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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
x[n]n−105152535…h[n]n−103142−334

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

h[n] = {3, 4, −3}x[n] = 5·u[n]
y[5] =
2.Easy
x[n]n−102142−23−24h[n]n−10−113234

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

x[n] = {2, 4, −2, −2}, n = 0…3h[n] = {−1, 3}, n = 0…1
y[1] =
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−0.5tu(t). Find the output y(2.5) at t = 2.5 s, after the pulse has ended.

x(t) = 5, 0 ≤ t < 1 sh(t) = e−0.5tu(t)t = 2.5 s
y(2.5) =
4.Medium
x(t)t5h(t)t0.51

A system has the impulse response h(t) = e−2tu(t). The input is a step x(t) = 5·u(t). Find the output y(0.5) at t = 0.5 s.

h(t) = e−2tu(t)x(t) = 5·u(t)t = 0.5 s
y(0.5) =
5.Hard
x(t)t (ms)1020401

A square wave switches between 1 and 0 with period T = 20 ms (half a period at each level). Find the amplitude c7 of its Fourier component at 350 Hz.

0 … 1T = 20 msf = 350 Hz
c7 =
6.Hard
2cos(15t)h(t) = 10e−20t​u(t)y(t)

The input x(t) = 2·cos(15t) is applied to a system with h(t) = 10·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 10e−20tu(t)x(t) = 2cos(15t)
φ = °

Answer key

  1. 1. y[5] = 20
  2. 2. y[1] = 2
  3. 3. y(2.5) = 1.859
  4. 4. y(0.5) = 1.58
  5. 5. c7 = 0.09095
  6. 6. φ = −36.87 °

Full step-by-step solutions: practise this topic on examproblems.com.