Characteristics of systems described differential and difference equations
It is informative to express the output of a system described by a differential or difference equation as the sum of two components, that is one associated only with the initial conditions, and the other due only to the input signal.
We will term the component of the output associated with the initial conditions as the natural response of the system and denote it as y^(n). The component of the output due only to the input is termed the forced response of the system and is denoted as y^(f).Thus, the complete output is y = y^(n)+y^(f).
THE NATURAL RESPONSE
The natural response is the system output for zero input and therefore describes the manner in which the system dissipates any stored energy or memory of the past represented by non-zero initial conditions. Since the natural response assumes zero input, it Is obtained from the homogeneous solution by choosing the coefficients ci so that the initial conditions are satisfied.
The natural response assumes zero input and hence does not involve a particular solution. Since the homogeneous solutions apply all the time, the natural response is determined without translating initial conditions forward in time.
THE FORCED RESPONSE
The forced response is the system output due to the input signal assuming zero initial conditions. Therefore, the forced response is of the same form as the complete solution of the differential or difference equation. A system with zero initial conditions is “at rest,” since there is no stored energy or memory in the system. The forced response describes the system behavior that is “forced “ by the input when the system is at rest.
The forced response depends on the particular solution, which is valid only for times t>0 or n>= 0. Accordingly, the at-rest conditions for a discrete-time system, y[-N]=0,…,y[-1]=0, must be translated forward to times n=0,1,…, N-1 before solving the undetermined coefficients, such as when one is determining the complete solution.
As before, we shall consider finding the forced response only for continuous-time systems and inputs that do not result in impulses on the right- hand side of the differential equation. This ensures the initial conditions at t=0+ are equal to the zero initial conditions at t=0
THE IMPULSE RESPONSE
Given the step response, the impulse response may be determined by exploiting the relationship between the two responses. The definition of the step response assumes that the system is at rest, so it represents the response of the system to a step input with zero initial conditions.
For a continuous-time system, the impulse response h(t) is related to the step response s(t) via the formula h(t) = ds(t)/dt. For a discrete-time system,h[n] =s[n]-s[n-1]. Thus, the impulse response is obtained by differentiating and differencing the step response.
There is no provision for initial conditions when one is using the impulse response; it applies only to systems that are initially at rest or when the inputs are known for all time. Differential- and difference equation system descriptions are more flexible in this respect since they apply to systems either at rest or with nonzero initial conditions
Linearity and time invariance
The forced response of an LTI system described by a differential or difference equation is linear with respect to the input. That is, if y1 is the forced response associated with an input x1 and y2 is the forced response associated with an input x2, then ax1 + Bx2 generates the forced response ay1+By2. Similarly, the natural response is linear with respect to the initial conditions I2 and y2 is the natural response associated with initial conditions aI1 + BI2 results in the natural response ay1 + By2.
The forced response is also time-invariant. A time shift in the input results in a time shift in the output since the system is initially at rest. By contrast, in general, the complete response of an LTI system described by a differential or difference equation is not time-variant, since the initial conditions will result in an output term that does not shift with a time shift of the input.
Finally, we observe that the forced response is also causal. Since the system is initially at rest, the output does not begin prior to the time at which the inputs are applied to the system.
ROOTS OF THE CHARACTERISTIC EQUATION
The forced response depends on both the input and the roots of the characteristic equation since it involves both the homogeneous solution and a particular solution of the differential or difference equation. The basic form of the natural response is dependent entirely on the roots of the characteristic equation.
The impulse response of an LTI system also depends on the roots of the characteristic equation, since it contains the same terms as the natural response. Therefore, the roots of the characteristic equation afford considerable information about LTI system behavior.
For example, the stability characteristics of an LTI system are directly related to the roots of the system’s characteristic equation. To see this, note that the output of a stable system in response to zero input must be bounded for any set of initial conditions.
This follows from the definition of EIBO stability and implies that the natural response of the system must be bounded. Thus, each term in the natural response must be bounded. In the discrete-time case, we must have bounded, or i < 1 for all. When – 2, the natural response does not decay, and the system is said to be on the verge of instability. For continuous-time LTI systems, we require that be bounded, which implies that
Rel) <0. Here again, when Re(r) -0, the system is said to be on the verge of instability. These results imply that a discrete-time LTI system is unstable if any root of the characteristic equation has a magnitude greater than unity and a continuous-time LTI system is unstable if the real part of any root of the characteristic equation is positive. This discussion leads to the idea that the roots of the characteristic equation indicate when an LTI system is unstable.
A discrete-time causal LTI system is stable if and only if all roots of the characteristic equation have magnitude less than unity, and a continuous-time causal LTI system is stable if and only if the real parts of all roots of the characteristic equation are negative. These stability conditions imply that the natural response of an LTI system goes to zero as time approaches infinity since each term in the natural response is a decaying exponential.
This “decay to zero” is consistent with our intuitive concept of an LTI system’s zero input behavior. We expect a zero output when the input is zero if all the stored energy in the system has dissipated. The initial conditions represent energy that is present in the system: in a stable LTI system with zero input, the stored energy eventually dissipates and the output approaches zero.
The response time of an LEI system is also determined by the roots of the characteristic equation. Once the natural response has decayed to zero, the system behavior is governed only by the particular solution, which is of the same form as the input.
Thus, the natural response component describes the transient behavior of the system; that is, it describes the transition of the system from its initial condition to an equilibrium condition determined by the input. Hence, the time it takes an LTI system to respond to a transient is determined by the time it takes for the natural response to decay to zero. Recall that the natural response contains terms of the form for a discrete-time LTI system and e’ for a continuous-time LTI system.
The response time of a discrete-time LTI system to a transient is therefore proportional to the root of the characteristic equation with the largest magnitude, while that of a continuous-time Il system is determined by the root with the largest real component. For a continuous-time LTI system to have a fast response time, all the roots of the characters must have large negative real parts.