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−10−31−32234h[n]n−1021−3234

The sequences x[n] = {−3, −3, 2} 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, 2}, n = 0…2h[n] = {2, −3}, n = 0…1
ymax =
2.Easy
x[n]n−10−3122−334h[n]n−10−3142134

The sequences x[n] = {−3, 2, −3} and h[n] = {−3, 4, 1} both start at n = 0. Find the sample y[1] of the convolution y[n] = x[n] ∗ h[n].

x[n] = {−3, 2, −3}, n = 0…2h[n] = {−3, 4, 1}, n = 0…2
y[1] =
3.Medium
x(t)t22h(t)t1

The pulse x(t) = 2 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with h(t) = e−5tu(t). Find the output y(4) at t = 4 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 2 sh(t) = e−5tu(t)t = 4 s
y(4) =
4.Medium
x(t)t41h(t)t42

The pulse x(t) = 1 for 0 ≤ t < 4 s is convolved with h(t), a triangle that falls linearly from 2 at t = 0 to 0 at t = 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 4 sh(0) = 2, h(4) = 0
Ay =
5.Hard
5cos(7.5t)h(t) = 2e−10t​u(t)y(t)

The input x(t) = 5·cos(7.5t) is applied to a system with h(t) = 2·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 2e−10tu(t)x(t) = 5cos(7.5t)
Ym =
6.Hard
x(t)t (ms)510205−5

A triangle wave rises and falls linearly between −5 and +5 with period T = 10 ms. Find the amplitude c1 of its Fourier component at 100 Hz.

±5T = 10 msf = 100 Hz
c1 =

Answer key

  1. 1. ymax = 13
  2. 2. y[1] = −18
  3. 3. y(4) = 1.816 × 10⁻⁵
  4. 4. Ay = 16
  5. 5. Ym = 0.8
  6. 6. c1 = 4.053

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