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−103112−13445h[n]n−100.51220.2534

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

b = {0.5, 2, 0.25}x[n] = {3, 1, −1, 4}, n = 0…3
y[2] =
2.Easy
x[n]n−104112−334h[n]n−103112−334

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

x[n] = {4, 1, −3}, n = 0…2h[n] = {3, 1, −3}, n = 0…2
y[3] =
3.Medium
x(t)t15h(t)t1

The pulse x(t) = 5 for 0 ≤ t < 1 s (zero elsewhere) is applied to a system with h(t) = e−1tu(t). Find the output y(1.5) at t = 1.5 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 1 sh(t) = e−1tu(t)t = 1.5 s
y(1.5) =
4.Medium
x(t)t33h(t)t45

The pulse x(t) = 3 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(5.5) at t = 5.5 s.

x(t) = 3, 0 ≤ t < 3 sh(t) = 5, 0 ≤ t < 4 st = 5.5 s
y(5.5) =
5.Hard
δ[n]3+h[n]0.4·z⁻¹+

A system obeys y[n] = 0.4·y[n − 1] + 3·x[n] and starts at rest. Find its impulse response sample h[5].

y[n] = 0.4y[n − 1] + 3x[n]x[n] = δ[n]
h[5] =
6.Hard
δ[n]0.5+h[n]0.4·z⁻¹+

A system obeys y[n] = 0.4·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[2].

y[n] = 0.4y[n − 1] + 0.5x[n]x[n] = δ[n]
h[2] =

Answer key

  1. 1. y[2] = 2.25
  2. 2. y[3] = −6
  3. 3. y(1.5) = 1.917
  4. 4. y(5.5) = 22.5
  5. 5. h[5] = 0.03072
  6. 6. h[2] = 0.08

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