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−1031−22−3334h[n]n−1041−3234

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

x[n] = {3, −2, −3, 3}, n = 0…3h[n] = {4, −3}, n = 0…1
y[2] =
2.Easy
x[n]n−102142−234h[n]n−1041−1234

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

x[n] = {2, 4, −2}, n = 0…2h[n] = {4, −1}, n = 0…1
y[1] =
3.Medium
x(t)t1h(t)t11

A system has the impulse response h(t) = e−1tu(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−1tu(t)x(t) = 1·u(t)t = 0.5 s
y(0.5) =
4.Medium
x(t)t2h(t)t0.51

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

h(t) = e−2tu(t)x(t) = 2·u(t)t = 1 s
y(1) =
5.Hard
δ[n]2+h[n]0.6·z⁻¹+

A system obeys y[n] = 0.6·y[n − 1] + 2·x[n] and starts at rest. Find its impulse response sample h[4].

y[n] = 0.6y[n − 1] + 2x[n]x[n] = δ[n]
h[4] =
6.Hard
x(t)t (ms)1.255101

A pulse train has height 1, pulse width 1.25 ms and period 5 ms. Find the amplitude c1 of harmonic number 1, at 200 Hz.

A = 1τ = 1.25 msT = 5 ms
c1 =

Answer key

  1. 1. y[2] = −6
  2. 2. y[1] = 14
  3. 3. y(0.5) = 0.3935
  4. 4. y(1) = 0.8647
  5. 5. h[4] = 0.2592
  6. 6. c1 = 0.4502

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