Understanding Discrete System A: Functions and Responses | Homework Statement

[1alph1]

Homework Statement

The relationship between the input x(n) and the output y(n) for the discrete System A is described by the expression

y(n)=2x(n)-3x(n-1)+x(n-1)

What is
(i) the impulse response function h(n)?
(ii) the frequency response function H(f)?
(iii) the amplitude response function A(f)?
(iv) the phase response function θ(f)?

Assume that the sample time T = 1 (units). Note that the sample frequency is then one sample per second and this means that the Nyquist limit is 0.5 (units–1). As a consequence our input signals x(t), before sampling, is assumed to have no frequency above f = 0.5 (units–1). If we expect to have higher frequencies then we should increase the sample frequency.

Homework Equations

impulse response function h(n) aka delta function δ(t) is the derivative of the unit step function. not sure if the is relevant or not if even accurate.

The Attempt at a Solution

The frequency response H(f) is equal to

y(jω)/x(jω)

and that the modulus of H(f) is equal to the amplitude response function A(f)

and the phase response function θ(f) is equal to the angle of H(f)

Im not too sure where to start this question, any pointers would be greatly appericated. thanks

 

[KataKoniK]

Hello,

Thank you for your question. Let me help you break down the problem and provide some guidance on how to approach it.

First, let's define the terms that are mentioned in the problem:

- Input x(n): This refers to the input signal to the system at discrete time points n. It can be represented as a sequence of values, with n being the index of the sequence.
- Output y(n): This refers to the output signal of the system at discrete time points n. It is also a sequence of values, with n being the index.
- Discrete System A: This is the system that is being described by the given expression.
- Impulse response function h(n): This is the output of the system when the input is an impulse signal, i.e. a signal that is 0 everywhere except at n=0 where it has a value of 1.
- Frequency response function H(f): This is the ratio of the output to the input in the frequency domain. It gives us information about how the system responds to different frequencies.
- Amplitude response function A(f): This is the magnitude of the frequency response function H(f).
- Phase response function θ(f): This is the phase of the frequency response function H(f).

Now, let's look at the given expression for the system:

y(n)=2x(n)-3x(n-1)+x(n-1)

This is a discrete system, which means that it operates on discrete values of the input. The output at a particular time point n is determined by the input at that time point (x(n)) and the input at the previous time point (x(n-1)). This can be represented in a block diagram as follows:

x(n) ---> [ System A ] ---> y(n)

Now, let's move on to the questions:

(i) The impulse response function h(n) is the output of the system when the input is an impulse signal. In other words, it is the output of the system when x(n)=1 and x(n-1)=0. From the given expression, we can see that this will result in a value of 2 at n=0 and a value of -3 at all other time points. Therefore, the impulse response function can be written as:

h(n) = 2δ(n) - 3δ(n-1)

where δ(n) is the impulse signal.

(ii) The frequency response function H(f

 

[Mene]

it is important to understand the concept of discrete systems and their functions and responses. In this case, the given expression describes the relationship between the input and output of System A. To answer the questions posed, we first need to understand the different functions involved.

(i) The impulse response function h(n) is a function that describes the response of a system to a brief input signal, or impulse. In this case, since the input is x(n), the impulse response function would be h(n) = 2δ(n) - 3δ(n-1) + δ(n-1).

(ii) The frequency response function H(f) is a function that describes how the system responds to different frequencies in the input signal. In this case, we can calculate it by taking the Fourier transform of the impulse response function h(n). So, H(f) = 2 - 3e^(-j2πf) + e^(-j2πf).

(iii) The amplitude response function A(f) is the modulus of the frequency response function, and it describes the magnitude of the output signal compared to the input signal. In this case, A(f) = |H(f)| = √(4 + 9 - 12cos(2πf)).

(iv) The phase response function θ(f) is the angle of the frequency response function, and it describes the phase shift of the output signal compared to the input signal. In this case, θ(f) = arctan(Im(H(f))/Re(H(f))) = arctan(-3sin(2πf)/4 - 3cos(2πf)).

To ensure accurate results, it is important to note the given assumptions about the sample time and frequency. If the input signal has frequencies above 0.5 (units^-1), the sample frequency should be increased accordingly. Additionally, the given equation for the impulse response function may not be accurate and should be verified.