# Question about an LTI System and its Frequency Response

• ntran26
In summary, this individual is trying to understand the basics of an LTI system and is having difficulty with the convolution.
ntran26
Hello All,
I am trying to learn about lti systems and what not for my research work but before I can apply my research I have to understand the fundamentals, which is what this question is kind of about!

I want to find h(t), the impulse response of an LTI system where my input and output are sinusoids. So I have made two functions.
Input(t)=4*SIN(3*t+PI/8)+9*SIN(4*t-PI()/6)
Output(t)=2*SIN(8*t+PI/3)+5*SIN(2*t-PI()/7)
I did use matlab's fast Fourier transform to bring results of n^10 data points to the frequency domain.
I then do output(w)/input(w) ,output and input in frequency domain, to get H(w), the frequency response. I then use MATLAB to apply the inverse fast Fourier transform to H(w) to obtain h(t).

I use matlabs convolution function which should result in I(t)*h(t)=o(t)

The problem is that this is not matching up.

Am i doing something wrong with my thought process? Thanks!

Your system is nonlinear, so it is not LTI. In particular, the output frequencies of a linear system must match those of the input (another way of saying that sines and cosines are the eigenfunctions of an LTI system).

It is invalid to arbitrarily make up an output function. I suggest you read and understand LTI systems, then you can generate an appropriate impulse response and watch what it does to your input signal. (Alternately, work the exercises in your textbook...)

marcusl said:
Your system is nonlinear, so it is not LTI. In particular, the output frequencies of a linear system must match those of the input (another way of saying that sines and cosines are the eigenfunctions of an LTI system).

It is invalid to arbitrarily make up an output function. I suggest you read and understand LTI systems, then you can generate an appropriate impulse response and watch what it does to your input signal. (Alternately, work the exercises in your textbook...)
I see, thank you! Unfortunately, I do not have any textbooks on this subject and have just been using online resources. I am using a reservoir simulator, where I apply a pressure pulse that is sinusoidal and observing pressures at another location. The reservoir is supposed to be an LTI system, but I cannot seem to get the convolution correctly. Thank you for your help!

## 1. What is an LTI system?

An LTI (linear time-invariant) system is a type of system in which the output is directly proportional to the input and does not change over time. This means that the system's behavior is consistent and predictable, making it easier to analyze and design.

## 2. How is the frequency response of an LTI system defined?

The frequency response of an LTI system is a mathematical representation of how the system responds to different frequencies of input signals. It is typically expressed as a complex function that relates the output to the input at each frequency.

## 3. What is the significance of the frequency response in an LTI system?

The frequency response is important because it allows us to understand how an LTI system behaves in the frequency domain. This information is useful for analyzing and designing filters, amplifiers, and other signal processing systems.

## 4. Can the frequency response of an LTI system be altered?

Yes, the frequency response of an LTI system can be altered by changing its components or parameters. For example, adding a capacitor or resistor to the system can change its frequency response and affect its behavior.

## 5. How can the frequency response of an LTI system be measured?

The frequency response of an LTI system can be measured by applying a sine wave input at different frequencies and measuring the corresponding output amplitudes and phases. This data can then be used to plot the frequency response curve and analyze the system's behavior.

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