# Find the Output of an LTI System Given Input and Impulse Response

• Captain1024
In summary, the conversation discusses a LTI system with filter coefficients of {1,2,1} and finding the output for a given input signal. Two frequencies in the composite signal are identified as pi/3 and pi/8. The last part is asking for the frequency response with respect to these two frequencies. The given h_k is the impulse response of the system.
Captain1024

## Homework Statement

Consider a LTI system for which the filter coefficients are ##{\mathrm{h}_k}=\{1,2,1\}##. Find the output when the input is ##\mathrm{x}[n]=3\mathrm{cos}(\frac{\pi}{3}n-\frac{\pi}{2})-3\mathrm{cos}(\frac{7\pi}{8}n)##. Identify two frequencies in this composite signal. Show the frequency response with respect to these two frequencies.

## The Attempt at a Solution

Input:
##\mathrm{x}[0]=3\mathrm{cos}(\frac{\pi}{2})-3\mathrm{cos}(0)=-3##
##\mathrm{x}[1]=3\mathrm{cos}(\frac{-\pi}{6})-3\mathrm{cos}(\frac{7\pi}{8})=-0.1736##
##\mathrm{x}[2]=3\mathrm{cos}(\frac{\pi}{6})-3\mathrm{cos}(\frac{7\pi}{4})=0.4768##

Is the output then ##\mathrm{y}[n]=\mathrm{h}_k*\mathrm{x}[n]=\{-3, -0.3472, 0.4768\}##?

Are two frequencies in this composite signal ##\frac{\pi}{3}## & ##\frac{\pi}{8}##?

I'm not sure what the last part is asking for. Are frequency response and output the same thing?

Also, is the given ##\mathrm{h}_k## called the impulse response? I'm trying to get my vocabulary down.

-Captain1024

I figured it out.

berkeman
Hey, could you tell us how you did?

## 1. What is an LTI system?

An LTI (linear time-invariant) system is a type of system in which the output is a linear combination of the input and the system's impulse response, and the system behaves the same regardless of when the input is applied.

## 2. How do you find the output of an LTI system given an input and impulse response?

To find the output of an LTI system, you can use the convolution integral, which involves multiplying the input and impulse response and integrating over a certain time period. This will give you the output as a function of time.

## 3. What is an impulse response?

An impulse response is the output of an LTI system when a unit impulse (a very short pulse) is applied as the input. It is a characteristic of the system that describes how the system responds to different inputs.

## 4. Can the output of an LTI system be calculated without using the convolution integral?

Yes, the output of an LTI system can also be calculated using the transfer function, which is the Laplace transform of the impulse response. The transfer function can be multiplied by the Laplace transform of the input to get the Laplace transform of the output, which can then be converted back to the time domain using the inverse Laplace transform.

## 5. Are there any limitations to using LTI systems?

Yes, LTI systems have some limitations. They only work for systems that are linear and time-invariant, which means they cannot handle nonlinear or time-varying systems. Additionally, LTI systems assume that the system is stable, meaning that the output does not grow infinitely with time. If these assumptions are not met, the output calculated using LTI methods may not accurately reflect the system's behavior.

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