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−103132333…h[n]n−10311243145

A system with the impulse response h[n] = {3, 1, 4, 1} (n = 0…3) receives the step input x[n] = 3·u[n]. Find the output y[2].

h[n] = {3, 1, 4, 1}x[n] = 3·u[n]
y[2] =
2.Easy
x[n]n−3−21−14031h[n]n−1041−3223

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

x[n]: n = -2…0h[n]: n = 0…2
nlast =
3.Medium
x(t)t−2−1h(t)t35

The signal x(t) is nonzero only between t = −2 s and t = −1 s, and h(t) only between t = 3 s and t = 5 s. Find the time tend at which y(t) = x(t) ∗ h(t) ends.

x: −2 … −1 sh: 3 … 5 s
tend = s
4.Medium
x(t)t5h(t)t11

A system has the impulse response h(t) = e−1tu(t). The input is a step x(t) = 5·u(t). Find the output y(4) at t = 4 s.

h(t) = e−1tu(t)x(t) = 5·u(t)t = 4 s
y(4) =
5.Hard
u[n]4+y[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 4·x[n] and starts at rest. The input is the unit step x[n] = u[n]. Find y[3].

y[n] = −0.5y[n − 1] + 4x[n]x[n] = u[n]
y[3] =
6.Hard
x(t)t (ms)420403

A pulse train has height 3, pulse width 4 ms and period 20 ms. Find its DC component (mean value) c0.

A = 3τ = 4 msT = 20 ms
c0 =

Answer key

  1. 1. y[2] = 24
  2. 2. nlast = 2
  3. 3. tend = 4 s
  4. 4. y(4) = 4.908
  5. 5. y[3] = 2.5
  6. 6. c0 = 0.6

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