There are five general properties of systems that are introduced in this homework. These include systems with and without memory, time invariant systems, linear systems, causal systems and stable systems. This post will detail how to check if a system exhibits these general properties.
1. Systems with and without memory:
- Def: A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent only on the input at that same time.
- Proving: This is very simple and can be done by visual inspection alone. If there is any kind of phase shift, the systems depends on values other than the input at that current time and is NOT memoryless. Also, if the system is described by an accumulator or summer again it depends of values other than the input at the current time and is NOT memoryless. Note that in a system comprised of the linear combination of sub-systems if any one of the sub-systems is memoryless than the entire system is memoryless.
2. Time Invariant Systems:
- Def: A system is time invariant if a time shift in the input signal results in an identical time shift in the output signal.
- Proving: In equation form, the system y(t) = x(t) is time invariant if y(t-to) = x(t-to). Plug in "t-to" for all "t's" in the system. Simplify. If the end result is just a time delay, then the system is time invariant. The easy way--if there are any "t's" outside the function x(t) [i.e. t*x(t)]? the system must NOT be time invariant.
3. Linear Systems:
- Def: A system is linear if any output can be derived from the sum of the products of an output and constant and another output and constant. In equation form for y(t) = x(t) with outputs y1, y2, and y3: $ y_3(t) = a\cdot y_1(t) + b\cdot y_2(t) $
- Proving: Use the equation above to prove. Example 1.17 in the text shows a nice overview. Basically, consider two arbitrary inputs and their respective outputs. A third input is considered to be a linear combination of the first two inputs. Write the output and substitute the third input for the linear combination. Separate the a and b variables. If you can arrange the equation so that the output of the third input is the linear combination of the first two outputs, then the system is linear.
4. Causal Systems:
- Def: A system is causal if the output at any time depends only on the values of the input at the present time and in the past.
- Proving: Consider each component of the system separately. If there is no time delay, the systems depends only on the present time and is causal. If there is a time delay, determine whether it is in past or future time. If it is past time, then the system is causal. When the systems is rotated over the y-axis [i.e. x(-t)]? then if is possible for some values of t that the system is in past time and for others in future. Determine these values. The system is causal for the values of past time as well as for the value of present time and else is NOT causal.
5. Stable Systems:
- Def: A system is stable if a bounded input function yields a bounded output function.
- Proving: Consider y(t) = x(t). If the input function is bounded then $ |x(t)| < \epsilon $. Consider the end behavior of all combinations of minimum and maximum values for x(t). If it is bounded, then y(t) IS stable. (Look at Example 1.13 in the text for further instruction).
... --brian.a.baumgartner.1, Wed, 05 Sep 2007 11:20:49
I was mistaken in article 4. Just because a system does not have a time delay does not PROVE that the system is causal. I wrote this without considering time scaling. I daresay that if a system does not have a time delay OR time scaling then it is causal; however I suggest doing the math to back this up. To find the values of when the system is causal and when it is not consider the system y(t) = x(at+b). Next set up the following inequality:
at+b <= t
The system IS causal as long as this inequality holds true. It is NOT causal for at+b > t.
