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−10−21−12134h[n]n−1011−1234

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

x[n] = {−2, −1, 1}, n = 0…2h[n] = {1, −1}, n = 0…1
y[2] =
2.Easy
x[n]n−105152535…h[n]n−1041−2243−345

A system with the impulse response h[n] = {4, −2, 4, −3} (n = 0…3) receives the step input x[n] = 5·u[n]. Find the output y[5].

h[n] = {4, −2, 4, −3}x[n] = 5·u[n]
y[5] =
3.Medium
x(t)t4h(t)t1

The input x(t) = 4·e−2tu(t) is applied to a system with h(t) = e−5tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 4·e−2tu(t)h(t) = e−5tu(t)t = 0.25 s
y(0.25) =
4.Medium
x(t)t1h(t)t1

The input x(t) = 1·e−3tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(1) at t = 1 s.

x(t) = 1·e−3tu(t)h(t) = e−2tu(t)t = 1 s
y(1) =
5.Hard
u[n]0.5+y[n]0.9·z⁻¹+

A system obeys y[n] = 0.9·y[n − 1] + 0.5·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[2].

y[n] = 0.9y[n − 1] + 0.5x[n]x[n] = u[n]
y[2] =
6.Hard
δ[n]4+h[n]0.6·z⁻¹+

A system obeys y[n] = 0.6·y[n − 1] + 4·x[n] and starts at rest. Find its impulse response sample h[3].

y[n] = 0.6y[n − 1] + 4x[n]x[n] = δ[n]
h[3] =

Answer key

  1. 1. y[2] = 2
  2. 2. y[5] = 15
  3. 3. y(0.25) = 0.4267
  4. 4. y(1) = 0.08555
  5. 5. y[2] = 1.355
  6. 6. h[3] = 0.864

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