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−102132234h[n]n−102112−134

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

x[n] = {2, 3, 2}, n = 0…2h[n] = {2, 1, −1}, n = 0…2
y[2] =
2.Easy
x[n]n−104132−234h[n]n−1041−22−334

The sequences x[n] = {4, 3, −2} and h[n] = {4, −2, −3} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {4, 3, −2}, n = 0…2h[n] = {4, −2, −3}
Σy[n] =
3.Medium
x(t)t14h(t)t43

The pulse x(t) = 4 for 0 ≤ t < 1 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 4 s
Ay =
4.Medium
x(t)t31h(t)t41

The pulse x(t) = 1 for 0 ≤ t < 3 s is convolved with the pulse h(t) = 1 for 0 ≤ t < 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 1, 0 ≤ t < 3 sh(t) = 1, 0 ≤ t < 4 s
Ay =
5.Hard
5cos(7.5t)h(t) = 10e−5t​u(t)y(t)

The input x(t) = 5·cos(7.5t) is applied to a system with h(t) = 10·e−5tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−5tu(t)x(t) = 5cos(7.5t)
Ym =
6.Hard
x(t)t (ms)510204

A square wave switches between 4 and 0 with period T = 10 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 100 Hz.

0 … 4T = 10 msf = 100 Hz
c1 =

Answer key

  1. 1. y[2] = 5
  2. 2. Σy[n] = −5
  3. 3. Ay = 48
  4. 4. Ay = 12
  5. 5. Ym = 5.547
  6. 6. c1 = 2.546

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