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−101112131…h[n]n−10−3132−33245

A system with the impulse response h[n] = {−3, 3, −3, 2} (n = 0…3) receives the step input x[n] = 1·u[n]. Find the output y[3].

h[n] = {−3, 3, −3, 2}x[n] = 1·u[n]
y[3] =
2.Easy
x[n]n−1031−12−334h[n]n−104122234

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

x[n] = {3, −1, −3}, n = 0…2h[n] = {4, 2, 2}, n = 0…2
ymax =
3.Medium
x(t)t1h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 1·u(t). Find the output y(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 1·u(t)t = 0.5 s
y(0.5) =
4.Medium
x(t)t210h(t)t44

The pulse x(t) = 10 for 0 ≤ t < 2 s is convolved with the pulse h(t) = 4 for 0 ≤ t < 4 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 2 sh(t) = 4, 0 ≤ t < 4 s
Ay =
5.Hard
δ(t)y′ + 2y = 3xh(t)

A system obeys y′(t) + 2·y(t) = 3·x(t). Find its impulse response at t = 0.05 s, h(0.05).

y′ + 2y = 3xt = 0.05 s
h(0.05) =
6.Hard
2cos(7.5t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 2·cos(7.5t) is applied to a system with h(t) = 10·e−10tu(t). Find the amplitude Ym of the steady-state output.

h(t) = 10e−10tu(t)x(t) = 2cos(7.5t)
Ym =

Answer key

  1. 1. y[3] = −1
  2. 2. ymax = 12
  3. 3. y(0.5) = 0.3161
  4. 4. Ay = 320
  5. 5. h(0.05) = 2.715
  6. 6. Ym = 1.6

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