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−10−1132134h[n]n−10−21−22−334

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

x[n] = {−1, 3, 1}, n = 0…2h[n] = {−2, −2, −3}
Σy[n] =
2.Easy
x[n]n−103132333…h[n]n−103122−23−145

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

h[n] = {3, 2, −2, −1}x[n] = 3·u[n]
y[5] =
3.Medium
x(t)t0.52h(t)t1

The pulse x(t) = 2 for 0 ≤ t < 0.5 s (zero elsewhere) is applied to a system with h(t) = e−2tu(t). Find the output y(1.5) at t = 1.5 s, after the pulse has ended.

x(t) = 2, 0 ≤ t < 0.5 sh(t) = e−2tu(t)t = 1.5 s
y(1.5) =
4.Medium
x(t)t10h(t)t1

The input x(t) = 10·e−3tu(t) is applied to a system with h(t) = e−0.5tu(t). Find the output y(0.25) at t = 0.25 s.

x(t) = 10·e−3tu(t)h(t) = e−0.5tu(t)t = 0.25 s
y(0.25) =
5.Hard
δ(t)y′ + 1y = 2xh(t)

A system obeys y′(t) + 1·y(t) = 2·x(t). Find its impulse response at t = 0.2 s, h(0.2).

y′ + 1y = 2xt = 0.2 s
h(0.2) =
6.Hard
δ(t)y′ + 1y = 2xh(t)

A system obeys y′(t) + 1·y(t) = 2·x(t). Find its impulse response at t = 0.1 s, h(0.1).

y′ + 1y = 2xt = 0.1 s
h(0.1) =

Answer key

  1. 1. Σy[n] = −21
  2. 2. y[5] = 6
  3. 3. y(1.5) = 0.08555
  4. 4. y(0.25) = 1.641
  5. 5. h(0.2) = 1.637
  6. 6. h(0.1) = 1.81

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