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−1031121324h[n]n−10−1122134

The sequences x[n] = {3, 1, 1, 2} and h[n] = {−1, 2, 1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {3, 1, 1, 2}, n = 0…3h[n] = {−1, 2, 1}
Σy[n] =
2.Easy
x[n]n−10−214213−145h[n]n−10210.2520.534

An FIR filter computes y[n] = 2·x[n] + 0.25·x[n − 1] + 0.5·x[n − 2]. The input is x[n] = {−2, 4, 1, −1} for n = 0…3 and zero elsewhere. Find the output y[2].

b = {2, 0.25, 0.5}x[n] = {−2, 4, 1, −1}, n = 0…3
y[2] =
3.Medium
x(t)t23h(t)t12

The signal x(t) is nonzero only between t = 2 s and t = 3 s, and h(t) only between t = 1 s and t = 2 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: 2 … 3 sh: 1 … 2 s
tstart = s
4.Medium
x(t)t−11h(t)t23

The signal x(t) is nonzero only between t = −1 s and t = 1 s, and h(t) only between t = 2 s and t = 3 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: −1 … 1 sh: 2 … 3 s
D = s
5.Hard
u[n]2+y[n]0.4·z⁻¹+

A system obeys y[n] = 0.4·y[n − 1] + 2·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[4].

y[n] = 0.4y[n − 1] + 2x[n]x[n] = u[n]
y[4] =
6.Hard
x(t)t (ms)1243−3

A triangle wave rises and falls linearly between −3 and +3 with period T = 2 ms. Find the amplitude c1 of its Fourier component at 500 Hz.

±3T = 2 msf = 500 Hz
c1 =

Answer key

  1. 1. Σy[n] = 14
  2. 2. y[2] = 2
  3. 3. tstart = 3 s
  4. 4. D = 3 s
  5. 5. y[4] = 3.299
  6. 6. c1 = 2.432

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