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.

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1.Easy
x[n]n−10−213233−24h[n]n−10−31−12334

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

x[n] = {−2, 3, 3, −2}, n = 0…3h[n] = {−3, −1, 3}, n = 0…2
y[3] =
2.Easy
x[n]n−10−2132434h[n]n−10112234

The sequences x[n] = {−2, 3, 4} and h[n] = {1, 2} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {−2, 3, 4}, n = 0…2h[n] = {1, 2}, n = 0…1
ymax =
3.Medium
x(t)t01h(t)t04

The signal x(t) is nonzero only between t = 0 s and t = 1 s, and h(t) only between t = 0 s and t = 4 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: 0 … 1 sh: 0 … 4 s
tend = s
4.Medium
x(t)t03h(t)t35

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

x: 0 … 3 sh: 3 … 5 s
tstart = s
5.Hard
2cos(40t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 2·cos(40t) is applied to a system with h(t) = 10·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−20tu(t)x(t) = 2cos(40t)
φ = °
6.Hard
10cos(4t)h(t) = 2e−2t​u(t)y(t)

The input x(t) = 10·cos(4t) is applied to a system with h(t) = 2·e−2tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 2e−2tu(t)x(t) = 10cos(4t)
Ym =

Answer key

  1. 1. y[3] = 12
  2. 2. ymax = 10
  3. 3. tend = 5 s
  4. 4. tstart = 3 s
  5. 5. φ = −63.43 °
  6. 6. Ym = 4.472

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