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−1012−33−1425h[n]n−101223−24−15

The sequence x[n] is nonzero for n = 2…4 and h[n] for n = 2…4, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 2…4h[n]: n = 2…4
nlast =
2.Easy
x[n]n−10411213145h[n]n−102112−0.534

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

b = {2, 1, −0.5}x[n] = {4, 1, 1, 1}, n = 0…3
y[1] =
3.Medium
x(t)t01h(t)t35

The signal x(t) is nonzero only between t = 0 s and t = 1 s, and h(t) only between t = 3 s and t = 5 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: 0 … 1 sh: 3 … 5 s
tend = s
4.Medium
x(t)t45h(t)t41

The pulse x(t) = 5 for 0 ≤ t < 4 s is convolved with h(t), a triangle that falls linearly from 1 at t = 0 to 0 at t = 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 5, 0 ≤ t < 4 sh(0) = 1, h(4) = 0
Ay =
5.Hard
δ(t)y′ + 1y = 2xh(t)

A system obeys y′(t) + 1·y(t) = 2·x(t). Find its impulse response at t = 0.2 s, h(0.2).

y′ + 1y = 2xt = 0.2 s
h(0.2) =
6.Hard
δ(t)y′ + 10y = 10xh(t)

A system obeys y′(t) + 10·y(t) = 10·x(t). Find its impulse response at t = 0.5 s, h(0.5).

y′ + 10y = 10xt = 0.5 s
h(0.5) =

Answer key

  1. 1. nlast = 8
  2. 2. y[1] = 6
  3. 3. tend = 6 s
  4. 4. Ay = 40
  5. 5. h(0.2) = 1.637
  6. 6. h(0.5) = 0.06738

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