A causal LTID system is described by the secondorder discret

A causal LTID system is described by the second-order discrete-time system as,

y[n] – 0.75y[n - 1] + 0.1y[n - 2] = x[n]

a) Use the method described in section M3.2 to compute the impulse response through filtering for n = (0:25).

b) Use Matlab function impz() to compute impulse response for the system, h[n], for n = (0:25). Plot h[n]. Did you get the same result as that in (a)? Plot h[n], and the difference between h[n]’s in part (a) and part (b).

c) Suppose the input for the system is

[n] = u[n] - u[n - 10]                                                    (2)

with initial condition y[-1] = y[-2] = 1, compute the zero-input response, y_zi, the zero-state response, y_zsand the total response, y[n].

Solution

Engineers and Physical scientists have for many years utilized the concept of a system to facilitate the study of the interaction between forces and matter. A system is a mathematical abstraction that is devised to serve as a model for a dynamic phenomenon. It represents the dynamic phenomenon in terms of mathematical relations among three sets of variables known as the input, the output, and the state. The input represents, in the form of a set of time functions or sequences, the external forces that are acting upon the dynamic phenomenon. In similar form, the output represents the measures of the directly observable behavior of the phenomenon. Input and output bear a cause-effect relation to each other; however, depending on the nature of the phenomenon, this relation may be strong or weak. A basic characteristic of any dynamic phenomenon is that the behavior at any time is traceable not only to the presently applied forces but also to those applied in the past. We may say that a dynamic phenomenon possesses a ”memory” in which the effect of past applied forces is stored. In formulating a system model, the state of the system represents, as a vector function of time, the instantaneous content of the ”cells” of this memory. Knowledge of the state at any time t, plus knowledge of the forces subsequently applied is sufficient to determine the output (and state) at any time t t0. As an example, a set of moving particles can be represented by a system in which the state describes the instantaneous position and momentum of each particle. Knowledge of position and momentum, together with knowledge of the external forces acting on the particles (i.e., the system input) is sufficient to determine the position and momentum at any future time. A system is, of course, not limited to modeling only physical dynamic phenomena; the concept is equally applicable to abstract dynamic phenomena such as those encountered in economics or other social sciences. If the time space is continuous, the system is known as a continuous-time system. However, if the input and state vectors are defined only for discrete instants of time k, where k ranges