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−103132333…h[n]n−1031−32−13445

A system with the impulse response h[n] = {3, −3, −1, 4} (n = 0…3) receives the step input x[n] = 3·u[n]. Find the output y[4].

h[n] = {3, −3, −1, 4}x[n] = 3·u[n]
y[4] =
2.Easy
x[n]n−105152535…h[n]n−103112234

A system with the impulse response h[n] = {3, 1, 2} (n = 0…2) receives the step input x[n] = 5·u[n]. Find the output y[3].

h[n] = {3, 1, 2}x[n] = 5·u[n]
y[3] =
3.Medium
x(t)t110h(t)t41

The pulse x(t) = 10 for 0 ≤ t < 1 s is convolved with h(t), a triangle that falls linearly from 1 at t = 0 to 0 at t = 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 1 sh(0) = 1, h(4) = 0
Ay =
4.Medium
x(t)t24h(t)t31

The pulse x(t) = 4 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(1) at t = 1 s.

x(t) = 4, 0 ≤ t < 2 sh(t) = 1, 0 ≤ t < 3 st = 1 s
y(1) =
5.Hard
x(t)t (ms)1243

A square wave switches between 3 and 0 with period T = 2 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 1500 Hz.

0 … 3T = 2 msf = 1500 Hz
c3 =
6.Hard
x(t)t (ms)2.55103−3

A triangle wave rises and falls linearly between −3 and +3 with period T = 5 ms. Find the amplitude c3 of its Fourier component at 600 Hz.

±3T = 5 msf = 600 Hz
c3 =

Answer key

  1. 1. y[4] = 9
  2. 2. y[3] = 30
  3. 3. Ay = 20
  4. 4. y(1) = 4
  5. 5. c3 = 0.6366
  6. 6. c3 = 0.2702

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