Friday, June 4, 2010

System Classifications and Properties

In this module some of the basic classifications of systems will be briefly introduced and the most important properties of these systems are explained. As can be seen, the properties of a system provide an easy way to separate one system from another. Understanding these basic differences between systems, and their properties, will be a fundamental concept used in all signal and system courses, such as digital signal processing (DSP). Once a set of systems can be identified as sharing particular properties, one no longer has to deal with proving a certain characteristic of a system each time, but it can simply be accepted do the systems classification. Also remember that this classification presented here is neither exclusive (systems can belong to several different classifications) nor is it unique (there are other methods of classification). Examples of simple systems can be found here.

Classification of Systems:
Along with the classification of systems below, it is also important to understand other Classification of Signals.

Continuous vs. Discrete:
This may be the simplest classification to understand as the idea of discrete-time and continuous time is one of the most fundamental properties to all of signals and system. A system where the input and output signals are continuous is a continuous system, and one where the input and output signals are discrete is a discrete system.

Linear vs. Nonlinear:
A linear system is any system that obeys the properties of scaling (homogeneity) and superposition (additivity), while a nonlinear system is any system that does not obey at least one of these. To show that a system H obeys the scaling property is to show that
H (k(f (t)) = kH (f (t))………………..(1)

Figure 1: A block diagram demonstrating the scaling property of linearity
To demonstrate that a system H obeys the superposition property of linearity is to show that
H (f1 (t) + f2 (t)) = H (f1 (t)) + H (f2 (t))…………(2)

Figure 2: A block diagram demonstrating the superposition property of linearity

It is possible to check a system for linearity in a single (though larger) step. To do this, simply combine the first two steps to get
H (k1f1 (t) + k2f2 (t)) = k2H (f1 (t)) + k2H (f2 (t)) ……..(3)

Time Invariant vs. Time Variant:
A time invariant system is one that does not depend on when it occurs: the shape of the output does not change with a delay of the input. That is to say that for a system H where H (f (t)) = y (t), H is time invariant if for all T
H (f (t – T) = y (t – T)…………..(4)

Figure 3
Figure 3: This block diagram shows what the condition for time invariance. The output is the same whether the delay is put on the input or the output.

When this property does not hold for a system, then it is said to be time variant, or time-varying.

Causal vs. Non-causal:
A causal system is one that is non-anticipative; that is, the output may depend on current and past inputs, but not future inputs. All "realtime" systems must be causal, since they can not have future inputs available to them.

One may think the idea of future inputs does not seem to make much physical sense; however, we have only been dealing with time as our dependent variable so far, which is not always the case. Imagine rather that we wanted to do image processing. Then the dependent variable might represent pixels to the left and right (the "future") of the current position on the image, and we would have a non-causal system.

Figure 4
Figure 4: (a) For a typical system to be causal... (b) ...the output at time t0(to), can only depend on the portion of the input signal before to.

Stable vs. Unstable:
A stable system is one where the output does not diverge as long as the input does not diverge. There are many ways to say that a signal "diverges"; for example it could have infinite energy. One particularly useful definition of divergence relates to whether the signal is bounded or not. Then a system is referred to as bounded input-bounded output (BIBO) stable if every possible bounded input produces a bounded output.

Representing this in a mathematical way, a stable system must have the following property, where x (t) is the input and y (t) is the output. The output must satisfy the condition
Іy (tMy < α …………..(5)

When we have an input to the system that can be described as
Іx (t) ІMx < α ………….. (6)

Mx and My both represent a set of finite positive numbers and these relationships hold for all of t.
If these conditions are not met, i.e. a system's output grows without limit (diverges) from a bounded input, then the system is unstable. Note that the BIBO stability of a linear time-invariant system (LTI) is neatly described in terms of whether or not its impulse response is absolutely integrable.

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