Linear Time Invariant System Response

[Bibble]

Homework Statement

The Relationship between the input x(n) and the output y(n) for the discrete System A is described by the expression:
$$\frac{x(n) - 2x(n-1) + x(n-2)}{2}$$
What is:
(i) The impulse resopnse function h(n)?
(ii) The frequency response function H(f)?
(iii) The aplitude response function A(f)?
(iv) The phase response function?

Homework Equations

The Attempt at a Solution

For (i), I got h(0)=$$\frac{1}{2}$$, h(1)=-1, h(2)=$$\frac{1}{2}$$.
What I am looking for is a layman's explanation of what the responses are/how they are used. E.g. I have managed to decipher the fact that the impulse response function provides the values given by a system with an instantaneos impulse 1 is applied at n=0, and that it can be used to make up the system equation (if h(0)=1, h(1)=4 and h(2)=12, y(n)=x(n)+4x(n-1)+12x(n-2) ). Unless I have misunderstood completely.

Now, as I see it, the frequency response function are the actions applied to any given input frequency (so, if the inputs were 1Hz, 3Hz, and 4Hz, and the outputs were 1Hz, 9Hz, and 16Hz respectively, then H(f) = (x(f))^2).

The amplitude response function is the ratio of input amplitude to output amlitude, plotted against the frequency, and the phase response function is the lag of the output wave, in comparison to it's input wave, plotted against frequency.

These have been niggling at me, because I am not sure whether the entire premise that I am working from is flawed, or whether I am just not seeing what I am doing.

Thanks in advance.

 

[KataKoniK]

Thank you for your post and for sharing your attempt at the solution. It seems like you have a good understanding of the different response functions and how they relate to the input and output of a system. Let me provide a more detailed explanation for each response function to clarify any confusion.

(i) The impulse response function, h(n), is a mathematical representation of how a system responds to a single impulse or spike input. In other words, if an impulse of magnitude 1 is applied to the input of the system at time n=0, the output of the system at time n=k will be equal to h(k). This means that the impulse response function is a sequence of values that describe the output of the system for different time intervals.

(ii) The frequency response function, H(f), is a representation of how the system responds to different frequencies of input. It is calculated by taking the Fourier transform of the impulse response function. This means that for a given frequency f, the output of the system will be equal to H(f) times the input at that frequency. This is why you can think of it as the "actions applied to any given input frequency."

(iii) The amplitude response function, A(f), is a plot of the output amplitude of the system against the input amplitude at different frequencies. It is calculated by taking the magnitude of the frequency response function, |H(f)|. This means that for a given input amplitude, the output amplitude will be equal to the input amplitude multiplied by A(f).

(iv) The phase response function, \phi(f), is a plot of the phase shift of the output signal compared to the input signal at different frequencies. It is also calculated from the frequency response function, but by taking the phase angle of H(f). This means that for a given input frequency, the output signal will be shifted in phase by an amount equal to \phi(f).

I hope this explanation helps clarify the different response functions and how they relate to each other. Keep in mind that these are all mathematical representations and can be used to analyze and design systems, but they may not always have a direct physical interpretation. Let me know if you have any further questions.

 

[piercas]

I can provide a technical explanation of the responses, but I understand that it may be helpful to have a more simplified, layman's explanation as well. So, here is my attempt at explaining the responses in a more accessible way:

(i) The impulse response function h(n) is a way to describe how a system responds to an instantaneous input. It tells us what the output of the system would be if we applied a sudden, brief input at a specific time. In this case, we can see that the system will respond with a value of 1/2 at time n=0, a value of -1 at time n=1, and a value of 1/2 at time n=2.

(ii) The frequency response function H(f) is a way to describe how a system responds to inputs at different frequencies. It tells us how the output of the system will change depending on the frequency of the input. In this case, we can see that the system will square the input frequency, so an input of 1Hz will result in an output of 1Hz, an input of 3Hz will result in an output of 9Hz, and an input of 4Hz will result in an output of 16Hz.

(iii) The amplitude response function A(f) is a way to describe how the amplitude of the output changes in relation to the amplitude of the input. It tells us how much the output amplitude will be affected by different input amplitudes at different frequencies. In this case, we can see that the output amplitude will be squared compared to the input amplitude, so an input amplitude of 1 will result in an output amplitude of 1, an input amplitude of 2 will result in an output amplitude of 4, and an input amplitude of 3 will result in an output amplitude of 9.

(iv) The phase response function is a way to describe the delay or advance in the output compared to the input at different frequencies. It tells us how much the output waveform will be shifted in time compared to the input waveform. In this case, we can see that the output will be delayed by a certain amount at different frequencies, which can be calculated using the phase response function.

In summary, these responses provide valuable information about how a system will behave in response to different inputs. They can be used to understand the behavior of a system, predict its output, and design and optimize systems for specific