Convolution & LTI systems worksheet with answers

Free printable Convolution & LTI systems worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
x[n]n−105152535…h[n]n−10−2112234

A system with the impulse response h[n] = {−2, 1, 2} (n = 0…2) receives the step input x[n] = 5·u[n]. Find the output y[1].

h[n] = {−2, 1, 2}x[n] = 5·u[n]
y[1] =
2.Easy
x[n]n−10−2122334h[n]n−10−114234

The sequences x[n] = {−2, 2, 3} and h[n] = {−1, 4} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {−2, 2, 3}, n = 0…2h[n] = {−1, 4}
Σy[n] =
3.Medium
x(t)t34h(t)t11

The pulse x(t) = 4 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 1 for 0 ≤ t < 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 3 sh(t) = 1, 0 ≤ t < 1 s
Ay =
4.Medium
x(t)t14h(t)t1

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−5tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 4, 0 ≤ t < 1 sh(t) = e−5tu(t)t = 2.5 s
y(2.5) =
5.Hard
δ(t)y′ + 0.5y = 3xh(t)

A system obeys y′(t) + 0.5·y(t) = 3·x(t). Find its impulse response at t = 8 s, h(8).

y′ + 0.5y = 3xt = 8 s
h(8) =
6.Hard
δ(t)y′ + 0.5y = 2xh(t)

A system obeys y′(t) + 0.5·y(t) = 2·x(t). Find its impulse response at t = 4 s, h(4).

y′ + 0.5y = 2xt = 4 s
h(4) =

Answer key

  1. 1. y[1] = −5
  2. 2. Σy[n] = 9
  3. 3. Ay = 12
  4. 4. y(2.5) = 0.0004395
  5. 5. h(8) = 0.05495
  6. 6. h(4) = 0.2707

Full step-by-step solutions: practise this topic on examproblems.com.