Recall that the natural solution to our differential equation was:

where *s*_{i} is the root of the characteristic
equation

The root of the characteristic
equation *s*_{i} can be either real or complex. If it is
complex, it must occur in conjugate pairs because the coefficients of the
characteristic equation are real.

• If s_{i} is complex, then |
is exponential in form. |

• If s_{i} is complex, then let |
and | and the conjugate pair of these terms | ||

can be expressed as: |

where |

The roots *s*_{i} will determine if the overall
system is BIBO stable.

Assume we have a *causal* LTI system.
The solution to our differential equation is of the form

where *y*(*t*) = 0 for *t* <
*t*_{0} (*t*_{0} is the start time). Since *y*_{p}(*t*)
is always of the same form
as the input *x*(*t*), if
the input *x*(*t*) is bounded, then
*y*_{p}(*t*) will also be bounded.
(Alternatively, since we are only considering BIBO stability,
our input is guaranteed to be stable.)

Thus, since the stability of the system will only depend on the system itself and not on the input, let's examine the solutions to the homogeneous equation:

Clearly, as long as the real part of all roots
(also called *poles*) of this equation,
*σ*_{i} < 0 (since we've assumed that the system is causal),
then each term in *y*_{c}(*t*) will be bounded.
Thus, we require for BIBO stability that:

Thus, a necessary and sufficient condition for stability
of a causal LTI system is that all roots of the system characteristic equation lie in the left half
plane of the *s*-plane.

__Ex.__ Given a causal LTI system described by the differential equation

determine if the system is BIBO stable.

Given an input
*x*(*t*) = *X e*^{st}
to a BIBO stable LTI system modeled by an
*n*th order linear differential equation with constant coefficients,
we will examine the steady-state system response. Assume that *X* and
*s* are complex.

The forced (particular or steady-state) response of the system to this input is of the same form of the input, i.e.

From the differential equation describing the system,

or,

Plugging in *x*(*t*)= *X e*^{st} and
*y*_{ss}(*t*)= *Y e*^{st} , we get

which we can write as
*Y* = *H*(*s*) *X*

where

is a *transfer function.*

So given an input | to an LTI system, the steady-state response is |

Similar to the differential equation and the impulse response, the transfer
function *H*(*s*) completely characterizes the LTI system (we can derive the differential
equation from *H*(*s*) and vice versa).

In general, by superposition, given an input | , the output of the system is | |

Eigenfunctions of CT LTI systems are complex exponentials:

Check: What is
*e*^{st}**h*(*t*)?

where | is the eigenvalue. This motivates the Laplace transform and the Fourier Transform: |

*H*(*s*) is known as the bilateral Laplace transform of*h*(*t*).- If
*s*=*jω*, then we get*H*(*jω*), the Fourier Transform of*h*(*t*).

__Ex.__Given an input
*x*(*t*) = *e*^{st}, and

*h*(*t*) = *e*^{3t} *u*(*t*)

find its steady-state output

*y*_{ss}(*t*) = *H*(*s*) *e*^{st}

YOU FINISH: