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−10−2142−234h[n]n−10−1122−134

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

x[n] = {−2, 4, −2}, n = 0…2h[n] = {−1, 2, −1}, n = 0…2
y[3] =
2.Easy
x[n]n−10113223145h[n]n−10210.520.534

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

b = {2, 0.5, 0.5}x[n] = {1, 3, 2, 1}, n = 0…3
y[4] =
3.Medium
x(t)t410h(t)t13

The pulse x(t) = 10 for 0 ≤ t < 4 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 4 sh(t) = 3, 0 ≤ t < 1 s
Ay =
4.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−2tu(t) is applied to a system with h(t) = e−4tu(t). Find the output y(0.5) at t = 0.5 s.

x(t) = 10·e−2tu(t)h(t) = e−4tu(t)t = 0.5 s
y(0.5) =
5.Hard
2cos(20t)h(t) = 10e−20t​u(t)y(t)

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

h(t) = 10e−20tu(t)x(t) = 2cos(20t)
φ = °
6.Hard
2cos(15t)h(t) = 10e−5t​u(t)y(t)

The input x(t) = 2·cos(15t) is applied to a system with h(t) = 10·e−5tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−5tu(t)x(t) = 2cos(15t)
Ym =

Answer key

  1. 1. y[3] = −8
  2. 2. y[4] = 1.5
  3. 3. Ay = 120
  4. 4. y(0.5) = 1.163
  5. 5. φ = −45 °
  6. 6. Ym = 1.265

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