Convolution integral with impulse and step functions

[lizm]

hey, i did my undergrad in music and english and now am lost beyond reckoning with the impulse response stuff in my postgrad. Could anyone help with these exercises? They're examples from my lecture notes and have no idea how my lecturer got the answers. Please help?

Ex.1) Show that x(t)*h(t) = h(t)*x(t)
EH? ANY IDEAS?

Ex.2) The unit response of a continuous system is h(t)=3e -2t-5e -4t/sup].
If the input is modeled as x(t)= [delta](t)-2[delta](t-1)+[delta](t-2.5), find the value of the output at time=1.5s.
Ans: y(t)=0.717 at t=1.5s.

Ex.3) An electrical system has unit-impulse response h(t)= 3te <sup>-4t . If a unit step function u(t) is applied to the system, use the convolution integral to determine the value of the output after 0.25s.
Ans: 4.95 x 10 -2 V
What's the unit step function for a start?

Ex.4) Evaluate the integral [integral infinity to minus infinity] f 1 (t)f 2 (t)dt
Where f 1 = 2sin (2000[pie]t) and f 2 (t) = [delta] (t-0.25 x 10 -3 )
Ans: 2
How do they get a pure whole number out of all those symbols?</sup>

 

[member 553083]

The unit step function u(t) is a function that is equal to 0 for t<0 and equal to 1 for t≥0. To evaluate the integral in Ex.3, you can use integration by parts. For Ex.4, this integral is an example of a convolution integral where you are taking the product of two functions and integrating over the domain of both functions. This integral simplifies to a pure number because the delta function is non-zero only at a single point, so the product of the two functions is 2 at that specific point, and the integral of a constant is itself times the area over which it is integrated.

 

[M98Ranger]

Impulse response and convolution are fundamental concepts in signal processing and are used to analyze and manipulate signals in various fields such as engineering, physics, and music.

The impulse response of a system is the output of the system when a delta function (or impulse) is applied as the input. It represents the characteristics of the system and is used to determine the output for any given input.

Convolution, on the other hand, is a mathematical operation that combines two signals to produce a third signal. It is represented by the symbol "*" and is used to model the output of a system when a given input signal is applied.

In Ex.1, we are asked to show that the order of convolution does not matter. This means that x(t)*h(t) is equal to h(t)*x(t). This can be shown by using the definition of convolution, which involves integrating the product of the two signals over all possible time values.

In Ex.2 and Ex.3, we are given specific functions for the input and the impulse response, and we are asked to find the output at a given time. This can be done by using the convolution integral, which involves integrating the product of the input and the impulse response over all possible time values.

In Ex.4, we are asked to evaluate an integral involving two signals. This can be done by using the properties of the delta function, which is defined as an impulse with an area of 1. By using this property, we can simplify the integral and obtain a pure number as the answer.

If you are having trouble with these exercises, it may be helpful to review the basics of convolution and impulse response, and also practice using the properties and definitions of these concepts. You can also seek help from your lecturer, classmates, or a tutor for further clarification and guidance.