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−10111243445h[n]n−100.51−12−0.534

An FIR filter computes y[n] = 0.5·x[n] − x[n − 1] − 0.5·x[n − 2]. The input is x[n] = {1, 1, 4, 4} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {0.5, −1, −0.5}x[n] = {1, 1, 4, 4}, n = 0…3
y[3] =
2.Easy
x[n]n−10−21−32−13−14h[n]n−10−21−12234

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

x[n] = {−2, −3, −1, −1}, n = 0…3h[n] = {−2, −1, 2}, n = 0…2
ymax =
3.Medium
x(t)t2h(t)t0.251

A system has the impulse response h(t) = e−4tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.75) at t = 0.75 s.

h(t) = e−4tu(t)x(t) = 2·u(t)t = 0.75 s
y(0.75) =
4.Medium
x(t)t24h(t)t13

The pulse x(t) = 4 for 0 ≤ t < 2 s is convolved with h(t), a triangle that falls linearly from 3 at t = 0 to 0 at t = 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 2 sh(0) = 3, h(1) = 0
Ay =
5.Hard
5cos(15t)h(t) = 2e−5t​u(t)y(t)

The input x(t) = 5·cos(15t) is applied to a system with h(t) = 2·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−5tu(t)x(t) = 5cos(15t)
φ = °
6.Hard
x(t)t (ms)2485−5

A triangle wave rises and falls linearly between −5 and +5 with period T = 4 ms. Find the amplitude c3 of its Fourier component at 750 Hz.

±5T = 4 msf = 750 Hz
c3 =

Answer key

  1. 1. y[3] = −2.5
  2. 2. ymax = 8
  3. 3. y(0.75) = 0.4751
  4. 4. Ay = 12
  5. 5. φ = −71.57 °
  6. 6. c3 = 0.4503

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