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−101112−33−14h[n]n−10−311234

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

x[n] = {1, 1, −3, −1}, n = 0…3h[n] = {−3, 1}
Σy[n] =
2.Easy
x[n]n−10111233145h[n]n−10210.520.2534

An FIR filter computes y[n] = 2·x[n] + 0.5·x[n − 1] + 0.25·x[n − 2]. The input is x[n] = {1, 1, 3, 1} for n = 0…3 and zero elsewhere. Find the output y[2].

b = {2, 0.5, 0.25}x[n] = {1, 1, 3, 1}, n = 0…3
y[2] =
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(0.1) at t = 0.1 s.

h(t) = e−4tu(t)x(t) = 2·u(t)t = 0.1 s
y(0.1) =
4.Medium
x(t)t22h(t)t14

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

x(t) = 2, 0 ≤ t < 2 sh(0) = 4, h(1) = 0
Ay =
5.Hard
2cos(20t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 2·cos(20t) is applied to a system with h(t) = 10·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−10tu(t)x(t) = 2cos(20t)
Ym =
6.Hard
2cos(7.5t)h(t) = 10e−5t​u(t)y(t)

The input x(t) = 2·cos(7.5t) is applied to a system with h(t) = 10·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−5tu(t)x(t) = 2cos(7.5t)
φ = °

Answer key

  1. 1. Σy[n] = 4
  2. 2. y[2] = 6.75
  3. 3. y(0.1) = 0.1648
  4. 4. Ay = 8
  5. 5. Ym = 0.8944
  6. 6. φ = −56.31 °

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