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−10123−3445−36h[n]n−10123−34−15

The sequence x[n] is nonzero for n = 3…5 and h[n] for n = 3…4, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 3…5h[n]: n = 3…4
nlast =
2.Easy
x[n]n−10313243−24h[n]n−10113234

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

x[n] = {3, 3, 4, −2}, n = 0…3h[n] = {1, 3}, n = 0…1
y[2] =
3.Medium
x(t)t−21h(t)t15

The signal x(t) is nonzero only between t = −2 s and t = 1 s, and h(t) only between t = 1 s and t = 5 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: −2 … 1 sh: 1 … 5 s
tend = s
4.Medium
x(t)t02h(t)t37

The signal x(t) is nonzero only between t = 0 s and t = 2 s, and h(t) only between t = 3 s and t = 7 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: 0 … 2 sh: 3 … 7 s
D = s
5.Hard
x(t)t (ms)0.5124−4

A square wave switches between +4 and −4 with period T = 1 ms (half a period at each level). Find the amplitude c3 of its Fourier component at 3000 Hz.

±4T = 1 msf = 3000 Hz
c3 =
6.Hard
x(t)t (ms)12410

A square wave switches between 10 and 0 with period T = 2 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 3500 Hz.

0 … 10T = 2 msf = 3500 Hz
c7 =

Answer key

  1. 1. nlast = 9
  2. 2. y[2] = 13
  3. 3. tend = 6 s
  4. 4. D = 6 s
  5. 5. c3 = 1.698
  6. 6. c7 = 0.9095

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