z-transform problems

[steveB]

Q1a I believe you are correct

Q1 (after part a) It seems to me different ways of saying the same thing, but I would have said it the same way as you did.

Q2
I don't see any reason why you can't use DTFT for IIR systems, if it exists for the system.

I believe you are correct that stable IIR systems will have a DTFT, for causal systems. In Q1 you noted an anti-causal system where the ROC did not include the unit circle, hence the DTFT does not exist in that case.

You use the terminology "handle" which is imprecise to me. The transforms do not handle the system. They only either exist or are undefined in certain cases. When they exist, we can used them and the engineer can "handle" the system. So in your question, for a causal system the DTFT can not "handle" the unstable IIR system because it does not exist in that case. The z-transform is in better position to "handle" this case because there still may be a region of convergence outside the unit circle. But, usually if the system is unstable, we can't really handle that system in practice.

DISCLAIMER: I hope I have these correct. I find the questions not so easy because I usually deal only with causal systems, and a lot of the answers to these questions might be different than our intuition would expect in the non-causal and anti-causal cases. Since 30 years has past since I studied this formally, i could easily give an answer that is correct for the causal case but not for the other cases.

 

[PG1995]

Thanks a lot, Steve. It's really kind of you. And as far as I'm concerned, many a time I just need general understanding which doesn't need to be 100% correct.

Best wishes
PG

 

[PG1995]

Hi

Could you please help me with these queries, Q1 and Q2? Thank you.

Regards
PG

 

[MrAl]

Hi,

To start, when i convert y=z/(1-z) to a difference equation i get:
y[n]=y[n-1]-x[n]

or:
y[n+1]=y[n]-x[n+1]

So it appears that the value of y[n+1] depends on the current value of y but also on a future value of x.

 

[PG1995]

Q1: I'm still struggling with it. Please help me. Thank you.

Q2: So, when pole is located at z=1 then the system is marginally stable or unstable. In such a case when pole is at z=1, one needs to look at the difference equation to see if the system is causal or anticausal. Please correct me if I'm wrong. Thanks.

 

[MrAl]

Hi,

I found a reference which states that it is causal if and only if the unit sample response is zero for n<0.

Looking at the diagram it looks causal, so looking back at the difference equation:
y[n+1]=y[n]-x[n+1]

if x[n+1]=0, then y[n+1]=y[n] and that's causal, and might be considered a "DC" value. If we straighten out the circle top and bottom and extend it to y equals plus and minus infinity, we end up with a straight line at x=1 that extends to plus and minus infinity, and then if we shift that one unit to the left we then have the system mapped to the complex 's' plane, where the point is now at the origin (0,0), which indicates a DC value (which could be zero).

The only thing i dont like about this is that where did that first value of yn come from if there was never any input. A value of yn that suddenly appears without any xn input means it is not causal. The answer could be that it depends on the input but unfortunately i cant find any better reference.

Another reference states that if we convert the 'sequence' into an odd and even sequence, if we can recover the original sequence from the odd and even sequences then the system is causal, but if not then it is not causal. Unfortunately this reference is skimpy as well with little other information other than how to calculate the two sequences and how to reconstruct the original from them. It too requires a sequence but since we have the sequence in that text maybe this would help.
It's been so so long since i had to use any of this myself and some of the references are from the 70's.

 

[MrAl]

Hi,

I just wanted to add that in the real life situation we find out soon enough if it is causal or not. I had to ask myself why i didnt have to review this too deeply in the past, and i found the answer was that when we go to design a real life circuit the stability issue comes up and is more important because we must insure that the system is stable. Then once we have that established, we figure out an implementation and that leads to some real world values which may or may not be starting values, but they must come from somewhere. Where they come from would tell us right away if the system is causal or not because if we had to use memory of some kind to get the values or a startup algorithm for generating values then it cant be causal.

There was a really good example of this in the field of robotics but unfortunately i cant remember where it might be found after so many years. The idea though was that we would like to know what a particular robot movement WOULD do if we activated a given control mechanism in a particular way. If we knew this advanced information beforehand, we could get the robot to respond faster and more accurately. One implementation idea then might be to build up a table of possible outcomes and use that table to look up the resulting responses so we could just pick the response we want and simply apply the applicable excitation.
This system would require memory so that we could look up future values and use them in the calculation, and this differs from using derivatives to predict outcomes because the derivatives only *imply* what the outcome would be while the actual stored value would be an actual absolute outcome precalculated or measured from a careful pretesting procedure.

I hope this isnt too off track, but the main point was that in the real life situation we find out all too soon if we have a causal system or not.

 

[steveB]

Mr Al,

Those are good points, and I feel the same way about this. I also typically deal with time domain systems, and causality is so embedded, necessary and implied in all work we do in this area. As a result, our intuition is very good for causal system and maybe not so good for non causal systems. We study non-causal systems, but by not dealing with them in practice, we forget some things and get biased towards causality rules/properties.

PG,

I thought it would be good to point out that an engineer/scientist/technologist may still encounter non-causal systems in practice. Images, which can involve 2D transforms do not require causality. We also sometimes deal with 1D images (scans), and this data is very similar to a time sequence, but causality need not apply there.

 

[MrAl]

Hi Steve,


I am glad you brought up the 2d picture transforms as that area is probably a good area to discuss in relation to causality.

It is interesting that whatever picture we take with our camera and upload to the computer can be regarded as having happened in the past. Thus when we go to perform some mathematical operations on the picture we might say that all of the data is in the past so we never have to look to the real future. Maybe this is why i read about an ongoing debate regarding what causality really is, or maybe it has to fall within a certain context in order to be classified correctly.
Taking that same picture and performing some sort of enhancement we have to read through the data scan line by scan line, but the human eye/mind sees the picture in true 2d so we cant just scan through each line one by one because the lines translate to the real world not only as being organized as successive lines stuck end to end but also as lines one very close to the other where there are important relationships between vertical pixels as wells as the horizontal (scan line) ones. This means we must look ahead, on the order of at least one more scan line, in order to render something which the 2d human eye/mind will interpret as somehow better or at least different. This of course suggests non causality. We cant just look at pixel x,y, we have to also look at pixel x+1,y, and even x,y+1 or x+1,y+1, which means we may be looking at pixels linearly far away from the first one which may be interpreted as in the future. That's because the human eye/mind wont be able to stand an algorithm which just operates on one pixel at a time (for this kind of enhancement). So i think we can se that is not causal. If we did have an algorithm that could operate on only one pixel at a time i think we could say it was causal. That would be maybe a color adjusting algorithm for example, where the human perception works on a point by point basis rather than spatial.

The above description where we consider the current pixel to be the present and any others to be in the future i think is the correct perspective. The perspective where the picture is completely in the past i think is more for the world philosophers than the engineers.

We could also consider a camera that takes the picture one scan line at a time and uploads each pixel one at a time in sequence. Now it becomes impossible to enhance the photo with only the current pixel in hand. We have to wait for future values to come in. But does that mean it is not causal? Once we do get those values we can do the required work, we just have to wait longer.

The other definition i have read now is that a causal system output does not change without a change to the input. There's no mention about future values. If the output does not change with no input but only changes when there is an input, then it is causal.

Your thoughts appreciated

 

[PG1995]

Thank you, Steve, MrAl.

I understand to some extent that issue of causality and non-causality. But at the moment I'm more worried about Q1 which is not actually related to causality. It would be nice if you could help me with that query. Thank you.

Regards
PG

 

[MrAl]

Hi again,

Those two solutions can be found from a certain difference equation where we solve it differently for each solution (such as different initial conditions). Looking at the unit sample response for successive values of n we note a pattern and thus a formula emerges for each solution, and that's where for example the negative sign can come from. However, they did not show that information so perhaps they did not want to bother you with that detail for now. They may have just wanted you to refer to the Table 4.1.2, so perhaps you should post that table so we can take a look.

 

[steveB]

Q1 is kind of a weird question. First, you are correct about the unit step function starting at -1 and going in the negative direction. That's clear.

However, the other parts of the question are basically "why is this mathematical answer the answer?". Isn't that a strange thing to ask. Maybe you are asking if there is a typo in the answer given, which is always possible. I would say, either there is a typo, or if the answer is correct then it is correct because it is correct. If the answer turns out to be -a instead of +a, then that's the way it is. If the answer turns out to start at n=-1 instead of n=0, then that's the way it is.

So, is it a typo? I don't know because I didn't try to work it out. Maybe later I'll get a chance to check it.

 

[steveB]

So, I had a chance to check the calculations for Q1. The book is correct and there is no typo. To check, I simply applied the definition of X(z) to the cases given. If x[n]=a^n u[n], then X(z)=z/(z-a) for |z|>a. If x[n]=-a^n u[-n-1], then X(z)=z/(z-a) for |z|<a.

In you are curious, the transform of x[n]=a^n u[-n] is X(z)=a/(a-z) for |z|<a.

 

[steveB]

... That's because the human eye/mind wont be able to stand an algorithm which just operates on one pixel at a time (for this kind of enhancement). So i think we can see that is not causal. If we did have an algorithm that could operate on only one pixel at a time i think we could say it was causal. That would be maybe a color adjusting algorithm for example, where the human perception works on a point by point basis rather than spatial. ...

Your thoughts appreciated

Hi MrAl,

Good thoughts and comments.

Of course the word "causality" has an implication of time in it, and successive images can be time based, as in a movie, for example. As you know, when we talk about causality for a 2D image signal which might be one "frame", then the variables are horizontal-x and vertical-y, and not time-t. Clearly causality can not apply to x and/or y because there should be no preferred direction in the x-y plane. So algorithms that we might apply are often symmetric with respect to rotation in the x-y plane. Of course, this is not absolutely required and we know from thinking about our own vision that algorithms in the brain seem to have orientation. For example if we compare two separate images with shapes to see if the shapes are similar, the time to know the answer is proportional to the angular difference between the two shapes. So our brains must be mentally rotating the image to make the comparison. I imagine that is a result of adapting (both from a species-evolution point of view and an individual-learning point of view) to a world with gravity and a concept of up and down that is created by that. - just speculating on that.

A good example or a symmetric algorithm is a Laplacian operator used for edge detection ...

https://www.cs.haifa.ac.il/~dkeren/ip/lecture9.pdf

https://www.owlnet.rice.edu/~elec539/Projects97/morphjrks/laplacian.html

... Here the second derivative operator, called the Laplacian, that can detect edges, is symmetric with respect to rotations. So past/future for x and past/future for y has no meaning. Also, the variable x and y are arbitrary since you could rotate the coordinate axis and create new axes (e.g. u and w).

 

[MrAl]

Hi Steve,

Yes that's an interesting view too, and you got me thinking about a simple 2d Fourier algorithm for picture filtering where we want to attenuate some image frequencies and amplify others. There we are sort of working in the frequency domain on image blocks. We have to have a whole block of information before we can even start.

I hope PG gets back here soon with that Table. I'd like to see how the text he is using wants to show this information. I think what is happening is that they want to skip some details so they can show others, that's all. That would be typical where they cant show every single detail of every single concept all at once or the book would be unfathomably long.

 

[PG1995]

I hope PG gets back here soon with that Table.

MrAl, I will try to post the table in next few hours. At the moment extremely busy! Thank you.

Regards
PG

Edit after several hours: Attached table. MrAl: I understand it now but you asked for the table so I have posted it. Thanks.

 

[PG1995]

Hi

What does time reversal property for z-transform mean? I had thought that it means that when a sequence x[n] having z-transform X(z) is changed to x[-n] then one simply needs to take multiplicative inverse of X(z) to find z-transform of x[-n]. But it look like I was wrong. Here, the difference between the sequences in parts (a) and (c) is that the sequence is part (c) is time reversal of the (a). But in (c), the only inverse taken from z-trasnform [1/(1-1/2z)] of (a) was of the term 1/2z to get [1/(1-2z)]. Could you please tell me what it really means? Thank you.

Regards
PG

 

[MrAl]

Hi,

Dont you just find X(z^-1) instead of X(z) ?