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−10−2122334h[n]n−103132134

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

x[n] = {−2, 2, 3}, n = 0…2h[n] = {3, 3, 1}, n = 0…2
ymax =
2.Easy
x[n]n−1011321324h[n]n−10−113234

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

x[n] = {1, 3, 1, 2}, n = 0…3h[n] = {−1, 3}, n = 0…1
y[3] =
3.Medium
x(t)t1h(t)t11

A system has the impulse response h(t) = e−1tu(t). The input is a step x(t) = 1·u(t). Find the output y(4) at t = 4 s.

h(t) = e−1tu(t)x(t) = 1·u(t)t = 4 s
y(4) =
4.Medium
x(t)t21h(t)t31

The pulse x(t) = 1 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with the impulse response h(t) = 1 for 0 ≤ t < 3 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 1, 0 ≤ t < 2 sh(t) = 1, 0 ≤ t < 3 st = 0.5 s
y(0.5) =
5.Hard
5cos(4t)h(t) = 2e−2t​u(t)y(t)

The input x(t) = 5·cos(4t) is applied to a system with h(t) = 2·e−2tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−2tu(t)x(t) = 5cos(4t)
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. ymax = 13
  2. 2. y[3] = 1
  3. 3. y(4) = 0.9817
  4. 4. y(0.5) = 0.5
  5. 5. φ = −63.43 °
  6. 6. Ym = 0.5547

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