Why Does This Equation Include Sums of h(t-ti)? Explained.

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In summary, the conversation discusses the use of a Dirac function in an equation and the confusion over the use of tau as a dummy variable in the integral. The equation is simplified to show that tau represents the same type of variable as t and ti.
  • #1
paalfis
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Can someone explain to me why in this equation (attached)
Captura de pantalla 2014-08-24 a la(s) 23.07.45.png


where ρ(t)=[itex]\sum[/itex]δ(t-ti) , dirac funtion.
in the left side we have the sum over h(t-ti) instead of the sum over h(ti) ?

It seems to me that the integral would work summing 1*h(t1)+1*h(t2)+...+1*h(ti) for all ti smaller than t.
 
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  • #2
$$\int\mathrm d\tau\, h(\tau)\rho(t-\tau) =\int\mathrm d\tau\, h(\tau)\sum_i\delta((t-\tau)-t_i) =\sum_i\int\mathrm d\tau\, h(\tau)\delta(\tau-(t-t_i)) =\sum_i h(t-t_i)$$
 
  • #3
Thanks, one more question, what does tau stands for in this type of integral. i.e. What is the meaning of (t-tau)?
 
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  • #4
It's a dummy variable (i.e. it can be replaced by any other variable symbol), so it's not really significant. But since you plug it into the same functions where you plug in ##t## and ##t_i##, it can be thought of as representing the same sort of thing as those guys do. (For example, time or temperature, or whatever t stands for in this context). This is especially clear when you approximate an integral by a Riemann sum, ##\int f(x)\mathrm dx\approx\sum_i f(x_i)(x_{i+1}-x_i)##.
 

1. Why do equations include sums of h(t-ti)?

Equations may include sums of h(t-ti) in order to represent a continuous function as a discrete sum of smaller functions. This allows for easier manipulation and analysis of the function.

2. What is the significance of the variable h(t-ti) in this equation?

The variable h(t-ti) represents a function that describes the behavior of the original function at a specific time ti. By summing up these smaller functions, we can approximate the behavior of the original function over a larger time interval.

3. How does the inclusion of sums of h(t-ti) affect the accuracy of the equation?

The accuracy of the equation depends on the number of smaller functions used in the summation. The more functions included, the closer the approximation will be to the original function. However, using too many functions can also lead to computational errors.

4. Can you provide an example of an equation that includes sums of h(t-ti)?

An example of an equation that includes sums of h(t-ti) is the Fourier series, which represents a periodic function as a sum of sines and cosines at different frequencies. Each term in the series contains a factor of h(t-ti) to represent the contribution of that particular frequency to the overall function.

5. How is the concept of sums of h(t-ti) used in real-world applications?

The concept of sums of h(t-ti) is used in various fields, such as signal processing, engineering, and physics. It allows scientists to model and analyze complex systems and phenomena, such as electrical circuits, vibrations, and heat transfer. It is also used in data compression techniques, where a continuous signal is represented by a series of discrete values.

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