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−10−111213−14h[n]n−10−3112434

The sequences x[n] = {−1, 1, 1, −1} and h[n] = {−3, 1, 4} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {−1, 1, 1, −1}, n = 0…3h[n] = {−3, 1, 4}, n = 0…2
ymax =
2.Easy
x[n]n−10−2122334h[n]n−10−3132134

The sequences x[n] = {−2, 2, 3} and h[n] = {−3, 3, 1} both start at n = 0. Find the sample y[1] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {−2, 2, 3}, n = 0…2h[n] = {−3, 3, 1}, n = 0…2
y[1] =
3.Medium
x(t)t5h(t)t1

The input x(t) = 5·e−2tu(t) is applied to a system with h(t) = e−0.5tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 5·e−2tu(t)h(t) = e−0.5tu(t)t = 0.25 s
y(0.25) =
4.Medium
x(t)t−10h(t)t26

The signal x(t) is nonzero only between t = −1 s and t = 0 s, and h(t) only between t = 2 s and t = 6 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: −1 … 0 sh: 2 … 6 s
tstart = s
5.Hard
5cos(7.5t)h(t) = 5e−10t​u(t)y(t)

The input x(t) = 5·cos(7.5t) is applied to a system with h(t) = 5·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 5e−10tu(t)x(t) = 5cos(7.5t)
Ym =
6.Hard
u[n]4+y[n]−0.8·z⁻¹+

A system obeys y[n] = −0.8·y[n − 1] + 4·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[3].

y[n] = −0.8y[n − 1] + 4x[n]x[n] = u[n]
y[3] =

Answer key

  1. 1. ymax = 8
  2. 2. y[1] = −12
  3. 3. y(0.25) = 0.9199
  4. 4. tstart = 1 s
  5. 5. Ym = 2
  6. 6. y[3] = 1.312

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