This is the first post in the blinkdagger signal processing series.

Introduction to Linear Time Invariant Systems

Fourier theory is predicated upon LTI (Linear Time Invariant) systems. Thus, before we even discuss signal processing, it is important to first understand what makes a particular system LTI. In this post, you will learn what classifies a system as LTI and why it plays a fundamental role in Fourier analysis.

There are three requirements for an LTI system which will be discussed in the following sections:

  1. Scalability
  2. Additivity
  3. Time Invariant

We will see how each property affects the following block diagram:

block diagram

Scalability

Scalability means that a change in the input signal’s amplitude results in a corresponding change in the output signal’s amplitude. Let’s look at a quick example.

In the following image, you can see the following input signal produces the following output signal when it is introduced to the LTI system. The input is simply shifted to the right by two sample points, and the input amplitude is multiplied by a factor of two. Now, lets multiply the input signal by a factor of 2 and run it through the LTI system. According to the rules of Scalability, the output would also increase by a factor of 2. You can see it illustrated below.

Scalability

Scalability 2

Additivity

For this example, we use the same LTI system as above wherein the input is simply shifted to the right by two sample points, and the input amplitude is multiplied by a factor of two. If you take a look at the series of images, you will understand the basic idea of additivity. The important thing to realize here is that added signals pass through the system without interacting. In other words, signals added at the input produce signals that are added at the output. The follow image provides an example of additivity.

Additive 1

Additive 2

Additive 3

Time Invariant

Being time invariant means that the characteristics of a system are not dependent on time. This means that a shift in the input signal will create an identical shift in the output signal. For example, if my system reacts in a particular way on day 1, it will act in an identical manner 1500 days from now! Thus, if I have an LTI system which takes this input signal and produces the following output signal:

Time Invariant

Then if I shift it over by two sample points, the output will also shift over by two sample points!

Time Invariant 2

Special Property of LTI Systems

One of the special properties of LTI systems is sinusoidal fidelity. What exactly does this mean? It means that if you input a sinusoid into an LTI system, then you will get a sinusoid as an output. Sinusoids have the special property wherein any sinusoidal input to a linear time-invariant system results in a sinusoidal output that differs only in amplitude and phase shift, while retaining the frequency and wave shape. As you will later learn, this is integral to Fourier Analysis.

Sinusoidal Fidelity

For a more indepth look at Linear Systems, visit dspguide.com for a very good explanation of LTI systems with many examples that are easy to understand. It is highly recommended that you read up on this if you want to learn more about LTI systems.

Divide and Conquer

Divide and Conquer

Photo taken from soldiersmediacenter

So now that we know exactly what it means to be an LTI system, why do we care? How does it relate to Fourier Theory?

When we deal with LTI systems, we can break down a very complicated signal to something that is more manageable (this is referred to as decomposition). This is possible because LTI systems have the property of being additive and scalable (incidentally, these two qualities can be lumped together and described simply as superposition). Superposition (summing up signals to form one signal) and Decomposition (breaking down a signal into simpler signals) are essentially two different sides to the same thing.

For instance, let’s say we have a very complicated signal. How can we go about analyzing it?

  1. First, we break that signal down into a series of simpler waveforms that are easy to analyze.
  2. Next, we analyze these signals individually.
  3. Finally, we sum up the outputs of each individual signal. This gives us the output of the original complex signal!

This is essentially what is done in Fourier Analysis. Stay tuned for the next post!