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.

Figure 1

Linear Scaling
Subfigure 1.1 Subfigure 1.2

In Subfigure 1.1 above, an input xx to the linear system LL gives the output yy. If xx is scaled by a value αα and passed through this same system, as in Subfigure 1.2, the output will also be scaled by αα.

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.

Figure 2

Subfigure 2.1 Subfigure 2.2

Figure 3: If Figure 2 is true, then the principle of superposition says that Figure 3 is true as well. This holds for linear systems.

Superposition Principle

That is, if Figure 2 is true, then Figure 3 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:

Figure 4

Subfigure 4.1 Subfigure 4.2

Figure 5: Given Figure 4 for a linear system, Figure 5 holds as well.

Superposition Principle with Linear Scaling

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 6: Subfigure 6.1 shows an input at time tt while Subfigure 6.2 shows the same input t0t0 seconds later. In a time-invariant system both outputs would be identical except that the one in Subfigure 6.2 would be delayed by t0t0.

Time-Invariant Systems
Subfigure 6.1 Subfigure 6.2

In this figure, xtxt and xt-t0 xtt0 are passed through the system TI. Because the system TI is time-invariant, the inputs xtxt and xt-t0 xtt0 produce the same output. The only difference is that the output due to xt-t0 xtt0 is shifted by a time t0t0.

Whether a system is time-invariant or time-varying can be seen in the differential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations. A constant coefficient differential (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 7: This is a combination of the two cases above. Since the input to Subfigure 7.2 is a scaled, time-shifted version of the input in Subfigure 7.1, so is the output.

Linear Time-Invariant Systems
Subfigure 7.1 Subfigure 7.2

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.

Figure 8

Subfigure 8.1 Subfigure 8.2

Figure 9: The principle of superposition applied to LTI systems

Superposition in Linear Time-Invariant Systems

LTI Systems in Series

If two or more LTI systems are in series with each other, their order can be interchanged without affecting the overall output of the system. Systems in series are also called cascaded systems.

Figure 10: The order of cascaded LTI systems can be interchanged without changing the overall effect.

Cascaded LTI Systems
Subfigure 10.1 Subfigure 10.2

LTI Systems in Parallel

If two or more LTI systems are in parallel with one another, an equivalent system is one that is defined as the sum of these individual systems.

Figure 11: Parallel systems can be condensed into the sum of systems.

Parallel LTI Systems
Subfigure 11.1 Subfigure 11.2

Causality

A system is causal if it does not depend on future values of the input to determine the output. This means that if the first input to a system comes at time t0t0, then the system should not give any output until that time. An example of a non-causal system would be one that "sensed" an input coming and gave an output before the input arrived:

Figure 12: In this non-causal system, an output is produced due to an input that occurs later in time.

Non-causal System

A causal system is also characterized by an impulse response htht that is zero for t<0t0.