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−10−3132−334h[n]n−10−1112234

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

x[n] = {−3, 3, −3}, n = 0…2h[n] = {−1, 1, 2}, n = 0…2
ymax =
2.Easy
x[n]n−10−11−22−234h[n]n−1021−1234

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

x[n] = {−1, −2, −2}, n = 0…2h[n] = {2, −1}
Σy[n] =
3.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−0.5tu(t). Find the output y(3) at t = 3 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 2 sh(t) = e−0.5tu(t)t = 3 s
y(3) =
4.Medium
x(t)t1h(t)t1

The input x(t) = 1·e−6tu(t) is applied to a system with h(t) = e−1tu(t). Find the output y(1.5) at t = 1.5 s.

x(t) = 1·e−6tu(t)h(t) = e−1tu(t)t = 1.5 s
y(1.5) =
5.Hard
u[n]0.5+y[n]0.5·z⁻¹+

A system obeys y[n] = 0.5·y[n − 1] + 0.5·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[4].

y[n] = 0.5y[n − 1] + 0.5x[n]x[n] = u[n]
y[4] =
6.Hard
δ[n]2+h[n]0.4·z⁻¹+

A system obeys y[n] = 0.4·y[n − 1] + 2·x[n] and starts at rest. Find its impulse response sample h[5].

y[n] = 0.4y[n − 1] + 2x[n]x[n] = δ[n]
h[5] =

Answer key

  1. 1. ymax = 3
  2. 2. Σy[n] = −5
  3. 3. y(3) = 1.534
  4. 4. y(1.5) = 0.0446
  5. 5. y[4] = 0.9688
  6. 6. h[5] = 0.02048

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