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−10211243145h[n]n−1011220.534

An FIR filter computes y[n] = x[n] + 2·x[n − 1] + 0.5·x[n − 2]. The input is x[n] = {2, 1, 4, 1} for n = 0…3 and zero elsewhere. Find the output y[3].

b = {1, 2, 0.5}x[n] = {2, 1, 4, 1}, n = 0…3
y[3] =
2.Easy
x[n]n−1011−22234h[n]n−10−314234

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

x[n] = {1, −2, 2}, n = 0…2h[n] = {−3, 4}, n = 0…1
ymax =
3.Medium
x(t)t4h(t)t1

The input x(t) = 4·e−1tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(1.5) at t = 1.5 s.

x(t) = 4·e−1tu(t)h(t) = e−2tu(t)t = 1.5 s
y(1.5) =
4.Medium
x(t)t22h(t)t12

The pulse x(t) = 2 for 0 ≤ t < 2 s is convolved with h(t), a triangle that falls linearly from 2 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) = 2, h(1) = 0
Ay =
5.Hard
x(t)t (ms)10204010

A square wave switches between 10 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 … 10T = 20 msf = 150 Hz
c3 =
6.Hard
x(t)t (ms)0.51210−10

A triangle wave rises and falls linearly between −10 and +10 with period T = 1 ms. Find the amplitude c5 of its Fourier component at 5000 Hz.

±10T = 1 msf = 5000 Hz
c5 =

Answer key

  1. 1. y[3] = 9.5
  2. 2. ymax = 10
  3. 3. y(1.5) = 0.6934
  4. 4. Ay = 4
  5. 5. c3 = 2.122
  6. 6. c5 = 0.3242

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