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−105152535…h[n]n−10−21−32−13145

A system with the impulse response h[n] = {−2, −3, −1, 1} (n = 0…3) receives the step input x[n] = 5·u[n]. Find the output y[4].

h[n] = {−2, −3, −1, 1}x[n] = 5·u[n]
y[4] =
2.Easy
x[n]n−101−32−33−24h[n]n−1013223

The sequence x[n] is nonzero for n = 1…3 and h[n] for n = 1…2, as shown. Find the first index nfirst at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 1…3h[n]: n = 1…2
nfirst =
3.Medium
x(t)t−2−1h(t)t−13

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

x: −2 … −1 sh: −1 … 3 s
tend = s
4.Medium
x(t)t12h(t)t−13

The signal x(t) is nonzero only between t = 1 s and t = 2 s, and h(t) only between t = −1 s and t = 3 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: 1 … 2 sh: −1 … 3 s
D = s
5.Hard
2cos(30t)h(t) = 2e−20t​u(t)y(t)

The input x(t) = 2·cos(30t) is applied to a system with h(t) = 2·e−20tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 2e−20tu(t)x(t) = 2cos(30t)
φ = °
6.Hard
δ[n]0.5+h[n]−0.5·z⁻¹+

A system obeys y[n] = −0.5·y[n − 1] + 0.5·x[n] and starts at rest. Find its impulse response sample h[2].

y[n] = −0.5y[n − 1] + 0.5x[n]x[n] = δ[n]
h[2] =

Answer key

  1. 1. y[4] = −25
  2. 2. nfirst = 2
  3. 3. tend = 2 s
  4. 4. D = 5 s
  5. 5. φ = −56.31 °
  6. 6. h[2] = 0.125

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