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−101121314−25h[n]n−1012324−25

The sequence x[n] is nonzero for n = 1…4 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 = 1…4h[n]: n = 3…4
nlast =
2.Easy
x[n]n−10−31−222344h[n]n−10−2122−334

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

x[n] = {−3, −2, 2, 4}, n = 0…3h[n] = {−2, 2, −3}, n = 0…2
ymax =
3.Medium
x(t)t−11h(t)t26

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 = 6 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: −1 … 1 sh: 2 … 6 s
tend = s
4.Medium
x(t)t11h(t)t23

The pulse x(t) = 1 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with the impulse response h(t) = 3 for 0 ≤ t < 2 s (zero elsewhere). Find the output y(0.5) at t = 0.5 s.

x(t) = 1, 0 ≤ t < 1 sh(t) = 3, 0 ≤ t < 2 st = 0.5 s
y(0.5) =
5.Hard
x(t)t (ms)0.5122−2

A triangle wave rises and falls linearly between −2 and +2 with period T = 1 ms. Find the amplitude c3 of its Fourier component at 3000 Hz.

±2T = 1 msf = 3000 Hz
c3 =
6.Hard
δ[n]4+h[n]0.8·z⁻¹+

A system obeys y[n] = 0.8·y[n − 1] + 4·x[n] and starts at rest. Find its impulse response sample h[5].

y[n] = 0.8y[n − 1] + 4x[n]x[n] = δ[n]
h[5] =

Answer key

  1. 1. nlast = 8
  2. 2. ymax = 6
  3. 3. tend = 7 s
  4. 4. y(0.5) = 1.5
  5. 5. c3 = 0.1801
  6. 6. h[5] = 1.311

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