A causal system? the biggest contradiction

In summary: I'm not sure if I'm following you, LM741. The system is causal for the reasons mentioned by D H and my earlier post. The system is not acausal. How are you defining the term "causal system"?In summary, the conversation discusses the confusion surrounding the determination of whether a given system is causal or not. One method involves delaying the signal by one unit and finding an expression involving past and present input/output values, while the other method involves shifting the entire system and finding an expression involving future input/output values. The confusion arises due to the lack of a clear definition of the "output" of the system. However, it is agreed upon that a system is causal if the unit-impulse
  • #1
LM741
130
0
a causal system?? the biggest contradiction!

Hey guys - can someone tell me why i get two contradicting asnwers...

i am asked to determine whether the following system is causal or not:

y[k+1] + 0.64y[k-1] = x[k] + 2x[k-1]
method 1:
put k = k-1 (delay the signal by one unit)
therefore , y[k] = -0.64y[k-2] +x[k-1] +2x[k-2]

it is cleary that this signal is casual because a present ouptut value is dependent on present and past input/output values.

method 2:
put k = k+1
therefore, 0.64y[k] = y[k+2] +x[k+1] +2x[k]

now the system is clearly not causal! whys is this happening!

i can't see any error in my maths. Also , shifting should not alter the signal characteristics!

please someone help!

calculating a tranfer function clearly shows the system IS causal-----so what is up with metho 2?/


thanks!
 
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  • #2
I think the confusion is coming about because you have not clearly defined what the "output" of the system is. If it is y[k], which is traditional notation, then your system is non-causal because it relies on a future value y[k+1]. If you are going to define the output of the system as y[k+1], which AFAIK is not traditional notation, then yes, the system would be causal.

Is the output of the system clearly defined for you?
 
  • #3
Er, isn't y[k] totally indepentent from y[k+1]?


Admittedly I don't know the definitions, but I would have assumed that "causal" means "y[k] can be computed without using any future values of x or y". In this case, finding an expression for y[k] involving future values of x and y doesn't invalidate the fact that you can compute y[k] without doing so.
 
  • #4
Hurkyl said:
Er, isn't y[k] totally indepentent from y[k+1]?


Admittedly I don't know the definitions, but I would have assumed that "causal" means "y[k] can be computed without using any future values of x or y". In this case, finding an expression for y[k] involving future values of x and y doesn't invalidate the fact that you can compute y[k] without doing so.

Yeah, this is confusing me. Since y[k] is not part of the equation of the "system", I"m not sure how to interpret the equation.

LM741 -- What is considered the "output" of the system described by that equation?
 
  • #5
the output of thie system is y[k].

Berkmen, with no disrespect, i fully disaggree with you.
You just can't have a system that is causal AND non causal. y[k+1] is merely the output in advance by one unit. Also, if you shift the entire system by the sam time unit - nothing has changed - thiss just allows you to view it from a different point of view...
I just can't seem to get around this 'strange' predicament...

thank
 
  • #6
There's no predicament.

Consider a simple smoothing IIR filter,
y[k] = 0.5*x[k] + 0.5*y[k-1].
This is clearly causal; the output depends only on past and current input and past output. Yet you can rewrite this as
y[k] = 2*y[k+1] - x[k-1]
and the filter appears to be acausal. It is not. Bottom line: You just have to be careful with feedback terms.
 
  • #7
LM741 said:
the output of thie system is y[k].

Berkmen, with no disrespect, i fully disaggree with you.
You just can't have a system that is causal AND non causal. y[k+1] is merely the output in advance by one unit. Also, if you shift the entire system by the sam time unit - nothing has changed - thiss just allows you to view it from a different point of view...
I just can't seem to get around this 'strange' predicament...

thank

No disrespect felt on my part at all, LM741. You should have seen my lame reply to Hurkyl's post before I deleted it and tried again. I misread the original equation my first couple times through.

But help me out here. The original equation does not contain a y[k] term, and you say what I thought, that y[k] is the output of the digital filter. So are you supposed to re-arrange the original equation with substitution like you did to make a y[k]? If so, then the system is causal as D H says. I'm just not used to being given a filter equation that does not specify the y[k] directly in terms of past (and possibly future) values of the input and output.

Guess I should pull out my old DSP book for a quick refresher, eh? :blushing:
 
  • #8
My trusty DSP book has the following:
A linear shift-invariant system is causal if and only if the unit-impulse response h[n] = 0 for n < 0.

In your case, work out the h[n] of the given difference equation and check if the above condition holds.

Btw, the definition of causal systems which appears in Wikipedia reads:
A causal system is a system where the output y(t) at some specific instant t0 only depends on the input x(t) for values of t less than or equal to t0. Therefore these kinds of systems have outputs and internal states that depends only on the current and previous input values. http://en.wikipedia.org/wiki/Causal_system" [Broken]

The definition applies only to "input x(t) for values of t less than or equal to t0" and not so much for the output terms which appear in the difference equation. Thus saying:
LM741 said:
it is cleary that this signal is casual because a present ouptut value is dependent on present and past input/output values.
is incorrect.
 
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  • #9
D H - you say the system APPEARS to be non causal - but it is not?
that's obviously the problem I'm having - this is a predicament!
y[k] = 2*y[k+1] - x[k-1] - how can you tell me this a causal system?

the current output is dependent on a future output value? - you see, this is really wired because it seems like we change the casuality property of the system by shifting it...
 

1. What is a causal system?

A causal system is a system in which the output depends only on the current and past inputs, and not on any future inputs. In other words, the output is causally related to the input.

2. How is causality defined in a causal system?

Causality in a causal system is defined by the principle that the output of the system cannot occur before the input is applied. This means that the system cannot have any future knowledge or anticipation of the input.

3. What are some examples of causal systems?

Examples of causal systems include a pendulum, a thermostat, and an electronic filter. In each of these examples, the output is determined solely by the current and past inputs, without any influence from future inputs.

4. How does a causal system differ from a non-causal system?

A causal system is defined by the fact that the output is determined by the input without any future knowledge or anticipation. In contrast, a non-causal system may have outputs that depend on future inputs, making it impossible to determine causality.

5. What is the biggest contradiction in a causal system?

The biggest contradiction in a causal system is the concept of time travel. If a system were to have knowledge of future inputs, it would violate the principle of causality and create a paradox. However, time travel is currently not possible in our understanding of physics.

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