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−10−11421334h[n]n−1031−32234

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

x[n] = {−1, 4, 1, 3}, n = 0…3h[n] = {3, −3, 2}, n = 0…2
ymax =
2.Easy
x[n]n−10−31−32−234h[n]n−10−311234

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

x[n] = {−3, −3, −2}, n = 0…2h[n] = {−3, 1}, n = 0…1
y[2] =
3.Medium
x(t)t2h(t)t0.21

A system has the impulse response h(t) = e−5tu(t). The input is a step x(t) = 2·u(t). Find the output y(0.4) at t = 0.4 s.

h(t) = e−5tu(t)x(t) = 2·u(t)t = 0.4 s
y(0.4) =
4.Medium
x(t)t22h(t)t1

The pulse x(t) = 2 for 0 ≤ t < 2 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(3) at t = 3 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 2 sh(t) = e−2tu(t)t = 3 s
y(3) =
5.Hard
δ[n]0.5+h[n]−0.8·z⁻¹+

A system obeys y[n] = −0.8·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[5].

y[n] = −0.8y[n − 1] + 0.5x[n]x[n] = δ[n]
h[5] =
6.Hard
x(t)t (ms)2.55103−3

A square wave switches between +3 and −3 with period T = 5 ms (half a period at each level). Find the amplitude c5 of its Fourier component at 1000 Hz.

±3T = 5 msf = 1000 Hz
c5 =

Answer key

  1. 1. ymax = 15
  2. 2. y[2] = 3
  3. 3. y(0.4) = 0.3459
  4. 4. y(3) = 0.1329
  5. 5. h[5] = −0.1638
  6. 6. c5 = 0.7639

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