Question

My question. Please sketch x(t), h(t), and y(t) from my question below. Your sketch should be reasonably accurate and the axes should be correctly labelled. Please do not use char GPT(Open AI).

This is my equation
x(t)=δ(t+2)+2δ(t+1)−3δ(t−1)
function, and
h(t)={ ​1, ​0≤t≤1
−1,1≤t≤2
0, elsewhere ​
this is my answer which is correct.
f(t)∗δ(t−to​)=f(t−to​)
y(t)=x(t)∗ h(t)
y(t)=(δ(t+2)+2δ(t+1)−3δ(t−1))∗ h(t)
y(t)=δ(t+2)∗ h(t)+2δ(t+1)∗ h(t)−3δ(t−1)∗ h(t)
y(t)=h(t+2)+2 h(t+1)−3 h(t−1)
Given h(t)=u[t]−2u[t−1]+u[t−2]
h(t+2)=u[t+2]−2u[t+1]+u[t]
h(t+1)=u[t+1]−2u[t]+u[t−1]
h(t−1)=u[t−1]−2u[t−2]+u[t−3]
y(t)=u[t+2]−3u[t]−u[t−1]+6u[t−2]−3u[t−3]​​

Answer 1

The student's question involves applying convolution to delta and step functions, and the task is to sketch x(t), h(t), and y(t) based on given definitions and convolution properties, with correctly labeled axes representing time and amplitude.

Step-by-step explanation:

The student is working on a problem involving convolution of two functions. The given functions are x(t), which is comprised of a series of Dirac delta functions, and a piecewise function h(t). By using the properties of convolution with the delta function, the student determined the output y(t) as a linear combination of shifted unit step functions, u(t).

To sketch the functions x(t), h(t), and y(t), you should follow these steps:

• For x(t), plot spikes at t = -2, t = -1, and t = 1 with corresponding magnitudes of 1, 2, and -3.
• For h(t), create a piecewise graph that is 1 from t = 0 to t = 1 and -1 from t = 1 to t = 2, and 0 elsewhere.
• For y(t), the convolution product, sketch the result based on the unit step function evaluations given. This will result in a graph that combines rising and falling edges at different time shifts.

Pay attention to properly labeling the axes for time (t) and amplitude for each of the sketches.