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−101112131…h[n]n−10−11−2223345

A system with the impulse response h[n] = {−1, −2, 2, 3} (n = 0…3) receives the step input x[n] = 1·u[n]. Find the output y[4].

h[n] = {−1, −2, 2, 3}x[n] = 1·u[n]
y[4] =
2.Easy
x[n]n−1021−22−334h[n]n−103132134

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

x[n] = {2, −2, −3}, n = 0…2h[n] = {3, 3, 1}
Σy[n] =
3.Medium
x(t)t44h(t)t22

The pulse x(t) = 4 for 0 ≤ t < 4 s is convolved with h(t), a triangle that falls linearly from 2 at t = 0 to 0 at t = 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 4 sh(0) = 2, h(2) = 0
Ay =
4.Medium
x(t)t03h(t)t−10

The signal x(t) is nonzero only between t = 0 s and t = 3 s, and h(t) only between t = −1 s and t = 0 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: 0 … 3 sh: −1 … 0 s
tstart = s
5.Hard
x(t)t (ms)0.6243

A pulse train has height 3, pulse width 0.6 ms and period 2 ms. Find the amplitude c3 of harmonic number 3, at 1500 Hz.

A = 3τ = 0.6 msT = 2 ms
c3 =
6.Hard
x(t)t (ms)2485−5

A triangle wave rises and falls linearly between −5 and +5 with period T = 4 ms. Find the amplitude c3 of its Fourier component at 750 Hz.

±5T = 4 msf = 750 Hz
c3 =

Answer key

  1. 1. y[4] = 2
  2. 2. Σy[n] = −21
  3. 3. Ay = 32
  4. 4. tstart = −1 s
  5. 5. c3 = 0.1967
  6. 6. c3 = 0.4503

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