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−101112131…h[n]n−10312213−145

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

h[n] = {3, 2, 1, −1}x[n] = 1·u[n]
y[2] =
2.Easy
x[n]n−1041−32−23−14h[n]n−103112−134

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

x[n] = {4, −3, −2, −1}, n = 0…3h[n] = {3, 1, −1}, n = 0…2
ymax =
3.Medium
x(t)t4h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 4·u(t). Find the output y(0.1) at t = 0.1 s.

h(t) = e−5tu(t)x(t) = 4·u(t)t = 0.1 s
y(0.1) =
4.Medium
x(t)t110h(t)t11

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 = 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 1 sh(0) = 1, h(1) = 0
Ay =
5.Hard
x(t)t (ms)1244−4

A triangle wave rises and falls linearly between −4 and +4 with period T = 2 ms. Find the amplitude c3 of its Fourier component at 1500 Hz.

±4T = 2 msf = 1500 Hz
c3 =
6.Hard
5cos(20t)h(t) = 5e−20t​u(t)y(t)

The input x(t) = 5·cos(20t) is applied to a system with h(t) = 5·e−20tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 5e−20tu(t)x(t) = 5cos(20t)
Ym =

Answer key

  1. 1. y[2] = 6
  2. 2. ymax = 12
  3. 3. y(0.1) = 0.3148
  4. 4. Ay = 5
  5. 5. c3 = 0.3603
  6. 6. Ym = 0.8839

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