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−101122334h[n]n−103132234

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

x[n] = {1, 2, 3}, n = 0…2h[n] = {3, 3, 2}, n = 0…2
ymax =
2.Easy
x[n]n−102122232…h[n]n−1031−12234

A system with the impulse response h[n] = {3, −1, 2} (n = 0…2) receives the step input x[n] = 2·u[n]. Find the output y[3].

h[n] = {3, −1, 2}x[n] = 2·u[n]
y[3] =
3.Medium
x(t)t32h(t)t45

The pulse x(t) = 2 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(3) at t = 3 s.

x(t) = 2, 0 ≤ t < 3 sh(t) = 5, 0 ≤ t < 4 st = 3 s
y(3) =
4.Medium
x(t)t110h(t)t1

The pulse x(t) = 10 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−4tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 10, 0 ≤ t < 1 sh(t) = e−4tu(t)t = 2.5 s
y(2.5) =
5.Hard
10cos(20t)h(t) = 2e−10t​u(t)y(t)

The input x(t) = 10·cos(20t) is applied to a system with h(t) = 2·e−10tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−10tu(t)x(t) = 10cos(20t)
φ = °
6.Hard
x(t)t (ms)1245−5

A triangle wave rises and falls linearly between −5 and +5 with period T = 2 ms. Find the amplitude c5 of its Fourier component at 2500 Hz.

±5T = 2 msf = 2500 Hz
c5 =

Answer key

  1. 1. ymax = 17
  2. 2. y[3] = 8
  3. 3. y(3) = 30
  4. 4. y(2.5) = 0.006083
  5. 5. φ = −63.43 °
  6. 6. c5 = 0.1621

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