Understanding Convolution: h(t) & x(t) w/ x(t)=u(t-1)

In summary, In trying to figure out how to convolve h(t) and x(t), the user has trouble understanding what to do when x(t) = u(t-1). They come up with two different expressions, but they are both correct. The first is x(T) = u(T-1) and the second is x(T) = u(t-T). The convolution of x and h is given by x \ast h(t) = \int_{-\infty}^{\infty}x(T)h(t - T)dT.
  • #1
seang
184
0
I'm having trouble understanding convolution. In particular, in convolving h(t) and x(t), I have no idea what to do when x(t) = u(t-1). So for example, if h(t) = exp(-at)u(t) an x(t) = u(t-1). Is this even the right set up? I don't think it is.

[tex]\int_o^t e^{-a(t-T)}u(t-1)dT[/tex]

the other idea I had was

[tex]\int_1^t e^{-a(t-T)}u(t)dT[/tex]

but I doubt this too.
Help?
 
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  • #2
lemme just be clear about your notation. Is this right?

[tex]x(t) = u(t-1)[/tex]

where [itex]u(t)[/itex] is the unit step function, ie:

[tex]
u(x) = \left \lbracket
\begin{array}{cc}
0 & \mbox{if } x<0 \\
1 & \mbox{if } x \geq 0 \\
\end{array}
\right.
[/tex]

and [tex]h(t) = e^{-at}u(t)[/tex].

the convolution of x and h, denoted [itex]x \ast h[/itex], is given by,
[tex] (x \ast h)(t) = \int_{-\infty}^{\infty}x(T)h(t - T)dT[/tex]

Also, convolution is commutative so [itex] x \ast h = h \ast x[/itex]. Therefore choose the one that makes your calculation easier. From your post it looks like you chose [itex] x \ast h[/itex]

your first goal is to come up with the correct expressions for

[tex] h(t - T) = \ ?? [/itex]
[tex]x(T) = \ ?? [/tex]
 
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  • #3
[tex] h(t - T) = e^{-a(t-T)} [/tex]
[tex]x(T) = u(T-1) [/tex]

Maybe?
 
  • #4
In your first post you said [itex] h(t) = e^{-at}u(t)[/itex]. What happened to [itex]u[/itex] in [itex]h(t-T)[/itex]?

Thats the correct expression for x(T).

What's [itex]u(t)[/itex]? Is it the unit step function?
 
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  • #5
yeah it is, and that's what's catching me a little i think; is the u(t) in the h(t) related to the u(t) which x(t) is equal to?
 
  • #6
yes. recall from algebra that the graph of [itex]u(t-1)[/itex] is the graph of [itex]u(t)[/itex] shifted to the right 1 unit.

If [itex]f,g,[/itex] and [itex]h[/itex] are functions such that [itex]f(t) = g(t)h(t)[/itex], and [itex]a[/itex] is a real number, then [itex]f(t-a) = g(t-a)h(t-a)[/itex]. Use this to come up with an expression for [itex]h(t-T)[/itex].
 
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  • #7
I feel like a n00bie.

So
[tex] h(t - T) = e^{-a(t-1)} [/tex]?
 
  • #8
[tex]h(t) = e^{-at}u(t)[/tex]

to get [itex]h(t-T)[/itex] replace all instances of [itex]t[/itex] with [itex]t-T[/itex]
 
  • #9
so h(t-T):

[tex]h(t-T) = e^{-a(t-T)}u(t-T)}[/tex]

forgive me its late and I'm having a really hard time understanding this.
 
  • #10
yes :smile:

now you can substitute the expresions for [itex]x(T)[/itex] and [itex]h(t-T)[/itex] in the convolution integral.
 
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1. What is convolution, and why is it important in scientific research?

Convolution is a mathematical operation that combines two functions to produce a third function. It is important in scientific research because it allows us to describe the relationship between two variables and analyze how they interact with each other.

2. What is the difference between h(t) and x(t) in convolution?

h(t) is known as the impulse response and represents the output of a system when an impulse is applied. x(t) is the input signal that is being convolved with the impulse response. In other words, h(t) describes the behavior of the system, while x(t) is the signal being processed by the system.

3. How does the time shift in x(t) affect the convolution result?

When x(t) is shifted by a time delay, the resulting convolution will also be shifted by the same amount of time. This is because the time shift in x(t) causes a delay in when the input signal reaches the system, resulting in a delayed output.

4. Can convolution be applied to signals other than continuous-time signals?

Yes, convolution can also be applied to discrete-time signals, such as those in digital signal processing. The process is the same, but the integral is replaced by a summation.

5. How is convolution used in real-world applications?

Convolution has many applications in various fields such as signal processing, image processing, and data analysis. For example, it can be used to filter noise from signals, enhance images, and analyze the effects of different inputs on a system.

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