# Linear Time-Invariant Systems

## Impulse Response of LTI systems

### The Discrete Case

Notice this operation is the convolution between the input and the impulse response.

### The Continuous Case

After applying the LTI system to it,

Notice this operation is the convolution between the input and the impulse response.

## Determining Properties of an LTI system

Because an LTI system is determined entirely by its impulse response, we can determine its properties from the impulse response.

### Causality

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### Memory

### Stability

so

## Frequency Response and Transfer Functions

If we pass a complex exponential into an LTI system, the output signal is the same signal but scaled. In otherwise, it is an eigenfunction of LTI systems.

The integral is a constant, and the original function is unchanged. The same analysis can be done in the discrete case.

We give these constant terms a special name called the transfer function.

**Notice:** The frequency response is the fourier transform of the impulse response! This means the Fourier Transform takes us from the impulse response of the system to the frequency response. There is no reason to limit ourselves to the Fourier Domain though

The transfer function is merely the Laplace Transform of the impulse response. In many ways, this can be more useful than the frequency response.

### Stability of transfer functions

Recall that an LTI system is stable if the impulse response is absolutely integrable. We can determine this from the transfer function.

### Bode Plots

Because transfer functions, and hence the frequency response, can be quite complex, we need a easy way to visualize how a system responds to different frequencies.

The log-log scale not only allows us to determine the behavior of Bode plots over a large range of frequencies, but they also let us easily figure out what the plot looks like because it converts the frequency response into piecewise linear components.

To see why, lets write our transfer function in polar form.

If we take the log of this, we get

Thus we can see at decades away from the poles and zeros, the magnitudes and the phases will have less of an effect. Let’s try constructing the Bode Plot for this transfer function.

## Special LTI Systems

### Linear Constant Coefficient Difference/Differential Equations

*Proof.* ** The Continuous Case**

Taking the Laplace Transform,

**The Discrete Case**

Taking the Z Transform

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### State Space Equations

When we have a LCCDE of the form

**Important: This is valid in Discrete Time as well!**

In general, state-space equations are useful because they allow us to find transfer functions of complex systems very easily.

Label the output of delay (discrete) or differentiation (continuous) blocks as the state variables.

Write the state equations using inputs and delays/derivatives. Express each as a weighted sum of states and inputs.

Use the formula above to find the transfer function.

### Second Order Systems

Most of the time, higher order systems only have 2 dominant poles. Accordingly, they can be approximated by second order systems (i.e systems with two poles). One way to write the transfer function of this system is

Using common laplace transform pairs, this corresponds to

There are several key features of this step response:

Using the step response, we can calculate some of these values.

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