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−10−11−32−334h[n]n−10−11−3234

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

x[n] = {−1, −3, −3}, n = 0…2h[n] = {−1, −3}
Σy[n] =
2.Easy
x[n]n−104142334h[n]n−10−311234

The sequences x[n] = {4, 4, 3} and h[n] = {−3, 1} are convolved. Find the sum Σy[n] of all samples of y[n] = x[n] ∗ h[n].

x[n] = {4, 4, 3}, n = 0…2h[n] = {−3, 1}
Σy[n] =
3.Medium
x(t)t33h(t)t45

The pulse x(t) = 3 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 5 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(4.5) at t = 4.5 s.

x(t) = 3, 0 ≤ t < 3 sh(t) = 5, 0 ≤ t < 4 st = 4.5 s
y(4.5) =
4.Medium
x(t)t24h(t)t23

The pulse x(t) = 4 for 0 ≤ t < 2 s is convolved with the pulse h(t) = 3 for 0 ≤ t < 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 4, 0 ≤ t < 2 sh(t) = 3, 0 ≤ t < 2 s
Ay =
5.Hard
δ(t)y′ + 5y = 3xh(t)

A system obeys y′(t) + 5·y(t) = 3·x(t). Find its impulse response at t = 0.25 s, h(0.25).

y′ + 5y = 3xt = 0.25 s
h(0.25) =
6.Hard
δ(t)y′ + 1y = 3xh(t)

A system obeys y′(t) + 1·y(t) = 3·x(t). Find its impulse response at t = 2 s, h(2).

y′ + 1y = 3xt = 2 s
h(2) =

Answer key

  1. 1. Σy[n] = 28
  2. 2. Σy[n] = −22
  3. 3. y(4.5) = 37.5
  4. 4. Ay = 48
  5. 5. h(0.25) = 0.8595
  6. 6. h(2) = 0.406

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