Terms in the Natural Response

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 si is complex, then

is exponential in form.

• If si is complex, then let		and		and the conjugate pair of these terms
can be expressed as:

where

Stability

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 < t₀ (t₀ 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.

System Response for Complex-Exponential Inputs

Given an input x(t) = X e^st to a BIBO stable LTI system modeled by an nth 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: