Systems&Signals Find and sketch the system output y(t)

[goaliejoe35]

Suppose that the system of figure (a) has the input x(t) given in Figure (b). The impulse response is the unit step function h(t) = u(t). Find and sketch the system output y(t).
http://qaboard.cramster.com/answer-board/image/a4d35f0d5eef154398161f9124b69a59.jpg

http://qaboard.cramster.com/answer-board/image/4e19bec6ce63932d6e85f12071ffe5ff.jpg

Could someone please explain this to me? I have no idea how to do it.

I thought maybe write an equation for x(t), but I am not sure. All I need is someone to push me in the right direction.

Thanks.

 

[jordan.nc]

Hi goaliejoe35...here's how u find out y(t) without having to write the equation for x(t). Basically what h(t) means is that if x(t)=input were an impulse (a function whose value is 1 at t=0 and 0 otherwise) instead of what is given in the figure, then the system represented by the box gives an output of h(t)=u(t)=step function. Since the actual input given in the pic can be decomposed as a sum of time shifted impulses, h(t)s of these time shifted impulses can be summed together to form y(t). Mathematically known as convolution. Check this out for an example on convolution:
http://cnx.org/content/m11541/latest/

To take the easier way, you could use a tool such as mathematica(although u have to write an equation to represent x(t) and h(t)) to perform the convolution and find out the output.
x(t) = 2u(t)-3u(t-1)+u(t+2);
h(t)=u(t);

 

[Mene]

I would approach this problem by first understanding the system depicted in the figure. Based on the given information, it is a linear time-invariant (LTI) system with an impulse response of the unit step function, h(t) = u(t). This means that the output of the system at any given time t will only depend on the input at that time and the previous inputs, and the output will be a scaled version of the input.

To find the system output y(t), we can use the convolution integral:

y(t) = x(t) * h(t)

Where * denotes convolution and x(t) is the input signal. Using the given input signal x(t) in Figure (b), we can rewrite the convolution integral as:

y(t) = (1 + u(t-1)) * u(t)

Now, we can solve this integral by splitting it into two parts:

y(t) = (1 * u(t)) + (u(t-1) * u(t))

The first part is simply the unit step function shifted by 1 unit to the right. The second part can be evaluated by using the definition of convolution:

u(t-1) * u(t) = ∫ u(t-1-τ)u(τ) dτ

= ∫ u(t-1-τ) dτ

= t - 1

Therefore, the system output y(t) can be written as:

y(t) = u(t) + (t-1)

To sketch the output, we can plot the two components separately. The first component, u(t), is a step function with a value of 1 for all t ≥ 0. The second component, (t-1), is a linear function with a slope of 1 and a y-intercept of -1. Combining these two components, we can sketch the system output y(t) as shown in the figure below:



I hope this explanation helps in understanding the problem and how to approach it.