# Linear Systems

A linear system implies that if two inputs are scaled and summed together to form a new input, the new output of the system is also a scaled sum of their individual outputs.

[Scaling] For scaling to hold, if

then

where is any scalar.

[Addition] When two such inputs are added together, the output should be the sum of their individual outputs as

A linear system combines the above two properties as

(1)

Below, we discuss examples of a linear and a non-linear system.

Example

Consider a system

The output of this system, as a response to an input , is

Similarly, response to a different signal is

When this system is given the input , the output is

Hence, it is a linear system.

On the other hand, when the same input is given to another system

and using the identity , the output is

Clearly, it is a non-linear system.

From above example, it is also clear that input sinusoids do not interact with each other in linear systems, and hence output frequencies were the same as the input frequencies. In a non-linear system, however, input sinusoids interacted with each other to produce frequencies that were not present in either of the input signal, as shown in Figure below for . Note from that actually consists of only two frequencies at bins and , but the output of the system is composed of other frequencies as well shown in .

The Discrete Fourier Transform, DFT, is a linear operation as it is evident from DFT definition that any scaling and addition of two or more input signals will result in a DFT output that is a scaled and summed version of their individual DFT outputs.