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?
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.
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. :)
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 occuring right now or occured in the past. Reusing our examples: 1: y(t)=x(t-2) is casual the output, y, only depends on x's which occured 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 in to 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.
Are you allowed to take t<0. I dont 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 occured 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?