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−4−33−2−2−120−11h[n]n−10123−342526

The sequence x[n] is nonzero for n = -3…0 and h[n] for n = 3…5, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = -3…0h[n]: n = 3…5
nlast =
2.Easy
x[n]n−10−31−32−2324h[n]n−10−2132234

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

x[n] = {−3, −3, −2, 2}, n = 0…3h[n] = {−2, 3, 2}, n = 0…2
ymax =
3.Medium
x(t)t15h(t)t23

The pulse x(t) = 5 for 0 ≤ t < 1 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 5, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 2 s
Ay =
4.Medium
x(t)t14h(t)t13

The signal x(t) is nonzero only between t = 1 s and t = 4 s, and h(t) only between t = 1 s and t = 3 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: 1 … 4 sh: 1 … 3 s
tstart = s
5.Hard
x(t)t (ms)10204010−10

A square wave switches between +10 and −10 with period T = 20 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 250 Hz.

±10T = 20 msf = 250 Hz
c5 =
6.Hard
x(t)t (ms)10204010−10

A triangle wave rises and falls linearly between −10 and +10 with period T = 20 ms. Find the amplitude c5 of its Fourier component at 250 Hz.

±10T = 20 msf = 250 Hz
c5 =

Answer key

  1. 1. nlast = 5
  2. 2. ymax = 6
  3. 3. Ay = 30
  4. 4. tstart = 2 s
  5. 5. c5 = 2.546
  6. 6. c5 = 0.3242

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