Are these equations describing causal or noncausal systems?

[ENGRstudent]

I'm working on homework, and I think I can explain the difference between causal and noncausal systems, but I don't know if I'm accurately recognizing them mathematically.

Here are my homework questions and my answers:

Are the systems described by the following equations with input x(t) and output y(t) causal or noncausal?

A) y(t)=x(t-2) : CAUSAL; involves a delay of the signal in real time.

B) y(t)=x(-t) : NONCAUSAL; causal signals do not exist before t=0 (I think?).

C) y(t)=x(at), a>1 : CAUSAL; a is never negative.

D) y(t)=x(at), a<1 : CAUSAL; a may be any real number between 0 and 1.

Can anyone verify or correct me on any of these?

 

[ENGRstudent]

Nobody can help me on this?

 

[es]

I think some of the answers you gave are wrong. As I understand it, and it has been a while since I took this class, a casual system is one whose current value only depends on the present and past values of the input.

So A is correct. The present value of y only depends on past values of x, or more specifically, x's that occurred 2 time units ago. i.e. for t=2, y(2)=x(0). Using this technique perhaps you can find the problems in your other answers.

Also, usually when I solve systems I don't restrict myself to t>=0. In general I will set the signal value to 0 when t<0 but this is not the same as saying the signal does not exist before t=0 and it does not mean t cannot be negative. Your instructor may have confined your range to t>=0 though.

 

[ENGRstudent]

Thank you, es. My answer for D was incorrect.

 

[es]

My pleasure however I think you may still have some errors hiding.
What are your opinions of the following points?
Equation C with a=5 and t=1
Equation D with a=-1
Equation D with a=1/2 and t=2

 

[ENGRstudent]

My pleasure however I think you may still have some errors hiding.
What are your opinions of the following points?
Equation C with a=5 and t=1
Equation D with a=-1
Equation D with a=1/2 and t=2

Well, I'm still learning, but here goes...

Equation C remains causal with a=5 and t=1 because the output still exists after t=0.

Equation D with a= -1 is noncausal because it places the signal before t=0.

Equation D with a=1/2 and t=2 becomes causal because once again, it places the resultant signal after t=0. But since the parameter given was a<1, the system is noncausal because it must hold true for all values of a<1.

Or something. :)

 

[es]

Ah. I understand where the confusion is now.

Causality does not reflect how events relate to t=0. Causailty relates times to each other. It means the output signal only depends on values that are occurring right now or occurred in the past.

Reusing our examples:

1: y(t)=x(t-2) is casual
the output, y, only depends on x's which occurred two time units ago, aka
in the past. for t=10, y(10)=x(8) which means the value of the output at
time 10 is the same what the output was at time 8. notice both values are
greater than t=0

2: y(t)=x(t) is casual
i just wanted to throw this into show that the output can depend on what
is happening right now and still be casual

3: y(t)=x(-t) is non-casual
when t<0, say -2, y(-2)=x(2). This means the value of the output at time
t=-2 is equal to the value of the input at time t=2. However t=2 occurs
after t=-2. This means the output depends on something which hasn't
happened yet, a future value. This is what makes equation three non-
casual.

4: y(t)=x(at), a>1 is non-casual
let's pick an example, a=5 t=2, here y(2)=x(10). Again, the output
depends on a values which occurs in the future, t=10 occurs after t=2.
All the t's here exist after zero but it is still non-casual.

Note, there are t's which make equation 3 casual. For instance t=0. However if only one point in the range is non-casual then the entire system is non-casual (at least this is how I always do it). If you require the system to become casual then you could try restricting the range. In our example you could say t is only allowed to be zero.

 

[Hells_Kitchen]

Are you allowed to take t<0. I don't think that would be physically correct or physically make sense.
For that matter for e.g. 4 that you have if we pick a = 5 and t = -2 (t<0) then we have y(-2) = x(-10) which means that the output at -2 depends on a value that occurred earlier -10 thus would make the system causal. What is the difference can you explain? Plus what would you think if a was constrained to only a<1?

 

[stewartcs]

Are you allowed to take t<0. I don't think that would be physically correct or physically make sense.
For that matter for e.g. 4 that you have if we pick a = 5 and t = -2 (t<0) then we have y(-2) = x(-10) which means that the output at -2 depends on a value that occurred earlier -10 thus would make the system causal. What is the difference can you explain? Plus what would you think if a was constrained to only a<1?

FYI...this thread is 4 years old.

CS

 

[Hells_Kitchen]

oops sorry i didnt even notice. thank you ;)