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−10−21−123314h[n]n−104142434

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

x[n] = {−2, −1, 3, 1}, n = 0…3h[n] = {4, 4, 4}, n = 0…2
y[3] =
2.Easy
x[n]n−10−1112334h[n]n−10−314234

The sequences x[n] = {−1, 1, 3} and h[n] = {−3, 4} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {−1, 1, 3}, n = 0…2h[n] = {−3, 4}
Σy[n] =
3.Medium
x(t)t11h(t)t22

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

x(t) = 1, 0 ≤ t < 1 sh(0) = 2, h(2) = 0
Ay =
4.Medium
x(t)t35h(t)t11

The pulse x(t) = 5 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 1 for 0 ≤ t < 1 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 5, 0 ≤ t < 3 sh(t) = 1, 0 ≤ t < 1 s
Ay =
5.Hard
x(t)t (ms)0.4122

A pulse train has height 2, pulse width 0.4 ms and period 1 ms. Find the amplitude c2 of harmonic number 2, at 2000 Hz.

A = 2τ = 0.4 msT = 1 ms
c2 =
6.Hard
2cos(6t)h(t) = 5e−2t​u(t)y(t)

The input x(t) = 2·cos(6t) is applied to a system with h(t) = 5·e−2tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 5e−2tu(t)x(t) = 2cos(6t)
φ = °

Answer key

  1. 1. y[3] = 12
  2. 2. Σy[n] = 3
  3. 3. Ay = 2
  4. 4. Ay = 15
  5. 5. c2 = 0.3742
  6. 6. φ = −71.57 °

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