Proving Energy of a Signal: g(t)

In summary, the conversation discusses a problem related to a generic energy signal and its various transformations. The question is to prove that the energy of any of the signals -g(t), g(-t), g(t-T) and g(at) is equal to the original energy Eg. The speaker is able to understand the concept intuitively but is struggling to prove it mathematically. They mention using the basic definition of energy signal and suggest using a change of variables to solve the problem.
  • #1
Infidel22
5
0
Hi, I am having a bit of a problem regarding a simple proof for a generic energy signal, the question reads as thus:
For an energy signal g(t) with energy Eg, show that the energy of anyone of the signals -g(t), g(-t) and g(t-T) is Eg. Show that the energy of g(at) is Eg/a.

While I can arrive at the answers intuitively, the total area under the curve is constant for the first parts and is being reduced or increased for the second part, I can't figure out how to mathematically prove any of these except the case of -g(t). I am starting with the basic definition of the energy signal Eg=integral(g(t)^2,t,-inf,inf) but I can't figure out a way to get any further without an actual function.Can anyone give me any guidance?

Thanks so much.
 
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  • #2
For all the other ones, you should use an appropriate change of variables. Look at the definition of Eg for the specific g given and think about what a good choice for a change of variable would be.
 
  • #3


I understand your struggle to mathematically prove the energy of a signal g(t). Let me provide some guidance to help you with your proof.

First, let's define the energy of a signal g(t) as Eg=integral(g(t)^2,t,-inf,inf). This means that the energy is equal to the integral of the square of the signal over all time.

Now, let's consider the first part of the question, where we need to show that the energy of -g(t), g(-t), and g(t-T) is also Eg. To prove this, we can use the properties of integrals. Since the integral is a linear operator, we can take the negative sign out of the integral for -g(t), resulting in Eg=integral((-g(t))^2,t,-inf,inf). This is equal to the integral of g(t)^2, which is Eg. Similarly, for g(-t), we can use the property of symmetry of integrals to change the limits of integration from -inf to inf to inf to -inf, resulting in Eg=integral(g(-t)^2,t,inf,-inf). Again, this is equal to the integral of g(t)^2, which is Eg. For g(t-T), we can use the property of translation of integrals to change the limits of integration from -inf to inf to (-inf+T) to (inf+T), resulting in Eg=integral(g(t-T)^2,t,-inf+T,inf+T). Again, this is equal to the integral of g(t)^2, which is Eg. Therefore, we have proven that the energy of -g(t), g(-t), and g(t-T) is Eg.

Moving on to the second part of the question, where we need to show that the energy of g(at) is Eg/a. To prove this, we can use a change of variable in the integral. Let's define u=at, which means that du=a*dt. Substituting these into the integral, we get Eg=integral(g(u)^2/a,u,-inf,inf). This is equal to (1/a)*integral(g(u)^2,u,-inf,inf), which is equal to (1/a)*Eg. Therefore, we have proven that the energy of g(at) is Eg/a.

I hope this helps guide you in your proof. Remember to always use the properties of integrals
 

1. What is the concept of energy in a signal?

The concept of energy in a signal refers to the amount of power or strength that a signal carries. It is a measure of the overall amplitude or magnitude of the signal.

2. How is the energy of a signal calculated?

The energy of a signal can be calculated by squaring the amplitude of the signal at each point in time and then integrating the result over the entire time duration of the signal.

3. What is the unit of measurement for signal energy?

The unit of measurement for signal energy is joules (J), which is the same unit used to measure energy in physics.

4. Why is it important to prove the energy of a signal?

Proving the energy of a signal is important because it allows us to accurately analyze and compare different signals, and make conclusions about their characteristics and behaviors. It also helps in designing and optimizing signal processing systems.

5. Can the energy of a signal be negative?

No, the energy of a signal cannot be negative. Since energy is a measure of power, it is always a positive quantity. However, the energy of a signal can be very small, close to zero, if the signal has a low amplitude or short duration.

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