Convolution-like change of variables

In summary, the conversation is about a person seeking help with analyzing linear invariant systems and proving an expression involving integrals. They have attempted to manipulate the expression and change variables, but are unsure of the next step. The expert provides a possible solution and explains the use of dummy variables in integrals.
  • #1
Taraborn
1
0

Homework Statement



Hi, this is not homework exactly, I'm doing some exercises as part of my personal study. I'm analizing linear invariant systems and I'm stuck in an apparently trivial step, please, help. I have these integrals:

Homework Equations



integral( x(tau)*dtau, from -infinity to t) - integral ( x(tau)*dtau, from -infinity to t-T). I must prove that the former expression is the same as integral( x(t)-x(t-T), from -infinity to t).

The Attempt at a Solution



I have manipulated the expression a little, tried the change of variables s=tau+T; ds=dtau and now I have -integral(x(s-T)ds, from -infinity to t) + integral(x(tau)dtau, -infinity to t) but I don't know how continue. I'm pretty sure it's really simple, but there's something I'm missing or something that I don't fully understand about dummy variables and so. Help, please.
 
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  • #2
Taraborn said:

Homework Statement



Hi, this is not homework exactly, I'm doing some exercises as part of my personal study. I'm analizing linear invariant systems and I'm stuck in an apparently trivial step, please, help. I have these integrals:

Homework Equations



integral( x(tau)*dtau, from -infinity to t) - integral ( x(tau)*dtau, from -infinity to t-T). I must prove that the former expression is the same as integral( x(t)-x(t-T), from -infinity to t).

The Attempt at a Solution



I have manipulated the expression a little, tried the change of variables s=tau+T; ds=dtau and now I have -integral(x(s-T)ds, from -infinity to t) + integral(x(tau)dtau, -infinity to t) but I don't know how continue. I'm pretty sure it's really simple, but there's something I'm missing or something that I don't fully understand about dummy variables and so. Help, please.

I assume you want

[tex]\int_{-\infty}^t x(\tau)d\tau-\int_{-\infty}^{t-T} x(\tau)d\tau=\int_{-\infty}^t [x(\tau)- x(\tau-T)]d\tau[/tex]

from where you are stuck...let [tex]s=\tau[/tex] and combine the integrals
 

1. What is "convolution-like change of variables"?

"Convolution-like change of variables" is a mathematical technique used to transform an equation or function into a new form. This technique is similar to convolution, in which two functions are combined to create a third function. However, in "convolution-like change of variables", the two functions may not necessarily be multiplied together.

2. How is "convolution-like change of variables" used in science?

"Convolution-like change of variables" is commonly used in science to simplify complex equations or functions. By transforming an equation into a new form, scientists can more easily analyze and manipulate the data. This technique is particularly useful in fields such as signal processing, image processing, and physics.

3. What are the benefits of using "convolution-like change of variables"?

One of the main benefits of using "convolution-like change of variables" is that it can make equations or functions easier to understand and work with. It can also help to reveal underlying patterns or relationships between different variables. Additionally, this technique can be used to solve difficult mathematical problems, often leading to more accurate and efficient solutions.

4. Are there any limitations to "convolution-like change of variables"?

While "convolution-like change of variables" can be a powerful tool, it may not always be applicable to every situation. This technique works best when dealing with linear equations or functions, and may not be as effective with non-linear equations. Additionally, the transformation process can sometimes be complex and time-consuming.

5. How can one learn more about "convolution-like change of variables"?

There are many resources available for learning about "convolution-like change of variables". These include textbooks, online tutorials, and courses. It is also helpful to have a strong understanding of basic calculus and linear algebra in order to fully grasp this technique. Practicing with different examples and problems can also help to improve understanding.

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