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−10−1122−13−34h[n]n−102122134

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

x[n] = {−1, 2, −1, −3}, n = 0…3h[n] = {2, 2, 1}, n = 0…2
y[1] =
2.Easy
x[n]n−1041−22−334h[n]n−1031−3234

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

x[n] = {4, −2, −3}, n = 0…2h[n] = {3, −3}, n = 0…1
ymax =
3.Medium
x(t)t25h(t)t22

The pulse x(t) = 5 for 0 ≤ t < 2 s is convolved with h(t), a triangle that falls linearly from 2 at t = 0 to 0 at t = 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 5, 0 ≤ t < 2 sh(0) = 2, h(2) = 0
Ay =
4.Medium
x(t)t2h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.1) at t = 0.1 s.

h(t) = e−5tu(t)x(t) = 2·u(t)t = 0.1 s
y(0.1) =
5.Hard
δ(t)y′ + 4y = 5xh(t)

A system obeys y′(t) + 4·y(t) = 5·x(t). Find its impulse response at t = 0.05 s, h(0.05).

y′ + 4y = 5xt = 0.05 s
h(0.05) =
6.Hard
x(t)t (ms)5102010

A square wave switches between 10 and 0 with period T = 10 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 100 Hz.

0 … 10T = 10 msf = 100 Hz
c1 =

Answer key

  1. 1. y[1] = 2
  2. 2. ymax = 12
  3. 3. Ay = 20
  4. 4. y(0.1) = 0.1574
  5. 5. h(0.05) = 4.094
  6. 6. c1 = 6.366

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