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−105152535…h[n]n−10312223−145

A system with the impulse response h[n] = {3, 2, 2, −1} (n = 0…3) receives the step input x[n] = 5·u[n]. Find the output y[2].

h[n] = {3, 2, 2, −1}x[n] = 5·u[n]
y[2] =
2.Easy
x[n]n−1021−32−2324h[n]n−1021−12434

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

x[n] = {2, −3, −2, 2}, n = 0…3h[n] = {2, −1, 4}, n = 0…2
y[3] =
3.Medium
x(t)t2h(t)t1

The input x(t) = 2·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) = 2·e−3tu(t)h(t) = e−2tu(t)t = 1 s
y(1) =
4.Medium
x(t)t0.55h(t)t1

The pulse x(t) = 5 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−4tu(t). Find the output y(2) at t = 2 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 0.5 sh(t) = e−4tu(t)t = 2 s
y(2) =
5.Hard
10cos(60t)h(t) = 2e−20t​u(t)y(t)

The input x(t) = 10·cos(60t) is applied to a system with h(t) = 2·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−20tu(t)x(t) = 10cos(60t)
φ = °
6.Hard
δ(t)y′ + 0.5y = 10xh(t)

A system obeys y′(t) + 0.5·y(t) = 10·x(t). Find its impulse response at t = 0.8 s, h(0.8).

y′ + 0.5y = 10xt = 0.8 s
h(0.8) =

Answer key

  1. 1. y[2] = 35
  2. 2. y[3] = −6
  3. 3. y(1) = 0.1711
  4. 4. y(2) = 0.002679
  5. 5. φ = −71.57 °
  6. 6. h(0.8) = 6.703

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