Transfer Functions

Written by Travis M. Moore
Last edited 30-Sep-2019

Linear Functions

A transfer function describes what a system does. It is a single plot that shows the output for a range of inputs into the system: in other words, what happens to a signal when it is transferred through the system.

What would such a useful plot look like? It's nothing fancy. There is an x axis that shows the input values, and a y axis that shows the output values. Take a look at Figure 1 to see an example.

Simple transfer function
FIG. 1. © Travis M. Moore (2019)

This is a very boring transfer function, but it's a good place to start. Figure 1 tells us that whatever input level we pass through the system, that output level is same as the input. In other words, this system does nothing to the input level. We can see this is true because if we look at an input level of say 20 dB on the x axis, the corresponding output on the y axis is also 20 dB. Drawing a line up from the x axis to the blue diagonal and then to the left to the y axis will always return the same number. This is a linear system, because the output always has a constant relationship to the input (in this case, the output is always the same as the input). Let's look at another example.

Linear transfer function
FIG. 2. © Travis M. Moore (2019)

Figure 2 shows a slightly more exciting transfer function. This time if we look at a 20-dB input on the x axis, the corresponding y axis value is 70 dB. In fact, if we look at any x-axis input value we can see the system always adds 50 dB. Just like in Figure 1, the output always has a constant relationship to the input, which means this system is also linear, and therefore also has the shape of a straight diagonal line. More formally, the line describes a particular function.

In other words, the "diagonal line" represents the function of how the output level changes based on the input level. Another way to say the same thing is that the diagonal line represents output as a function of input. We could write an equation that describes the shape of the function (i.e., line) for any given input. The equation for the system in Figure 2 is: f(x) = x + 50 dB. The "f(x)" part just stands for "the function of x is." The actual equation is extremely simple: x + 50 dB.

Let's recap:

  • The function for Figure 2 is f(x) = x + 50 dB
  • In words, that means when any given input level is transferred through the system, the result is that number (x) plus 50 dB.
  • A transfer function is a plot that shows the outputs for a range of inputs. In other words, it describes the function of the entire system (i.e., what the system does).

Nonlinear Functions

So far the only type of function we have described is a linear one. But what happens if the relationship between the input and the output changes? For example, a hearing aid algorithm would be pretty useless if it didn't increase the level of quiet sounds. Let's imagine the function of a hearing aid can be described by f(x) = x + 50 dB, just like Figure 2 above. That means a very soft input level, such as 10 dB would be amplified to 60 dB. Depending on the degree of hearing loss, this might be exactly what a patient needs in order to detect that 10 dB sound. BUT, let's say an ambulance drives by while our patient is walking outside. The ambulance's siren is 110 dB, which is already loud enough to cause some discomfort, but that linear hearing aid does the only thing it can do: f(110) = 110 + 50. That's 160 dB! The hearing aid would make an already loud sound intense enough to cause significant damage to our patient's hearing.

The solution is a "smarter" system, with a function changes how it modifies the output based on specific input levels. That type of function would not always have the same relationship between input and output, so we can no longer call it linear. The term used to describe a transfer function that changes is nonlinear. Makes sense if "linear" describes a function that doesn't change that our new dynamic function is "not linear" (i.e. nonlinear). Let's take a look at an example of a system with a nonlinear transfer function.

Nonlinear transfer function
FIG. 3. © Travis M. Moore (2019)

You can see how the function behaves differently depending on the input level by drawing two sets of lines on the x axis. If we draw one set of lines under the linear portion of the function and one set of lines under the nonlinear portion, we will be able to see how the function changes what it does depending on the input level.

The yellow lines are spaced 10 dB apart on the x axis and are under the linear section of the transfer function. Just like we have seen before, the relationship between input and output level is the same: x + 50 dB. That means inputs of 10 and 20 dB are output as 60 and 70 dB. A 10-dB change in input level leads to a 10-dB change in output level.

The rust-colored lines are also spaced 10 dB apart on the x axis, but are under the nonlinear section of the transfer function. The output levels for 40 and 50 dB inputs are roughly 80 and 83 dB, respectively. Now a 10-dB change in input level only leads to a 3-dB change in output level. We no longer see the x + 50 dB relationship.

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