A SYSTEM is an operation for which cause-and-effect relations exist.

Here, we discuss some of the properties that a continuous-time system could
have. We will use *x*(*t*) for the input to the system, *y*(*t*) as its output, and
use the notation:

*y*(*t*) = T[*x*(*t*)]

or

*y*(*t*) = S[*x*(*t*)]

or

*x*(*t*) → *y*(*t*)

Systems whose output
*y*(*t*_{0}) at time
*t*_{0} depends on values of the input other than just
*x*(*t*_{0}) have memory.

In other words, a system *y*(*t*_{0}) has memory if its output at time *t*_{0} depends on
the input *x*(*t*) for *t* > *t*_{0} or *t* < *t*_{0},
i.e. it depends on values of the input other than *x*(*t*_{0}).

Otherwise, the system is MEMORYLESS

Example of a memoryless system:

Resistor *v*(*t*_{0}) = *R i*(*t*_{0});
the voltage at time *t*_{0}
depends only on the current at time *t*_{0}.

Example of System with Memory:

Capacitor |
; the voltage depends on past values of the current so a capacitor has memory. |

__Ex 1.__ Does *y*(*t*) = *x*(*t*)
+ 5 have memory?

__Ex 2.__ Does *z*(*t*) = *x*(*t* + 5) have memory?

__Ex 3.__ Does *y*(*t*) = (*t* + 5)*x*(*t*) have memory?

__Ex 4.__ Does *z*(*t*) = [*x*(*t* + 5)]^{2} have memory?

__Ex 5.__ Does *a*(*t*) = *x*(5) have memory?

__Ex 6.__ Does *v*(*t*) = *x*(2*t*) have memory?

A system is invertible if you can determine the input uniquely from the
output, i.e. there is a one-to-one relationship between the input and output.
In this case, we would write that the inverse of the system *S* is
*S _{I}*:

A resistor is invertible because you
can recover the current from the voltage:
*x*(*t*) = *i*(*t*) ,
*y*(*t*) = *v*(*t*),
*x*(*t*) = *y*(*t*)/*R*.

*y*(*t*) = *x*^{5}(*t*) is an invertible system since it is one-to-one.

Examples of some systems that are not invertible:

*y*(*t*) = *x*(*t*)*u*(*t*)
→ zeros out much of the input

*y*(*t*) = *x*^{2}(*t*) → don't know sign

*y*(*t*) = cos[*x*(*t*)] → add 2π to
*x*(*t*)

Output *y*(*t*) depends only on past and present inputs and
**not on the future.**

All physical real-time systems are causal because we can not anticipate the future. Clearly, the stock market is a causal system.

Physical systems that are noncausal are not real-time. For example, a system in which music is recorded and processed later is noncausal but it is not real-time

If a system is memoryless, it is also causal. However, being causal does not necessarily imply that a system is memoryless. In fact, most causal systems do have memory.

__Examples:__
Resistors and capacitors are causal
systems:

Ex 7. |

Is this Causal? You fill in.

__Ex 8.__ Let *y*(*t*) = *x*(-*t*).

Is this causal? Try letting *t* be a negative number.

We will consider Bounded Input - Bounded Output (BIBO) Stability

We say that a system is
BIBO stable if an input *x*(*t*)
that is bounded (finite) for all time produces an output *y*(*t*)
that is also bounded or finite for all time.

Mathematically,
we write if |*x*(*t*)|
*B*_{1} →
|*y*(*t*)|
*B*_{2}, where
*B*_{1} and *B*_{2} are finite constants and
*y*(*t*) is the output.

In this example, the input *x*(*t*)
is bounded by the constant
*B*_{1} and the output *y*(*t*)
is bounded by a second constant *B*_{2}:

__Example 9:__ A
resistor is stable because:

If | *i*(*t*) |
*B*_{1} → | *v*(*t*) |
*RB*_{1}

Example: 10 |
Capacitor: |

Is this stable?

Let *i*(*t*) = *B*_{1}*u*(*t*), where *B*_{1}≠0

grows linearly with t and as |
, | . So a capacitor is not BIBO stable. |