Finding impulse of a CT system

In summary, the conversation discusses finding the impulse response of a continuous-time system with a given input/output relationship. The suggested method involves expressing the convolution as an integral and comparing it to the given equation to obtain a function of the form h(t-\lambda), which can then be used to determine h(t).
  • #1
l46kok
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Homework Statement


A continuous-time system has the input/output relationship

[tex]y(t) = \int_{-\infty}^{t} (t - \lambda + 2)x(\lambda) d\lambda[/tex]

Determine the impulse response h(t) of the system

Homework Equations


Convolution theorems

[tex]y(t) = x(t) * h(t)[/tex]
Where y(t) is the output, x(t) is the input and h(t) is the impulse response


The Attempt at a Solution


I have absolutely no clue how to obtain the impulse response, going backwards in a CT domain from the given equation above. Can anyone give me some hints to start me off?
 
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  • #2
Express the convolution as an integral. Compare this integral with the problem statement and obtain a function of the form [itex]h(t-\lambda)[/itex]. It should not be too difficult then to determine [itex]h(t)[/itex].
 
  • #3


it is important to approach problems with a systematic and logical mindset. In this case, we can start by analyzing the given equation and breaking it down into smaller components.

First, let's look at the integral in the equation. This indicates that the system is a linear time-invariant (LTI) system, as it follows the convolution theorem. This means that the impulse response can be determined by finding the inverse Laplace transform of the transfer function.

Next, we can identify that the input signal is x(t) and the output signal is y(t). This aligns with the general form of the convolution theorem, where y(t) is equal to the convolution of x(t) and h(t).

To find h(t), we can use the convolution theorem and rewrite the equation as:

y(t) = x(t) * h(t) = \int_{-\infty}^{t} (t - \lambda + 2)h(\lambda) d\lambda

From this, we can see that the impulse response h(t) is equal to (t - \lambda + 2). This can be further simplified by taking the derivative of both sides with respect to t, giving us:

h(t) = \frac{d}{dt} \int_{-\infty}^{t} (t - \lambda + 2)h(\lambda) d\lambda = \frac{d}{dt}y(t)

Therefore, the impulse response of the given CT system is h(t) = \frac{d}{dt}y(t). We can confirm this by taking the Laplace transform of both sides and seeing that they are equal.

In summary, to find the impulse response of a CT system, we can use the convolution theorem and take the derivative of the output signal with respect to time. This approach can be applied to other types of systems as well, as long as they follow the convolution theorem.
 

1. What is the definition of impulse in a CT system?

In a CT (computed tomography) system, impulse refers to the sudden change in momentum caused by an external force acting on the system. In other words, it is the amount of force applied over a short period of time.

2. How is impulse measured in a CT system?

The impulse of a CT system is typically measured in units of Newton-seconds (N·s). This unit is derived from the SI base units of Newtons (N) for force and seconds (s) for time.

3. What factors affect the impulse of a CT system?

The impulse of a CT system is affected by various factors such as the mass of the object, the velocity of the object, and the duration of the force applied. Additionally, the direction and angle of the force can also impact the impulse.

4. How is impulse related to the motion of a CT system?

In a CT system, impulse is directly related to the change in velocity of the object. This is described by the impulse-momentum theorem, which states that the impulse of a force is equal to the change in momentum of an object.

5. Why is it important to calculate the impulse of a CT system?

Calculating the impulse of a CT system is important for understanding the motion and dynamics of the system. It can also help in determining the forces acting on the system and how to optimize its performance. In medical imaging, measuring the impulse of a CT system can provide valuable information for diagnosing diseases and injuries.

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