Impulse response in the LTI system

 

 

 

Property of LTI system: Predictable 

For example, I will get one drink if I insert 2 dollars into the banding machine.

Then, when I put 20 dollars in that machine, how many drinks do I get? We can get the answer without something practical. This property is called "Predictable".

 

We can denote such a system mathematically like this:

 

$$ y(t) = f(x(t) \times h(t)) $$

 

From that equation, $h(t)$ is the impulse response. 

Before impulse response, we need to check about impulse function $\delta[n]$

 

$$ \delta[n]=\begin{cases} 1, & \mbox{if }n\mbox{=0} \\ 0, &\mbox{Other wise} \end{cases}$$

The impulse function has such property, therefore, it has two statements.

 

1. $\sum_{k=-\infty}^\infty \delta[n]=1$

 

2. $\sum_{k=-\infty}^\infty \delta[n]\times f[n] = f[0] $

 

Similarly, in the continuous time area, it can be defined. 

$$ \delta(t)=\begin{cases} \infty, & \mbox{if }t\mbox{=0} \\ 0, &\mbox{Other wise} \end{cases}$$

There $\delta (n)$ impulse function has a similar property this :

 

1. $\int_{-\infty}^{\infty} \delta(t)\, dt = 1$

 

2. $\int_{-\infty}^{\infty} f(t)\times \delta(t)\, dt = f(0)$

 

Figure1. Impulse function in DT area

 

Figure2. impulse function in CT area

In the continuous-time impulse signal, it isn't discrete Discontinuous does not mean discrete. It is defined as distinctively in the continuous area. 

 Then, we can get Impulse response $h(t)$ by putting impulse functions into any system.

In other words, we can define impulse response in any kind of function and system. such as LTI, casual, non-linear, etc.

 

We can calculate the output y[n] by using a sequence of x[n] and an impulse function.

Figure3. sequence of x[n]

Let's assume that discrete function x[n] is plotted. Therefore, the impulse function is $ \delta[n]$.

By using the property of impulse function: it has a value of 1 if and only if the time domain equals 0. and otherwise is 0.

Mathematically, we can represent it like this :

 

 

 

$$ x[n] = ... x[-2]\delta[n+2] + x[-1]\delta[n+1] + x[0]\delta[n] + x[1]\delta[n-1] + x[2]\delta[n-2]...$$

On the same principle, in order to obtain the output of the signal, it is possible to replace the position of the impulse function with the impulse response function.

 

$$ y[n] = ... x[-2]h[n+2] + x[-1]h[n+1] + x[0]h[n] + x[1]h[n-1] + x[2]h[n-2]...$$

Finally, we can simple relation between input and output.

 

1. $ x[n]= \sum_{k=-\infty}^\infty x[k]\times \delta[n-k]$

 

2. $ y[n]= \sum_{k=-\infty}^\infty x[k]\times h[n-k]$

 

At the point of equation 2, in the LTI systems, $\sum_{k=-\infty}^\infty$ means additibility, $x[n]$ means scale, and $h[n-k]$ means time-invariant.

 

simple exercise -LTI system-

Let's represent discrete-time function $x[n]$ by the sum of the impulse function $\delta[n]$  $\because$ LTI systems satisfy that a function can be expressed by the sum of functions.

 

Figure4. DT signal x[n]

We know that function $x[n]$ is :

$$ x[n] = 3\times\delta[n] + 2\times\delta[n-1] + 1\times\delta[n-2]$$ 

 

in conclusion, we can find the relation between impulse response and impulse function.

Figure5

As you can see in Figure 5, by using the LTI system property, we make the equation.

LTI systems satisfy the following:

$ x[n] = 3\times\delta[n] + 2\times\delta[n-1] + 1\times\delta[n-2]$, and also that $ y[n] = 3\times h[n] + 2\times h[n-1] + 1\times h[n-2]$