BME 343 - Invertible Systems

A system is invertible if distinct inputs lead to distinct outputs. As a result, it is possible to inverse the output to get the original input.

Analogy of invertible system:

Talking - we turn words into data and back into words. For a good communication system, we need an invertible system that can turn the data back into words.

Non-invertible system:

Encryption - Now, let's say we don’t want the boys group chat to be seen. An encryption is important because if someone has the output signal of the system, they can not inverse the signal to get the original input signal.

The system can become invertible with a key code, so systems with the key code will be able to invert the system (ie. read the messages) but the systems without the key code won’t be able to.

Lets look at a simple system:

y(t) = 5 + x(t)

If x(t) is 5, then y(t) = 10. If x(t) is -5, then y(t) = 0. This system is invertible.

Lets look at a simple system:

y(t) = 5 + x^2(t)

If x(t) is 5, then y(t) = 30. If x(t) is -5, then y(t) = 30.

This system is non invertible because both 5 and -5 makes 30.

To identify a system, we can look for obvious excitations or calculate the inverse system.

Problem 1

Is this system invertible or non-invertible?

It’s non-invertible!

If we visualize the excitation, then we can see that anything before t=0 will be 0. There is no unique way to invert multiple 0s, so this is non-invertible.

Problem 2

Is this system invertible or non-invertible?

It’s non-invertible!

There isn’t an obvious excitation, so we need to calculate the inverse system. Luckily, this is simple.

Because anything divided by 0 is undefined, this system is non-invertible.