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

• goaliejoe35
In summary, the system output y(t) is equal to 2u(t)-3u(t-1)+u(t+2) where u(t) is the unit step function. This can be found by convolving the input x(t) with the unit step function h(t) = u(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).

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.

Last edited by a moderator:
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);

Last edited by a moderator:

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.

## 1. What is a system output?

A system output is the response or result of a system to an input. It is the output of a system that is determined by the input and any internal components of the system.

## 2. How is the system output determined?

The system output is determined by applying the input signal to the system and observing the resulting output signal. This can be done through mathematical analysis or by experimenting with the system.

## 3. What is the difference between a continuous and discrete system output?

A continuous system output is a function of time and can take on any value within a given range. A discrete system output is a function of time but can only take on specific values at specific points in time.

## 4. What is the significance of sketching the system output?

Sketching the system output helps to visualize and understand how the input signal is affected by the system. It can also be used to analyze the behavior of the system and make predictions about its performance.

## 5. Are there any limitations to sketching the system output?

Yes, there are limitations to sketching the system output. It is a simplified representation of the actual output and may not accurately reflect the behavior of the system in all cases. Additionally, it may not account for external influences on the system.

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