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−104122−134h[n]n−1011−12−134

The sequences x[n] = {4, 2, −1} and h[n] = {1, −1, −1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {4, 2, −1}, n = 0…2h[n] = {1, −1, −1}
Σy[n] =
2.Easy
x[n]n−10212233h[n]n−1012−23−3435

The sequence x[n] is nonzero for n = 0…2 and h[n] for n = 2…4, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 0…2h[n]: n = 2…4
nfirst =
3.Medium
x(t)t4h(t)t0.251

A system has the impulse response h(t) = e−4tu(t). The input is a step x(t) = 4·u(t). Find the output y(0.25) at t = 0.25 s.

h(t) = e−4tu(t)x(t) = 4·u(t)t = 0.25 s
y(0.25) =
4.Medium
x(t)t4h(t)t1

The input x(t) = 4·e−6tu(t) is applied to a system with h(t) = e−2tu(t). Find the output y(0.5) at t = 0.5 s.

x(t) = 4·e−6tu(t)h(t) = e−2tu(t)t = 0.5 s
y(0.5) =
5.Hard
x(t)t (ms)1020403−3

A square wave switches between +3 and −3 with period T = 20 ms (half a period at each level). Find the amplitude c1 of its Fourier component at 50 Hz.

±3T = 20 msf = 50 Hz
c1 =
6.Hard
5cos(3.75t)h(t) = 2e−5t​u(t)y(t)

The input x(t) = 5·cos(3.75t) is applied to a system with h(t) = 2·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−5tu(t)x(t) = 5cos(3.75t)
φ = °

Answer key

  1. 1. Σy[n] = −5
  2. 2. nfirst = 2
  3. 3. y(0.25) = 0.6321
  4. 4. y(0.5) = 0.3181
  5. 5. c1 = 3.82
  6. 6. φ = −36.87 °

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