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−10−3122234h[n]n−10−3142334

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

x[n] = {−3, 2, 2}, n = 0…2h[n] = {−3, 4, 3}
Σy[n] =
2.Easy
x[n]n−10−31−12134h[n]n−103112−134

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

x[n] = {−3, −1, 1}, n = 0…2h[n] = {3, 1, −1}, n = 0…2
ymax =
3.Medium
x(t)t21h(t)t21

The pulse x(t) = 1 for 0 ≤ t < 2 s is convolved with the pulse h(t) = 1 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 2 sh(t) = 1, 0 ≤ t < 2 s
Ay =
4.Medium
x(t)t12h(t)t23

The pulse x(t) = 2 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(1.5) at t = 1.5 s.

x(t) = 2, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 2 st = 1.5 s
y(1.5) =
5.Hard
5cos(30t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 5·cos(30t) is applied to a system with h(t) = 10·e−10tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−10tu(t)x(t) = 5cos(30t)
φ = °
6.Hard
x(t)t (ms)2.55101−1

A square wave switches between +1 and −1 with period T = 5 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 1000 Hz.

±1T = 5 msf = 1000 Hz
c5 =

Answer key

  1. 1. Σy[n] = 4
  2. 2. ymax = 5
  3. 3. Ay = 4
  4. 4. y(1.5) = 6
  5. 5. φ = −71.57 °
  6. 6. c5 = 0.2546

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