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−10−113213−24h[n]n−10−114234

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

x[n] = {−1, 3, 1, −2}, n = 0…3h[n] = {−1, 4}, n = 0…1
ymax =
2.Easy
x[n]n−101142−1344h[n]n−10411234

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

x[n] = {1, 4, −1, 4}, n = 0…3h[n] = {4, 1}, n = 0…1
y[3] =
3.Medium
x(t)t02h(t)t04

The signal x(t) is nonzero only between t = 0 s and t = 2 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 … 2 sh: 0 … 4 s
tend = s
4.Medium
x(t)t02h(t)t−10

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

x: 0 … 2 sh: −1 … 0 s
tstart = s
5.Hard
u[n]0.5+y[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 0.5·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[2].

y[n] = −0.5y[n − 1] + 0.5x[n]x[n] = u[n]
y[2] =
6.Hard
2cos(10t)h(t) = 5e−5t​u(t)y(t)

The input x(t) = 2·cos(10t) is applied to a system with h(t) = 5·e−5tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 5e−5tu(t)x(t) = 2cos(10t)
Ym =

Answer key

  1. 1. ymax = 11
  2. 2. y[3] = 15
  3. 3. tend = 6 s
  4. 4. tstart = −1 s
  5. 5. y[2] = 0.375
  6. 6. Ym = 0.8944

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