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−102132−134h[n]n−10111234

The sequences x[n] = {2, 3, −1} and h[n] = {1, 1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {2, 3, −1}, n = 0…2h[n] = {1, 1}
Σy[n] =
2.Easy
x[n]n−10−11322344h[n]n−10−31−1234

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

x[n] = {−1, 3, 2, 4}, n = 0…3h[n] = {−3, −1}, n = 0…1
ymax =
3.Medium
x(t)t5h(t)t1

The input x(t) = 5·e−6tu(t) is applied to a system with h(t) = e−4tu(t). Find the output y(2) at t = 2 s.

x(t) = 5·e−6tu(t)h(t) = e−4tu(t)t = 2 s
y(2) =
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
x(t)t (ms)10204010

A square wave switches between 10 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 … 10T = 20 msf = 350 Hz
c7 =
6.Hard
2cos(15t)h(t) = 10e−10t​u(t)y(t)

The input x(t) = 2·cos(15t) 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(15t)
Ym =

Answer key

  1. 1. Σy[n] = 8
  2. 2. ymax = 3
  3. 3. y(2) = 0.0008233
  4. 4. y(1.5) = 0.0446
  5. 5. c7 = 0.9095
  6. 6. Ym = 1.109

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