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−1021124324h[n]n−10−111234

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

x[n] = {2, 1, 4, 2}, n = 0…3h[n] = {−1, 1}, n = 0…1
y[2] =
2.Easy
x[n]n−10−2142234h[n]n−10−313234

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

x[n] = {−2, 4, 2}, n = 0…2h[n] = {−3, 3}, n = 0…1
y[1] =
3.Medium
x(t)t14h(t)t1

The pulse x(t) = 4 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−4tu(t). Find the output y(1.5) at t = 1.5 s, after the pulse has ended.

x(t) = 4, 0 ≤ t < 1 sh(t) = e−4tu(t)t = 1.5 s
y(1.5) =
4.Medium
x(t)t34h(t)t14

The pulse x(t) = 4 for 0 ≤ t < 3 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) = 4, 0 ≤ t < 3 sh(0) = 4, h(1) = 0
Ay =
5.Hard
x(t)t (ms)1020403

A square wave switches between 3 and 0 with period T = 20 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 150 Hz.

0 … 3T = 20 msf = 150 Hz
c3 =
6.Hard
u[n]3+y[n]0.5·z⁻¹+

A system obeys y[n] = 0.5·y[n − 1] + 3·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[3].

y[n] = 0.5y[n − 1] + 3x[n]x[n] = u[n]
y[3] =

Answer key

  1. 1. y[2] = −3
  2. 2. y[1] = −18
  3. 3. y(1.5) = 0.1329
  4. 4. Ay = 24
  5. 5. c3 = 0.6366
  6. 6. y[3] = 5.625

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