Thursday, June 3, 2010

Properties of Systems

Linear Systems:

If a system is linear, this means that when an input to a given system is scaled by a value, the output of the system is scaled by the same amount.

In Figure 1(a) above, an input x to the linear system L gives the output y. If x is scaled by a value α and passed through this same system, as in Figure 1(b), the output will also be scaled by α.

A linear system also obeys the principle of superposition. This means that if two inputs are added together and passed through a linear system, the output will be the sum of the individual inputs' outputs.

If Figure 1 is true, then the principle of superposition says that Figure 2 (Superposition

Principle) is true as well. This holds for linear systems.

That is, if Figure 1 is true, then Figure 2 (Superposition Principle) is also true for a linear system.

The scaling property mentioned above still holds in conjunction with the superposition principle. Therefore, if the inputs x and y are scaled by factors α and β, respectively, then the sum of these scaled inputs will give the sum of the individual scaled outputs:

Time-Invariant Systems:

A time-invariant system has the property that a certain input will always give the same output, without regard to when the input was applied to the system.

Figure (a) shows an input at time t while Figure (b) shows the same input t0 seconds later. In a time-invariant system both outputs would be identical except that the one in Figure (b) would be delayed by t0.

In this figure, x (t) and x (t - t0) are passed through the system TI. Because the system TI is time invariant, the inputs x (t) and x (t - t0) produce the same output. The only difference is that the output due to x (t - t0) is shifted by a time t0.

Whether a system is time-invariant or time-varying can be seen in the deferential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations. A constant coefficient deferential (or difference) equation means that the parameters of the system are not changing over time and an input now will give the same result as the same input later.

Linear Time-Invariant (LTI) Systems:

Certain systems are both linear and time-invariant, and are thus referred to as LTI systems.

Figure: This is a combination of the two cases above. Since the input to Figure (b) is a scaled, time-shifted version of the input in Figure (a), so is the output.

As LTI systems are a subset of linear systems, they obey the principle of superposition. In the figure below, we see the effect of applying time-invariance to the superposition definition in the linear systems section above.

Superposition in Linear Time-Invariant Systems