to find poles and zeros,
but what else?

Consider an ARMA filter: 
y(t)+a1*y(t-1)+a2*y(t-2)=x(t)+c1*x(t-1)+c2*x(t-2)

After z-tranformation it can be written:  Y(z)=H(z)*X(z)  
                                         
                                          where H(z) is the transfer
function

As I understand, you never simulate time series in the z-plane, i.e.,
obtaining a series Y(z) which is transform to y(t) afterwards. It this
true? And what would X(z) in that case be?

Final question: Are the parameters usually estimated in the time-plane
or in the z-plane and what about advantages or disadvantages (if there
are any)?
 

I would very much appreciate any help or comments.

Sincerely
/Martin Haglund, 
Division of Structural Engineering, Lund, Sweden

Martin wrote:
> Hello!
> 
> I have some questions about the z-transform and what to use it for in
> for example ARMA-filters. I know it is used to find poles and zeros,
> but what else?
> 
> Consider an ARMA filter: 
> y(t)+a1*y(t-1)+a2*y(t-2)=x(t)+c1*x(t-1)+c2*x(t-2)
> 
> After z-tranformation it can be written:  Y(z)=H(z)*X(z)  
>                                          
>                                           where H(z) is the transfer
> function
> 
> As I understand, you never simulate time series in the z-plane, i.e.,
> obtaining a series Y(z) which is transform to y(t) afterwards. It this
> true? And what would X(z) in that case be?
> 
> Final question: Are the parameters usually estimated in the time-plane
> or in the z-plane and what about advantages or disadvantages (if there
> are any)?
>  
> 
> I would very much appreciate any help or comments.
> 
> Sincerely
> /Martin Haglund, 
> Division of Structural Engineering, Lund, Sweden

As far as I understand one of the applications of z-plane is to model or 
to design a filter that approximate H(z). Since H(z)=Y(z)/X(z) 
correspond to an IIR filter which can have poles that causes instability 
in the filter. By checking the z-plane those poles can be controlled so 
you can obtain an approximation of the real H(z) by using few 
coefficients (a0,a1 ... and c0,c1 ...) which is good in terms of 
practical implementation.
Usually one is interested in the transfer function of a system, H(z) 
which in time domain is known as the impulse response h(t) because, x(t) 
correspond to an impulse which in frequency domain excites all 
frequencies ideally. This signal can also be approximated by a white 
noise signal.

I hope that can help you.

martin_haglund@hotmail.com (Martin) wrote in message news:<b6435120.0311260144.235bff19@posting.google.com>...
> Hello!
> 
> I have some questions about the z-transform and what to use it for in
> for example ARMA-filters. I know it is used to find poles and zeros,
> but what else?
> 
> Consider an ARMA filter: 
> y(t)+a1*y(t-1)+a2*y(t-2)=x(t)+c1*x(t-1)+c2*x(t-2)
> 
> After z-tranformation it can be written:  Y(z)=H(z)*X(z)  
>                                          
>                                           where H(z) is the transfer
> function
> 
> As I understand, you never simulate time series in the z-plane, i.e.,
> obtaining a series Y(z) which is transform to y(t) afterwards. It this
> true? And what would X(z) in that case be?
> 
> Final question: Are the parameters usually estimated in the time-plane
> or in the z-plane and what about advantages or disadvantages (if there
> are any)?

Hi Martin. 

There appears to be a confusion in terminology in your post. I'll try to 
define some of the terms before I attemt to answer your questions. 

First, the filter you descripe is called (at least in the context of DSP) 
an Infinite Impulse Response (IIR) filter. The terms AutoRegressive 
Moving Average (ARMA) is usually used for describing the statistical 
properties of some stochastic process that generates random data. 
The importance is that data from an ARMA process is quite a bit more 
difficult to deal with, mathematically speaking, than data from a process
that is "only" AR. The difference is important if you deal with 
systems identification and want to use the data to say something 
about the process that generated those data.

Now, I assume you are not into systems indentification, so I'll use the 
term "IIR filter" for what you call an "ARMA filter".

As for the Z transform, I  generally  tend to see it as an analytical 
mathematical tool and not an implementational technique. There is one
exception, though: The Discrete Fourier Transform (DFT). 

The DFT is essentially the Z transform evaluated for the special case
z=exp(-jw), where j is square root of -1 and w (omega) is the angular 
frequency. The DFT is, in fact, one of the most versatile tools available 
in the whole field of the DSP. So if you regard the DFT as a special 
case of the Z transform, I think you will find that very little can 
be done *without* using the Z transform. 

I suggest you take a look at 

Lyons: "Understanding Digital Signal Processing"

to read more about the details and more elaborate explanations of these
things. 

As for where parameters are estimated, some times the systems specification
are formulated in time domain, some times the specs are formulated in 
frequency domain, i.e. in z domain. It really depends on the application 
what is most convenient.
Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0311260613.2a3db5cc@posting.google.com>...
> martin_haglund@hotmail.com (Martin) wrote in message news:<b6435120.0311260144.235bff19@posting.google.com>...
> > Hello!
> > 
> > I have some questions about the z-transform and what to use it for in
> > for example ARMA-filters. I know it is used to find poles and zeros,
> > but what else?
> > 
> > Consider an ARMA filter: 
> > y(t)+a1*y(t-1)+a2*y(t-2)=x(t)+c1*x(t-1)+c2*x(t-2)
> > 
> > After z-tranformation it can be written:  Y(z)=H(z)*X(z)  
> >                                          
> >                                           where H(z) is the transfer
> > function
> > 
> > As I understand, you never simulate time series in the z-plane, i.e.,
> > obtaining a series Y(z) which is transform to y(t) afterwards. It this
> > true? And what would X(z) in that case be?
> > 
> > Final question: Are the parameters usually estimated in the time-plane
> > or in the z-plane and what about advantages or disadvantages (if there
> > are any)?
> 
> Hi Martin. 
> 
> There appears to be a confusion in terminology in your post. I'll try to 
> define some of the terms before I attemt to answer your questions. 
> 
> First, the filter you descripe is called (at least in the context of DSP) 
> an Infinite Impulse Response (IIR) filter. The terms AutoRegressive 
> Moving Average (ARMA) is usually used for describing the statistical 
> properties of some stochastic process that generates random data. 
> The importance is that data from an ARMA process is quite a bit more 
> difficult to deal with, mathematically speaking, than data from a process
> that is "only" AR. The difference is important if you deal with 
> systems identification and want to use the data to say something 
> about the process that generated those data.
> 
> Now, I assume you are not into systems indentification, so I'll use the 
> term "IIR filter" for what you call an "ARMA filter".
> 
> As for the Z transform, I  generally  tend to see it as an analytical 
> mathematical tool and not an implementational technique. There is one
> exception, though: The Discrete Fourier Transform (DFT). 
> 
> The DFT is essentially the Z transform evaluated for the special case
> z=exp(-jw), where j is square root of -1 and w (omega) is the angular 
> frequency. The DFT is, in fact, one of the most versatile tools available 
> in the whole field of the DSP. So if you regard the DFT as a special 
> case of the Z transform, I think you will find that very little can 
> be done *without* using the Z transform. 
> 
> I suggest you take a look at 
> 
> Lyons: "Understanding Digital Signal Processing"
> 
> to read more about the details and more elaborate explanations of these
> things. 
> 
> As for where parameters are estimated, some times the systems specification
> are formulated in time domain, some times the specs are formulated in 
> frequency domain, i.e. in z domain. It really depends on the application 
> what is most convenient.
> Rune


Thanks for your answer, it was helpful. I know that the terminology
can be confusing since different books use different terms. But since
I am doing time series analysis, the term ARMA filter is used every
now and then in literature on time series modelling.


/martin

martin_haglund@hotmail.com (Martin) wrote in message news:<b6435120.0311280716.6a3775d3@posting.google.com>...
> Thanks for your answer, it was helpful. I know that the terminology
> can be confusing since different books use different terms. But since
> I am doing time series analysis, the term ARMA filter is used every
> now and then in literature on time series modelling.

I'm glad I can help. In my experence, sorting out terminology can 
account for as much as 40-60% of the work when learning a new field. 

I've been looking into some time series analysis recently, i.e. signal 
processing from a general point, not merely confined to the communications/
physics applications DSPers usually talk about. Do you have any suggestions
for a good book? I have the one by Shumway (Applied Statistical Time Series 
Analysis), butwould like to find more.

Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0311290406.4484b2e0@posting.google.com>...
> martin_haglund@hotmail.com (Martin) wrote in message news:<b6435120.0311280716.6a3775d3@posting.google.com>...
> > Thanks for your answer, it was helpful. I know that the terminology
> > can be confusing since different books use different terms. But since
> > I am doing time series analysis, the term ARMA filter is used every
> > now and then in literature on time series modelling.
> 
> I'm glad I can help. In my experence, sorting out terminology can 
> account for as much as 40-60% of the work when learning a new field. 
> 
> I've been looking into some time series analysis recently, i.e. signal 
> processing from a general point, not merely confined to the communications/
> physics applications DSPers usually talk about. Do you have any suggestions
> for a good book? I have the one by Shumway (Applied Statistical Time Series 
> Analysis), butwould like to find more.
> 
> Rune



I'm glad to hear that I am not the only one that finds sorting out
terminology time consuming. As for suggestions on literature I would
suggest "Introdution to Time series and forecasting" by Brockwell &
Davis and "Time series analysis: Forecasting and Control" by Box &
Jenkins (famous). Ljung, Lennart have also written som books on system
identification that is good but with focus on system identification in
general. L&#4294967295;tkepohl, Helmut have a book on Multiple time series
analysis. Perhaps some of these books will be useful for you.

/Martin