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−102142−334h[n]n−10−31−3234

The sequences x[n] = {2, 4, −3} and h[n] = {−3, −3} both start at n = 0. Find the sample y[2] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {2, 4, −3}, n = 0…2h[n] = {−3, −3}, n = 0…1
y[2] =
2.Easy
x[n]n−102142−13145h[n]n−10110.52134

An FIR filter computes y[n] = x[n] + 0.5·x[n − 1] + x[n − 2]. The input is x[n] = {2, 4, −1, 1} for n = 0…3 and zero elsewhere. Find the output y[1].

b = {1, 0.5, 1}x[n] = {2, 4, −1, 1}, n = 0…3
y[1] =
3.Medium
x(t)t02h(t)t23

The signal x(t) is nonzero only between t = 0 s and t = 2 s, and h(t) only between t = 2 s and t = 3 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: 0 … 2 sh: 2 … 3 s
D = s
4.Medium
x(t)t5h(t)t1

The input x(t) = 5·e−3tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 5·e−3tu(t)h(t) = e−2tu(t)t = 0.25 s
y(0.25) =
5.Hard
5cos(7.5t)h(t) = 2e−5t​u(t)y(t)

The input x(t) = 5·cos(7.5t) is applied to a system with h(t) = 2·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−5tu(t)x(t) = 5cos(7.5t)
φ = °
6.Hard
2cos(15t)h(t) = 5e−10t​u(t)y(t)

The input x(t) = 2·cos(15t) 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) = 2cos(15t)
Ym =

Answer key

  1. 1. y[2] = −3
  2. 2. y[1] = 5
  3. 3. D = 3 s
  4. 4. y(0.25) = 0.6708
  5. 5. φ = −56.31 °
  6. 6. Ym = 0.5547

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