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−10−21−32−3334h[n]n−10412234

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

x[n] = {−2, −3, −3, 3}, n = 0…3h[n] = {4, 2}, n = 0…1
y[3] =
2.Easy
x[n]n−103132−134h[n]n−10−213234

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

x[n] = {3, 3, −1}, n = 0…2h[n] = {−2, 3}, n = 0…1
ymax =
3.Medium
x(t)t5h(t)t21

A system has the impulse response h(t) = e−0.5tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.4) at t = 0.4 s.

h(t) = e−0.5tu(t)x(t) = 5·u(t)t = 0.4 s
y(0.4) =
4.Medium
x(t)t31h(t)t23

The pulse x(t) = 1 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 3 sh(t) = 3, 0 ≤ t < 2 s
Ay =
5.Hard
δ(t)y′ + 10y = 2xh(t)

A system obeys y′(t) + 10·y(t) = 2·x(t). Find its impulse response at t = 0.1 s, h(0.1).

y′ + 10y = 2xt = 0.1 s
h(0.1) =
6.Hard
x(t)t (ms)520401

A pulse train has height 1, pulse width 5 ms and period 20 ms. Find the amplitude c3 of harmonic number 3, at 150 Hz.

A = 1τ = 5 msT = 20 ms
c3 =

Answer key

  1. 1. y[3] = 6
  2. 2. ymax = 11
  3. 3. y(0.4) = 1.813
  4. 4. Ay = 18
  5. 5. h(0.1) = 0.7358
  6. 6. c3 = 0.1501

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