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−10−11−12234h[n]n−10114234

The sequences x[n] = {−1, −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, −1, 2}, n = 0…2h[n] = {1, 4}, n = 0…1
ymax =
2.Easy
x[n]n−10−212233−24h[n]n−10311234

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

x[n] = {−2, 2, 3, −2}, n = 0…3h[n] = {3, 1}, n = 0…1
ymax =
3.Medium
x(t)t12h(t)t41

The pulse x(t) = 2 for 0 ≤ t < 1 s is convolved with the pulse h(t) = 1 for 0 ≤ t < 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 2, 0 ≤ t < 1 sh(t) = 1, 0 ≤ t < 4 s
Ay =
4.Medium
x(t)t0.55h(t)t1

The pulse x(t) = 5 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−1tu(t). Find the output y(1.5) at t = 1.5 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 0.5 sh(t) = e−1tu(t)t = 1.5 s
y(1.5) =
5.Hard
2cos(60t)h(t) = 2e−20t​u(t)y(t)

The input x(t) = 2·cos(60t) is applied to a system with h(t) = 2·e−20tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 2e−20tu(t)x(t) = 2cos(60t)
Ym =
6.Hard
x(t)t (ms)1020402−2

A triangle wave rises and falls linearly between −2 and +2 with period T = 20 ms. Find the amplitude c3 of its Fourier component at 150 Hz.

±2T = 20 msf = 150 Hz
c3 =

Answer key

  1. 1. ymax = 8
  2. 2. ymax = 11
  3. 3. Ay = 8
  4. 4. y(1.5) = 0.7237
  5. 5. Ym = 0.06325
  6. 6. c3 = 0.1801

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