Question 4.13: Finding a bounded excitation that produces an unbounded resp......
Finding a bounded excitation that produces an unbounded response
Consider an accumulator for which \mathbf{y}[n]_{=}\sum_{m=-\infty}^{n}\mathbf{x}[m]. Find the eigenvalues of the solution of this equation and find a bounded excitation that will produce an unbounded response.
We can take the first backward difference of both sides of the difference equation yielding \mathbf{y}[n]-\mathbf{y}[n-1]=\mathbf{x}[n]. This is a very simple difference equation with one eigenvalue, and the homogeneous solution is a constant because the eigenvalue is one. Therefore this system should be BIBO unstable. The bounded excitation that produces an unbounded response has the same functional form as the homogeneous solution. In this case a constant excitation produces an unbounded response. Since the response is the accumulation of the excitation, it should be clear that as discrete time n passes, the magnitude of the response to a constant excitation grows linearly without an upper bound.
The concepts of memory, causality, static nonlinearity, and invertibility are the same for discrete-time systems as for continuous-time systems. Figure 4.49 is an example of a static system.
One example of a statically nonlinear system would be a two-input OR gate in a digital logic system. Suppose the logic levels are 0V for a logical 0 and 5V for a logical 1. If we apply 5V to either of the two inputs, with 0V on the other, the response is 5V. If we then apply 5V to both inputs simultaneously, the response is still 5V. If the system were linear, the response to 5V on both inputs simultaneously would be 10V. This is also a noninvertible system. If the output signal is 5V, we do not know which of three possible input-signal combinations caused it, and therefore knowledge of the output signal is insufficient to determine the input signals.
Even though all real physical systems must be causal in the strict sense that they cannot respond before being excited, there are real signal-processing systems that are sometimes described, in a superficial sense, as noncausal. These are data-processing systems in which signals are recorded and then processed “off-line” at a later time to produce a computed response. Since the whole history of the input signals has been recorded, the computed response at some designated time in the data stream can be based on values of the already-recorded input signals that occurred later in time (Figure 4.50). But, since the whole data processing operation occurs after the input signals have been recorded, this kind of system is still causal in the strict sense.
